diff --git a/FCI/README.txt b/FCI/README.txt new file mode 100644 index 0000000..de604ce --- /dev/null +++ b/FCI/README.txt @@ -0,0 +1,13 @@ +On 19 Dec 2013, Peter Knowles sent me various files to allow me to do Full CI calculations. + +To compile, I needed to do: + + gfortran -lblas fci.f standard.f + +To run, I needed to do: + + ./a.out < hf.dat + +The resulting output compared well with the file hf.out that Peter K provided. + +The program uses single-bar, not double-bar, integrals. diff --git a/FCI/boxints.txt b/FCI/boxints.txt new file mode 100644 index 0000000..b4e8600 --- /dev/null +++ b/FCI/boxints.txt @@ -0,0 +1,56513 @@ +&FCI NORB = 30, NELEC = 7, MS2 = 7, ISYM = 1, NROOT = 1, MAXIT = 200, + ORBSYM = 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, + MEMORY = 300000000 / +0.4473211247724916 1 1 1 1 +-0.40635977105343735 2 1 2 1 +0.6259919502912837 2 2 1 1 +0.12977728134069944 2 2 2 2 +-0.17867082551879201 3 1 1 1 +0.4962146689505843 3 1 2 2 +-0.6748854944693764 3 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+-0.8357436242766103 30 30 30 30 +0.5 1 1 0 0 +2. 2 2 0 0 +4.5 3 3 0 0 +8. 4 4 0 0 +12.5 5 5 0 0 +18. 6 6 0 0 +24.5 7 7 0 0 +32. 8 8 0 0 +40.5 9 9 0 0 +50. 10 10 0 0 +60.5 11 11 0 0 +72. 12 12 0 0 +84.5 13 13 0 0 +98. 14 14 0 0 +112.5 15 15 0 0 +128. 16 16 0 0 +144.5 17 17 0 0 +162. 18 18 0 0 +180.5 19 19 0 0 +200. 20 20 0 0 +220.5 21 21 0 0 +242. 22 22 0 0 +264.5 23 23 0 0 +288. 24 24 0 0 +312.5 25 25 0 0 +338. 26 26 0 0 +364.5 27 27 0 0 +392. 28 28 0 0 +420.5 29 29 0 0 +450. 30 30 0 0 diff --git a/FCI/fci.f b/FCI/fci.f new file mode 100644 index 0000000..358a060 --- /dev/null +++ b/FCI/fci.f @@ -0,0 +1,1722 @@ + PROGRAM FCI + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /CHAMIL/ ISYM,NNCI + COMMON /TAPES / INP,IOUT + DIMENSION MULTT(8,8) + DATA MULTT /1,2,3,4,5,6,7,8, 2,1,4,3,6,5,8,7, + 2 3,4,1,2,7,8,5,6, 4,3,2,1,8,7,6,5, + 4 5,6,7,8,1,2,3,4, 6,5,8,7,2,1,4,3, + 6 7,8,5,6,3,4,1,2, 8,7,6,5,4,3,2,1/ + INP = 5 + IOUT = 6 + WRITE (IOUT,35) +35 FORMAT(' PROGRAM * FCI (Determinant based Full CI)', + > 5X,'Author: P.J. Knowles, 1984') +C... SET UP DEFAULT OR INITIAL VALUES + MEMORY = 100 000 + IPRTIM = -1 + IPRINT = -1 + IPRDIA = -1 + IPRHAM = -1 + IOPTIO = 0 + MAXIT = 25 + THR = 1D-5 + THRRES = 0.05 + NROOT = 1 + NACT = 0 + NELEC = 0 + MS2 = 0 + ISYM = 1 + DO 2 I=1,8 + NOCC(I) = 0 +C... D2H MULTIPLICATION TABLE + DO 2 J=1,8 +2 MULT(J,I) = MULTT(J,I) + DO 3 I=1,256 +3 ITYPEA(I)=1 +C +C... READ INPUT + CALL INPDAT +C +C... INITIALIZE MEMORY HANDLING + I = INICOR(MEMORY) +C +C... NUMBERS OF ALPHA, BETA ELECTRONS + NA = (NELEC+MS2)/2 + NB = (NELEC-MS2)/2 + DO 36 I=1,NACT +36 NOCC(ITYPEA(I)) = NOCC(ITYPEA(I))+1 + CALL INITC +C + WRITE (IOUT,25) ISYM +25 FORMAT(/' Space symmetry:',T30,I3) + WRITE (IOUT,21) MAXIT,THR,THRRES,NROOT +21 FORMAT( + 4' Maximum iterations:', T30,I3/ + 5' Convergence threshold:',T30,F12.7/ + 8' Output threshold:', T30,F12.7/ + A' Number of roots:', T30,I3) + NNCI = NCI(ISYM) + IV1 = ICORR(NNCI) + IV2 = ICORR(NNCI) + CALL DAVIDS (Q(IV1),Q(IV2),IOPTIO) + CALL CORLSR(IV1) +C + CALL ENDCOR + END + SUBROUTINE FCMXM (D,Z,E,NK,NIJ,NOPCNT) +CSUBR MATRIX MULTIPLICATION KERNEL FOR FULL CI PROGRAM +CSUBR E = D * Z +CSUBR Z IS INTEGRAL MATRIX AND NOT SPARSE, DIMENSION NIJ=NUMBER OF +CSUBR ORBITAL PAIRS OF GIVEN SYMMETRY +CSUBR D,E LEADING DIMENSION NK=NUMBER OF DETERMINANTS OF A SYMMETRY +CSUBR BLOCK, OR SUB-BATCH OF THIS +CSUBR D IS VERY SPARSE IN 1ST ITERATION OF CI, GRADUALLY RISES TO +CSUBR USUALLY 50-80% POPULATED +CSUBR OPTIMUM FORM OF THIS ROUTINE IS DEPENDENT ON MACHINE ARCHITECTURE +CSUBR USUALLY NK>NIJ, SO BETTER TO VECTORISE ALONG NK, BUT SOMETIMES +CSUBR BETTER TO EXPLOIT SPARSITY OF D +CSUBR ROUTINE SHOULD RETURN NOPCNT -- NUMBER OF FLOATING POINT OPS -- +CSUBR IF ADDITIONAL TIMING INFORMATION IS REQUIRED + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + PARAMETER (NHALF=30) + DIMENSION D(NK,NIJ),Z(NIJ,NIJ),E(NK,NIJ) +c pmwg replaced all code by a dgemm in Jan 2014 + call dgemm('n','n',nk,nij,nij,1d0,d,nk,z,nij,0d0,e,nk) +c DO 1 J=1,NK*NIJ +c1 E(J,1) = 0D0 +C... EVALUATE SPARSITY OF D +c IDS = 0 +c DO 7 J=1,NK*NIJ +c7 IF (D(J,1).NE.0D0) IDS=IDS+1 +c IZS = NIJ**2 +c IF (IDS*(NIJ+NHALF).LT.IZS*(NK+NHALF)) THEN +C... USE SPARSITY OF D +c NOPCNT = NOPCNT + 2*IDS*NIJ +C CALL MXMA (Z,1,NIJ, D,NK,1, E,NK,1, NIJ,NIJ,NK) +c DO 2 K=1,NK +c DO 2 J=1,NIJ +c IF (D(K,J).EQ.0D0) GOTO 2 +c DO 21 I=1,NIJ +c21 E(K,I) = E(K,I) + D(K,J)*Z(J,I) +c2 CONTINUE +c ELSE +C... VECTORISE ALONG D +c NOPCNT = NOPCNT + 2*IZS*NK +C CALL MXMA (D,1,NK, Z,NIJ,1, E,1,NK, NK,NIJ,NIJ) +c DO 3 I=1,NIJ +c DO 3 J=1,NIJ +C IF (Z(J,I).EQ.0D0) GOTO 3 +c DO 31 K=1,NK +c31 E(K,I) = E(K,I) + D(K,J)*Z(J,I) +c3 CONTINUE +c END IF + RETURN + END + SUBROUTINE INPDAT +CSUBR DATA INPUT FOR PORTABLE FULL CI PROGRAM +CSUBR NAMELIST INPUT MAY REQUIRE ADJUSTMENT ON SOME MACHINES + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /CHAMIL/ ISYM,NNCI + COMMON /TAPES / INP,IOUT + INTEGER ORBSYM(255) + EQUIVALENCE (ORBSYM(1),ITYPEA(1)) + NAMELIST /FCI / NORB,NELEC,MS2,ISYM,ORBSYM + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM + > ,INT,MEMORY,CORE + > ,MAXIT,THR,THRRES,NROOT + INT = -1 + READ (INP,FCI) + IF (INT.NE.-1) INP=INT + NACT = NORB + RETURN + END + SUBROUTINE INPINT (Z1,Z2) +CSUBR INTEGRAL INPUT FOR PORTABLE FULL CI PROGRAM + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /TAPES / INP,IOUT + DIMENSION Z2(*),Z1(*) +cpmg +c rs = 2.0d0 +c R = sqrt(dfloat(na+nb))/2 * rs + R = 0.25d0 + write(*,*) 'Scaling integrals for R =',R +cpmg + CALL FZERO (Z1,NPAIR(1)) + CALL FZERO (Z2,NINT2) +1 READ (INP,*,END=99) Z,I,J,K,L + IF (MAX(I,J,K,L).GT.NACT) GOTO 1 + IF (K.NE.0) THEN +C... TWO ELECTRON INTEGRAL +cpmg + z = z/R +cpmg + ISYMIJ = MULT(ITYPEA(I),ITYPEA(J)) + IJ = IQ(INTADT+I-1+(J-1)*NACT) + KL = IQ(INTADT+K-1+(L-1)*NACT) + Z2(IJ+(KL-1)*NPAIR(ISYMIJ)+INTOFF(ISYMIJ)) = Z + Z2(KL+(IJ-1)*NPAIR(ISYMIJ)+INTOFF(ISYMIJ)) = Z + ELSE IF (I.NE.0) THEN +C... ONE ELECTRON INTEGRAL +cpmg + z = z/R/R +cpmg + IJ = IQ(INTADT+I-1+(J-1)*NACT) + Z1(IJ) = Z + ELSE +C... CORE REPULSION ENERGY + CORE = Z + END IF + GOTO 1 +99 RETURN + END + SUBROUTINE DAVIDS (V1,V2,IOPTIO) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /TAPES / INP,IOUT +C +C... DAVIDSON DIAGONALISER, SINGLE ROOT +C... WARNING, NOT NECESSARILY SAFE WITH DEGENERACIES, ETC. + PARAMETER (IFILD=92,IFIL1=93,IFIL2=94,MAXITR=200) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /CHAMIL/ ISYMH,N + DIMENSION V1(N),V2(N) + DIMENSION ZM(MAXITR,MAXITR),ZV(MAXITR,MAXITR),R(MAXITR) +C + TRIAL = 0 + RESULT = 0 + MAXITS = MIN(MAXIT,MAXITR,N) + CALL FCDIAG (V2) + IF (IOPTIO.GE.5) CALL OUTVEC (V2,N,'DIAGONAL ELEMENTS') +C + OPEN (IFILD,STATUS='SCRATCH',FORM='UNFORMATTED') + OPEN (IFIL1,STATUS='SCRATCH',FORM='UNFORMATTED') + OPEN (IFIL2,STATUS='SCRATCH',FORM='UNFORMATTED') + REWIND IFILD + REWIND IFIL1 + REWIND IFIL2 +C + WRITE (IFILD) V2 + REWIND IFILD + CALL TRIALV (V1,V2) +C + WRITE (IOUT,'(/'' It Tr CPU'',2X,''Convergence'',6X, + >''Energy''/)') + DO 190 IT=1,MAXITS + WRITE (IFIL1) V1 +C... CALL THE CI PROGRAM V2 = H . V1 + V2 + DO 30 I=1,N +30 V2(I)=0.0D0 + CALL FSIGMA (V1,V2) + WRITE (IFIL2) V2 +C + REWIND IFIL1 + DO 50 JT=1,IT + READ (IFIL1) V1 + ZM(JT,IT) = 0.0D0 + DO 40 I=1,N +40 ZM(JT,IT) = ZM(JT,IT) + V1(I)*V2(I) +50 ZM(IT,JT) = ZM(JT,IT) + REWIND IFIL1 + REWIND IFIL2 +C +C... SMALL MATRIX DIAGONALISER + DO 60 JT=1,IT + DO 60 KT=1,IT +60 ZV(KT,JT) = ZM(KT,JT) + CALL DIAG2 (MAXITR,IT,R,ZV) +C + TEST = 0 + DO 61 IROOT=1,MIN(IT,NROOT) + TES = ABS(ZV(IT,IROOT)) + IF (TES.LT.TEST) GOTO 61 + ITRACK = IROOT + TEST = TES +61 CONTINUE + IF (IT.EQ.1) EREF=R(1) + WRITE (IOUT,70) IT,ITRACK,SECOND(),TEST + >,(R(IROOT)+CORE,IROOT=1,MIN(NROOT,IT)) +70 FORMAT(I3,I4,F7.1,F12.8,10F20.9) + IF (TEST.LT.THR) GOTO 210 +C +C... MAKE RESIDUAL + DO 80 I=1,N +80 V1(I) = 0.0D0 + DO 100 JT=1,IT + READ (IFIL1) V2 + DO 90 I=1,N +90 V1(I) = V1(I) + (-R(ITRACK)*ZV(JT,ITRACK))*V2(I) + READ (IFIL2) V2 + DO 100 I=1,N +100 V1(I) = V1(I) + ZV(JT,ITRACK) * V2(I) + READ (IFILD) V2 + IF (IT.EQ.1) THEN + DO 110 JJ = 1,N + IF (ABS(V2(JJ)-EREF).LT.1D-9) V2(JJ)=1D20 +110 CONTINUE + END IF + REWIND IFILD + DO 130 I=1,N +130 V1(I) = V1(I) / (V2(I)-R(1)+1.0D-10) +C +C... ORTHOGONALISE +140 REWIND IFIL1 + DO 160 JT=1,IT + READ (IFIL1) V2 + ZZ = 0.0D0 + DO 150 I=1,N +150 ZZ = ZZ + V1(I)*V2(I) + DO 160 I=1,N +160 V1(I) = V1(I) + (-ZZ) * V2(I) + ZZ = 0.0D0 + DO 170 I=1,N +170 ZZ = ZZ + V1(I)**2 + IF (ZZ.LT.1D-20) GOTO 210 + ZZ = 1.0 / SQRT(ZZ) + DO 180 I=1,N +180 V1(I) = V1(I) * ZZ +C... REORTHOGONALISE IF A LOT HAS BEEN ANNIHILATED + IF (ZZ.GT.1D3) GOTO 140 +C +190 CONTINUE + IT = MAXITS + WRITE (IOUT,200) +200 FORMAT(/1X,'*** Convergence not achieved in max iterations') +C +210 DO 280 IROOT=1,NROOT + REWIND IFIL1 + DO 220 I=1,N +220 V1(I) = 0.0D0 + DO 230 JT=1,IT + READ (IFIL1) V2 + DO 230 I=1,N +230 V1(I) = V1(I) + ZV(JT,IROOT) * V2(I) + WRITE (IOUT,'('' State'',I2,5X,''Energy'',F23.12)') + > IROOT,R(IROOT)+CORE + IF (IROOT.EQ.1) WRITE (IOUT,'('' Correlation energy'',F23.12)') + > R(1)-EREF + WRITE (IOUT,'(/'' Final CI vector''/)') + CALL FCVECO (V1,ISYMH,THRRES) +280 CONTINUE +C + CLOSE (IFILD,STATUS='DELETE') + CLOSE (IFIL1,STATUS='DELETE') + CLOSE (IFIL2,STATUS='DELETE') +C + RETURN + END + SUBROUTINE TRIALV (V1,V2) +CSUBR SUBROUTINE TO GENERATE TRIAL CI VECTOR +CSUBR COULD BE REPLACED BY E.G. READING VECTOR IF REQUIRED + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /TAPES / INP,IOUT + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /CHAMIL/ ISYMH,N + DIMENSION IMIN(64) + DIMENSION V1(N),V2(N) + TRIAL = 0 + IF (NINT(TRIAL).EQ.0) THEN +C... FIND THE DETERMINANT WITH LOWEST ENERGY + VMIN = 1D20 + DO 10 I=1,N + IF (V2(I).GT.VMIN) GOTO 10 + IMIN(1) = I + VMIN = V2(I) +10 CONTINUE +C... CONSTRUCT A SPIN EIGENFUNCTION CONTAINING THIS DETERMINANT + CALL FCSPAD (V1,IMIN(1)) + WRITE (IOUT,20) +20 FORMAT(/1X,'Initial configuration generated:') + NREF = 0 + DO 40 I=1,N + IF (V1(I).EQ.0.0D0) GOTO 40 + WRITE (IOUT,30) I,V1(I),V2(I)+CORE + NREF = NREF+1 + IMIN(NREF) = I +30 FORMAT(I8,2F18.7) +40 CONTINUE + ELSE +C... READ FROM FILE + CONTINUE + END IF + RETURN + END + SUBROUTINE FCVECO (V,ISYM,THRESH) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /TAPES / INP,IOUT + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + CHARACTER*130 TEST + DIMENSION V(*) + COMMON /CLOCAL/ ICGA(32),ICGB(32) + N = NCI(ISYM) + IBASE = ICORR(0) + INTERA = ICORI(NA*NACT) + CALL INITI (IQ(INTERA),NA,NACT) + INTERB = ICORI(NB*NACT) + CALL INITI (IQ(INTERB),NA,NACT) + IC = ICORI(NSTRAA) + ITZ = ICORI(NSTRBB) + CALL FCMIC (IQ(IC),IQ(ITZ),ISYM) + NFROZ=0 + DO 60 II=1,NCI(ISYM) + IF (ABS(V(II)).LT.THRESH) GOTO 60 + CALL FCSTRG (II,ISYM,ICGA,ICGB) + WRITE (IOUT,'(1X,F20.12,64I3)') V(II) + 1,(ICGA(I)+NFROZ,I=1,NA) + 2,(ICGB(I)+NFROZ,I=1,NB) +60 CONTINUE + WRITE (IOUT,'(1X,''/EOF'')') + CALL CORLSR(IBASE) + RETURN + END + SUBROUTINE INITC + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /TAPES / INP,IOUT + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + DIMENSION ICG(32) +C... LENGTHS OF CI EXPANSIONS +C... NSTRA, NSTRB ARE NUMBERS OF ALPHA,BETA STRINGS OF EACH SYMMETRY +C... NCI ARE NUMBERS OF DETERMINANTS OF EACH SYMMETRY +C... NPAIR ARE NUMBERS OF ORBITAL PAIRS OF EACH SYMMETRY +C.... INTOFF OFFSET FOR SYMMETRY BLOCKS OF TWO ELECTRON INTEGRALS + NSTRAA = 0 + NSTRBB = 0 + MAXAA = 0 + MAXBB = 0 + DO 100 ISYM=1,8 + ICG(1) = 0 + NSTRA(ISYM) = -1 +80 NSTRA(ISYM) = NSTRA(ISYM)+1 + CALL FCSTRS (NA,ICG,ISYM,ITYPEA) + IF (ICG(1).NE.0) GOTO 80 + NSTRAA = NSTRAA + NSTRA(ISYM) + MAXAA = MAX(MAXAA,NSTRA(ISYM)) + ICG(1) = 0 + NSTRB(ISYM) = -1 +90 NSTRB(ISYM) = NSTRB(ISYM)+1 + CALL FCSTRS (NB,ICG,ISYM,ITYPEA) + IF (ICG(1).NE.0) GOTO 90 + NSTRBB = NSTRBB + NSTRB(ISYM) + MAXBB = MAX(MAXBB,NSTRB(ISYM)) +100 CONTINUE + DO 110 ISYM=1,8 + NCI(ISYM) = 0 + DO 110 ISYMB=1,8 +110 NCI(ISYM) = NCI(ISYM) + NSTRA(MULT(ISYMB,ISYM))*NSTRB(ISYMB) +C +C... ORBITAL PAIR ADDRESSING, ETC.. + DO 120 I=1,256 +120 ICF(I) = (I*(I-1))/2 + INTADT = ICORI(NACT*NACT) + NINT2=0 + MAXPAR = 0 + DO 140 ISYM=1,8 + INTOFF(ISYM) = NINT2 + NPAIR(ISYM)=0 + NPB = 0 + DO 130 I=1,NACT + DO 130 J=1,I + IF (ISYM.EQ.MULT(ITYPEA(I),ITYPEA(J))) THEN + NPAIR(ISYM) = NPAIR(ISYM)+1 + IQ(INTADT+(I-1)*NACT+(J-1)) = NPAIR(ISYM) + IQ(INTADT+(I-1)+NACT*(J-1)) = NPAIR(ISYM) + END IF +130 CONTINUE + MAXPAR = MAX(NPAIR(ISYM),MAXPAR) +140 NINT2 = NINT2 + NPAIR(ISYM)**2 +C... UPPER BOUND ON NUMBER OF SINGLE REPLACEMENTS FROM ANY STRING + MAXREP = MIN(MAXPAR,MAX(NA*(NACT+1-NA),NB*(NACT+1-NB))) +C + WRITE (IOUT,141) + > NACT,(NOCC(I),I=1,8) + >,NA+NB,(NA-NB)*0.5,NPAIR,NSTRA,NSTRB,NCI +141 FORMAT( + 1' Active orbitals:', T30,I3,'(',8I3,')'/ + 2' Active electrons:', T30,I3/ + 3' Spin quantum number:', T30,F5.1/ + 4' Orbital pairs:', T30,8I9/ + 5' Strings:', T30,8I9/T30,8I9/ + 6' Determinants:', T30,4I12/T30,4I12) +C WRITE (IOUT,'('' ACTIVE ORBITAL SYMMETRIES:'')') +C WRITE (IOUT,150) (ITYPEA(I),I=1,NACT) +C150 FORMAT(1X,40I2) +C +C... LOAD INTEGRALS + INTAA = ICORR(NINT2) + INTHAA = ICORR(NACT) + INTJAA = ICORR(NACT**2) + INTKAA = ICORR(NACT**2) + IBASE = ICORR(0) + IONEA = ICORR(NPAIR(1)) + CALL INPINT (Q(IONEA),Q(INTAA)) +C +C... INTEGRALS FOR DIAGONAL ELEMENTS OF HAMILTONIAN + DO 10 I=1,NACT +10 Q(INTHAA-1+I) = Q(IONEA-1+IQ(INTADT+(I-1)*(NACT+1))) + DO 11 I=1,NACT + DO 11 J=1,NACT + IJ = IQ(INTADT+(I-1)*NACT+J-1) + II = IQ(INTADT+(I-1)*NACT+I-1) + JJ = IQ(INTADT+(J-1)*NACT+J-1) + ISYMIJ = MULT(ITYPEA(I),ITYPEA(J)) + Q(INTJAA+(I-1)*NACT+J-1) = Q(INTAA+(II-1)*NPAIR(1)+JJ-1) +11 Q(INTKAA+(I-1)*NACT+J-1) = + > Q(INTAA+INTOFF(ISYMIJ)+(IJ-1)*(NPAIR(ISYMIJ)+1)) +C +C... MODIFY INTEGRALS SUCH THAT HAMILTONIAN BECOMES +C SUM(IJKL) E(IJ) E(KL) (IJ|KL)" +C (NO 1 ELECTRON TERM REMAINS) +C +C... (IJ|KL)' = 0.5 * (IJ|KL) + DO 20 IJKL=1,NINT2 +20 Q(INTAA-1+IJKL) = Q(INTAA-1+IJKL)*0.5D0 +C +C... H'(IJ) = H(IJ) - SUM(K) (IK|KJ)' + IJ = IONEA-1 + DO 40 I=1,NACT + DO 40 J=1,I + IF (ITYPEA(I).NE.ITYPEA(J)) GOTO 40 + IJ = IJ+1 + DO 30 K=1,NACT + ISYMIK = MULT(ITYPEA(I),ITYPEA(K)) + IK =IQ(INTADT+(I-1)*NACT+K-1)-1 + JK =IQ(INTADT+(J-1)*NACT+K-1)-1 +30 Q(IJ) = Q(IJ) - Q(INTAA+INTOFF(ISYMIK)+IK*NPAIR(ISYMIK)+JK) +40 CONTINUE +C +C... (IJ|KL)" = (IJ|KL)'+(1/2*NELEC)*(DEL(IJ)*H'(KL)+DEL(KL)*H'(IJ)) + FAC = 2*(NA+NB) + FAC = 1D0/FAC + DO 70 K=1,NACT + KK = IQ(INTADT+(K-1)*(NACT+1)) + DO 50 IJ=1,NPAIR(1) +50 Q(INTAA+(KK-1)*NPAIR(1)+IJ-1) = Q(INTAA+(KK-1)*NPAIR(1)+IJ-1) + > + FAC*Q(IONEA-1+IJ) + DO 60 IJ=1,NPAIR(1) +60 Q(INTAA+(IJ-1)*NPAIR(1)+KK-1) = Q(INTAA+(IJ-1)*NPAIR(1)+KK-1) + > + FAC*Q(IONEA-1+IJ) +70 CONTINUE + CALL CORLSR(IBASE) + WRITE (IOUT,'('' Core energy:'',T30,F20.12)') CORE + RETURN + END + SUBROUTINE FSIGMA (C,S) +CSUBR RETURNS S = H * C + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /TAPES / INP,IOUT + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + LOGICAL TIMING + DIMENSION IOFFCI(8),C(*),S(*) + COMMON /CHAMIL/ ISYMH,NNCI + TIMING = IPRTIM.GE.0 + NOPMXM = 0 + NOPMX = 0 + NOPOTH = 0 + NOPCCA = 0 + NOPCCB = 0 + NLOCCA = 0 + NLOCCB = 0 + NOPONE = 0 + T0 = SECOND() + TMXM = 0 + TCCA = 0 + TCCB = 0 + TONE = 0 + IBASE = ICORR(0) + NNCI = 0 + DO 10 ISYMB=1,8 + ISYMA = MULT(ISYMH,ISYMB) + IOFFCI(ISYMB) = NNCI + IF (IPRHAM.GT.0) WRITE (IOUT,*)'ISYMB,ISYMA,NNCI,NSTRA,NSTRB ' + 1,ISYMB,ISYMA,NNCI,NSTRA(ISYMA),NSTRB(ISYMB) +10 NNCI = NNCI + NSTRA(ISYMA)*NSTRB(ISYMB) + IF (IPRHAM.GT.0) CALL OUTVEC (C,NNCI,'c passed to sigma') + IC = ICORI(NSTRAA) + ITZ = ICORI(NSTRBB) + CALL FCMIC (IQ(IC),IQ(ITZ),ISYMH) + DO 170 ISYMA=1,8 + DO 170 ISYMB=1,8 + IBASES = ICORR(0) + ISYMR = MULT(ISYMH,MULT(ISYMA,ISYMB)) + ISYMBE = MULT(ISYMB,ISYMR) + NSBE = NSTRB(ISYMBE) + NSA = NSTRA(ISYMA) + NSB = NSTRB(ISYMB) + NPAIRR = NPAIR(ISYMR) + IF (NSA.EQ.0.OR.NSB.EQ.0.OR.NPAIRR.EQ.0) GOTO 170 + MAXREA = MIN(NA*(NACT-NA+1),NPAIRR) + MAXREB = MAX(1,MIN(NB*(NACT-NB+1),NPAIRR)) + MAXR2A = MAXREA * 2 + MAXR3A = MAXREA * 3 + MAXR2B = MAXREB * 2 + MAXR3B = MAXREB * 3 + IBB = ICORR(0) + IWA = ICORI(MAXR3A*NSA) + NWA = ICORI(NSA) + IWB = ICORI(MAXR3B*NSB) + NWB = ICORI(NSB) +C... CONSTRUCT ALL SINGLE REPLACEMENTS + T1 = SECOND() + INTER = ICORI(NA*NACT) + CALL INITI (IQ(INTER),NA,NACT) + CALL ONELAL (NA,ITYPEA,IQ(INTER),IQ(INTADT),ISYMA,ISYMR + 1 ,IQ(IWA),IQ(NWA),IQ(IC),MAXREA) + CALL CORLSI(INTER) + INTER = ICORI(NB*NACT) + CALL INITI (IQ(INTER),NB,NACT) + CALL ONELAL (NB,ITYPEA,IQ(INTER),IQ(INTADT),ISYMB,ISYMR + 1 ,IQ(IWB),IQ(NWB),IQ(ITZ),MAXREB) + TONE = TONE+SECOND()-T1 + CALL CORLSI(INTER) +C... DETERMINE BLOCKING + NMAT = 2 + MWORD = ICORRM()/NMAT + WORDS = MWORD + NWORD = SQRT(WORDS/NPAIRR) + NABLK = (MIN(NWORD,NSA)/2)*2+1 + DO 20 NBBLK=(NSB/2)*2+1,1,-2 + IF (NABLK*NBBLK*NPAIRR.LE.MWORD) GOTO 30 +20 CONTINUE +30 IDA = ICORR(NABLK*NBBLK*NPAIRR) + IEA = ICORR(NABLK*NBBLK*NPAIRR) + IDB = IDA + IEB = IEA +C... LOOP OVER BLOCKS + DO 160 IOFFA=0,NSA-1,NABLK + NAA = MIN(NABLK,NSA-IOFFA) + DO 160 IOFFB=0,NSB-1,NBBLK + NBB = MIN(NBBLK,NSB-IOFFB) + NAABB = NAA*NBB + IF (TIMING) NOPOTH = NOPOTH+NAABB*NPAIRR + CALL FZERO (Q(IDA),NAABB*NPAIRR) +C... ALPHA REPLACEMENTS + IF (TIMING) THEN + T1 = SECOND() + ICNT = 0 + END IF + IATABB = IWA+IOFFA*MAXR3A + IDAA = IDA-1-NAABB + DO 61 IA=1,NAA + IATAB = IATABB + IF (TIMING) ICNT = ICNT+IQ(NWA+IOFFA-1+IA) + DO 60 IWW=1,IQ(NWA+IOFFA-1+IA) + ICCC = IOFFB + IQ(IATAB) + IAAA = IDAA+IQ(IATAB+MAXR2A)*NAABB + IF (IQ(IATAB+MAXREA).LT.0) THEN +C... EQUATION (18) + DO 40 IB=1,NBB +40 Q(IAAA+IB) = - C(ICCC+IB) + ELSE +C... EQUATION (18) + DO 50 IB=1,NBB +50 Q(IAAA+IB) = C(ICCC+IB) + END IF +60 IATAB = IATAB+1 + IDAA = IDAA + NBB +61 IATABB = IATABB + MAXR3A + IF (TIMING) THEN + TCCA = TCCA + SECOND()-T1 + NOPCCA = NOPCCA + ICNT*NBB + NLOCCA = NLOCCA + ICNT + END IF +C... BETA REPLACEMENTS + IF (TIMING) THEN + ICNT = 0 + T1 = SECOND() + END IF + IBTABB = IWB+IOFFB*MAXR3B + ICCCC = IOFFCI(ISYMBE)+(IOFFA-1)*NSBE + IDBB = IDB-NBB-NAABB + DO 91 IB=1,NBB + IBTAB = IBTABB + IF (TIMING) ICNT = ICNT+IQ(NWB+IOFFB-1+IB) + DO 90 IWW=1,IQ(NWB-1+IOFFB+IB) + ICCC = ICCCC+IQ(IBTAB) + IBBB = IDBB+IQ(IBTAB+MAXR2B)*NAABB + IF (IQ(IBTAB+MAXREB).LT.0) THEN +C... EQUATION (19) + DO 70 IA=1,NAA +70 Q(IBBB+IA*NBB) = Q(IBBB+IA*NBB) - C(ICCC+IA*NSBE) + ELSE +C... EQUATION (19) + DO 80 IA=1,NAA +80 Q(IBBB+IA*NBB) = Q(IBBB+IA*NBB) + C(ICCC+IA*NSBE) + END IF +90 IBTAB = IBTAB + 1 + IDBB = IDBB + 1 +91 IBTABB = IBTABB + MAXR3B + IF (TIMING) THEN + TCCB = TCCB + SECOND()-T1 + NOPCCB = NOPCCB + ICNT*NAA + NLOCCB = NLOCCB + ICNT + END IF +C +C... MATRIX MULTIPLICATION WITH INTEGRALS +C... EQUATION (20) + IF (TIMING) T1 = SECOND() + IF (IPRHAM.GT.3) + 1CALL OUTSQR (Q(IDA),NAABB,NAABB,NPAIRR,'DA') + IF (IPRHAM.GT.3) + 1CALL OUTSQR (Q(INTAA+INTOFF(ISYMR)),NPAIRR,NPAIRR,NPAIRR,'INTAA') + CALL FCMXM (Q(IDA),Q(INTAA+INTOFF(ISYMR)),Q(IEA),NAABB,NPAIRR, + >NOPMXM) + NOPMX = NOPMX+2*NAABB*NPAIRR**2 + IF (IPRHAM.GT.3) + 1CALL OUTSQR (Q(IEA),NAABB,NAABB,NPAIRR,'EA') + IF (TIMING) TMXM = TMXM+SECOND()-T1 +C +C... ALPHA REPLACEMENTS + IF (TIMING) THEN + T1 = SECOND() + ICNT = 0 + END IF + IATABB = IWA+IOFFA*MAXR3A + IEAA = IEA-1-NAABB + DO 121 IA=1,NAA + IATAB = IATABB + IF (TIMING) ICNT = ICNT+IQ(NWA+IOFFA-1+IA) + DO 120 IWW=1,IQ(NWA+IOFFA-1+IA) + ICCC = IOFFB + IQ(IATAB) + IAAA = IEAA+IQ(IATAB+MAXR2A)*NAABB + IF (IQ(IATAB+MAXREA).LT.0) THEN +C... EQUATION (21) + DO 100 IB=1,NBB +100 S(ICCC+IB) = S(ICCC+IB) - Q(IAAA+IB) + ELSE +C... EQUATION (21) + DO 110 IB=1,NBB +110 S(ICCC+IB) = S(ICCC+IB) + Q(IAAA+IB) + END IF +120 IATAB = IATAB + 1 + IEAA = IEAA + NBB +121 IATABB = IATABB + MAXR3A + IF (TIMING) THEN + TCCA = TCCA + SECOND()-T1 + NOPCCA = NOPCCA + ICNT*NBB + NLOCCA = NLOCCA + ICNT + END IF +C... BETA REPLACEMENTS + IF (TIMING) THEN + T1 = SECOND() + ICNT = 0 + END IF + IBTABB = IWB+IOFFB*MAXR3B + ICCCC = IOFFCI(ISYMBE)+(IOFFA-1)*NSBE + IEBB = IEB-NBB-NAABB + DO 151 IB=1,NBB + IBTAB = IBTABB + IF (TIMING) ICNT = ICNT+IQ(NWB+IOFFB-1+IB) + DO 150 IWW=1,IQ(NWB-1+IOFFB+IB) + ICCC = ICCCC+IQ(IBTAB) + IBBB = IEBB+IQ(IBTAB+MAXR2B)*NAABB + IF (IQ(IBTAB+MAXREB).LT.0) THEN +C... EQUATION (22) + DO 130 IA=1,NAA +130 S(ICCC+IA*NSBE) = S(ICCC+IA*NSBE) - Q(IBBB+IA*NBB) + ELSE +C... EQUATION (22) + DO 140 IA=1,NAA +140 S(ICCC+IA*NSBE) = S(ICCC+IA*NSBE) + Q(IBBB+IA*NBB) + END IF +150 IBTAB = IBTAB + 1 + IEBB = IEBB + 1 +151 IBTABB = IBTABB + MAXR3B + IF (TIMING) THEN + TCCB = TCCB + SECOND()-T1 + NOPCCB = NOPCCB + ICNT*NAA + NLOCCB = NLOCCB + ICNT + END IF +160 CONTINUE + CALL CORLSR(IBASES) +170 CONTINUE + CALL CORLSR(IBASE) + IF (TIMING) THEN + TTOT = SECOND()-T0 + TOTH = TTOT-TMXM-TCCA-TCCB-TONE + NOPTOT = NOPOTH+NOPMXM+NOPCCA+NOPCCB+NOPONE + CALL TPRINT ('String c. coeff',NOPONE,TONE,' ',0) + CALL TPRINT ('Coupl coeffs(A)',NOPCCA,TCCA,'Av. v length' + >,NOPCCA/NLOCCA) + CALL TPRINT ('Coupl coeffs(B)',NOPCCB,TCCB,'Av. v length' + >,NOPCCB/NLOCCB) + IAVSP = NINT(1D2-(1D2*NOPMXM)/NOPMX) + CALL TPRINT ('Matrix multiply',NOPMXM,TMXM,'Av. % sparsity',IAVSP) + CALL TPRINT ('Other ',NOPOTH,TOTH,' ',0) + CALL TPRINT ('Total ',NOPTOT,TTOT,' ',0) + END IF + IF (IPRHAM.GT.0) + 1CALL OUTVEC (S,NNCI,'S RETURNED BY SIGMA') + RETURN + END + SUBROUTINE ONELAL (N,ITYPE,INTER,INTAD,ISYM,ISYMR,IW,NW,ITZ,MAXRE) +CSUBR CONSTRUCT AND STORE ALL ONE ELECTRON REPLACEMENTS + IMPLICIT DOUBLE PRECISION (A-H,O-Z) +C... MAXIMUM NUMBERS OF ORBITALS IN ANY SYMMETRY MUST BE LESS THANTHIS! + PARAMETER (MXORB=64) + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + DIMENSION ITYPE(*),INTER(N,*),INTAD(*) + 1 ,IW(*),NW(*),ITZ(*) +C 1 ,IW(MAXRE,3,*),NW(*),ITZ(*) + DIMENSION ICG(32),ICGI(33),JSTOR1(MXORB),JSTOR2(MXORB) + >,JSTOR3(MXORB) + nw(1) = 0 + if (n.le.0) return + MAXRE2 = MAXRE*2 + MAXRE3 = MAXRE*3 +C... SET COUNTERS TO ZERO AND SET UP LOOK-UP FOR BRAS + NINSYM = IBINOM(NACT,N) + ITZI = ICORI(NINSYM) + NI = 0 + ICGI(1) = 0 + DO 23 II=1,2999999 + IISYM = 0 + CALL FCSTRS (N,ICGI,IISYM,ITYPE) + IF (ICGI(1).EQ.0) GOTO 24 + IF (IISYM.NE.ISYM) GOTO 23 + NI=NI+1 + IQ(ITZI-1+II) = NI +23 CONTINUE +24 CONTINUE + DO 25 I=1,NI +25 NW(I) = 0 + IF (NI.EQ.0) GOTO 999 +C... LOOP OVER INTERMEDIATE N-1 ELECTRON STRINGS K + ICG(1) = 0 + DO 50 ISK=1,2999999 + ISYMK = 0 + CALL FCSTRS (N-1,ICG,ISYMK,ITYPE) + IF (ICG(1).EQ.0) GOTO 999 + ICG(N) = NACT+1 + ISYMIO = MULT(ISYMK,ISYM) + ISYMJO = MULT(ISYMR,ISYMIO) + ISYMJ = MULT(ISYM,ISYMR) +C... CONSTRUCT CREATIONS TO KETS J (SYMMETRY ISYMJ) +C DO 777 IREP=1,10 + IREPL = 1 + IADR = 1 + IPARIT = 1 + DO 10 J=1,N-1 +10 IADR = IADR + INTER(J+1,ICG(J)) + NJ = 0 + DO 30 JO=1,NACT + IF(JO.EQ.ICG(IREPL))THEN + IADR = IADR+INTER(IREPL,JO)-INTER(IREPL+1,JO) + IREPL = IREPL+1 + IPARIT = -IPARIT + ELSE IF (ITYPE(JO).EQ.ISYMJO) THEN +C... GET CONNECTED N ELECTRON DETERMINANT J +C... STORE INFO + NJ = NJ+1 + JSTOR1(NJ) = ITZ(IADR+INTER(IREPL,JO)) + JSTOR2(NJ) = IPARIT + JSTOR3(NJ) = JO + END IF +30 CONTINUE +C777 CONTINUE + IF (NJ.EQ.0) GOTO 50 +C... CONSTRUCT CREATIONS TO BRAS I (SYMMETRY ISYM) + IREPL = 1 + IADR = ITZI + IPARIT = 1 + DO 110 J=1,N-1 +110 IADR = IADR + INTER(J+1,ICG(J)) + DO 130 IO=1,NACT + IF(IO.EQ.ICG(IREPL))THEN + IADR = IADR+INTER(IREPL,IO)-INTER(IREPL+1,IO) + IREPL = IREPL+1 + IPARIT = -IPARIT + ELSE IF (ITYPE(IO).EQ.ISYMIO) THEN +C... GET CONNECTED N ELECTRON DETERMINANT I + I = IQ(IADR+INTER(IREPL,IO)) + IOFFS = (I-1)*MAXRE3+NW(I) + IOO = (IO-1)*NACT +C... STORE INFO FOR EACH J + IF (IPARIT.GT.0) THEN +CDIR$ SHORTLOOP +CDIR$ IVDEP +C$DIR NO_RECURRENCE +*VOPTION INDEP VEC + DO 45 J=1,NJ + IW(IOFFS +J) = JSTOR1(J) + IW(IOFFS+MAXRE2+J) = JSTOR3(J)+IOO +C IW(IOFFS+MAXRE2+J) = INTAD(JSTOR3(J)+IOO) +45 IW(IOFFS+MAXRE +J) = JSTOR2(J) + ELSE +CDIR$ SHORTLOOP +CDIR$ IVDEP +C$DIR NO_RECURRENCE +*VOPTION INDEP VEC + DO 46 J=1,NJ + IW(IOFFS +J) = JSTOR1(J) + IW(IOFFS+MAXRE2+J) = JSTOR3(J)+IOO +C IW(IOFFS+MAXRE2+J) = INTAD(JSTOR3(J)+IOO) +46 IW(IOFFS+MAXRE +J) = -JSTOR2(J) + END IF + NW(I) = NW(I) + NJ + END IF +130 CONTINUE +50 CONTINUE +999 CALL CORLSI(ITZI) +C... GATHER LOOP FOR PACKED INTEGRAL ADDRESSES + IOFFS = MAXRE2 + DO 147 I=1,NI + DO 148 J=1,NW(I) +148 IW(IOFFS+J) = INTAD(IW(IOFFS+J)) +147 IOFFS = IOFFS + MAXRE3 + RETURN + END + SUBROUTINE FCDIAG (S) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /TAPES / INP,IOUT + COMMON /CHAMIL/ ISYMH,NNCI + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + DIMENSION S(*) + COMMON /CLOCAL/ ICGA(32),ICGB(32) + IF (IPRTIM.GE.0) T0 = SECOND() + NOP = 0 + IF (IPRDIA.GT.0) THEN +C WRITE (IOUT,*) 'COMPUTE DIAGONAL ELEMENTS' + CALL OUTVEC (Q(INTHAA),NACT,'HAA') + CALL OUTSQR (Q(INTJAA),NACT,NACT,NACT,'INTJAA') + CALL OUTSQR (Q(INTKAA),NACT,NACT,NACT,'INTKAA') + END IF +C IF (IPRDIA.GT.1) WRITE (IOUT,*) 'MAXAA,MAXBB ',MAXAA,MAXBB + IZB = ICORR(NACT*MAXBB) + IZA = ICORR(NACT*MAXAA) + IZ = ICORR(NACT*MAXBB) + IF = ICORR(MAXBB) + ICCC=0 + DO 220 ISYMB=1,8 + ISYMA=MULT(ISYMB,ISYMH) + NSA = NSTRA(ISYMA) + NSB = NSTRB(ISYMB) + IF (IPRDIA.GT.1) WRITE (IOUT,*) 'ISYMB,ISYMA,NSA,NSB ',ISYMB + 1,ISYMA,NSA,NSB + IF (NSA*NSB.LE.0) GOTO 220 + CALL FZERO (Q(IZB),NACT*NSB) +C.. STORE BETA STRING OCCUPANCIES + ICGB(1) = 0 + DO 20 IB=1,NSB + CALL FCSTRS (NB,ICGB,ISYMB,ITYPEA) + DO 10 II=1,NB +10 Q(IZB-1+IB+(ICGB(II)-1)*NSB)=1 +20 CONTINUE + IF (IPRDIA.GT.2) CALL OUTSQR (Q(IZB),NSB,NSB,NACT,'ZB') +C +C.. STORE ALPHA STRING OCCUPANCIES + ICGA(1) = 0 + CALL FZERO (Q(IZA),NACT*NSA) + DO 40 IA=1,NSA + CALL FCSTRS (NA,ICGA,ISYMA,ITYPEA) + DO 40 II=1,NA +40 Q(IZA-1+IA+(ICGA(II)-1)*NSA) = 1 + IF (IPRDIA.GT.2) CALL OUTSQR (Q(IZA),NSA,NSA,NACT,'ZA') +C + DO 50 IAB=1,NSA*NSB +50 S(ICCC+IAB) = 0 +C +C... SPECIAL CODE FOR RHF ORBITALS + DO 210 IA=1,NSA +C... OCCUPATION NUMBERS OF DETERMINANTS + DO 100 I=1,NACT + DO 100 IB=1,NSB +100 Q(IZ-1+IB+(I-1)*NSB) = Q(IZB-1+IB+(I-1)*NSB) + 1 + Q(IZA-1+IA+(I-1)*NSA) + IF (IPRDIA.GT.3) CALL OUTSQR (Q(IZ),NSB,NSB,NACT,'Z') +C +C... CONSTANT EXCHANGE CONTRIBUTION & COULOMB +C... 1/2 SUM(IJ) N(I) N(J) ((II|JJ)-0.5(IJ|JI)) +C... EQUATIONS (23), (25) + IJ = -1 + DO 130 I=1,NACT + ZZ = Q(INTHAA-1+I) + IF (IPRDIA.GT.3) WRITE (IOUT,*) '1567;I,HAA(I) ',I,ZZ + DO 110 IB=1,NSB +110 S(ICCC+IB) = S(ICCC+IB) + ZZ*Q(IZ-1+IB+(I-1)*NSB) + DO 130 J=1,I + IJ = (I-1)*NACT-1+J + ZZ = Q(INTJAA+IJ)-0.5D0*Q(INTKAA+IJ) + IF (I.EQ.J) ZZ = ZZ*0.5D0 + IF (IPRDIA.GT.3) WRITE (IOUT,*) '1567; I,J,ZZ ',I,J,ZZ + DO 120 IB=1,NSB +120 S(ICCC+IB) = S(ICCC+IB) + 1 + ZZ*Q(IZ-1+IB+(I-1)*NSB)*Q(IZ-1+IB+(J-1)*NSB) +130 CONTINUE +C + IF (IPRDIA.GT.3) CALL OUTVEC (S(ICCC+1),NSB,'S AFTER CONSTS') +C... VV = 4*S**2, SEE EQUATION (26) + VV = (NA-NB)**2 +C.. Z TO HOLD 1 IF SINGLY OCC, 0 OTHERWISE +C... (CAPITAL N = N(2-N), EQUATION (25)) + CALL FZERO (Q(IF),NSB) + DO 150 I=1,NACT + DO 140 IB=1,NSB +140 Q(IZ-1+IB+(I-1)*NSB) = Q(IZ-1+IB+(I-1)*NSB) + 1 * (2.0D0-Q(IZ-1+IB+(I-1)*NSB)) + DO 150 IB=1,NSB +150 Q(IF-1+IB) = Q(IF-1+IB) + Q(IZ-1+IB+(I-1)*NSB) + DO 160 IB=1,NSB +160 Q(IF-1+IB) = MAX(Q(IF-1+IB),1.1D0) + DO 170 IB=1,NSB +170 Q(IF-1+IB) = (VV-Q(IF-1+IB))/( Q(IF-1+IB) * (Q(IF-1+IB)-1.0D0) ) +C.. F NOW HOLDS F FACTOR, EQUATION (26) +C + IF (IPRDIA.GT.3) CALL OUTVEC (Q(IF),NSB,'F') + IJ = -1 + DO 200 I=1,NACT + IF (I.EQ.1) GOTO 190 + DO 180 J=1,I-1 + IJ = (I-1)*NACT-1+J + ZZ = -0.5D0*Q(INTKAA+IJ) + DO 180 IB=1,NSB +180 S(ICCC+IB) = S(ICCC+IB) + 1 + ZZ*Q(IF-1+IB)*Q(IZ-1+IB+(I-1)*NSB)*Q(IZ-1+IB+(J-1)*NSB) +190 IJ = (I-1)*NACT-1+I + ZZ = -0.25D0*Q(INTKAA+IJ) + IF (IPRDIA.GT.3) WRITE (IOUT,*) '1875; I,ZZ ',I,ZZ + DO 200 IB=1,NSB +200 S(ICCC+IB) = S(ICCC+IB) + ZZ*Q(IZ-1+IB+(I-1)*NSB) +210 ICCC = ICCC+NSB + NOP = NOP+NSA*NSB*(NACT*7+3*NACT*(NACT+1)+5) +220 CONTINUE + CALL CORLSR(IZB) + IF (IPRDIA.GE.0) CALL OUTVEC (S,ICCC,'DIAGONAL ELEMENTS') + IF (IPRTIM.GE.0) CALL TPRINT ('Hamiltonian diag. els.' + >,NOP,SECOND()-T0,' ',0) + RETURN + END + FUNCTION INICOR(MEMREQ) +C + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + INTEGER CORMON,CORLSR,CORLSI,ENDCOR + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /TAPES / INP,IOUT + COMMON /CORCTL/ INTREL,LWORD,LTOP,LMAX,LMIN,MREAL,MXBFF,IBASE,MEM + SAVE ICOUNT + DATA ICOUNT/0/ +C +C... EXECUTE ENDCOR CODE IF NOT FIRST CALL TO THIS ROUTINE + IF (ICOUNT.NE.0) CALL FMAIN (Q(IBASE+1),MEM) + ICOUNT = ICOUNT+1 +C + MEM = MEMREQ + LENN = MEM+2 +C +C + CALL GMAINV (Q,IBASE,LENN) + IBASE = IBASE-1 + IF (LENN.LT.MEM) THEN + WRITE (IOUT,'('' Request for'',I9,'' words refused'', + > '', '',I9,'' available'')') MEM,LENN + STOP 'Insufficient memory' + END IF + WRITE (IOUT,'('' Variable memory set to '',I10,'' words'')') MEM + LWORD = MEM+IBASE + LTOP = IBASE + LMAX = IBASE + LMIN = IBASE + INICOR = LWORD-IBASE + RETURN +C + ENTRY ICORIM () + ICORIM = (LWORD - LTOP) * INTREL + RETURN +C + ENTRY ICORRM () + ICORRM = LWORD - LTOP + RETURN +C + ENTRY ICORI (NWOR) + NWORD = IABS(NWOR) + NW = (NWORD+INTREL-1)/INTREL + MFR = LWORD - LTOP + IF (NW .GT. MFR) GOTO 20 + ICORI = LTOP*INTREL+1 + LTOP=LTOP+NW + IF (NWOR.LT.0) LMAX = MAX0(LMAX,LTOP) + IF (NWOR.GT.0) LMIN = MAX0(LMIN,LTOP) + RETURN +C + ENTRY ICORR (NWOR) + NWORD = IABS(NWOR) + NW = NWORD + MFR = LWORD - LTOP + IF (NW .GT. MFR) GOTO 20 + ICORR = LTOP+1 + LTOP=LTOP+NW + IF (NWOR.LT.0) LMAX = MAX0(LTOP,LMAX) + IF (NWOR.GT.0) LMIN = MAX0(LTOP,LMIN) + RETURN +C +20 CONTINUE + WRITE (IOUT,*) 'insufficient memory available - require ',NW, + > ' have ',MFR + IF (NW .NE. NWORD) THEN + WRITE(IOUT,*) 'the request was for ',NWORD,' integer words' + ELSE + WRITE(IOUT,*) 'the request was for real words' + END IF + STOP 'Insufficient memory' + END + SUBROUTINE CORLSI (IAD) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + COMMON /TAPES / INP,IOUT + COMMON /CORCTL/ INTREL,LWORD,LTOP,LMAX,LMIN,MREAL,MXBFF,IBASE,MEM + LTOP = (IAD-2+INTREL)/INTREL + RETURN +C + ENTRY CORLSR (IAD) + LTOP = IAD - 1 + RETURN +C + ENTRY CORMON() + WRITE (IOUT,40) LMIN-IBASE,LWORD-LMIN +40 FORMAT(/1X,'Minimal core high water =',I8, + 1' real numbers (spare core =',I8,')') + RETURN +C + ENTRY ENDCOR() + CALL FMAIN (Q(IBASE+1),MEM) + WRITE (IOUT,'('' Variable memory released'')') + RETURN + END + FUNCTION FCISTR (IPAR,INTER,ICG,N) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + DIMENSION ICG(N),INTER(N,*) + DIMENSION ICGG(32) + DO 10 J=1,N +10 ICGG(J) = ICG(J) + IF (N.LT.2) GOTO 40 + DO 30 JJ=2,N + JJM = JJ-1 + I = JJM + DO 20 II=1,JJM + IF (ICGG(I).LT.ICGG(I+1)) GOTO 20 + J1=ICGG(I) + ICGG(I)=ICGG(I+1) + ICGG(I+1)=J1 + IPAR=-IPAR +20 I=I-1 +30 CONTINUE +40 CONTINUE + FCISTR=1 + DO 50 J=1,N +50 FCISTR = FCISTR+INTER(J,ICGG(J)) + RETURN + END + SUBROUTINE FCMIC (IC,ITZ,ISYM) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + DIMENSION ICG(32) + DIMENSION IC(*),ITZ(*) + DO 1 I=1,8 + NSTRA(I) = 0 +1 NSTRB(I) = 0 + NSTRAT = IBINOM(NACT,NA) + NSTRBT = IBINOM(NACT,NB) +C... ALPHA STRING INFORMATION + ICG(1) = 0 + DO 11 MT=1,NSTRAT + JMT = 0 + CALL FCSTRS (NA,ICG,JMT,ITYPEA) + NSTRA(JMT) = NSTRA(JMT)+1 +C... STORE SYMMETRY FOR LATER PROCESSING +11 IC(MT) = -JMT +C... BETA STRING INFORMATION + ICG(1) = 0 + DO 10 MT=1,NSTRBT + JMT = 0 + CALL FCSTRS (NB,ICG,JMT,ITYPEA) + NSTRB(JMT) = NSTRB(JMT)+1 +10 ITZ(MT) = NSTRB(JMT) +C +C... LOOP OVER SYMMETRIES BUILDING TOTAL OFFSETS + NNCI = 0 + DO 110 ISYMB=1,8 + ISYMA = MULT(ISYM,ISYMB) + DO 120 MT=1,NSTRAT + IF (IC(MT).NE.-ISYMA) GOTO 120 + IC(MT) = NNCI + NNCI = NNCI + NSTRB(ISYMB) +120 CONTINUE +110 CONTINUE + RETURN + END + SUBROUTINE FCSPAD (C,IREF) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + COMMON /BIG/ Q(1) + INTEGER IQ(1) + EQUIVALENCE (Q(1),IQ(1)) + DIMENSION C(*) + COMMON /CHAMIL/ ISYMH,NNCI + DIMENSION ICGA(32),ICGB(32),IDOCA(32) + 1 ,IDOCB(32),IALP(32),IBET(32) + DO 10 I=1,NNCI +10 C(I)=0 + IBASE = ICORR(0) + INTERA = ICORI(NA*NACT) + INTERB = ICORI(NB*NACT) + IC = ICORI(NSTRAA) + ITZ = ICORI(NSTRBB) + CALL INITI (IQ(INTERA),NA,NACT) + CALL INITI (IQ(INTERB),NB,NACT) + CALL FCMIC (IQ(IC),IQ(ITZ),ISYMH) + CALL FCSTRG (IREF,ISYMH,ICGA,ICGB) + NDOC=0 + NALP=0 + NBET=0 + DO 70 I=1,NACT + ICASE=1 + DO 20 J=1,NA + IF (ICGA(J).NE.I) GOTO 20 + ICASE=3 + JA=J +20 CONTINUE + DO 30 J=1,NB + IF (ICGB(J).NE.I) GOTO 30 + ICASE=ICASE+1 + JB=J +30 CONTINUE + GOTO (70,40,50,60),ICASE +40 NBET=NBET+1 + IBET(NBET)=JB + GOTO 70 +50 NALP=NALP+1 + IALP(NALP)=JA + GOTO 70 +60 NDOC=NDOC+1 + IDOCA(NDOC)=JA + IDOCB(NDOC)=JB +70 CONTINUE + FAC = 1.0D0/SQRT(DBLE(2**NBET)) + IF (NBET.GT.5) STOP 5 + DO 120 I5=1,2 + DO 110 I4=1,2 + DO 100 I3=1,2 + DO 90 I2=1,2 + DO 80 I1=1,2 + IPAR = 1 + MTA = FCISTR (IPAR,IQ(INTERA),ICGA,NA) + MTB = FCISTR (IPAR,IQ(INTERB),ICGB,NB) + C(IQ(IC-1+MTA)+IQ(ITZ-1+MTB)) = DBLE(IPAR)*FAC + IF (NBET.LT.1) GOTO 130 + J1=ICGB(IBET(1)) + ICGB(IBET(1))=ICGA(IALP(1)) + ICGA(IALP(1))=J1 +80 CONTINUE + IF (NBET.LT.2) GOTO 130 + J1=ICGB(IBET(2)) + ICGB(IBET(2))=ICGA(IALP(2)) + ICGA(IALP(2))=J1 +90 CONTINUE + IF(NBET.LT.3) GOTO 130 + J1=ICGB(IBET(3)) + ICGB(IBET(3))=ICGA(IALP(3)) + ICGA(IALP(3))=J1 +100 CONTINUE + IF (NBET.LT.4) GOTO 130 + J1=ICGB(IBET(4)) + ICGB(IBET(4))=ICGA(IALP(4)) + ICGA(IALP(4))=J1 +110 CONTINUE + IF (NBET.LT.5) GOTO 130 + J1=ICGB(IBET(5)) + ICGB(IBET(5))=ICGA(IALP(5)) + ICGA(IALP(5))=J1 +120 CONTINUE +130 CONTINUE + CALL CORLSR (IBASE) + RETURN + END + SUBROUTINE FCSTRG (II,ISYM,ICGA,ICGB) +CSUBR EXTRACT STRINGS ICGA,ICGB FOR CONFIGURATION II IN SYMMETRY ISYM + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + DIMENSION ICGA(*),ICGB(*) + III = II + DO 10 ISYMB=1,8 + ISYMA = MULT(ISYMB,ISYM) + NSB = NSTRB(ISYMB) + NSA = NSTRA(ISYMA) + IF (III.LE.NSB*NSA) GOTO 20 +10 III = III - NSB*NSA +20 ISA = (III-1)/NSB+1 + ISB = III - (ISA-1)*NSB + ICGA(1) = 0 + DO 30 JSA=1,ISA +30 CALL FCSTRS (NA,ICGA,ISYMA,ITYPEA) + ICGB(1) = 0 + DO 40 JSB=1,ISB +40 CALL FCSTRS (NB,ICGB,ISYMB,ITYPEA) + RETURN + END + SUBROUTINE FCSTRS (N,ICG,ISYM,ITYPE) +C... GENERATE A NEW ALPHA OR BETA STRING WITH SYMMETRY ISYM +C... ICG SHOULD BE PRESERVED BETWEEN CALLS +C... IF (ISYM.EQ.0) ON ENTRY, THEN JUST NEXT STRING GENERATED +C... ITS SYMMETRY RETURNED IN ISYM +C + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + DIMENSION ICG(*),ITYPE(*) +C + IF (ICG(1).LE.0) THEN + DO 10 I=1,N +10 ICG(I)=I + IF (N.LE.0) THEN + DO 111 JJ=1,NACT + IF (ITYPE(JJ).EQ.1) GOTO 112 +111 CONTINUE +112 ICG(1) = JJ + END IF + GOTO 40 + END IF +C +21 DO 20 IU=N,1,-1 + ICG(IU) = ICG(IU) + 1 + IF (ICG(IU).LE.NACT+IU-N) GOTO 30 +20 CONTINUE + GOTO 60 +CDIR$ NEXTSCALAR +*VOPTION NOVEC +30 DO 31 JU=IU+1,N +31 ICG(JU) = ICG(IU)+JU-IU +C +40 JMT = ITYPE(ICG(1)) + DO 50 J=2,N +50 JMT = MULT(JMT,ITYPE(ICG(J))) + IF (JMT.NE.ISYM.AND.ISYM.NE.0) GOTO 21 + IF (ISYM.EQ.0) ISYM = JMT + RETURN +60 ICG(1) = 0 + RETURN + END + FUNCTION IBINOM (M,N) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + TOP = M + NN = MIN0(M-N,N) + IF (NN.LT.0) GOTO 20 + BOT = NN + BINOM = 1.0D0 + DO 10 I=1,NN + BINOM = BINOM * TOP/BOT + TOP = TOP - 1.0D0 +10 BOT = BOT - 1.0D0 + IBINOM = BINOM + 0.1D0 + IF (DABS(BINOM-DBLE(IBINOM)).GT.0.5D0) + 1 STOP 'ARITHMETIC PROBLEM IN COMPUTING BINOMIAL COEFFICIENT' + RETURN +20 IBINOM = 0 + RETURN + END + SUBROUTINE INITI (INTER,NITEM,NBOX) +CSUBR SETS UP PARTIAL WEIGHT ARRAY FOR ADDRESSING BINOMIAL DISTRIBUTIONS + DIMENSION INTER(NITEM,*) + N1=NITEM+1 + DO 30 K=1,NITEM + DO 10 L=1,NBOX +10 INTER(K,L)=0 + DO 20 L=K,NBOX-NITEM+K-1 + INTER(K,L+1) = IBINOM(NBOX-L,NITEM-K)+INTER(K,L) +20 CONTINUE +30 CONTINUE + DO 40 K=1,NITEM-1 + DO 40 L=K,NBOX-NITEM+K +40 INTER(K,L) = INTER(K,L) - INTER(K+1,L+1) + DO 50 L=NITEM,NBOX +50 INTER(NITEM,L) = L-NITEM + RETURN + END + SUBROUTINE FZERO (A,N) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + DIMENSION A(*) + DO 1 I=1,N +1 A(I) = 0D0 + RETURN + END + SUBROUTINE OUTSQR (Q,IDIM,IA,IB,TITLE) + IMPLICIT DOUBLE PRECISION(A-H,O-Z) + CHARACTER*(*) TITLE + DIMENSION Q(IDIM,1) + CHARACTER*10 FORMAT + DATA FORMAT /'(/1X,A000)'/ + WRITE (FORMAT(7:9),'(I3)') 55+LEN(TITLE)/2 + WRITE (6,FORMAT) TITLE + M=1 + N=8 +10 IF (IB.LT.M) RETURN + N=MIN0(N,IB) + WRITE (6,30) (I,I=M,N) + WRITE (6,40) + DO 20 J=1,IA + WRITE (6,50) J,(Q (J,I) ,I=M,N) +20 CONTINUE + M=M+8 + N=N+8 + GOTO 10 +30 FORMAT(//3X,8I14) +40 FORMAT(/) +50 FORMAT(7X,I3,8F14.7) + END + SUBROUTINE OUTVEC (P,IB,TITLE) + IMPLICIT DOUBLE PRECISION(A-H,O-Z) + COMMON /TAPES / INP,IOUT + CHARACTER*(*) TITLE + DIMENSION P(1) + WRITE (IOUT,'(/1X,(A))') TITLE + M=1 + N=10 +10 IF (IB.LT.M) RETURN + N=MIN0(N,IB) + WRITE(IOUT,20) M,N,(P (I) ,I=M,N) + M=M+10 + N=N+10 + GOTO 10 +20 FORMAT(I4,'-',I3,10F12.6) + END + SUBROUTINE DIAG2(M,N,D,X) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + PARAMETER (MAXDIM=200) + DIMENSION D(M), X(M,M) + DIMENSION E(MAXDIM) + IF(M.GT.MAXDIM) THEN + WRITE(6,10) M,MAXDIM +10 FORMAT(' DIMENSION TOO LARGE IN DIAG2:',2I8) + STOP + END IF +CSTART VAX UNIX-HP UNIX-CONVEX CRAY IBM UNIVAC + EPS=2.5E-13 + TOL=2.5D-18 +CEND +C + IF(N.EQ.1) GO TO 400 +C +C HOUSEHOLDER'S REDUCTION +C SIMULATION OF LOOP DO 150 I=N,2,(-1) +C + DO 150 NI=2,N + I=N+2-NI + L=I-2 + H=0.0 + G=X(I,I-1) + IF(L) 140,140,20 + 20 DO 30 K=1,L + 30 H=H+X(I,K)**2 + S=H+G*G + IF(S.GE.TOL) GO TO 50 + 40 H=0.0 + GO TO 140 + 50 IF(H) 140,140,60 + 60 L=L+1 + F=G + G=DSQRT(S) + IF(F) 75,75,70 + 70 G=-G + 75 H=S-F*G + X(I,I-1)=F-G + F=0.0 +C + DO 110 J=1,L + X(J,I)=X(I,J)/H + S=0.0 + DO 80 K=1,J + 80 S=S+X(J,K)*X(I,K) + J1=J+1 + IF(J1.GT.L) GO TO 100 + DO 90 K=J1,L + 90 S=S+X(K,J)*X(I,K) + 100 E(J)=S/H + 110 F=F+S*X(J,I) +C + F=F/(H+H) +C + DO 120 J=1,L + 120 E(J)=E(J)-F*X(I,J) +C + DO 130 J=1,L + F=X(I,J) + S=E(J) + DO 130 K=1,J + 130 X(J,K)=X(J,K)-F*E(K)-X(I,K)*S +C + 140 D(I)=H + 150 E(I-1)=G +C +C ACCUMULATION OF TRANSFORMATION MATRICES +C + 160 D(1)=X(1,1) + X(1,1)=1.0 + DO 220 I=2,N + L=I-1 + IF(D(I)) 200,200,170 + 170 DO 190 J=1,L + S=0.0 + DO 180 K=1,L + 180 S=S+X(I,K)*X(K,J) + DO 190 K=1,L + 190 X(K,J)=X(K,J)-S*X(K,I) + 200 D(I)=X(I,I) + X(I,I)=1.0 + 210 DO 220 J=1,L + X(I,J)=0.0 + 220 X(J,I)=0.0 +C +C DIAGONALIZATION OF THE TRIDIAGONAL MATRIX +C + B=0.0 + F=0.0 + E(N)=0.0 +C + DO 340 L=1,N + H=EPS*(DABS(D(L))+DABS(E(L))) + IF (H.GT.B) B=H +C +C TEST FOR SPLITTING +C + DO 240 J=L,N + IF (DABS(E(J)).LE.B) GOTO 250 + 240 CONTINUE +C +C TEST FOR CONVERGENCE +C + 250 IF(J.EQ.L) GO TO 340 +C +C SHIFT FROM UPPER 2*2 MINOR +C + 260 P=(D(L+1)-D(L))*0.5/E(L) + R=DSQRT(P*P+1.0) + IF(P) 270,280,280 + 270 P=P-R + GO TO 290 + 280 P=P+R + 290 H=D(L)-E(L)/P + DO 300 I=L,N + 300 D(I)=D(I)-H + F=F+H +C +C QR TRANSFORMATION +C + P=D(J) + C=1.0 + S=0.0 +C +C SIMULATION OF LOOP DO 330 I=J-1,L,(-1) +C + J1=J-1 + DO 330 NI=L,J1 + I=L+J1-NI + G=C*E(I) + H=C*P +C +C PROTECTION AGAINST UNDERFLOW OF EXPONENTS +C + IF (DABS(P).LT.DABS(E(I))) GOTO 310 + C=E(I)/P + R=DSQRT(C*C+1.0) + E(I+1)=S*P*R + S=C/R + C=1.0/R + GO TO 320 + 310 C=P/E(I) + R=DSQRT(C*C+1.0) + E(I+1)=S*E(I)*R + S=1.0/R + C=C/R + 320 P=C*D(I)-S*G + D(I+1)=H+S*(C*G+S*D(I)) + DO 330 K=1,N + H=X(K,I+1) + X(K,I+1)=X(K,I)*S+H*C + 330 X(K,I)=X(K,I)*C-H*S +C + E(L)=S*P + D(L)=C*P + IF (DABS(E(L)).GT.B) GO TO 260 +C +C CONVERGENCE +C + 340 D(L)=D(L)+F +C +C ORDERING OF EIGENVALUES +C + NI=N-1 + 350 DO 380I=1,NI + K=I + P=D(I) + J1=I+1 + DO 360J=J1,N + IF(D(J).GE.P) GOTO 360 + K=J + P=D(J) + 360 CONTINUE + IF (K.EQ.I) GOTO 380 + D(K) =D(I) + D(I)=P + DO 370 J=1,N + P=X(J,I) + X(J,I)=X(J,K) + 370 X(J,K)=P + 380 CONTINUE +C +C FIXING OF SIGN +C + DO 385 I=1,N + PM=0 + DO 386 J=1,N + IF(PM.GT.DABS(X(J,I))) GOTO 386 + PM =DABS(X(J,I)) + K=J + 386 CONTINUE + IF(X(K,I).GE.0) GOTO 385 + DO 387 J=1,N + 387 X(J,I)=-X(J,I) + 385 CONTINUE + 390 GO TO 410 +C +C SPECIAL TREATMENT OF CASE N = 1 +C + 400 D(1)=X(1,1) + X(1,1)=1.0 + 410 RETURN + END + SUBROUTINE TPRINT (TITLE,NOP,T,STITLE,NS) + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + COMMON /TAPES / INP,IOUT + COMMON /CFULLR/ CORE,THR,THRRES + COMMON /CFULL / MULT(8,8),ICF(256),NPAIR(8),NSTRA(8),NSTRB(8) + > ,INTOFF(8),ITYPEA(256) + > ,NOCC(8),NCI(8),NELEC,MS2,NA,NB,NACT + > ,INTAA,INTHAA,INTJAA,INTKAA,NINT2 + > ,NSTRAA,NSTRBB,MAXAA,MAXBB,MAXREP,MAXPAR,INTADT + > ,IPRINT,IPRDIA,IPRHAM,IPRTIM,MAXIT,NROOT,MEMORY + CHARACTER*(*) TITLE,STITLE + FLOP = 0 + IF (T.NE.0D0) FLOP = 1D-6 * NOP/T + IF (STITLE.EQ.' ') THEN + WRITE (IOUT,666) TITLE,NOP,T,FLOP + ELSE + WRITE (IOUT,666) TITLE,NOP,T,FLOP,STITLE,NS + END IF +666 FORMAT(' Timing: ',A,5X,'Operations',I12,5X,'Time',F10.2, + >5X,'Mflops',F10.3,:,5X,A,I6) + RETURN + END diff --git a/FCI/hookints.txt b/FCI/hookints.txt new file mode 100644 index 0000000..7222f08 --- /dev/null +++ b/FCI/hookints.txt @@ -0,0 +1,56273 @@ +&FCI NORB = 30, NELEC = 4, MS2 = 4, ISYM = 2, NROOT = 10, MAXIT = 200, + ORBSYM = 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, + MEMORY = 40000000 / +-0.3989422804 2 1 2 1 +0.3989422804 2 2 1 1 +0.2820947918 3 1 1 1 +-0.2820947918 3 1 2 2 +-0.3989422804 3 1 3 1 +-0.03526184897 3 1 3 3 +-0.3490744954 3 2 3 2 +0.3989422804 3 3 1 1 +0.3490744954 3 3 2 2 +-0.03526184897 3 3 3 1 +0.3257350079 4 1 2 1 +-0.2015382539 4 1 3 2 +-0.3823196854 4 1 4 1 +-0.02350789931 4 1 4 3 +0.162867504 4 2 1 1 +0.122150628 4 2 2 2 +-0.02879117912 4 2 3 1 +-0.3053765699 4 2 3 3 +-0.3656970904 4 2 4 2 +-0.04580648549 4 2 4 4 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+++ b/FCI/standard.f @@ -0,0 +1,27 @@ + SUBROUTINE GMAINV (QQ,IBASE,LENN) +C.....SUBSTITUTE FOR DYNAMIC MEMORY ALLOCATION +C.... THIS ROUTINE MUST BE CALLED WITH QQ FIRST WORD IN COMMON/BIG/ +C.... THIS ROUTINE IS NOT FOOLPROOF + IMPLICIT DOUBLE PRECISION (A-H,O-Z) + PARAMETER (MAXM = 1 100 000 000) + COMMON/BIG/Q(MAXM) + COMMON /CORCTL/ INTREL,ICORCT(8) +C... INTREL MUST BE NUMBER OF INTEGERS PER REAL + INTREL = 2 +C + LENN = MIN(MAXM,LENN) + IBASE = 1 + RETURN + ENTRY FMAIN (QQ,LENN) +C.....SUBSITUTE FOR MEMORY RELEASE + RETURN + END + FUNCTION SECOND() + IMPLICIT DOUBLE PRECISION (A-H,O-Z) +C.....SHOUld RETURN CPU TIME IN SECONDS + SAVE T + DATA T/0D0/ + SECOND=T + T=T+.01D0 + RETURN + END diff --git a/Integrals/Boxium_ERI.dat b/Integrals/Boxium_ERI.dat new file mode 100644 index 0000000..fac7954 --- /dev/null +++ b/Integrals/Boxium_ERI.dat @@ -0,0 +1,56480 @@ +0.4473211247724916 1 1 1 1 +-0.40635977105343735 2 1 2 1 +0.6259919502912837 2 2 1 1 +0.12977728134069944 2 2 2 2 +-0.17867082551879201 3 1 1 1 +0.4962146689505843 3 1 2 2 +-0.6748854944693764 3 1 3 1 +0.060188132909130804 3 1 3 3 +0.07032877126888289 3 2 2 1 +-0.6016520863522647 3 2 3 2 +0.6148459997525558 3 3 1 1 +0.5546578668434251 3 3 2 2 +0.060188132909130804 3 3 3 1 +-0.03470695586952757 3 3 3 3 +-0.3360309997845544 4 1 2 1 +-0.5313233150833817 4 1 3 2 +-0.8673543148679361 4 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+\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable} + +\usepackage{mathpazo,libertine} +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{RGB}{0, 180, 0} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} + +%useful stuff +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\mr}{\multirow} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\la}{\lambda} +\newcommand{\si}{\sigma} + +% functionals, potentials, densities, etc +\newcommand{\eps}{\epsilon} +\newcommand{\e}[2]{\eps_\text{#1}^{#2}} +\renewcommand{\v}[2]{v_\text{#1}^{#2}} +\newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}} +\newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}} +\newcommand{\n}[1]{n^{#1}} +\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}} +\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}} + + +% energies +\newcommand{\EHF}{E_\text{HF}} +\newcommand{\Ec}{E_\text{c}} +\newcommand{\Ecat}{E_\text{cat}} +\newcommand{\Eneu}{E_\text{neu}} +\newcommand{\Eani}{E_\text{ani}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EFCI}{E_\text{FCI}} + +% matrices +\newcommand{\br}{\bm{r}} +\newcommand{\bw}{\bm{w}} +\newcommand{\bG}{\bm{G}} +\newcommand{\bS}{\bm{S}} +\newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}} +\newcommand{\bH}{\bm{H}} +\newcommand{\bHc}{\bm{H}^\text{c}} +\newcommand{\bF}[1]{\bm{F}^{#1}} +\newcommand{\Ex}[1]{\Omega^{#1}} +\newcommand{\E}[1]{E^{#1}} + +% elements +\newcommand{\ew}[1]{w_{#1}} +\newcommand{\eG}[1]{G_{#1}} +\newcommand{\eS}[1]{S_{#1}} +\newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}} +\newcommand{\eHc}[1]{H_{#1}^\text{c}} +\newcommand{\eF}[2]{F_{#1}^{#2}} + +% Numbers +\newcommand{\Nel}{N} +\newcommand{\Nbas}{K} + +% Ao and MO basis +\newcommand{\MO}[2]{\phi_{#1}^{#2}} +\newcommand{\cMO}[2]{c_{#1}^{#2}} +\newcommand{\AO}[1]{\chi_{#1}} + + +% units +\newcommand{\IneV}[1]{#1~eV} +\newcommand{\InAU}[1]{#1~a.u.} +\newcommand{\InAA}[1]{#1~\AA} + +\newcommand{\SI}{\textcolor{blue}{supplementary material}} + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France} + + +%%%% added by Manu %%%%% +\newcommand{\manu}[1]{{\textcolor{blue}{ Manu: #1 }} } +\newcommand{\beq}{\begin{eqnarray}} +\newcommand{\eeq}{\end{eqnarray}} +% +\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector +\newcommand{\bmg}{\bm{\gamma}} % orbital rotation vector +\newcommand{\bfx}{\bf{x}} +\newcommand{\bfr}{\bf{r}} +%%%% + +\begin{document} + +\title{Supplementary Material for ``Weight-dependent local density-functional approximations for ensembles''} + +\author{Pierre-Fran\c{c}ois Loos} +\email{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} +\author{Emmanuel Fromager} +\email{fromagere@unistra.fr} +\affiliation{\LCQ} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{abstract} +\end{abstract} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\maketitle +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Ensemble Hartree--Fock method} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +The Hartree--Fock (HF) ensemble energy can be written +as +\beq\label{eq:eHF_ener} +&&E^{\bw}_{\rm +HF}({\bm\kappa})= +\sum_{pq}\langle +\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{pq} +\nonumber\\ +&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert +\varphi_r(\bmk)\varphi_s(\bmk)\rangle +%\times +\sum^M_{K=0}w^{(K)}D^{(K)}_{pr}D^{(K)}_{qs}, +\nonumber\\ +\eeq +where the one- and antisymmetrized two-electron integrals read, +\beq +\langle +\varphi_p({\bmk})\vert\hat{h}\vert +\varphi_q({\bmk})\rangle=\int d{\bfx}\; +\varphi_p({\bmk},{\bfx})\hat{h}\varphi_q({\bmk},{\bfx}) +\eeq +with $\hat{h}\equiv-\frac{\nabla_{\bfr}^2}{2}+v_{\rm +ext}(\bfr)$ +and +\beq +&&\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert +\varphi_r(\bmk)\varphi_s(\bmk)\rangle= +\nonumber\\ +&&\int d{\bfx}_1\int d{\bfx}_2\; +\varphi_p({\bmk},{\bfx}_1)\varphi_q({\bmk},{\bfx}_2)\frac{1}{\vert +{\bfr}_1-{\bfr}_2\vert} +\nonumber +\\ +&&\times\Big[\varphi_r({\bmk},{\bfx}_1)\varphi_s({\bmk},{\bfx}_2) +-\varphi_s({\bmk},{\bfx}_1)\varphi_r({\bmk},{\bfx}_2)\Big] +, +\eeq +respectively. Note that we use {\it real algebra} and the shorthand +notation $\int d{\bfx}\equiv\int +d{\bfr}\sum_{\sigma}$ for integration over space and +summation over spin. +% normalization condition +%$\sum^M_{K=0}w^{(K)}=1$ +%. +The antihermitian $\bmk\equiv\{\kappa_{pq}\}_{p>q}$ matrix which appears in the integrals controls +the rotation of the spin-orbitals as follows, +\beq\label{eq:orb_taylor_expansion} +&&\varphi_p({\bmk},{\bfx})=\sum_q\left[e^{-{\bmk}}\right]_{qp}\varphi_q({\bfx}) +\nonumber\\ +&&= +\varphi_p({\bfx})+\sum_{q
p}\kappa_{qp}\varphi_q({\bfx})
++\mathcal{O}\left({\bmk}^2\right).
+\eeq
+The (${\bmk}$-independent) one-electron reduced density matrices (1RDMs)
+in Eq.~(\ref{eq:eHF_ener}) are defined in the unrotated molecular
+spin-orbital basis for each (unrotated) determinant $\Phi^{(K)}$ belonging to
+the ensemble as follows: $D^{(K)}_{pr}=\delta_{pr}$ if $\varphi_p$ and
+$\varphi_r$ are both
+occupied in $\Phi^{(K)}$, otherwise $D^{(K)}_{pr}=0$. If the unrotated
+spin-orbitals are the minimizing ensemble HF ones, then the following
+stationarity condition is fulfilled,
+\beq\label{eq:station_cond}
+\left.\dfrac{\partial E^{\bw}_{\rm
+HF}({\bm\kappa})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=0,
+\eeq
+with $l>m$. Since, according to Eq.~(\ref{eq:orb_taylor_expansion}),
+\beq
+\left.\dfrac{\partial
+\varphi_p({\bmk},{\bfx})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=\delta_{lp}\varphi_m({\bfx})-\delta_{mp}\varphi_l({\bfx}),
+\eeq
+Eq.~(\ref{eq:station_cond}) can be written more explicitly as the
+following commutation relation,
+\iffalse%%%%%
+%%%%%% intermediate steps ... %%%%
+\beq
+\sum^M_{K=0}w^{(K)}\sum_qD^{(K)}_{mq}f^{(K)}_{lq}
+-
+\sum^M_{K=0}w^{(K)}\sum_qf^{(K)}_{mq}D^{(K)}_{lq}=0
+\eeq
+% original %%
+\iffalse%%%
+\beq
+&&
+2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
+\nonumber\\
+&&-
+2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
+\nonumber\\
+&&+2\sum_{qrs}\langle \varphi_m\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{lr}D^{(K)}_{qs}
+\nonumber\\
+&&-2\sum_{qrs}\langle \varphi_l\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{mr}D^{(K)}_{qs}
+\nonumber\\
+&&=0\eeq
+\fi%%%%
+%%%%%%%
+\beq
+&&
+2\sum_q\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}
+\nonumber\\
+&&-
+2\sum_q\langle\varphi_l\vert\hat{h}\vert \varphi_q\rangle\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}
+\nonumber\\
+&&+2\sum_{qrs}\langle \varphi_m\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{lq}D^{(K)}_{rs}
+\nonumber\\
+&&-2\sum_{qrs}\langle \varphi_l\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle
+%\times
+\sum^M_{K=0}w^{(K)}D^{(K)}_{mq}D^{(K)}_{rs}
+\nonumber\\
+&&=0\eeq
+%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%
+\fi%%%%
+%%%%%%%
+\beq\label{eq:stat_cond_commut_ind}
+\sum^M_{K=0}w^{(K)}\left[{\bm f}^{(K)},{\bm D}^{(K)}\right]=0,
+\eeq
+where the $K$th Fock matrix elements read
+\beq
+f^{(K)}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
++
+\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle D^{(K)}_{rs}.
+\eeq
+In the minimizing ensemble HF spin-orbital basis, Eq.~(\ref{eq:stat_cond_commut_ind}) reads
+\beq
+\sum^M_{K=0}w^{(K)}\Big(\nu^{(K)}_m-\nu_l^{(K)}\Big)f^{(K)}_{lm}=0,
+\eeq
+where $\nu^{(K)}_m$ is the occupation of the spin-orbital $\varphi_m$ in the
+determinant $\Phi^{(K)}$.\\
+
+Note that, in more conventional ensemble calculations, the following HF
+energy expression is employed,
+\beq\label{eq:GI_ensHF_ener}
+&&\tilde{E}^{\bw}_{\rm
+HF}({\bm\kappa})=
+\sum_{pq}\langle
+\varphi_p(\bmk)\vert\hat{h}\vert \varphi_q(\bmk)\rangle D^{\bw}_{pq}
+\nonumber\\
+&&
++\dfrac{1}{2}\sum_{pqrs}\langle \varphi_p(\bmk)\varphi_q(\bmk)\vert\vert
+\varphi_r(\bmk)\varphi_s(\bmk)\rangle
+%\times
+D^{\bw}_{pr}D^{\bw}_{qs},
+\eeq
+where ${\bm D}^{\bw}=\sum^M_{K=0}w^{(K)}{\bm D}^{(K)}$ is the ensemble
+1RDM. In this case, the stationarity condition simply reads
+\beq
+\left[{\bm f}^{\bw},{\bm D}^{\bw}\right]=0,
+\eeq
+where the ensemble Fock matrix elements are defined as follows,
+\beq
+f^{\bw}_{mq}=\langle\varphi_m\vert\hat{h}\vert \varphi_q\rangle
++
+\sum_{rs}\langle \varphi_m\varphi_r\vert\vert
+\varphi_q\varphi_s\rangle D^{\bw}_{rs}.
+\eeq
+The major issue with the expression of the ensemble energy in
+Eq.~(\ref{eq:GI_ensHF_ener}) is the
+ghost-interaction error from which our expression (see
+Eq.~(\ref{eq:eHF_ener})) is free. Note also
+that, by construction, the ensemble energy in Eq.~(\ref{eq:GI_ensHF_ener}) is quadratic in the
+ensemble weights while ours, like the exact one, varies linearly with
+the weights.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Ensemble Hartree--Fock exchange and density-functional
+ghost-interaction correction}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\beq
+F^{\bw}_{\rm HF}[n]&=&
+\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]\right\}
+\nonumber\\
+&=&{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]
+\eeq
+where
+$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\gamma}^{(K)}$ is an ensemble density matrix operator constructed
+from Slater determinants, the ensemble 1RDM elements are $\gamma_{pq}^{\bw}={\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
+and $W_{\rm
+HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\gamma_{pr}\gamma_{qs}$.\\
+
+In-principle-exact decomposition:
+
+\beq
+F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
+Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
+\eeq
+
+The complementary ensemble Hx energy removes the ghost-interaction
+errors introduced in $W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]$:
+\beq
+\overline{E}^{{\bw}}_{\rm
+Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
+HF}\left[{\bmg}^{(K)}[n]\right]
+-W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right],
+\eeq
+which gives in the canonical orbital basis
+\beq
+&&\overline{E}^{{\bw}}_{\rm
+Hx}[n]=
+\dfrac{1}{2}\sum_{pq}
+\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
+\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
+\nonumber\\
+&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
+-\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
+.\eeq
+\manu{I would guess that, in a uniform system, the GOK-DFT and our
+canonical orbitals are the same. This is nice since we can construct
+in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
+Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
+
+Variational expression for the ensemble energy:
+\beq
+E^{{\bw}}=\underset{\hat{\gamma}^{{\bw}}}{\rm min}\Big\{
+&&{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
++
+\overline{E}^{{\bw}}_{\rm
+Hxc}\left[n_{\hat{\gamma}^{{\bw}}}\right]
+%+E^{{\bw}}_{\rm c}\left[n_{\hat{\gamma}^{{\bw}}}\right]
+\nonumber\\
+&&
++\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\gamma}^{{\bw}}}({\bfr})
+\Big\}
+\eeq
+
+Note that, if we use orbital rotations, the gradient of the DFT energy
+contributions can be expressed as follows,
+\beq
+\left.\dfrac{\partial
+\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
+}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0},
+\eeq
+where
+\beq
+n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw}
+\eeq
+thus leading to
+\beq
+&&\left.\dfrac{\partial
+\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
+}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=
+\sum_{pq}\gamma_{pq}^{\bw}
+\nonumber\\
+&&\times\left.\dfrac{\partial}
+{\partial \kappa_{lm}}
+\Big[\left\langle\varphi_p(\bmk)\middle\vert\hat{\overline{v}}^{{\bw}}_{\rm
+Hxc}
+\middle\vert \varphi_q(\bmk)\right\rangle
+\Big]
+\right|_{{\bmk}=0}.
+\eeq
+
+In conclusion, the minimizing canonical orbitals fulfill the following
+hybrid HF/GOK-DFT equation,
+\beq
+&&\left(-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
+ext}({\bfr})+\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]
++\dfrac{\delta \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}\right)\varphi^{{\bw}}_p({\bfx})
+\nonumber
+\\
+&&=\varepsilon^{{\bw}}_p\varphi^{{\bw}}_p({\bfx}).
+\eeq
+
+
+Since $\partial \gamma_{pq}^{\bw}/\partial
+w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
+
+\manu{just for me ...
+\beq
+&&+\dfrac{1}{2}
+\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
+\nonumber\\
+&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_q\varphi_p\vert\vert
+\varphi_s\varphi_r\rangle
+%\times
+ \gamma^{\bw}_{pr}\left(\gamma_{qs}^{(I)}-\gamma_{qs}^{(0)}\right)
+\nonumber\\
+&&=
+\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
+\nonumber\\
+&&=
+\sum_{pr}\left[\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]\right]_{pr}\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)
+\nonumber\\
+&&=
+\sum_p\left[\hat{u}_{\rm
+HF}\left[\gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right)
+\eeq
+}
+
+\beq
+\dfrac{dE^{\bw}}{dw^{(I)}}=\sum_p\varepsilon^{{\bw}}_p\left(\nu_p^{(I)}-\nu_p^{(0)}\right)+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
+\eeq
+
+LZ shift in this context: $\varepsilon^{{\bw}}_p\rightarrow
+\overline{\varepsilon}^{{\bw}}_p=\varepsilon^{{\bw}}_p+\overline{\Delta}_{\rm
+LZ}^{{\bw}}$ where
+
+\beq
+N\overline{\Delta}_{\rm
+LZ}^{{\bw}}&=&\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]
+-\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}n^{{\bw}}({\bfr})
+\nonumber\\
+&&
+-W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
+\eeq
+
+such that
+\beq
+E^{{\bw}}=\sum^M_{K=0}w^{(K)}\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p.
+\eeq
+
+Thus we conclude that individual energies can be expressed in principle
+exactly as follows,
+
+\beq
+E^{(K)}=\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p+\sum^M_{I>0}\left(\delta_{IK}-w^{(I)}\right)\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
+\eeq
+%%%%%%%%%%%%%%
+\iffalse%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Generalized GOK-DFT}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The energy to be minimized in a generalized GOK-DFT approach can be
+written as
+\beq
+\eeq
+\fi%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Construction of the density-functional approximations}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The density-functional approximations designed in this manuscript are based on highly-accurate energies for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
+We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
+
+The reduced (i.e.~per electron) HF energy for these three states is:
+\begin{subequations}
+\begin{align}
+ \e{HF}{(0)}(n) & = \frac{\pi^2}{8} n^2 + n,
+ \\
+ \e{HF}{(1)}(n) & = \frac{\pi^2}{2}n^2 + \frac{4}{3} n,
+ \\
+ \e{HF}{(2)}(n) & = \frac{9\pi^2}{8}n^2 + \frac{23}{15} n.
+\end{align}
+\end{subequations}
+All these states have the same (uniform) density $n = 2/(2\pi R)$ where $R$ is the radius of the ring on which the electrons are confined.
+
+The total energy of the ground and doubly-excited states are given by the two lowest eigenvalues of the Hamiltonian $\bH$ with elements
+\begin{equation}
+\begin{split}
+ H_{ij}
+ & = \int_0^\pi \qty[ \frac{\psi_i(\omega)}{R} \frac{\psi_j(\omega)}{R} + \frac{\psi_i(\omega)\psi_j(\omega)}{2R\sin(\omega/2)} ] d\omega
+ \\
+ & = \frac{\sqrt{\pi}}{2 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{2})} + \frac{ij}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+2}{2})} ],
+\end{split}
+\end{equation}
+where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function \cite{NISTbook}, and
+\begin{equation}
+ \psi_i(\omega) = \sin(\omega/2) \sin^{i-1}(\omega/2), \quad i=1,\ldots,M
+\end{equation}
+are (non-orthogonal) explicitly-correlated basis functions with overlap matrix elements
+\begin{equation}
+ S_{ij}
+ = \int_0^\pi \psi_i(\omega)\psi_j(\omega) d\omega
+ = \sqrt{\pi} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+2}{2})}.
+\end{equation}
+Thanks to this explicitly-correlated basis, the convergence rate of the energy is exponential with respect to $M$.
+Therefore, high accuracy is reached with a very small number of basis functions.
+Here, we typically use $M=10$.
+For the singly-excited state, one has to modify the basis functions as
+\begin{equation}
+ \psi_i(\omega) = \cos(\omega/2) \sin^{i-1} (\omega/2),
+\end{equation}
+and its energy is obtained by the lowest root of the Hamiltonian in this basis, and the matrix elements reads
+\begin{align}
+ H_{ij} & = \frac{\sqrt{\pi}}{4 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{3})} + \frac{3ij+i+j-1}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+4}{2})} ],
+ \\
+ S_{ij} & = \frac{\sqrt{\pi}}{2} \frac{\Gamma\qty(\frac{i+j+1}{2})}{\Gamma\qty(\frac{i+j+4}{2})}.
+\end{align}
+The numerical values of the correlation energy for various $R$ are reported in Table \ref{tab:Ref} for the three states of interest.
+
+%%% FIG 1 %%%
+\begin{figure}
+ \includegraphics[width=\linewidth]{Ec}
+ \caption{
+ $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi n)$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
+ The data gathered in Table \ref{tab:Ref} are also reported.
+ }
+ \label{fig:Ec}
+\end{figure}
+%%% %%% %%%
+
+%%% TABLE I %%%
+\begin{turnpage}
+\begin{squeezetable}
+\begin{table*}
+ \caption{
+ \label{tab:Ref}
+ $-\e{c}{(I)}$ as a function of the radius of the ring $R$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lcddddddddddd}
+ State & $I$ & \mc{11}{c}{Ring's radius $R = 1/(\pi n)$} \\
+ \cline{3-13}
+ & & \tabc{$0$} & \tabc{$1/10$} & \tabc{$1/5$} & \tabc{$1/2$} & \tabc{$1$} & \tabc{$2$} & \tabc{$5$} & \tabc{$10$} & \tabc{$20$} & \tabc{$50$} & \tabc{$100$} \\
+ \hline
+ Ground state & $0$ & 0.013708 & 0.012859 & 0.012525 & 0.011620 & 0.010374 & 0.008558 & 0.005673 & 0.003697 & 0.002226 & 0.001046 & 0.000567 \\
+ Singly-excited state & $1$ & 0.0238184 & 0.023392 & 0.022979 & 0.021817 & 0.020109 & 0.017371 & 0.012359 & 0.008436 & 0.005257 & 0.002546 & 0.001399 \\
+ Doubly-excited state & $2$ & 0.018715 & 0.018653 & 0.018576 & 0.018300 & 0.017743 & 0.016491 & 0.013145 & 0.009670 & 0.006365 & 0.003231 & 0.001816 \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+\end{squeezetable}
+\end{turnpage}
+
+Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
+\begin{equation}
+\label{eq:ec}
+ \e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
+\end{equation}
+where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
+The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
+Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
+
+
+%%% FIG 2 %%%
+\begin{figure*}
+ \includegraphics[height=0.25\linewidth]{EvsL_2}
+ \includegraphics[height=0.25\linewidth]{EvsL_3}
+ \includegraphics[height=0.25\linewidth]{EvsL_4}
+ \includegraphics[height=0.25\linewidth]{EvsL_5}
+ \includegraphics[height=0.25\linewidth]{EvsL_6}
+ \includegraphics[height=0.25\linewidth]{EvsL_7}
+ \caption{
+ Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the box length $L$ for various methods.
+ }
+ \label{fig:EvsL}
+\end{figure*}
+%%% %%% %%%
+
+%%% FIG 1 %%%
+\begin{figure*}
+ \includegraphics[height=0.25\linewidth]{EvsN_0125}
+ \includegraphics[height=0.25\linewidth]{EvsN_025}
+ \includegraphics[height=0.25\linewidth]{EvsN_05}
+ \includegraphics[height=0.25\linewidth]{EvsN_1}
+ \includegraphics[height=0.25\linewidth]{EvsN_2}
+ \includegraphics[height=0.25\linewidth]{EvsN_4}
+ \includegraphics[height=0.25\linewidth]{EvsN_8}
+ \caption{
+ Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the number of electrons $N$ for various methods and box length $L$.
+ }
+ \label{fig:EvsL}
+\end{figure*}
+%%% %%% %%%
+
+
+%%% TABLE I %%%
+\begin{table*}
+ \caption{
+ \label{tab:OptGap}
+ Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
+ The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lclddddddd}
+ & & & \mc{7}{c}{$L/\pi$} \\
+ \cline{4-10}
+ Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
+ \hline
+ FCI & & $\E{(0)}$ & 168.1946 & 44.0662 & 12.0035 & 3.4747 & 1.0896 & 0.3719 & 0.1367 \\
+ & & $\E{(1)}$ & 330.2471 & 85.0890 & 22.5112 & 6.2247 & 1.8355 & 0.5845 & 0.2006 \\
+ & & $\E{(2)}$ & 809.9972 & 204.9840 & 52.4777 & 13.7252 & 3.7248 & 1.0696 & 0.3300 \\
+ & & $\Ex{(1)}$ & 162.0525 & 41.0228 & 10.5078 & 2.7500 & 0.7458 & 0.2125 & 0.0639 \\
+ & & $\Ex{(2)}$ & 641.8026 & 160.9177 & 40.4743 & 10.2505 & 2.6352 & 0.6977 & 0.1933 \\
+ \\
+ CIS & & $\Ex{(1)}$ & 0.0104 & 0.0102 & 0.0099 & 0.0092 & 0.0077 & 0.0051 & 0.0022 \\
+ \\
+ TDHF & & $\Ex{(1)}$ & 0.0019 & 0.0021 & 0.0023 & 0.0027 & 0.0029 & 0.0023 & 0.0011 \\
+ \\
+ TDA-TDLDA& & $\Ex{(1)}$ & 0.0099 & 0.0088 & 0.0058 & -0.0041 & -0.0316 & -0.0467 & \\
+ \\
+ TDLDA & & $\Ex{(1)}$ & 0.0015 & 0.0006 & -0.0018 & -0.0106 & -0.0370 & -0.0518 & \\
+ \\
+ eDFT & $(0,0)$ & $\E{(0)}$ & -0.0397 & -0.0391 & -0.0380 & -0.0361 & -0.0323 & -0.0236 \\
+ & & $\E{(1)}$ & 0.0215 & 0.0213 & 0.0210 & 0.0200 & 0.0159 & 0.0102 \\
+ & & $\E{(2)}$ & -0.0426 & -0.0425 & -0.0419 & -0.0386 & -0.0249 & -0.0044 \\
+ & & $\Ex{(1)}$ & 0.0612 & 0.0604 & 0.0590 & 0.0561 & 0.0483 & 0.0338 \\
+ & & $\Ex{(2)}$ & -0.0029 & -0.0034 & -0.0039 & -0.0025 & 0.0074 & 0.0191 \\
+ & & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
+ & & $\DD{c}{(1)}$ & 0.0064 & 0.0056 & 0.0043 & 0.0022 & -0.0007 & -0.0037 & \\
+ & & $\DD{c}{(2)}$ & 0.0159 & 0.0147 & 0.0126 & 0.0093 & 0.0046 & -0.0009 & \\
+ & & $\LZ{c}{}$ & 0.0013 & 0.0024 & 0.0043 & 0.0070 & 0.0100 & 0.0121 & \\
+ \\
+ eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0031 & 0.0036 & 0.0044 & 0.0054 & 0.0042 & -0.0025 \\
+ & & $\E{(1)}$ & 0.0090 & 0.0087 & 0.0083 & 0.0076 & 0.0070 & 0.0071 \\
+ & & $\E{(2)}$ & -0.0005 & -0.0009 & -0.0015 & -0.0023 & -0.0030 & -0.0026 \\
+ & & $\Ex{(1)}$ & 0.0058 & 0.0052 & 0.0039 & 0.0022 & 0.0028 & 0.0096 \\
+ & & $\Ex{(2)}$ & -0.0036 & -0.0045 & -0.0058 & -0.0077 & -0.0072 & 0.0000 \\
+ & & $\DD{c}{(0)}$ & -0.0074 & -0.0067 & -0.0055 & -0.0036 & -0.0010 & 0.0019 & \\
+ & & $\DD{c}{(1)}$ & -0.0010 & -0.0011 & -0.0014 & -0.0017 & -0.0021 & -0.0022& \\
+ & & $\DD{c}{(2)}$ & 0.0084 & 0.0079 & 0.0069 & 0.0053 & 0.0031 & 0.0003 & \\
+ & & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0023 & 0.0040 & 0.0063 & 0.0087 & \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+
+%%% TABLE II %%%
+\begin{table*}
+ \caption{
+ \label{tab:OptGap}
+ Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
+ The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lclddddddd}
+ & & & \mc{7}{c}{$L/\pi$} \\
+ \cline{4-10}
+ Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
+ \hline
+ FCI & & $\E{(0)}$ & 475.6891 & 125.7776 & 34.8248 & 10.3536 & 3.3766 & 1.2126 & 0.4721 \\
+ & & $\E{(1)}$ & 702.8330 & 183.3370 & 49.5922 & 14.2255 & 4.4269 & 1.5105 & 0.5606 \\
+ & & $\E{(2)}$ & 1379.3128 & 353.5967 & 92.7398 & 25.3135 & 7.3546 & 2.3203 & 0.7990 \\
+ & & $\Ex{(1)}$ & 227.1438 & 57.5594 & 14.7674 & 3.8720 & 1.0504 & 0.2979 & 0.0885 \\
+ & & $\Ex{(2)}$ & 903.6236 & 227.8191 & 57.9150 & 14.9599 & 3.9780 & 1.1077 & 0.3269 \\
+ \\
+ CIS & & $\Ex{(1)}$ & 0.0163 & 0.0161 & 0.0157 & 0.0149 & 0.0133 & 0.0102 & 0.0057 \\
+ \\
+ TDHF & & $\Ex{(1)}$ & 0.0013 & 0.0013 & 0.0014 & 0.0014 & 0.0013 & 0.0010 & 0.0007 \\
+ \\
+ TDA-TDLDA& & $\Ex{(1)}$ & 0.0162 & 0.0157 & 0.0146 & 0.0110 & -0.0049 & -0.0344 & -0.0378 \\
+ \\
+ TDLDA & & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0264 & 0.0269 & 0.0273 & 0.0206 & -0.0116 \\
+ \\
+ eDFT & $(0,0)$ & $\E{(0)}$ & -0.0481 & -0.0478 & -0.0473 & -0.0463 & -0.0446 & -0.0387 & -0.0257 \\
+ & & $\E{(1)}$ & 0.0343 & 0.0336 & 0.0321 & 0.0292 & 0.0220 & 0.0084 & 0.0008 \\
+ & & $\E{(2)}$ & 0.0277 & 0.0267 & 0.0247 & 0.0216 & 0.0187 & 0.0208 & 0.0209 \\
+ & & $\Ex{(1)}$ & 0.0824 & 0.0814 & 0.0794 & 0.0755 & 0.0666 & 0.0471 & 0.0266 \\
+ & & $\Ex{(2)}$ & 0.0759 & 0.0745 & 0.0720 & 0.0679 & 0.0633 & 0.0595 & 0.0467 \\
+ & & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
+ & & $\DD{c}{(1)}$ & 0.0100 & 0.0092 & 0.0077 & 0.0051 & 0.0012 & -0.0034 & -0.0072 \\
+ & & $\DD{c}{(2)}$ & 0.0244 & 0.0231 & 0.0208 & 0.0168 & 0.0108 & 0.0029 & -0.0050 \\
+ & & $\LZ{c}{}$ & 0.0014 & 0.0026 & 0.0048 & 0.0083 & 0.0127 & 0.0169 & 0.01860 \\
+ \\
+ eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0078 & 0.0080 & 0.0082 & 0.0085 & 0.0081 & 0.0024 & -0.0022 \\
+ & & $\E{(1)}$ & 0.0172 & 0.0162 & 0.0144 & 0.0112 & 0.0064 & 0.0019 & 0.0004 \\
+ & & $\E{(2)}$ & 0.0645 & 0.0636 & 0.0621 & 0.0590 & 0.0530 & 0.0420 & 0.0300 \\
+ & & $\Ex{(1)}$ & 0.0094 & 0.0083 & 0.0062 & 0.0027 & -0.0018 & -0.0004 & 0.0026 \\
+ & & $\Ex{(2)}$ & 0.0567 & 0.0557 & 0.0539 & 0.0506 & 0.0449 & 0.0397 & 0.0323 \\
+ & & $\DD{c}{(0)}$ & -0.0115 & -0.0107 & -0.0094 & -0.0072 & -0.0038 & 0.0005 & 0.0045 \\
+ & & $\DD{c}{(1)}$ & -0.0015 & -0.0016 & -0.0018 & -0.0022 & -0.0028 & -0.0033 & -0.0032 \\
+ & & $\DD{c}{(2)}$ & 0.0129 & 0.0123 & 0.0113 & 0.0094 & 0.0066 & 0.0028 & -0.0013 \\
+ & & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0025 & 0.0044 & 0.0074 & 0.0110 & 0.0143 \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+
+
+%%% TABLE III %%%
+\begin{table*}
+ \caption{
+ \label{tab:OptGap}
+ Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
+ The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lclddddddd}
+ & & & \mc{7}{c}{$L/\pi$} \\
+ \cline{4-10}
+ Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
+ \hline
+ FCI & & $\E{(0)}$ & 1020.3778 & 270.0849 & 74.9426 & 22.3790 & 7.3595 & 2.6798 & 1.0633 \\
+ & & $\E{(1)}$ & 1312.2776 & 344.0184 & 93.8936 & 27.3398 & 8.7021 & 3.0600 & 1.1764 \\
+ & & $\E{(2)}$ & 2183.4399 & 563.5949 & 149.6753 & 41.7213 & 12.5052 & 4.1033 & 1.4749 \\
+ & & $\Ex{(1)}$ & 291.8998 & 73.9335 & 18.9510 & 4.9608 & 1.3426 & 0.3802 & 0.1131 \\
+ & & $\Ex{(2)}$ & 1163.0621 & 293.5099 & 74.7326 & 19.3423 & 5.1457 & 1.4235 & 0.4116 \\
+ \\
+ CIS & & $\Ex{(1)}$ & 0.0203 & 0.0202 & 0.0200 & 0.0195 & 0.0187 & 0.0167 & 0.0116 \\
+ \\
+ TDHF & & $\Ex{(1)}$ & 0.0008 & 0.0008 & 0.0009 & 0.0009 & 0.0008 & 0.0008 & 0.0007 \\
+ \\
+ TDA-TDLDA& & $\Ex{(1)}$ & 0.0203 & 0.0201 & 0.0195 & 0.0181 & 0.0106 & -0.0178 & -0.0369 \\
+ \\
+ TDLDA & & $\Ex{(1)}$ & 0.0008 & 0.0007 & 0.0004 & -0.0006 & -0.0074 & -0.0360 & -0.0653 \\
+ \\
+ eDFT & $(0,0)$ & $\E{(0)}$ & -0.0541 & -0.0539 & -0.0537 & -0.0534 & -0.0529 & -0.0504 & -0.0386 \\
+ & & $\E{(1)}$ & 0.0413 & 0.0406 & 0.0390 & 0.0362 & 0.0304 & 0.0159 & 0.0008 \\
+ & & $\E{(2)}$ & 0.0642 & 0.0622 & 0.0586 & 0.0517 & 0.0399 & 0.0254 & 0.0149 \\
+ & & $\Ex{(1)}$ & 0.0954 & 0.0945 & 0.0927 & 0.0896 & 0.0833 & 0.0663 & 0.0394 \\
+ & & $\Ex{(2)}$ & 0.1182 & 0.1162 & 0.1123 & 0.1051 & 0.0928 & 0.0758 & 0.0534 \\
+ & & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
+ & & $\DD{c}{(1)}$ & 0.0136 & 0.0127 & 0.0111 & 0.0083 & 0.0038 & -0.0022 & -0.0080 \\
+ & & $\DD{c}{(2)}$ & 0.0330 & 0.0316 & 0.0291 & 0.0248 & 0.0178 & 0.0080 & -0.0028 \\
+ & & $\LZ{c}{}$ & 0.0014 & 0.0027 & 0.0051 & 0.0091 & 0.0147 & 0.0207 & 0.0245 \\
+ \\
+ eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0085 & 0.0085 & 0.0084 & 0.0082 & 0.0072 & 0.0021 & -0.0015 \\
+ & & $\E{(1)}$ & 0.0164 & 0.0152 & 0.0129 & 0.0087 & 0.0020 & -0.0050 & -0.0044 \\
+ & & $\E{(2)}$ & 0.0936 & 0.0917 & 0.0880 & 0.0807 & 0.0664 & 0.0434 & 0.0300 \\
+ & & $\Ex{(1)}$ & 0.0079 & 0.0067 & 0.0045 & 0.0006 & -0.0051 & -0.0071 & -0.0029 \\
+ & & $\Ex{(2)}$ & 0.0851 & 0.0832 & 0.0796 & 0.0725 & 0.0593 & 0.0413 & 0.0315 \\
+ & & $\DD{c}{(0)}$ & -0.0155 & -0.0148 & -0.0134 & -0.0110 & -0.0071 & -0.0017 & 0.0040 \\
+ & & $\DD{c}{(1)}$ & -0.0020 & -0.0021 & -0.0023 & -0.0027 & -0.0034 & -0.0042 & -0.0044 \\
+ & & $\DD{c}{(2)}$ & 0.0175 & 0.0168 & 0.0157 & 0.0137 & 0.0105 & 0.0059 & 0.0004 \\
+ & & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0025 & 0.0047 & 0.0081 & 0.0127 & 0.0175 \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+
+%%% TABLE IV %%%
+\begin{table*}
+ \caption{
+ \label{tab:OptGap}
+ Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
+ The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lclddddddd}
+ & & & \mc{7}{c}{$L/\pi$} \\
+ \cline{4-10}
+ Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
+ \hline
+ FCI & & $\E{(0)}$ & 1867.6344 & 493.6760 & 136.7020 & 40.7244 & 13.3763 & 4.8811 & 1.9492 \\
+ & & $\E{(1)}$ & 2224.11488 & 583.8981 & 159.7957 & 46.7553 & 15.0029 & 5.3399 & 2.0855 \\
+ & & $\E{(2)}$ & 3289.2022 & 852.4249 & 228.0415 & 64.3597 & 19.6613 & 6.6206 & 2.4547 \\
+ & & $\Ex{(1)}$ & 356.4804 & 90.2221 & 23.0937 & 6.0308 & 1.6266 & 0.4588 & 0.1363 \\
+ & & $\Ex{(2)}$ & 1421.56773 & 358.7489 & 91.3395 & 23.6352 & 6.2850 & 1.7395 & 0.5055 \\
+ \\
+ CIS & & $\Ex{(1)}$ & 0.0230 & 0.0230 & 0.0229 & 0.0229 & 0.0230 & 0.0225 & 0.0182 \\
+ \\
+ TDHF & & $\Ex{(1)}$ & 0.0005 & 0.0005 & 0.0005 & 0.0005 & 0.0005 & 0.0005 & 0.0006 \\
+ \\
+ TDA-TDLDA& & $\Ex{(1)}$ & 0.0230 & 0.0230 & 0.0228 & 0.0223 & 0.0192 & -0.0015 & -0.0309 \\
+ \\
+ TDLDA & & $\Ex{(1)}$ & 0.0005 & 0.0005 & 0.0004 & 0.0000 & -0.0033 & -0.0248 & -0.0650 \\
+ \\
+ eDFT & $(0,0)$ & $\E{(0)}$ & -0.0587 & -0.0586 & -0.0587 & -0.0588 & -0.0591 & -0.0590 & -0.0506 \\
+ & & $\E{(1)}$ & 0.0457 & 0.0450 & 0.0435 & 0.0409 & 0.0362 & 0.0241 & 0.0033 \\
+ & & $\E{(2)}$ & 0.0861 & 0.0838 & 0.0793 & 0.0712 & 0.0571 & 0.0377 & 0.0196 \\
+ & & $\Ex{(1)}$ & 0.1044 & 0.1036 & 0.1022 & 0.0997 & 0.0953 & 0.0830 & 0.0540 \\
+ & & $\Ex{(2)}$ & 0.1447 & 0.1424 & 0.1380 & 0.1300 & 0.1162 & 0.0966 & 0.0703 \\
+ & & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
+ & & $\DD{c}{(1)}$ & 0.0172 & 0.0163 & 0.0147 & 0.0117 & 0.0067 & -0.0004 & -0.0080 \\
+ & & $\DD{c}{(2)}$ & 0.0416 & 0.0402 & 0.0376 & 0.0329 & 0.0253 & 0.0140 & 0.0005 \\
+ & & $\LZ{c}{}$ & 0.0015 & 0.0028 & 0.0053 & 0.0096 & 0.0161 & 0.0238 & 0.0297 \\
+ \\
+ eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0070 & 0.0070 & 0.0068 & 0.0063 & 0.0053 & 0.0015 & -0.0049 \\
+ & & $\E{(1)}$ & 0.0162 & 0.0151 & 0.0128 & 0.0086 & 0.0018 & -0.0066 & -0.0095 \\
+ & & $\E{(2)}$ & 0.1080 & 0.1056 & 0.1011 & 0.0925 & 0.0772 & 0.0538 & 0.0325 \\
+ & & $\Ex{(1)}$ & 0.0092 & 0.0081 & 0.0060 & 0.0022 & -0.0035 & -0.0081 & -0.0047 \\
+ & & $\Ex{(2)}$ & 0.1010 & 0.0986 & 0.0943 & 0.0862 & 0.0719 & 0.0523 & 0.0373 \\
+ & & $\DD{c}{(0)}$ & -0.0196 & -0.0188 & -0.0174 & -0.0148 & -0.0106 & -0.0044 & 0.0029 \\
+ & & $\DD{c}{(1)}$ & -0.0024 & -0.0025 & -0.0027 & -0.0032 & -0.0040 & -0.0050 & -0.0056 \\
+ & & $\DD{c}{(2)}$ & 0.0220 & 0.0213 & 0.0201 & 0.0180 & 0.0146 & 0.0093 & 0.0027 \\
+ & & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0026 & 0.0049 & 0.0087 & 0.0141 & 0.0202 \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+
+%%% TABLE V %%%
+\begin{table*}
+ \caption{
+ \label{tab:OptGap}
+ Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
+ The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lclddddddd}
+ & & & \mc{7}{c}{$L/\pi$} \\
+ \cline{4-10}
+ Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
+ \hline
+ FCI & & $\E{(0)}$ & 3082.5386 & 813.0910 & 224.3734 & 66.5257 & 21.7454 & 7.9136 & 3.1633 \\
+ & & $\E{(1)}$ & 3503.4911 & 919.5487 & 251.5842 & 73.6145 & 23.6504 & 8.4487 & 3.3217 \\
+ & & $\E{(2)}$ & 4762.0921 & 1236.8257 & 332.1993 & 94.3988 & 29.1455 & 9.9582 & 3.7572 \\
+ & & $\Ex{(1)}$ & 420.9525 & 106.4577 & 27.2108 & 7.0888 & 1.9050 & 0.5351 & 0.1583 \\
+ & & $\Ex{(2)}$ & 1679.5536 & 423.7347 & 107.8259 & 27.8731 & 7.4001 & 2.0446 & 0.5938 \\
+ \\
+ CIS & & $\Ex{(1)}$ & 0.0249 & 0.0248 & 0.0250 & 0.0253 & 0.0261 & 0.0272 & 0.0248 \\
+ \\
+ TDHF & & $\Ex{(1)}$ & 0.0002 & 0.0000 & 0.0003 & 0.0003 & 0.0003 & 0.0003 & 0.0004 \\
+ \\
+ TDA-TDLDA& & $\Ex{(1)}$ & 0.0249 & 0.0248 & 0.0250 & 0.0250 & 0.0242 & 0.0114 & -0.0223 \\
+ \\
+ TDLDA & & $\Ex{(1)}$ & 0.0002 & 0.0000 & 0.0002 & 0.0000 & -0.0016 & -0.0162 & -0.0612 \\
+ \\
+ eDFT & $(0,0)$ & $\E{(0)}$ & -0.0626 & -0.0627 & -0.0628 & -0.0632 & -0.0641 & -0.0654 & -0.0612 \\
+ & & $\E{(1)}$ & 0.0486 & 0.0477 & 0.0465 & 0.0440 & 0.0400 & 0.0308 & 0.0078 \\
+ & & $\E{(2)}$ & 0.1017 & 0.0992 & 0.0946 & 0.0862 & 0.0718 & 0.0507 & 0.0271 \\
+ & & $\Ex{(1)}$ & 0.1112 & 0.1104 & 0.1093 & 0.1072 & 0.1041 & 0.0962 & 0.0690 \\
+ & & $\Ex{(2)}$ & 0.1643 & 0.1619 & 0.1575 & 0.1494 & 0.1358 & 0.1162 & 0.0884 \\
+ & & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
+ & & $\DD{c}{(1)}$ & 0.0208 & 0.0199 & 0.0182 & 0.0151 & 0.0098 & 0.0018 & -0.0075 \\
+ & & $\DD{c}{(2)}$ & 0.0503 & 0.0488 & 0.0460 & 0.0412 & 0.0330 & 0.0205 & 0.0046 \\
+ & & $\LZ{c}{}$ & 0.0015 & 0.0029 & 0.0054 & 0.0100 & 0.0172 & 0.0264 & 0.0344 \\
+ \\
+ eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0046 & 0.0045 & 0.0043 & 0.0039 & 0.0031 & 0.0006 & -0.0067 \\
+ & & $\E{(1)}$ & 0.0157 & 0.0144 & 0.0123 & 0.0080 & 0.0009 & -0.0091 & -0.0160 \\
+ & & $\E{(2)}$ & 0.1167 & 0.1142 & 0.1095 & 0.1007 & 0.0853 & 0.0616 & 0.0355 \\
+ & & $\Ex{(1)}$ & 0.0112 & 0.0099 & 0.0080 & 0.0041 & -0.0022 & -0.0097 & -0.0093 \\
+ & & $\Ex{(2)}$ & 0.1121 & 0.1097 & 0.1051 & 0.0968 & 0.0822 & 0.0610 & 0.0423 \\
+ & & $\DD{c}{(0)}$ & -0.0237 & -0.0229 & -0.0214 & -0.0188 & -0.0142 & -0.0073 & 0.0013 \\
+ & & $\DD{c}{(1)}$ & -0.0029 & -0.0030 & -0.0032 & -0.0037 & -0.0045 & -0.0057 & -0.0066 \\
+ & & $\DD{c}{(2)}$ & 0.0266 & 0.0259 & 0.0246 & 0.0224 & 0.0187 & 0.0130 & 0.0053 \\
+ & & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0026 & 0.0050 & 0.0091 & 0.0151 & 0.0224 \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+
+
+%%% TABLE VI %%%
+\begin{table*}
+ \caption{
+ \label{tab:OptGap}
+ Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of $\Nel = 4$ electrons in a box of length $L$.
+ The values of the derivative discontinuity $\DD{c}{(I)}$ and the Levy-Zahariev shift $\LZ{c}{}$ are also reported.
+ }
+ \begin{ruledtabular}
+ \begin{tabular}{lclddddddd}
+ & & & \mc{7}{c}{$L/\pi$} \\
+ \cline{4-10}
+ Method & $\bw$ & State & 1/8 & 1/4 & 1/2 & 1 & 2 & 4 & 8 \\
+ \hline
+ FCI & & $\E{(0)}$ & 4729.98018 & 1244.7753 & 342.1796 & 100.8943 & 32.7728 & 11.8683 & 4.7359 \\
+ & & $\E{(1)}$ & 5215.3307 & 1367.4316 & 373.4897 & 109.0326 & 34.9524 & 12.4779 & 4.9156 \\
+ & & $\E{(2)}$ & 6667.18516 & 1733.3319 & 466.4133 & 132.9686 & 41.2715 & 14.2096 & 5.4146 \\
+ & & $\Ex{(1)}$ & 485.3505 & 122.6563 & 31.3101 & 8.1382 & 2.1796 & 0.6096 & 0.1797 \\
+ & & $\Ex{(2)}$ & 1937.2050 & 488.5566 & 124.2336 & 32.0743 & 8.4987 & 2.3413 & 0.6787 \\
+ \\
+ CIS & & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0265 & 0.0270 & 0.0283 & 0.0308 & 0.0309 \\
+ \\
+ TDHF & & $\Ex{(1)}$ & 0.0000 & 0.0001 & 0.0000 & 0.0000 & 0.0000 & 0.0001 & 0.0003 \\
+ \\
+ TDA-TDLDA& & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0264 & 0.0269 & 0.0273 & 0.0206 & -0.0116 \\
+ \\
+ TDLDA & & $\Ex{(1)}$ & 0.0000 & 0.0001 & 0.0000 & -0.0001 & -0.0009 & -0.0107 & -0.0539 \\
+ \\
+ eDFT & $(0,0)$ & $\E{(0)}$ & -0.0664 & -0.0666 & -0.0667 & -0.0672 & -0.0684 & -0.0707 & -0.0702 \\
+ & & $\E{(1)}$ & 0.0502 & 0.0495 & 0.0482 & 0.0459 & 0.0423 & 0.0355 & 0.0131 \\
+ & & $\E{(2)}$ & 0.1122 & 0.1104 & 0.1061 & 0.0979 & 0.0836 & 0.0635 & 0.0360 \\
+ & & $\Ex{(1)}$ & 0.1165 & 0.1161 & 0.1149 & 0.1131 & 0.1108 & 0.1062 & 0.0834 \\
+ & & $\Ex{(2)}$ & 0.1785 & 0.1769 & 0.1728 & 0.1652 & 0.1520 & 0.1342 & 0.1063 \\
+ & & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
+ & & $\DD{c}{(1)}$ & 0.0244 & 0.0235 & 0.0218 & 0.0186 & 0.0130 & 0.0043 & -0.0065 \\
+ & & $\DD{c}{(2)}$ & 0.0589 & 0.0574 & 0.0546 & 0.0496 & 0.0410 & 0.0275 & 0.0095 \\
+ & & $\LZ{c}{}$ & 0.0015 & 0.0029 & 0.0055 & 0.0103 & 0.0180 & 0.0284 & 0.038555 \\
+ \\
+ eDFT & $(1/3,1/3)$ & $\E{(0)}$ & 0.0014 & 0.0013 & 0.0012 & 0.0009 & 0.0003 & -0.0013 & -0.0079 \\
+ & & $\E{(1)}$ & 0.0149 & 0.0138 & 0.0115 & 0.0072 & -0.0001 & -0.0110 & -0.0209 \\
+ & & $\E{(2)}$ & 0.1217 & 0.1198 & 0.1154 & 0.1069 & 0.0917 & 0.0691 & 0.0389 \\
+ & & $\Ex{(1)}$ & 0.0135 & 0.0125 & 0.0103 & 0.0063 & -0.0005 & -0.0096 & -0.0130 \\
+ & & $\Ex{(2)}$ & 0.1203 & 0.1185 & 0.1142 & 0.1060 & 0.0914 & 0.0705 & 0.0469 \\
+ & & $\DD{c}{(0)}$ & -0.0278 & -0.0270 & -0.0255 & -0.0227 & -0.0180 & -0.0105 & -0.0007 \\
+ & & $\DD{c}{(1)}$ & -0.0034 & -0.0034 & -0.0037 & -0.0041 & -0.0050 & -0.0063 & -0.0076 \\
+ & & $\DD{c}{(2)}$ & 0.0311 & 0.0304 & 0.0291 & 0.0268 & 0.0230 & 0.0168 & 0.0083 \\
+ & & $\LZ{c}{}$ & 0.0007 & 0.0013 & 0.0027 & 0.0051 & 0.0094 & 0.0159 & 0.0242 \\
+ \end{tabular}
+ \end{ruledtabular}
+\end{table*}
+
+
+\bibliography{../eDFT}
+
+\end{document}
diff --git a/Manuscript/eDFT.bib b/Manuscript/eDFT.bib
new file mode 100644
index 0000000..0b4da10
--- /dev/null
+++ b/Manuscript/eDFT.bib
@@ -0,0 +1,8658 @@
+%% This BibTeX bibliography file was created using BibDesk.
+%% http://bibdesk.sourceforge.net/
+
+%% Created for Pierre-Francois Loos at 2019-01-28 08:55:21 +0100
+
+
+%% Saved with string encoding Unicode (UTF-8)
+
+
+
+@article{Schulz_1993,
+ Author = {H. J. Schulz},
+ Date-Added = {2018-12-11 15:14:49 +0100},
+ Date-Modified = {2018-12-11 15:16:49 +0100},
+ Doi = {10.1103/PhysRevLett.71.1864},
+ Journal = {Phys. Rev. Lett.},
+ Pages = {1864},
+ Title = {Wigner crystal in one dimension},
+ Volume = {71},
+ Year = {1993},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.71.1864}}
+
+@article{Meyer_2009,
+ Author = {J. S. Meyer and K. A. Matveev},
+ Date-Added = {2018-12-11 15:13:12 +0100},
+ Date-Modified = {2018-12-11 15:19:37 +0100},
+ Doi = {10.1088/0953-8984/21/2/023203},
+ Journal = {J. Phys.: Condens. Matter},
+ Pages = {023203},
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+ Bdsk-Url-1 = {https://doi.org/10.1088/0953-8984/21/2/023203}}
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+@article{Deshpande_2010,
+ Author = {Vikram V. Deshpande and Marc Bockrath and Leonid I. Glazman and Amir Yacoby},
+ Date-Added = {2018-12-11 15:13:07 +0100},
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+ Volume = {464},
+ Year = {2010},
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+
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+ Date-Modified = {2018-12-11 15:19:13 +0100},
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+ Author = {K. Deur and E. Fromager},
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+ Date-Modified = {2018-12-09 14:19:11 +0100},
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+ Year = {2018}}
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+ Date-Modified = {2018-12-08 17:56:38 +0100},
+ Editor = {F. W. J. Olver and D. W. Lozier and R. F. Boisvert and C. W. Clark},
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+ Publisher = {Cambridge University Press},
+ Title = {NIST Handbook of Mathematical Functions},
+ Year = {2010}}
+
+@article{Agboola_2015,
+ Author = {Agboola, Davids and Knol, Anneke L. and Gill, Peter M. W. and Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4929353},
+ File = {/Users/loos/Zotero/storage/YFD785CA/46.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {084114},
+ Title = {Uniform Electron Gases. {{III}}. {{Low}}-Density Gases on Three-Dimensional Spheres},
+ Volume = {143},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4929353}}
+
+@article{Ambrosek_2010,
+ Author = {Ambrosek, David and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier and Daniel, Chantal},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.jinorgbio.2010.04.002},
+ File = {/Users/loos/Zotero/storage/WA85FMNN/22.pdf},
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+ Journal = {J. Inorg. Biochem.},
+ Language = {en},
+ Month = sep,
+ Number = {9},
+ Pages = {893-901},
+ Title = {A Theoretical Study of {{Ru}}({{II}}) Polypyridyl {{DNA intercalatorsStructure}} and Electronic Absorption Spectroscopy of [{{Ru}}(Phen)2(Dppz)]2+ and [{{Ru}}(Tap)2(Dppz)]2+ Complexes Intercalated in Guanine\textendash{}Cytosine Base Pairs},
+ Volume = {104},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.jinorgbio.2010.04.002}}
+
+@article{Ball_2017,
+ Author = {Ball, Caleb J. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1039/C6CP06801D},
+ File = {/Users/loos/Zotero/storage/7X5YE6WH/52.pdf},
+ Issn = {1463-9076, 1463-9084},
+ Journal = {Phys. Chem. Chem. Phys.},
+ Language = {en},
+ Number = {5},
+ Pages = {3987-3998},
+ Title = {Molecular Electronic Structure in One-Dimensional {{Coulomb}} Systems},
+ Volume = {19},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1039/C6CP06801D}}
+
+@article{Barca_2016,
+ Author = {Barca, Giuseppe M. J. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1021/acs.jctc.6b00130},
+ File = {/Users/loos/Zotero/storage/JS793AVF/49.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Pages = {1735-1740},
+ Title = {Many-{{Electron Integrals}} over {{Gaussian Basis Functions}}. {{I}}. {{Recurrence Relations}} for {{Three}}-{{Electron Integrals}}},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00130}}
+
+@article{Barca_2017,
+ Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ File = {/Users/loos/Zotero/storage/DCFUMHWZ/56.pdf},
+ Journal = {J. Chem. Phys.},
+ Number = {2},
+ Pages = {024103},
+ Shorttitle = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals},
+ Title = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals: {{Fundamental}} Integrals, Upper Bounds, and Recurrence Relations},
+ Volume = {147},
+ Year = {2017}}
+
+@incollection{Barca_2018,
+ Author = {Barca, Giuseppe M.J. and Loos, Pierre-Fran{\c c}ois},
+ Booktitle = {Advances in {{Quantum Chemistry}}},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/bs.aiq.2017.03.004},
+ File = {/Users/loos/Zotero/storage/JQKXB9RU/58.pdf},
+ Isbn = {978-0-12-813002-5},
+ Language = {en},
+ Pages = {147-165},
+ Publisher = {{Elsevier}},
+ Title = {Recurrence {{Relations}} for {{Four}}-{{Electron Integrals Over Gaussian Basis Functions}}},
+ Volume = {76},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1016/bs.aiq.2017.03.004}}
+
+@article{Bernard_2013,
+ Author = {Bernard, Yves A. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M.W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1080/00268976.2013.811302},
+ File = {/Users/loos/Zotero/storage/8XRA54GV/39.pdf},
+ Issn = {0026-8976, 1362-3028},
+ Journal = {Mol. Phys.},
+ Language = {en},
+ Month = sep,
+ Number = {16-17},
+ Pages = {2414-2426},
+ Title = {Distribution of r {\textsubscript{12}} $\cdot$ p {\textsubscript{12}} in Quantum Systems},
+ Volume = {111},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2013.811302}}
+
+@article{Dumont_2008,
+ Author = {Dumont, \'Elise and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.cplett.2008.05.010},
+ File = {/Users/loos/Zotero/storage/5Q5WP253/12.pdf},
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+ Journal = {Chem. Phys. Lett.},
+ Language = {en},
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+ Title = {Effect of Ring Strain on Disulfide Electron Attachment},
+ Volume = {458},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2008.05.010}}
+
+@article{Dumont_2008a,
+ Author = {Dumont, \'Elise and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1021/jp806465e},
+ File = {/Users/loos/Zotero/storage/5FHSI4FF/14.pdf},
+ Issn = {1520-6106, 1520-5207},
+ Journal = {J. Phys. Chem. B},
+ Language = {en},
+ Month = oct,
+ Number = {43},
+ Pages = {13661-13669},
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+ Volume = {112},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1021/jp806465e}}
+
+@article{Dumont_2008b,
+ Author = {Dumont, Elise and Loos, Pierre-Fran{\c c}ois and Laurent, Ad\`ele D. and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1021/ct800161m},
+ File = {/Users/loos/Zotero/storage/AU5GAG48/13.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {1171-1173},
+ Title = {Huge {{Disulfide}}-{{Linkage}}'{{S Electron Capture Variation Induced}} by $\alpha$-{{Helix Orientation}}},
+ Volume = {4},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct800161m}}
+
+@article{Dumont_2009,
+ Author = {Dumont, \'Elise and Laurent, Ad\`ele D. and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1021/ct900093h},
+ File = {/Users/loos/Zotero/storage/DXBEWEYC/15.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = jun,
+ Number = {6},
+ Pages = {1700-1708},
+ Title = {Analyzing the {{Selectivity}} and {{Successiveness}} of a {{Two}}-{{Electron Capture}} on a {{Multiply Disulfide}}-{{Linked Protein}}},
+ Volume = {5},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct900093h}}
+
+@article{Dumont_2010,
+ Author = {Dumont, \~Alise and Loos, Pierre-Fran\~A\textsection{}ois and Laurent, Ad\~A\textasciidieresis{}le D. and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1002/qua.22072},
+ File = {/Users/loos/Zotero/storage/4VG63LZM/20.pdf},
+ Issn = {00207608, 1097461X},
+ Journal = {Int. J. Quantum Chem.},
+ Language = {en},
+ Month = mar,
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+ Title = {Electronic Effects and Ring Strain Influences on the Electron Uptake by Selenium-Containing Bonds},
+ Volume = {110},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1002/qua.22072}}
+
+@article{Fornili_2006,
+ Author = {Fornili, Arianna and Loos, Pierre-Fran{\c c}ois and Sironi, Maurizio and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.cplett.2006.06.095},
+ File = {/Users/loos/Zotero/storage/28RWJAZ8/2.pdf},
+ Issn = {00092614},
+ Journal = {Chem. Phys. Lett.},
+ Language = {en},
+ Month = aug,
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+ Title = {Frozen Core Orbitals as an Alternative to Specific Frontier Bond Potential in Hybrid {{Quantum Mechanics}}/{{Molecular Mechanics}} Methods},
+ Volume = {427},
+ Year = {2006},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2006.06.095}}
+
+@article{Garniron_2017,
+ Author = {Garniron, Yann and Scemama, Anthony and Loos, Pierre-Fran{\c c}ois and Caffarel, Michel},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4992127},
+ File = {/Users/loos/Zotero/storage/NHU2BWMZ/57.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = jul,
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+ Volume = {147},
+ Year = {2017},
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+
+@article{Gill_2012,
+ Author = {Gill, Peter M. W. and Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:11:33 +0100},
+ Doi = {10.1007/s00214-011-1069-7},
+ File = {/Users/loos/Zotero/storage/A85Y7E9B/32.pdf},
+ Issn = {1432-881X, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {1069},
+ Title = {Uniform Electron Gases},
+ Volume = {131},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1007/s00214-011-1069-7}}
+
+@article{Gill_2014,
+ Author = {Gill, Peter M. W. and Loos, Pierre-Fran{\c c}ois and Agboola, Davids},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4903984},
+ File = {/Users/loos/Zotero/storage/YM5Q8ST9/43.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Pages = {244102},
+ Title = {Basis Functions for Electronic Structure Calculations on Spheres},
+ Volume = {141},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4903984}}
+
+@article{Loos_2007,
+ Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1002/qua.21410},
+ File = {/Users/loos/Zotero/storage/64MPU2ZJ/8.pdf},
+ Issn = {00207608, 1097461X},
+ Journal = {Int. J. Quantum Chem.},
+ Language = {en},
+ Number = {12},
+ Pages = {2243-2252},
+ Title = {Core-Ionized and Core-Excited States of Macromolecules},
+ Volume = {107},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1002/qua.21410}}
+
+@article{Loos_2007a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier and Rivail, Jean-Louis},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1007/s00214-007-0258-x},
+ File = {/Users/loos/Zotero/storage/S6FBKP2M/5.pdf},
+ Issn = {1432-881X, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Language = {en},
+ Month = jun,
+ Number = {1},
+ Pages = {165-171},
+ Title = {Intramolecular Interactions and Cis Peptidic Bonds},
+ Volume = {118},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1007/s00214-007-0258-x}}
+
+@inproceedings{Loos_2007b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier},
+ Booktitle = {{{AIP Conference Proceedings}}},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ File = {/Users/loos/Zotero/storage/HGKGXAMT/10.pdf},
+ Pages = {308--315},
+ Publisher = {{AIP}},
+ Title = {On {{The Frontier Bond Location In The QM}}/{{MM Description Of Peptides And Proteins}}.},
+ Volume = {963},
+ Year = {2007}}
+
+@article{Loos_2007c,
+ Author = {Loos, P.-F. and Fornili, A. and Sironi, M. and Assfeld, X.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ File = {/Users/loos/Zotero/storage/CAHLR8H8/9.pdf},
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+ Number = {2-4},
+ Pages = {2--4},
+ Shorttitle = {Removing Extra Frontier Parameters in {{QM}}/{{MM}} Methods},
+ Title = {Removing Extra Frontier Parameters in {{QM}}/{{MM}} Methods: A Tentative with the {{Local Self}}-{{Consistent Field}} Approach},
+ Volume = {3},
+ Year = {2007}}
+
+@article{Loos_2007d,
+ Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1021/ct6003214},
+ File = {/Users/loos/Zotero/storage/A9TSDVGQ/6.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = may,
+ Number = {3},
+ Pages = {1047-1053},
+ Title = {Self-{{Consistent Strictly Localized Orbitals}}},
+ Volume = {3},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct6003214}}
+
+@article{Loos_2008,
+ Author = {Loos, Pierre-Fran{\c c}ois and Preat, Julien and Laurent, Ad\`ele D. and Michaux, Catherine and Jacquemin, Denis and Perp\`ete, Eric A. and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1021/ct700188w},
+ File = {/Users/loos/Zotero/storage/LSU72HGK/11.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Pages = {637-645},
+ Shorttitle = {Theoretical {{Investigation}} of the {{Geometries}} and {{UV}}-vis {{Spectra}} of {{Poly}}(},
+ Title = {Theoretical {{Investigation}} of the {{Geometries}} and {{UV}}-vis {{Spectra}} of {{Poly}}( {\textsc{l}} -Glutamic Acid) {{Featuring}} a {{Photochromic Azobenzene Side Chain}}},
+ Volume = {4},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct700188w}}
+
+@article{Loos_2009,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.3275519},
+ File = {/Users/loos/Zotero/storage/E9MV4JCJ/19.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Pages = {241101},
+ Title = {Correlation Energy of Two Electrons in the High-Density Limit},
+ Volume = {131},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3275519}}
+
+@article{Loos_2009a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:02:44 +0100},
+ Doi = {10.1103/PhysRevA.79.062517},
+ File = {/Users/loos/Zotero/storage/R9N5FN2F/17.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = jun,
+ Number = {6},
+ Pages = {062517},
+ Title = {Ground State of Two Electrons on a Sphere},
+ Volume = {79},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.79.062517}}
+
+@article{Loos_2009b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Dumont, Elise and Laurent, Ad\`ele D. and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.cplett.2009.05.041},
+ File = {/Users/loos/Zotero/storage/QDEXZPUE/16.pdf},
+ Issn = {00092614},
+ Journal = {Chem. Phys. Lett.},
+ Language = {en},
+ Month = jun,
+ Number = {1-3},
+ Pages = {120-123},
+ Title = {Important Effects of Neighbouring Nucleotides on Electron Induced {{DNA}} Single-Strand Breaks},
+ Volume = {475},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2009.05.041}}
+
+@article{Loos_2009c,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:02:54 +0100},
+ Doi = {10.1103/PhysRevLett.103.123008},
+ File = {/Users/loos/Zotero/storage/GRFHDW5F/18.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = sep,
+ Number = {12},
+ Pages = {123008},
+ Shorttitle = {Two {{Electrons}} on a {{Hypersphere}}},
+ Title = {Two {{Electrons}} on a {{Hypersphere}}: {{A Quasiexactly Solvable Model}}},
+ Volume = {103},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.123008}}
+
+@article{Loos_2010,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 15:39:43 +0100},
+ Doi = {10.1016/j.cplett.2010.09.019},
+ File = {/Users/loos/Zotero/storage/DJKJNV2H/27.pdf},
+ Issn = {00092614},
+ Journal = {Chem. Phys. Lett.},
+ Language = {en},
+ Month = nov,
+ Number = {1-3},
+ Pages = {1-8},
+ Shorttitle = {A Tale of Two Electrons},
+ Title = {A Tale of Two Electrons: {{Correlation}} at High Density},
+ Volume = {500},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2010.09.019}}
+
+@article{Loos_2010a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.3455706},
+ File = {/Users/loos/Zotero/storage/QWSE9HHH/24.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = jun,
+ Number = {23},
+ Pages = {234111},
+ Title = {Correlation Energy of Two Electrons in a Ball},
+ Volume = {132},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3455706}}
+
+@article{Loos_2010b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:10:58 +0100},
+ Doi = {10.1103/PhysRevA.81.052510},
+ File = {/Users/loos/Zotero/storage/3MZKVWT9/23.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = may,
+ Number = {5},
+ Pages = {052510},
+ Title = {Ground State of Two Electrons on Concentric Spheres},
+ Volume = {81},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.052510}}
+
+@article{Loos_2010c,
+ Author = {Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:10:50 +0100},
+ Doi = {10.1103/PhysRevA.81.032510},
+ File = {/Users/loos/Zotero/storage/KJEJ5D77/21.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = mar,
+ Number = {3},
+ Pages = {032510},
+ Title = {Hooke's Law Correlation in Two-Electron Systems},
+ Volume = {81},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.032510}}
+
+@article{Loos_2010d,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:03:11 +0100},
+ Doi = {10.1103/PhysRevLett.105.113001},
+ File = {/Users/loos/Zotero/storage/I2UH4CU5/26.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = sep,
+ Number = {11},
+ Pages = {113001},
+ Title = {Invariance of the {{Correlation Energy}} at {{High Density}} and {{Large Dimension}} in {{Two}}-{{Electron Systems}}},
+ Volume = {105},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.105.113001}}
+
+@article{Loos_2010e,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M.W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1080/00268976.2010.508472},
+ File = {/Users/loos/Zotero/storage/TLWJZ3HQ/25.pdf},
+ Issn = {0026-8976, 1362-3028},
+ Journal = {Mol. Phys.},
+ Language = {en},
+ Month = oct,
+ Number = {19-20},
+ Pages = {2527-2532},
+ Title = {Excited States of Spherium},
+ Volume = {108},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2010.508472}}
+
+@article{Loos_2011,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:10:39 +0100},
+ Doi = {10.1103/PhysRevB.84.033103},
+ File = {/Users/loos/Zotero/storage/NK4SJ24B/29.pdf},
+ Issn = {1098-0121, 1550-235X},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = jul,
+ Number = {3},
+ Pages = {033103},
+ Title = {Correlation Energy of the Spin-Polarized Uniform Electron Gas at High Density},
+ Volume = {84},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.84.033103}}
+
+@article{Loos_2011a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:10:34 +0100},
+ Doi = {10.1103/PhysRevB.83.233102},
+ File = {/Users/loos/Zotero/storage/2QNIQXM5/28.pdf},
+ Issn = {1098-0121, 1550-235X},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = jun,
+ Number = {23},
+ Pages = {233102},
+ Title = {Exact Energy of the Spin-Polarized Two-Dimensional Electron Gas at High Density},
+ Volume = {83},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.83.233102}}
+
+@article{Loos_2011b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.3665393},
+ File = {/Users/loos/Zotero/storage/R8SHV7JI/31.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {21},
+ Pages = {214111},
+ Shorttitle = {Thinking Outside the Box},
+ Title = {Thinking Outside the Box: {{The}} Uniform Electron Gas on a Hypersphere},
+ Volume = {135},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3665393}}
+
+@article{Loos_2012,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:10:25 +0100},
+ Doi = {10.1103/PhysRevLett.108.083002},
+ File = {/Users/loos/Zotero/storage/D5YVLEB5/34.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = feb,
+ Number = {8},
+ Pages = {083002},
+ Title = {Exact {{Wave Functions}} of {{Two}}-{{Electron Quantum Rings}}},
+ Volume = {108},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.108.083002}}
+
+@article{Loos_2012a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M.W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1080/00268976.2012.679634},
+ File = {/Users/loos/Zotero/storage/H82N8B89/35.pdf},
+ Issn = {0026-8976, 1362-3028},
+ Journal = {Mol. Phys.},
+ Language = {en},
+ Month = oct,
+ Number = {19-20},
+ Pages = {2337-2342},
+ Title = {Harmonically Trapped Jellium},
+ Volume = {110},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2012.679634}}
+
+@article{Loos_2012b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1002/qua.23155},
+ File = {/Users/loos/Zotero/storage/PMVR53GF/33.pdf},
+ Issn = {00207608},
+ Journal = {Int. J. Quantum Chem.},
+ Language = {en},
+ Month = mar,
+ Number = {6},
+ Pages = {1712-1716},
+ Title = {Leading-Order Behavior of the Correlation Energy in the Uniform Electron Gas},
+ Volume = {112},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1002/qua.23155}}
+
+@article{Loos_2012c,
+ Author = {Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.physleta.2012.05.010},
+ File = {/Users/loos/Zotero/storage/JUPS4RAD/36.pdf},
+ Issn = {03759601},
+ Journal = {Phys. Lett. A},
+ Language = {en},
+ Month = may,
+ Number = {26-27},
+ Pages = {1997-2000},
+ Title = {Understanding Excitons Using Spherical Geometry},
+ Volume = {376},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.physleta.2012.05.010}}
+
+@article{Loos_2013,
+ Author = {Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4790613},
+ File = {/Users/loos/Zotero/storage/47LGX76U/37.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = feb,
+ Number = {6},
+ Pages = {064108},
+ Title = {High-Density Correlation Energy Expansion of the One-Dimensional Uniform Electron Gas},
+ Volume = {138},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4790613}}
+
+@article{Loos_2013a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4802589},
+ File = {/Users/loos/Zotero/storage/5Q2YJGTX/38.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = apr,
+ Number = {16},
+ Pages = {164124},
+ Title = {Uniform Electron Gases. {{I}}. {{Electrons}} on a Ring},
+ Volume = {138},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4802589}}
+
+@article{Loos_2014,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M.W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.physleta.2013.11.045},
+ File = {/Users/loos/Zotero/storage/BRNDKC4F/40.pdf},
+ Issn = {03759601},
+ Journal = {Phys. Lett. A},
+ Language = {en},
+ Month = jan,
+ Number = {4},
+ Pages = {329-333},
+ Title = {Exact Wave Functions for Concentric Two-Electron Systems},
+ Volume = {378},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.physleta.2013.11.045}}
+
+@article{Loos_2014a,
+ Author = {Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-12-11 14:02:09 +0100},
+ Doi = {10.1103/PhysRevA.89.052523},
+ File = {/Users/loos/Zotero/storage/E8XZBSLG/42.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = may,
+ Number = {5},
+ Pages = {052523},
+ Title = {Generalized Local-Density Approximation and One-Dimensional Finite Uniform Electron Gases},
+ Volume = {89},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.89.052523}}
+
+@article{Loos_2014b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Ball, Caleb J. and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4867910},
+ File = {/Users/loos/Zotero/storage/QD667QG2/41.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {18},
+ Pages = {18A524},
+ Title = {Uniform Electron Gases. {{II}}. {{The}} Generalized Local Density Approximation in One Dimension},
+ Volume = {140},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4867910}}
+
+@article{Loos_2015a,
+ Author = {Loos, Pierre-Fran{\c c}ois and Ball, Caleb J. and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:21 +0200},
+ Doi = {10.1039/C4CP03571B},
+ File = {/Users/loos/Zotero/storage/4JXMCN8L/44.pdf},
+ Issn = {1463-9076, 1463-9084},
+ Journal = {Phys. Chem. Chem. Phys.},
+ Language = {en},
+ Number = {5},
+ Pages = {3196-3206},
+ Title = {Chemistry in One Dimension},
+ Volume = {17},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1039/C4CP03571B}}
+
+@article{Loos_2015d,
+ Author = {Loos, Pierre-Fran{\c c}ois and Bloomfield, Nathaniel J. and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:35 +0200},
+ Doi = {10.1063/1.4935374},
+ File = {/Users/loos/Zotero/storage/HEH82IUH/47.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = nov,
+ Number = {18},
+ Pages = {181101},
+ Shorttitle = {Communication},
+ Title = {Communication: {{Three}}-Electron Coalescence Points in Two and Three Dimensions},
+ Volume = {143},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4935374}}
+
+@article{Loos_2015b,
+ Author = {Loos, Pierre-Fran{\c c}ois and Bressanini, Dario},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4922159},
+ File = {/Users/loos/Zotero/storage/UBTEMR7G/45.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = jun,
+ Number = {21},
+ Pages = {214112},
+ Title = {Nodal Surfaces and Interdimensional Degeneracies},
+ Volume = {142},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4922159}}
+
+@article{Loos_2016,
+ Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 23:04:36 +0200},
+ Doi = {10.1002/wcms.1257},
+ File = {/Users/loos/Zotero/storage/HEXYAMEN/50.pdf},
+ Issn = {17590876},
+ Journal = {Wiley Interdiscip. Rev. Comput. Mol. Sci.},
+ Language = {en},
+ Month = jul,
+ Number = {4},
+ Pages = {410-429},
+ Shorttitle = {The Uniform Electron Gas},
+ Title = {The Uniform Electron Gas},
+ Volume = {6},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1002/wcms.1257}}
+
+@article{Loos_2017,
+ Author = {Loos, Pierre-Fran{\c c}ois and Rivail, Jean-Louis and Assfeld, Xavier},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1007/s00894-017-3347-3},
+ File = {/Users/loos/Zotero/storage/BFLICA2H/55.pdf},
+ Issn = {1610-2940, 0948-5023},
+ Journal = {J. Mol. Model.},
+ Language = {en},
+ Month = jun,
+ Number = {6},
+ Title = {Iterative Stochastic Subspace Self-Consistent Field Method},
+ Volume = {23},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1007/s00894-017-3347-3}}
+
+@article{Loos_2017a,
+ Author = {Loos, Pierre-Fran{\c c}ois},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1063/1.4978409},
+ File = {/Users/loos/Zotero/storage/NJ3WTVZZ/54.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = mar,
+ Number = {11},
+ Pages = {114108},
+ Title = {Exchange Functionals Based on Finite Uniform Electron Gases},
+ Volume = {146},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4978409}}
+
+@article{Moreau_2004,
+ Author = {Moreau, Y. and Loos, P.-F. and Assfeld, X.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1007/s00214-004-0581-4},
+ File = {/Users/loos/Zotero/storage/I2GBGPYP/1.pdf},
+ Issn = {1432-881X, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Language = {en},
+ Month = sep,
+ Number = {4},
+ Shorttitle = {Solvent Effects on the Asymmetric {{Diels}}?},
+ Title = {Solvent Effects on the Asymmetric {{Diels}}?{{Alder}} Reaction between Cyclopentadiene and (?)-Menthyl Acrylate Revisited with the Three-Layer Hybrid Local Self-Consistent Field/Molecular Mechanics/Self-Consistent Reaction Field Method},
+ Volume = {112},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1007/s00214-004-0581-4}}
+
+@article{Preat_2007,
+ Author = {Preat, Julien and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier and Jacquemin, Denis and Perp\`ete, Eric A.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1016/j.theochem.2006.12.047},
+ File = {/Users/loos/Zotero/storage/4BZSHB49/7.pdf},
+ Issn = {01661280},
+ Journal = {J. Mol. Struct. THEOCHEM},
+ Language = {en},
+ Month = apr,
+ Number = {1-3},
+ Pages = {85-91},
+ Shorttitle = {A {{TD}}-{{DFT}} Investigation of {{UV}} Spectra of Pyrano\"idic Dyes},
+ Title = {A {{TD}}-{{DFT}} Investigation of {{UV}} Spectra of Pyrano\"idic Dyes: {{A NCM}} vs {{PCM}} Comparison},
+ Volume = {808},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.theochem.2006.12.047}}
+
+@article{Preat_2007a,
+ Author = {Preat, Julien and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier and Jacquemin, Denis and Perp\`ete, Eric A.},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ Doi = {10.1002/qua.21182},
+ File = {/Users/loos/Zotero/storage/D5F3ED8Z/4.pdf},
+ Issn = {00207608, 1097461X},
+ Journal = {Int. J. Quantum Chem.},
+ Language = {en},
+ Number = {3},
+ Pages = {574-585},
+ Shorttitle = {{{DFT}} and {{TD}}-{{DFT}} Investigation of {{IR}} and {{UV}} Spectra of Solvated Molecules},
+ Title = {{{DFT}} and {{TD}}-{{DFT}} Investigation of {{IR}} and {{UV}} Spectra of Solvated Molecules: {{Comparison}} of Two {{SCRF}} Continuum Models},
+ Volume = {107},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1002/qua.21182}}
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+@inproceedings{Rivail_2006,
+ Author = {Rivail, J. L. and Bouchy, A. and Loos, P. F.},
+ Booktitle = {Anales de La {{Asociaci\'on Qu\'imica Argentina}}},
+ Date-Added = {2018-10-24 22:57:04 +0200},
+ Date-Modified = {2018-10-24 22:57:04 +0200},
+ File = {/Users/loos/Zotero/storage/XGMNTL9R/3.pdf},
+ Pages = {19--26},
+ Publisher = {{SciELO Argentina}},
+ Title = {Electronic Factors Favouring the Cis Conformation in Proline Peptidic Bonds},
+ Volume = {94},
+ Year = {2006}}
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+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.93.235114}}
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+@article{Rogers_2017,
+ Author = {Rogers, Fergus JM and Loos, Pierre-Fran{\c c}ois},
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+ Pages = {6401-6405},
+ Title = {Infinitesimal differential diffusion quantum Monte Carlo study of CuH spectroscopic constants},
+ Url = {https://doi.org/10.1063/1.464838},
+ Volume = {98},
+ Year = {1993},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.464838},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.464838}}
+
+@article{Bender_1969,
+ Author = {Bender, Charles F. and Davidson, Ernest R.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/physrev.183.23},
+ Issn = {0031-899X},
+ Journal = {Phys. Rev.},
+ Month = {Jul},
+ Number = {1},
+ Pages = {23--30},
+ Publisher = {American Physical Society (APS)},
+ Title = {Studies in Configuration Interaction: The First-Row Diatomic Hydrides},
+ Url = {http://dx.doi.org/10.1103/physrev.183.23},
+ Volume = {183},
+ Year = {1969},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1103/physrev.183.23}}
+
+@article{Berge_s_2008,
+ Author = {Berg{\`e}s, Jacqueline and Varmenot, Nicolas and Scemama, Anthony and Abedinzadeh, Zohreh and Bobrowski, Krzysztof},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/jp711944v},
+ Issn = {1520-5215},
+ Journal = {J. Phys. Chem. A},
+ Month = {Jul},
+ Number = {30},
+ Pages = {7015--7026},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Energies, Stability and Structure Properties of Radicals Derived from Organic Sulfides Containing an Acetyl Group after the*OH Attack: ab Initio and DFT Calculations vs Experiment},
+ Url = {http://dx.doi.org/10.1021/jp711944v},
+ Volume = {112},
+ Year = {2008},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/jp711944v}}
+
+@article{Bernardi_1990,
+ Author = {Bernardi, Fernando and De, Sushovan and Olivucci, Massimo and Robb, Michael A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ja00161a013},
+ Issn = {0002-7863},
+ Journal = {Journal of the American Chemical Society},
+ Language = {en},
+ Month = feb,
+ Number = {5},
+ Pages = {1737-1744},
+ Shorttitle = {The Mechanism of Ground-State-Forbidden Photochemical Pericyclic Reactions},
+ Title = {The Mechanism of Ground-State-Forbidden Photochemical Pericyclic Reactions: Evidence for Real Conical Intersections},
+ Volume = {112},
+ Year = {1990},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ja00161a013}}
+
+@article{Bernardi_1996,
+ Author = {Bernardi, Fernando and Olivucci, Massimo and Robb, Michael A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1039/cs9962500321},
+ Issn = {0306-0012, 1460-4744},
+ Journal = {Chemical Society Reviews},
+ Language = {en},
+ Number = {5},
+ Pages = {321},
+ Title = {Potential Energy Surface Crossings in Organic Photochemistry},
+ Volume = {25},
+ Year = {1996},
+ Bdsk-Url-1 = {https://doi.org/10.1039/cs9962500321}}
+
+@article{Blunt_2015,
+ Author = {Blunt, N. S. and Smart, Simon D. and Booth, George H. and Alavi, Ali},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4932595},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = oct,
+ Number = {13},
+ Pages = {134117},
+ Title = {An Excited-State Approach within Full Configuration Interaction Quantum {{Monte Carlo}}},
+ Volume = {143},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4932595}}
+
+@article{Blunt_2017,
+ Author = {Blunt, N. S. and Neuscamman, Eric},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4998197},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = nov,
+ Number = {19},
+ Pages = {194101},
+ Shorttitle = {Charge-Transfer Excited States},
+ Title = {Charge-Transfer Excited States: {{Seeking}} a Balanced and Efficient Wave Function Ansatz in Variational {{Monte Carlo}}},
+ Volume = {147},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4998197}}
+
+@article{Blunt_2018,
+ Author = {N. S. Blunt},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.5037923},
+ Journal = {J. Chem. Phys.},
+ Pages = {221101},
+ Title = {An efficient and accurate perturbative correction to initiator full configuration interaction quantum Monte Carlo},
+ Volume = {148},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.5037923}}
+
+@article{Boggio-Pasqua_2000,
+ Abstract = {Realistic analytical representations of the twelve lowest singlet and triplet electronic adiabatic potential energy curves of C2 molecule are given in this article. The corresponding electronic states are correlated with C atoms both in their 3P state. A new set of high level MRCI calculations coupled with a double many-body expansion analytical \textregistered{}tting based on the extended Hartree$\pm$Fock approximate correlation energy model have been used in this work. Using RKR data available in the literature, comparison is made between our results and RKR turning points concerning the four lowest singlet states X1Sg1, A1Pu, B1Dg and BH 1Sg1 of C2. The agreement is very satisfying. q 2000 Elsevier Science B.V. All rights reserved.},
+ Author = {{Boggio-Pasqua}, M. and Voronin, A.I. and Halvick, Ph. and Rayez, J.-C.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/S0166-1280(00)00442-5},
+ File = {/Users/loos/Zotero/storage/8BP7KA4E/Boggio-Pasqua et al. - 2000 - Analytical representations of high level ab initio.pdf},
+ Issn = {01661280},
+ Journal = {Journal of Molecular Structure: THEOCHEM},
+ Language = {en},
+ Month = oct,
+ Number = {1-3},
+ Pages = {159-167},
+ Title = {Analytical Representations of High Level Ab Initio Potential Energy Curves of the {{C}} 2 Molecule},
+ Volume = {531},
+ Year = {2000},
+ Bdsk-Url-1 = {https://doi.org/10.1016/S0166-1280(00)00442-5}}
+
+@article{Boggio-Pasqua_2004,
+ Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Klene, Michael and Robb, Michael A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1690756},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {17},
+ Pages = {7849-7860},
+ Title = {A Computational Strategy for Geometry Optimization of Ionic and Covalent Excited States, Applied to Butadiene and Hexatriene},
+ Volume = {120},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1690756}}
+
+@article{Boggio-Pasqua_2007,
+ Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/jo070452v},
+ Issn = {0022-3263, 1520-6904},
+ Journal = {The Journal of Organic Chemistry},
+ Language = {en},
+ Month = jun,
+ Number = {12},
+ Pages = {4497-4503},
+ Title = {Toward a {{Mechanistic Understanding}} of the {{Photochromism}} of {{Dimethyldihydropyrenes}}},
+ Volume = {72},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1021/jo070452v}}
+
+@article{Bomble_2004,
+ Author = {Bomble, Yannick J. and Sattelmeyer, Kurt W. and Stanton, John F. and Gauss, J\"urgen},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1780159},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = sep,
+ Number = {11},
+ Pages = {5236-5240},
+ Title = {On the Vertical Excitation Energy of Cyclopentadiene},
+ Volume = {121},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1780159}}
+
+@article{Booth_2009,
+ Author = {Booth, George H. and Thom, Alex J. W. and Alavi, Ali},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3193710},
+ File = {Full Text PDF:/home/scemama/Dropbox/Zotero/storage/2MNQC3DS/Booth et al. - 2009 - Fermion Monte Carlo without fixed nodes A game of.pdf:application/pdf;JChemPhys_131_054106.pdf:/home/scemama/Dropbox/Zotero/storage/AYB9I4U9/JChemPhys_131_054106.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/U56UGSZM/Booth et al. - 2009 - Fermion Monte Carlo without fixed nodes A game of.html:text/html},
+ Issn = {0021-9606},
+ Journal = {J. Chem. Phys.},
+ Month = aug,
+ Number = {5},
+ Pages = {054106},
+ Shorttitle = {Fermion {Monte} {Carlo} without fixed nodes},
+ Title = {Fermion {Monte} {Carlo} without fixed nodes: {A} game of life, death, and annihilation in {Slater} determinant space},
+ Url = {http://aip.scitation.org/doi/full/10.1063/1.3193710},
+ Urldate = {2017-11-13},
+ Volume = {131},
+ Year = {2009},
+ Bdsk-Url-1 = {http://aip.scitation.org/doi/full/10.1063/1.3193710},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.3193710}}
+
+@article{Booth_2011,
+ Author = {Booth, George H. and Cleland, Deidre and Thom, Alex J. W. and Alavi, Ali},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3624383},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {084104},
+ Shorttitle = {Breaking the Carbon Dimer},
+ Title = {Breaking the Carbon Dimer: {{The}} Challenges of Multiple Bond Dissociation with Full Configuration Interaction Quantum {{Monte Carlo}} Methods},
+ Volume = {135},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3624383}}
+
+@article{Boschen_2014,
+ Author = {Boschen, Jeffery S. and Theis, Daniel and Ruedenberg, Klaus and Windus, Theresa L.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/s00214-013-1425-x},
+ Issn = {1432-881X, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Language = {en},
+ Month = feb,
+ Number = {2},
+ Title = {Accurate Ab Initio Potential Energy Curves and Spectroscopic Properties of the Four Lowest Singlet States of {{C2}}},
+ Volume = {133},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1007/s00214-013-1425-x}}
+
+@article{Bouabca_2009,
+ Author = {Bouab{\c c}a, Thomas and Ben Amor, Nadia and Maynau, Daniel and Caffarel, Michel},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3086023},
+ File = {/Users/loos/Zotero/storage/3KDG45R5/Bouab{\c c}a et al. - 2009 - A study of the fixed-node error in quantum Monte C.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = mar,
+ Number = {11},
+ Pages = {114107},
+ Shorttitle = {A Study of the Fixed-Node Error in Quantum {{Monte Carlo}} Calculations of Electronic Transitions},
+ Title = {A Study of the Fixed-Node Error in Quantum {{Monte Carlo}} Calculations of Electronic Transitions: {{The}} Case of the Singlet N$\rightarrow\pi{_\ast}$ ({{CO}}) Transition of the Acrolein},
+ Volume = {130},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3086023}}
+
+@incollection{Bressanini_2001,
+ Author = {D. Bressanini and D. M. Ceperley and P. Reynolds},
+ Booktitle = {Recent Advances in Quantum Monte Carlo Methods},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Editor = {W. A. {Lester Jr.} and S. M. Rothstein and S. Tanaka},
+ Publisher = {World Scientfic},
+ Title = {What do we know about wave function nodes?},
+ Volume = {2},
+ Year = {2001}}
+
+@article{Bressanini_2005a,
+ Author = {D. Bressanini and P. J. Reynolds},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {Phys. Rev. Lett.},
+ Pages = {110201},
+ Title = {Unexpected Symmetry in the Nodal Structure of the He Atom},
+ Volume = {95},
+ Year = {2005}}
+
+@article{Bressanini_2005b,
+ Author = {D. Bressanini and G. Morosi and S. Tarasco},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {J. Chem. Phys.},
+ Pages = {204109},
+ Title = {An investigation of nodal structures and the construction of trial wave functions},
+ Volume = {123},
+ Year = {2005}}
+
+@article{Bressanini_2008,
+ Author = {D. Bressanini and G. Morosi},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {J. Chem. Phys.},
+ Pages = {054103},
+ Title = {On the nodal structure of single-particle approximation based atomic wave functions},
+ Volume = {129},
+ Year = {2008}}
+
+@article{Bressanini_2012,
+ Author = {D. Bressanini},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {Phys. Rev. B},
+ Pages = {115120},
+ Title = {Implications of the two nodal domains conjecture for ground state fermionic wave functions},
+ Volume = {86},
+ Year = {2012}}
+
+@article{Bridgeman_2000,
+ Author = {Bridgeman, Adam J. and Rothery, Joanne},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1039/a906523g},
+ Issn = {1364-5447},
+ Journal = {J. Chem. Soc. Dalton Trans.},
+ Number = {2},
+ Pages = {211--218},
+ Publisher = {Royal Society of Chemistry (RSC)},
+ Title = {Periodic trends in the diatomic monoxides and monosulfides of the 3d transition metals},
+ Url = {http://dx.doi.org/10.1039/a906523g},
+ Year = {2000},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1039/a906523g}}
+
+@article{Budzak_2017,
+ Author = {Budz{\'a}k, {\v S}. and Scalmani, G. and Jacquemin, D.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {J. Chem. Theory Comput.},
+ Pages = {6237--6252},
+ Title = {Accurate Excited-State Geometries: a CASPT2 and Coupled-Cluster Reference Database for Small Molecules},
+ Volume = {13},
+ Year = {2017}}
+
+@article{Buendia_2006,
+ Author = {E. Buend{\'\i}a and F.J. G{\'a}lvez and A. Sarsa},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {https://doi.org/10.1016/j.cplett.2006.07.027},
+ Issn = {0009-2614},
+ Journal = {Chem. Phys. Lett.},
+ Number = {4},
+ Pages = {241 - 244},
+ Title = {Correlated wave functions for the ground state of the atoms Li through Kr},
+ Url = {http://www.sciencedirect.com/science/article/pii/S0009261406010372},
+ Volume = {428},
+ Year = {2006},
+ Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261406010372},
+ Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2006.07.027}}
+
+@article{Buendia_2013,
+ Author = {E. Buend{\'\i}a and F.J. G{\'a}lvez and P. Maldonado and A. Sarsa},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {https://doi.org/10.1016/j.cplett.2012.12.055},
+ Issn = {0009-2614},
+ Journal = {Chem. Phys. Lett.},
+ Number = {Supplement C},
+ Pages = {12 - 17},
+ Title = {Quantum Monte Carlo ionization potential and electron affinity for transition metal atoms},
+ Url = {http://www.sciencedirect.com/science/article/pii/S0009261413000079},
+ Volume = {559},
+ Year = {2013},
+ Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261413000079},
+ Bdsk-Url-2 = {https://doi.org/10.1016/j.cplett.2012.12.055}}
+
+@article{Buenker_1968,
+ Author = {Buenker, Robert J. and Peyerimhoff, Sigrid D.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/BF00528266},
+ Issn = {0040-5744, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Language = {en},
+ Number = {3},
+ Pages = {183-199},
+ Title = {{{CI}} Method for the Study of General Molecular Potentials},
+ Volume = {12},
+ Year = {1968},
+ Bdsk-Url-1 = {https://doi.org/10.1007/BF00528266}}
+
+@article{Buenker_2014,
+ Author = {Buenker, Robert J. and Phillips, Robin A. and Krebs, Stefan and Liebermann, Heinz-Peter and Alekseyev, Aleksey B. and Funke, Peter},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/s00214-014-1468-7},
+ File = {Full Text PDF:/home/scemama/Dropbox/Zotero/storage/9C9YFWDL/Buenker et al. - 2014 - The Wuppertal multireference configuration interac.pdf:application/pdf;Snapshot:/home/scemama/Dropbox/Zotero/storage/8KCIVRJS/10.html:text/html},
+ Issn = {1432-881X, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Pages = {1468},
+ Title = {The {Wuppertal} multireference configuration interaction ({MRD}-{CI}) program system},
+ Url = {https://link.springer.com/article/10.1007/s00214-014-1468-7},
+ Urldate = {2017-11-13},
+ Volume = {133},
+ Year = {2014},
+ Bdsk-Url-1 = {https://link.springer.com/article/10.1007/s00214-014-1468-7},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1007/s00214-014-1468-7}}
+
+@article{Bunge_2006,
+ Author = {Bunge, Carlos F. and Carb{\'o}-Dorca, Ramon},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.2207621},
+ File = {Snapshot:/home/scemama/Dropbox/Zotero/storage/SB8DJGT3/1.html:text/html},
+ Issn = {0021-9606},
+ Journal = {J. Chem. Phys.},
+ Month = jul,
+ Number = {1},
+ Pages = {014108},
+ Title = {Select-divide-and-conquer method for large-scale configuration interaction},
+ Url = {http://aip.scitation.org/doi/abs/10.1063/1.2207621},
+ Urldate = {2017-11-17},
+ Volume = {125},
+ Year = {2006},
+ Bdsk-Url-1 = {http://aip.scitation.org/doi/abs/10.1063/1.2207621},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.2207621}}
+
+@article{Burkatzki_2007,
+ Author = {Burkatzki, M. and Filippi, C. and Dolg, M.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.2741534},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Jun},
+ Number = {23},
+ Pages = {234105},
+ Publisher = {AIP Publishing},
+ Title = {Energy-consistent pseudopotentials for quantum Monte Carlo calculations},
+ Url = {http://dx.doi.org/10.1063/1.2741534},
+ Volume = {126},
+ Year = {2007},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2741534}}
+
+@article{Burkatzki_2008,
+ Author = {Burkatzki, M. and Filippi, Claudia and Dolg, M.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.2987872},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Oct},
+ Number = {16},
+ Pages = {164115},
+ Publisher = {AIP Publishing},
+ Title = {Energy-consistent small-core pseudopotentials for 3d-transition metals adapted to quantum Monte Carlo calculations},
+ Url = {http://dx.doi.org/10.1063/1.2987872},
+ Volume = {129},
+ Year = {2008},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2987872}}
+
+@article{Bytautas_2009,
+ Author = {Bytautas, Laimutis and Ruedenberg, Klaus},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/j.chemphys.2008.11.021},
+ Issn = {0301-0104},
+ Journal = {Chem. Phys.},
+ Month = {Feb},
+ Number = {1-3},
+ Pages = {64--75},
+ Publisher = {Elsevier BV},
+ Title = {A priori identification of configurational deadwood},
+ Url = {http://dx.doi.org/10.1016/j.chemphys.2008.11.021},
+ Volume = {356},
+ Year = {2009},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.chemphys.2008.11.021}}
+
+@article{Caffarel_2005,
+ Author = {Michel Caffarel and Jean-Pierre Daudey and Jean-Louis Heully and Alejandro Ram{\'\i}rez-Sol{\'\i}s},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.2011393},
+ Journal = {J. Chem. Phys.},
+ Number = {9},
+ Pages = {094102},
+ Title = {Towards accurate all-electron quantum Monte Carlo calculations of transition-metal systems: Spectroscopy of the copper atom},
+ Url = {https://doi.org/10.1063/1.2011393},
+ Volume = {123},
+ Year = {2005},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.2011393},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.2011393}}
+
+@article{Caffarel_2007,
+ Author = {Caffarel, Michel and Hern{\'a}ndez-Lamoneda, Ram{\'o}n and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/physrevlett.99.153001},
+ Issn = {1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Month = {Oct},
+ Number = {15},
+ Publisher = {American Physical Society (APS)},
+ Title = {Multireference Quantum Monte Carlo Study of theO4Molecule},
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+ Volume = {99},
+ Year = {2007},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.99.153001},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevlett.99.153001}}
+
+@article{Caffarel_2009,
+ Author = {Caffarel, Michel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/s00214-009-0713-y},
+ Issn = {1432-2234},
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+ Number = {3-4},
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+ Publisher = {Springer Nature},
+ Title = {The lithium--thiophene interaction: a critical study using highly correlated electronic structure approaches of quantum chemistry},
+ Url = {http://dx.doi.org/10.1007/s00214-009-0713-y},
+ Volume = {126},
+ Year = {2009},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1007/s00214-009-0713-y}}
+
+@article{Caffarel_2010,
+ Author = {Thomas Bouab\c{c}a and Beno\^{i}t Bra\"{\i}da and Michel Caffarel},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3457364},
+ Journal = {J. Chem. Phys.},
+ Number = {4},
+ Pages = {044111},
+ Title = {Multi-Jastrow trial wavefunctions for electronic structure calculations with quantum Monte Carlo},
+ Url = {https://doi.org/10.1063/1.3457364},
+ Volume = {133},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3457364},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.3457364}}
+
+@article{Caffarel_2014,
+ Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct5004252},
+ Issn = {1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Month = {Dec},
+ Number = {12},
+ Pages = {5286--5296},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Spin Density Distribution in Open-Shell Transition Metal Systems: A Comparative Post-Hartree--Fock, Density Functional Theory, and Quantum Monte Carlo Study of the CuCl2Molecule},
+ Url = {http://dx.doi.org/10.1021/ct5004252},
+ Volume = {10},
+ Year = {2014},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct5004252}}
+
+@article{Caffarel_2016,
+ Author = {Caffarel, Michel and Applencourt, Thomas and Giner, Emmanuel and Scemama, Anthony},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4947093},
+ Issn = {1089-7690},
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+ Month = {Apr},
+ Number = {15},
+ Pages = {151103},
+ Publisher = {AIP Publishing},
+ Title = {Communication: Toward an improved control of the fixed-node error in quantum Monte Carlo: The case of the water molecule},
+ Url = {http://dx.doi.org/10.1063/1.4947093},
+ Volume = {144},
+ Year = {2016},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4947093}}
+
+@inbook{Caffarel_2016b,
+ Author = {Michel Caffarel and Thomas Applencourt and Emmanuel Giner and Anthony Scemama},
+ Booktitle = {Recent Progress in Quantum Monte Carlo},
+ Chapter = {2},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/bk-2016-1234.ch002},
+ Pages = {15-46},
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+ Year = {2016},
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+ Bdsk-Url-2 = {http://dx.doi.org/10.1021/bk-2016-1234.ch002}}
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Publisher = {American Physical Society (APS)},
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+ Volume = {74},
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+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ File = {28a1e2d5fd9eba4d0cba5227b6acd8463ff941ffaa24949d1437dfca2bc8f3c7.pdf:/home/scemama/Dropbox/Zotero/storage/4L9PPHEJ/28a1e2d5fd9eba4d0cba5227b6acd8463ff941ffaa24949d1437dfca2bc8f3c7.pdf:application/pdf;540080105_ftp.pdf:/home/scemama/Dropbox/Zotero/storage/MZIFQQ9W/540080105_ftp.pdf:application/pdf},
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+ Year = {1987}}
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+ Month = {Mar},
+ Number = {4-6},
+ Pages = {341--345},
+ Publisher = {Elsevier BV},
+ Title = {Photoelectron spectra of FeS− explained by a CASPT2 ab initio study},
+ Url = {http://dx.doi.org/10.1016/j.cplett.2007.01.073},
+ Volume = {436},
+ Year = {2007},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2007.01.073}}
+
+@article{Crack_2014,
+ Author = {Crack, Jason C. and Green, Jeffrey and Thomson, Andrew J. and Brun, Nick E. Le},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ar5002507},
+ Issn = {1520-4898},
+ Journal = {Acc. Chem. Res.},
+ Month = {Oct},
+ Number = {10},
+ Pages = {3196--3205},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Iron--Sulfur Clusters as Biological Sensors: The Chemistry of Reactions with Molecular Oxygen and Nitric Oxide},
+ Url = {http://dx.doi.org/10.1021/ar5002507},
+ Volume = {47},
+ Year = {2014},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/ar5002507}}
+
+@article{Daday_2012,
+ Author = {Daday, Csaba and Smart, Simon and Booth, George H. and Alavi, Ali and Filippi, Claudia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct300486d},
+ File = {/Users/loos/Zotero/storage/APCJKTM8/Daday et al. - 2012 - Full Configuration Interaction Excitations of Ethe.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory. Comput.},
+ Language = {en},
+ Month = nov,
+ Number = {11},
+ Pages = {4441-4451},
+ Shorttitle = {Full {{Configuration Interaction Excitations}} of {{Ethene}} and {{Butadiene}}},
+ Title = {Full {{Configuration Interaction Excitations}} of {{Ethene}} and {{Butadiene}}: {{Resolution}} of an {{Ancient Question}}},
+ Volume = {8},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct300486d}}
+
+@article{Dallos_2004,
+ Author = {Dallos, Michal and Lischka, Hans},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/s00214-003-0557-9},
+ Issn = {1432-881X, 1432-2234},
+ Journal = {Theor. Chem. Acc.},
+ Month = apr,
+ Number = {1},
+ Pages = {16-26},
+ Title = {A Systematic Theoretical Investigation of the Lowest Valence- and {{Rydberg}}-Excited Singlet States of Trans-Butadiene. {{The}} Character of the 1 1 {{B}} u ({{V}}) State Revisited},
+ Volume = {112},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1007/s00214-003-0557-9}}
+
+@article{Dash_2018,
+ Author = {M. Dash and S. Moroni and A. Scemama and C. Filippi},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {arXiv:1804.09610},
+ Title = {Perturbatively selected configuration-interaction wave functions for efficient geometry optimization in quantum Monte Carlo},
+ Year = {2018}}
+
+@article{Delgado_2010,
+ Author = {Delgado, Juan Luis and Bouit, Pierre-Antoine and Filippone, Salvatore and Herranz, Ma\'Angeles and Mart\'in, Nazario},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1039/c003088k},
+ Issn = {1359-7345, 1364-548X},
+ Journal = {Chem. Comm.},
+ Language = {en},
+ Number = {27},
+ Pages = {4853},
+ Shorttitle = {Organic Photovoltaics},
+ Title = {Organic Photovoltaics: A Chemical Approach},
+ Volume = {46},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1039/c003088k}}
+
+@article{Deur_2017,
+ Author = {Deur, Killian and Mazouin, Laurent and Fromager, Emmanuel},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-12-11 14:05:37 +0100},
+ Doi = {10.1103/PhysRevB.95.035120},
+ File = {/Users/loos/Zotero/storage/966B9AIB/Deur et al. - 2017 - Exact ensemble density functional theory for excit.pdf},
+ Issn = {2469-9950, 2469-9969},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = jan,
+ Number = {3},
+ Pages = {035120},
+ Shorttitle = {Exact Ensemble Density Functional Theory for Excited States in a Model System},
+ Title = {Exact Ensemble Density Functional Theory for Excited States in a Model System: {{Investigating}} the Weight Dependence of the Correlation Energy},
+ Volume = {95},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.95.035120}}
+
+@article{Deur_2018a,
+ Abstract = {Gross\textendash{}Oliveira\textendash{}Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excitedstate densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [Phys. Rev. B 95, 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre\textendash{}Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.},
+ Archiveprefix = {arXiv},
+ Author = {Deur, Killian and Mazouin, Laurent and Senjean, Bruno and Fromager, Emmanuel},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-12-11 14:07:04 +0100},
+ Doi = {10.1140/epjb/e2018-90124-7},
+ File = {/Users/loos/Zotero/storage/2398CIXN/Deur et al. - 2018 - Exploring weight-dependent density-functional appr.pdf},
+ Journal = {Eur. Phys. J. B},
+ Keywords = {Physics - Chemical Physics,Condensed Matter - Strongly Correlated Electrons},
+ Language = {en},
+ Month = jul,
+ Number = {7},
+ Pages = {162},
+ Title = {Exploring Weight-Dependent Density-Functional Approximations for Ensembles in the {{Hubbard}} Dimer},
+ Volume = {91},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1140/epjb/e2018-90124-7}}
+
+@article{Diedrich_2005,
+ Author = {Diedrich, Christian and L{\"u}chow, Arne and Grimme, Stefan},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1846654},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Jan},
+ Number = {2},
+ Pages = {021101},
+ Publisher = {AIP Publishing},
+ Title = {Performance of diffusion Monte Carlo for the first dissociation energies of transition metal carbonyls},
+ Url = {http://dx.doi.org/10.1063/1.1846654},
+ Volume = {122},
+ Year = {2005},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.1846654}}
+
+@article{Dierksen_2004,
+ Author = {Dierksen, M. and Grimme, S.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {J. Chem. Phys.},
+ Pages = {3544--3554},
+ Title = {A density functional calculation of the vibronic structure of electronic absorption spectra},
+ Volume = 120,
+ Year = 2004}
+
+@article{Doblhoff_Dier_2016,
+ Author = {Doblhoff-Dier, Katharina and Meyer, J{\"o}rg and Hoggan, Philip E. and Kroes, Geert-Jan and Wagner, Lucas K.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.jctc.6b00160},
+ Issn = {1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Month = {Jun},
+ Number = {6},
+ Pages = {2583--2597},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Diffusion Monte Carlo for Accurate Dissociation Energies of 3d Transition Metal Containing Molecules},
+ Url = {http://dx.doi.org/10.1021/acs.jctc.6b00160},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jctc.6b00160}}
+
+@article{Dolgov_2004,
+ Author = {Dolgov, Eugeniy K. and Bataev, Vadim A. and Pupyshev, Vladimir I. and Godunov, Igor A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/qua.10745},
+ Issn = {1097-461X},
+ Journal = {Int. J. Quantum Chem.},
+ Keywords = {nitrosomethane, excited states, torsion potential, CASSCF, MR-AQCC},
+ Number = {6},
+ Pages = {589--597},
+ Publisher = {Wiley Subscription Services, Inc., A Wiley Company},
+ Title = {Ab Initio Description of the Structure and Dynamics of the Nitrosomethane Molecule in the First Excited Singlet and Triplet Electronic States},
+ Url = {http://dx.doi.org/10.1002/qua.10745},
+ Volume = {96},
+ Year = {2004},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1002/qua.10745}}
+
+@article{Dolgov_2004b,
+ Author = {Dolgov, Eugeniy K. and Bataev, Vadim A. and Godunov, Igor A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/qua.10683},
+ Issn = {1097-461X},
+ Journal = {Int. J. Quantum Chem.},
+ Number = {3},
+ Pages = {193--201},
+ Title = {Structure of the Nitrosomethane Molecule (CH$_3$NO) in the Ground Electronic State: Testing of Ab Initio Methods for the Description of Potential Energy Surface},
+ Url = {http://dx.doi.org/10.1002/qua.10683},
+ Volume = {96},
+ Year = {2004},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1002/qua.10683}}
+
+@article{Dolgov_2004c,
+ Author = {Dolgov, Eugeniy K. and Bataev, Vadim A. and Pupyshev, Vladimir I. and Godunov, Igor A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/qua.20186},
+ Issn = {1097-461X},
+ Journal = {Int. J. Quantum Chem.},
+ Keywords = {excited electronic states, CASSCF, MR-CI, MR-AQCC, VibSCF},
+ Number = {4},
+ Pages = {509--518},
+ Publisher = {John Wiley & Sons, Inc.},
+ Title = {Structure and Vibrations of the CF$_3$NO Molecule in the Ground and Lowest Excited Electronic States: A Test of Ab Initio Methods},
+ Url = {http://dx.doi.org/10.1002/qua.20186},
+ Volume = {100},
+ Year = {2004},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1002/qua.20186}}
+
+@article{Dreuw_2003,
+ Author = {Dreuw, Andreas and Weisman, Jennifer L. and {Head-Gordon}, Martin},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1590951},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {6},
+ Pages = {2943-2946},
+ Title = {Long-Range Charge-Transfer Excited States in Time-Dependent Density Functional Theory Require Non-Local Exchange},
+ Volume = {119},
+ Year = {2003},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1590951}}
+
+@article{Dreuw_2004,
+ Author = {Dreuw, Andreas and Head-Gordon, Martin},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ja039556n},
+ File = {/Users/loos/Zotero/storage/TSG4VHAB/Dreuw and Head-Gordon - 2004 - Failure of Time-Dependent Density Functional Theor.pdf},
+ Issn = {0002-7863, 1520-5126},
+ Journal = {J. Am. Chem. Soc.},
+ Language = {en},
+ Month = mar,
+ Number = {12},
+ Pages = {4007-4016},
+ Shorttitle = {Failure of {{Time}}-{{Dependent Density Functional Theory}} for {{Long}}-{{Range Charge}}-{{Transfer Excited States}}},
+ Title = {Failure of {{Time}}-{{Dependent Density Functional Theory}} for {{Long}}-{{Range Charge}}-{{Transfer Excited States}}: {{The Zincbacteriochlorin}}-{{Bacteriochlorin}} and {{Bacteriochlorophyll}}-{{Spheroidene Complexes}}},
+ Volume = {126},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ja039556n}}
+
+@article{Dreuw_2015,
+ Author = {Dreuw, Andreas and Wormit, Michael},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/wcms.1206},
+ Issn = {1759-0884},
+ Journal = {WIREs Comput. Mol. Sci.},
+ Number = {1},
+ Pages = {82--95},
+ Publisher = {Wiley Periodicals, Inc.},
+ Title = {The Algebraic Diagrammatic Construction Scheme for the Polarization Propagator for the Calculation of Excited States},
+ Url = {http://dx.doi.org/10.1002/wcms.1206},
+ Volume = {5},
+ Year = {2015},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1002/wcms.1206}}
+
+@article{Drowart_1967,
+ Author = {J. Drowart and A. Pattoret and S. Smoes},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {Proc. Br. Ceram. Soc.},
+ Pages = {67−88},
+ Title = {Mass Spectrometric Studies of the Vaporization of Refractory Compounds},
+ Volume = {8},
+ Year = {1967}}
+
+@article{Dubeck__2016,
+ Author = {Dubeck{\'y}, Mat{\'u}{\v s} and Mit\'a\v{s}, Lubos and Jure{\v c}ka, Petr},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.chemrev.5b00577},
+ Issn = {1520-6890},
+ Journal = {Chem. Rev.},
+ Month = {May},
+ Number = {9},
+ Pages = {5188--5215},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Noncovalent Interactions by Quantum Monte Carlo},
+ Url = {http://dx.doi.org/10.1021/acs.chemrev.5b00577},
+ Volume = {116},
+ Year = {2016},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.chemrev.5b00577}}
+
+@article{Dubecky_2010,
+ Author = {Dubeck{\'y}, M. and Derian, R. and Mitas, L. and {\v S}tich, I.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3506028},
+ File = {/Users/loos/Zotero/storage/CRWWWYSF/Dubeck{\'y} et al. - 2010 - Ground and excited electronic states of azobenzene.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Pages = {244301},
+ Shorttitle = {Ground and Excited Electronic States of Azobenzene},
+ Title = {Ground and Excited Electronic States of Azobenzene: {{A}} Quantum {{Monte Carlo}} Study},
+ Volume = {133},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3506028}}
+
+@article{Dupuy_2015,
+ Author = {Dupuy, Nicolas and Bouaouli, Samira and Mauri, Francesco and Sorella, Sandro and Casula, Michele},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4922048},
+ File = {/Users/loos/Zotero/storage/EQRHVUV2/Dupuy et al. - 2015 - Vertical and adiabatic excitations in anthracene f.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = jun,
+ Number = {21},
+ Pages = {214109},
+ Shorttitle = {Vertical and Adiabatic Excitations in Anthracene from Quantum {{Monte Carlo}}},
+ Title = {Vertical and Adiabatic Excitations in Anthracene from Quantum {{Monte Carlo}}: {{Constrained}} Energy Minimization for Structural and Electronic Excited-State Properties in the {{JAGP}} Ansatz},
+ Volume = {142},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4922048}}
+
+@article{Durig_1984,
+ Author = {Durig, J.R. and Whang, C.M. and Attia, G.M. and Li, Y.S.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/0022-2852(84)90182-6},
+ Issn = {00222852},
+ Journal = {J. Mol. Spectrosc.},
+ Language = {en},
+ Month = dec,
+ Number = {2},
+ Pages = {240-248},
+ Title = {Microwave Spectra, Structure, and Barrier to Internal Rotation of {{CH3SnH2D}}, {{CH3SnHD2}} and {{CH3SnD3}}},
+ Volume = {108},
+ Year = {1984},
+ Bdsk-Url-1 = {https://doi.org/10.1016/0022-2852(84)90182-6}}
+
+@article{Dykstra_1977,
+ Author = {Dykstra, Clifford E. and Lucchese, Robert R. and Schaefer, Henry F.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.435214},
+ Issn = {00219606},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Number = {6},
+ Pages = {2422},
+ Title = {Electron Correlation Effects on the Excitation Energies of the Lowest Triplet States of Glyoxal},
+ Volume = {67},
+ Year = {1977},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.435214}}
+
+@article{Dykstra_1977a,
+ Author = {Dykstra, Clifford E. and Lucchese, Robert R. and Schaefer, Henry F.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.435214},
+ Issn = {00219606},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Number = {6},
+ Pages = {2422},
+ Title = {Electron Correlation Effects on the Excitation Energies of the Lowest Triplet States of Glyoxal},
+ Volume = {67},
+ Year = {1977},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.435214}}
+
+@article{Dzubak_2017,
+ Author = {Dzubak, Allison L. and Krogel, Jaron T. and Reboredo, Fernando A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4991414},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Jul},
+ Number = {2},
+ Pages = {024102},
+ Publisher = {AIP Publishing},
+ Title = {Quantitative estimation of localization errors of 3d transition metal pseudopotentials in diffusion Monte Carlo},
+ Url = {http://dx.doi.org/10.1063/1.4991414},
+ Volume = {147},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4991414}}
+
+@article{Ehara_2011,
+ Author = {Ehara, Masahiro and Oyagi, Fumito and Abe, Yoko and Fukuda, Ryoichi and Nakatsuji, Hiroshi},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3617233},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = jul,
+ Number = {4},
+ Pages = {044316},
+ Title = {Excited-State Geometries and Vibrational Frequencies Studied Using the Analytical Energy Gradients of the Direct Symmetry-Adapted Cluster\textendash{}Configuration Interaction Method. {{I}}. {{HAX}}-Type Molecules},
+ Volume = {135},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3617233}}
+
+@article{Elliott_2011,
+ Author = {Elliott, Peter and Goldson, Sharma and Canahui, Chris and Maitra, Neepa T.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/j.chemphys.2011.03.020},
+ File = {/Users/loos/Zotero/storage/U6T3LQ8L/Elliott et al. - 2011 - Perspectives on double-excitations in TDDFT.pdf},
+ Issn = {03010104},
+ Journal = {Chem. Phys.},
+ Language = {en},
+ Month = nov,
+ Number = {1},
+ Pages = {110-119},
+ Title = {Perspectives on Double-Excitations in {{TDDFT}}},
+ Volume = {391},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.chemphys.2011.03.020}}
+
+@article{Ernsting_1978,
+ Author = {Ernsting, Nikolaus P. and Pfab, Josef and Romelt, Joachim},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1039/F29787402286},
+ Issue = {0},
+ Journal = {J. Chem. Soc.{,} Faraday Trans. 2},
+ Pages = {2286--2294},
+ Publisher = {The Royal Society of Chemistry},
+ Title = {Geometry Changes Accompanying Electronic Excitation of Nitrosomethane in the 650 nm Region},
+ Url = {http://dx.doi.org/10.1039/F29787402286},
+ Volume = {74},
+ Year = {1978},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1039/F29787402286}}
+
+@article{Evangelista_2014,
+ Author = {Evangelista, Francesco A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4869192},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Mar},
+ Number = {12},
+ Pages = {124114},
+ Publisher = {AIP Publishing},
+ Title = {Adaptive multiconfigurational wave functions},
+ Url = {http://dx.doi.org/10.1063/1.4869192},
+ Volume = {140},
+ Year = {2014},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4869192}}
+
+@article{Evangelisti_1983,
+ Author = {Evangelisti, Stefano and Daudey, Jean-Pierre and Malrieu, Jean-Paul},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/0301-0104(83)85011-3},
+ Issn = {0301-0104},
+ Journal = {Chem. Phys.},
+ Month = {Feb},
+ Number = {1},
+ Pages = {91--102},
+ Publisher = {Elsevier BV},
+ Title = {Convergence of an improved CIPSI algorithm},
+ Url = {http://dx.doi.org/10.1016/0301-0104(83)85011-3},
+ Volume = {75},
+ Year = {1983},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Journal = {Phys. Rev. B},
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+ Number = {6},
+ Pages = {3503--3522},
+ Shorttitle = {Variational Quantum {{Monte Carlo}} Nonlocal Pseudopotential Approach to Solids},
+ Timestamp = {2017-11-16T10:54:00Z},
+ Title = {Variational Quantum {{Monte Carlo}} Nonlocal Pseudopotential Approach to Solids: {{Formulation}} and Application to Diamond, Graphite, and Silicon},
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+
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+ Author = {David Feller and Kirk A. Peterson and Ernest R. Davidson},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ Volume = {141},
+ Year = {2014},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4894482}}
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+@article{Feng_2003,
+ Author = {Feng, Liang and van der Hart, Hugo W},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1088/0953-4075/36/1/101},
+ File = {/Users/loos/Zotero/storage/LKYCYUI2/Feng and Hart - 2003 - Two-photon double ionization of He.pdf},
+ Issn = {0953-4075},
+ Journal = {J. Phys. B},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {L1-L7},
+ Title = {Two-Photon Double Ionization of {{He}}},
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+
+@article{Filatov_2015,
+ Author = {Filatov, Michael and {Huix-Rotllant}, Miquel and Burghardt, Irene},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4919773},
+ File = {/Users/loos/Zotero/storage/PC4HY8T5/Filatov et al. - 2015 - Ensemble density functional theory method correctl.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {18},
+ Pages = {184104},
+ Title = {Ensemble Density Functional Theory Method Correctly Describes Bond Dissociation, Excited State Electron Transfer, and Double Excitations},
+ Volume = {142},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4919773}}
+
+@inbook{Filatov_2015b,
+ Abstract = {Ensemble density functional theory (DFT) is a novel time-independent formalism for obtaining excitation energies of many-body fermionic systems. A considerable advantage of ensemble DFT over the more common Kohn\textendash{}Sham (KS) DFT and time-dependent DFT formalisms is that it enables one to account for strong non-dynamic electron correlation in the ground and excited states of molecular systems in a transparent and accurate fashion. Despite its positive aspects, ensemble DFT has not so far found its way into the repertoire of methods of modern computational chemistry, probably because of the perceived lack of practically affordable implementations of the theory. The spin-restricted ensemble-referenced KS (REKS) method is perhaps the first computationally feasible implementation of the ideas behind ensemble DFT which enables one to describe accurately electronic transitions in a wide class of molecular systems, including strongly correlated molecules (biradicals, molecules undergoing bond breaking/formation), extended $\pi$-conjugated systems, donor\textendash{}acceptor charge transfer adducts, etc.},
+ Address = {Cham},
+ Author = {Filatov, Michael},
+ Booktitle = {Density-{{Functional Methods}} for {{Excited States}}},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/128_2015_630},
+ Editor = {Ferr\'e, Nicolas and Filatov, Michael and {Huix-Rotllant}, Miquel},
+ File = {/Users/loos/Zotero/storage/IL7CHRFF/Filatov - 2015 - Ensemble DFT Approach to Excited States of Strongl.pdf},
+ Isbn = {978-3-319-22080-2 978-3-319-22081-9},
+ Pages = {97-124},
+ Publisher = {{Springer International Publishing}},
+ Title = {Ensemble {{DFT Approach}} to {{Excited States}} of {{Strongly Correlated Molecular Systems}}},
+ Volume = {368},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1007/128_2015_630}}
+
+@article{Filatov_2015c,
+ Author = {Filatov, Michael},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/wcms.1209},
+ File = {/Users/loos/Zotero/storage/5BSWZ5ZF/Filatov - 2015 - Spin-restricted ensemble-referenced Kohn-Sham meth.pdf},
+ Issn = {17590876},
+ Journal = {WIREs Comput. Mol. Sci.},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {146-167},
+ Shorttitle = {Spin-Restricted Ensemble-Referenced {{Kohn}}-{{Sham}} Method},
+ Title = {Spin-Restricted Ensemble-Referenced {{Kohn}}-{{Sham}} Method: Basic Principles and Application to Strongly Correlated Ground and Excited States of Molecules},
+ Volume = {5},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1002/wcms.1209}}
+
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+ Issn = {0021-9606, 1089-7690},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Apr},
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+ Publisher = {AIP Publishing},
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+ Volume = {146},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4980034}}
+
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+ Author = {Garniron, Yann and Scemama, Anthony and Loos, Pierre-Fran{\c c}ois and Caffarel, Michel},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {1089-7690},
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+ Volume = {147},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4992127}}
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+ Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ Month = jul,
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+ Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel},
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+ Author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {1089-7690},
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+ Volume = {142},
+ Year = {2015},
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+ Issn = {1089-7690},
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+ Year = {2016},
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+ Author = {Giner, Emmanuel and Angeli, Celestino and Garniron, Yann and Scemama, Anthony and Malrieu, Jean-Paul},
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+ Issn = {1089-7690},
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+ Month = {Jun},
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+ Publisher = {AIP Publishing},
+ Title = {A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves},
+ Url = {http://dx.doi.org/10.1063/1.4984616},
+ Volume = {146},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4984616}}
+
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+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = jan,
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+ Shorttitle = {The Flexible Nature of Exchange, Correlation, and {{Hartree}} Physics},
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+ Author = {Gould, Tim and Toulouse, Julien},
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+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Pages = {243001},
+ Shorttitle = {Hartree and {{Exchange}} in {{Ensemble Density Functional Theory}}},
+ Title = {Hartree and {{Exchange}} in {{Ensemble Density Functional Theory}}: {{Avoiding}} the {{Nonuniqueness Disaster}}},
+ Volume = {119},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.119.243001}}
+
+@article{Gould_2018a,
+ Author = {Gould, Tim and Kronik, Leeor and Pittalis, Stefano},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-12-09 11:39:36 +0100},
+ Doi = {10.1063/1.5022832},
+ File = {/Users/loos/Zotero/storage/C5DEDGG2/Gould et al. - 2018 - Charge transfer excitations from exact and approxi.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {17},
+ Pages = {174101},
+ Title = {Charge Transfer Excitations from Exact and Approximate Ensemble {{Kohn}}-{{Sham}} Theory},
+ Volume = {148},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.5022832}}
+
+@article{Gozem_2012,
+ Abstract = {The ground state potential energy surface of the retinal chromophore of visual pigments (e.g., bovine rhodopsin) features a low-lying conical intersection surrounded by regions with variable charge-transfer and diradical electronic structures. This implies that dynamic electron correlation may have a large effect on the shape of the force fields driving its reactivity. To investigate this effect, we focus on mapping the potential energy for three paths located along the ground state CASSCF potential energy surface of the penta-2,4-dieniminium cation taken as a minimal model of the retinal chromophore. The first path spans the bond length alternation coordinate and intercepts a conical intersection point. The other two are minimum energy paths along two distinct but kinetically competitive thermal isomerization coordinates. We show that the effect of introducing the missing dynamic electron correlation variationally (with MRCISD) and perturbatively (with the CASPT2, NEVPT2, and XMCQDPT2 methods) leads, invariably, to a stabilization of the regions with charge transfer character and to a significant reshaping of the reference CASSCF potential energy surface and suggesting a change in the dominating isomerization mechanism. The possible impact of such a correction on the photoisomerization of the retinal chromophore is discussed.},
+ Author = {Gozem, Samer and Huntress, Mark and Schapiro, Igor and Lindh, Roland and Granovsky, Alexander A. and Angeli, Celestino and Olivucci, Massimo},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct3003139},
+ File = {/Users/loos/Zotero/storage/BBENY2RV/Gozem et al. - 2012 - Dynamic Electron Correlation Effects on the Ground.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = nov,
+ Number = {11},
+ Pages = {4069-4080},
+ Title = {Dynamic {{Electron Correlation Effects}} on the {{Ground State Potential Energy Surface}} of a {{Retinal Chromophore Model}}},
+ Volume = {8},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct3003139}}
+
+@article{Gozem_2013,
+ Abstract = {This work investigates the performance of equation-ofmotion coupled-cluster (EOM-CC) methods for describing the changes in the potential energy surfaces of the penta-2,4-dieniminium cation, a reduced model of the retinal chromophore of visual pigments, due to dynamical electron correlation effects. The groundstate wave function of this model includes charge-transfer and diradical configurations whose weights vary along different displacements and are rapidly changing at the conical intersection between the ground and the first excited states, making the shape of the potential energy surface sensitive to a balanced description of nondynamical and dynamical correlation. Recently, variational (MRCISD) and perturbative (MRPT2) approaches for including dynamical correlation in CASSCF-based calculations were tested along three representative ground state paths. Here, we use the same three paths to compare the performance of single-reference EOM-CC methods against MRCISD and MRCISD+Q. We find that the spin-flip variant of EOM-CCSD with perturbative inclusion of triple excitations (dT or fT) produces potential energy profiles of the two lowest electronic states in quantitative agreement with MRCISD+Q (our highest-quality reference method). The nonparallelity errors and differences in vertical energy differences of the two surfaces along these scans are less than 1.4 kcal/mol (EOM-SF-CCSD(dT) versus MRCISD+Q). For comparison, the largest error of MRCISD versus MRCISD+Q is 1.7 kcal/mol. Our results show that the EOM-CC methods provide an alternative to multireference approaches and may be used to study photochemical systems like the one used in this work.},
+ Author = {Gozem, Samer and Krylov, Anna I. and Olivucci, Massimo},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct300759z},
+ File = {/Users/loos/Zotero/storage/DG2HDM7R/Gozem et al. - 2013 - Conical Intersection and Potential Energy Surface .pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {284-292},
+ Shorttitle = {Conical {{Intersection}} and {{Potential Energy Surface Features}} of a {{Model Retinal Chromophore}}},
+ Title = {Conical {{Intersection}} and {{Potential Energy Surface Features}} of a {{Model Retinal Chromophore}}: {{Comparison}} of {{EOM}}-{{CC}} and {{Multireference Methods}}},
+ Volume = {9},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct300759z}}
+
+@article{Gozem_2013a,
+ Author = {Gozem, Samer and Melaccio, Federico and Lindh, Roland and Krylov, Anna I. and Granovsky, Alexander A. and Angeli, Celestino and Olivucci, Massimo},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct400460h},
+ File = {/Users/loos/Zotero/storage/PNMIDT7G/Gozem et al. - 2013 - Mapping the Excited State Potential Energy Surface.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = oct,
+ Number = {10},
+ Pages = {4495-4506},
+ Title = {Mapping the {{Excited State Potential Energy Surface}} of a {{Retinal Chromophore Model}} with {{Multireference}} and {{Equation}}-of-{{Motion Coupled}}-{{Cluster Methods}}},
+ Volume = {9},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct400460h}}
+
+@article{Gozem_2014,
+ Abstract = {We report and characterize ground-state and excited-state potential energy profiles using a variety of electronic structure methods along a loop lying on the branching plane associated with a conical intersection (CI) of a reduced retinal model, the penta-2,4-dieniminium cation (PSB3). Whereas the performance of the equation-of-motion coupled-cluster, density functional theory, and multireference methods had been tested along the excited- and ground-state paths of PSB3 in our earlier work, the ability of these methods to correctly describe the potential energy surface shape along a CI branching plane has not yet been investigated. This is the focus of the present contribution. We find, in agreement with earlier studies by others, that standard time-dependent DFT (TDDFT) does not yield the correct two-dimensional (i.e., conical) crossing along the branching plane but rather a one-dimensional (i.e., linear) crossing along the same plane. The same type of behavior is found for SS-CASPT2(IPEA=0), SS-CASPT2(IPEA=0.25), spin-projected SF-TDDFT, EOM-SF-CCSD, and, finally, for the reference MRCISD+Q method. In contrast, we found that MRCISD, CASSCF, MS-CASPT2(IPEA=0), MS-CASPT2(IPEA=0.25), XMCQDPT2, QD-NEVPT2, non-spin-projected SF-TDDFT, and SI-SA-REKS yield the expected conical crossing. To assess the effect of the different crossing topologies (i.e., linear or conical) on the PSB3 photoisomerization efficiency, we discuss the results of 100 semiclassical trajectories computed by CASSCF and SS-CASPT2(IPEA=0.25) for a PSB3 derivative. We show that for the same initial conditions, the two methods yield similar dynamics leading to isomerization quantum yields that differ by only a few percent.},
+ Author = {Gozem, Samer and Melaccio, Federico and Valentini, Alessio and Filatov, Michael and {Huix-Rotllant}, Miquel and Ferr\'e, Nicolas and Frutos, Luis Manuel and Angeli, Celestino and Krylov, Anna I. and Granovsky, Alexander A. and Lindh, Roland and Olivucci, Massimo},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct500154k},
+ File = {/Users/loos/Zotero/storage/VABYG9ND/Gozem et al. - 2014 - Shape of Multireference, Equation-of-Motion Couple.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {3074-3084},
+ Title = {Shape of {{Multireference}}, {{Equation}}-of-{{Motion Coupled}}-{{Cluster}}, and {{Density Functional Theory Potential Energy Surfaces}} at a {{Conical Intersection}}},
+ Volume = {10},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct500154k}}
+
+@article{Grimme_2004,
+ Author = {Grimme, S. and Izgorodina, E. I.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {Chem. Phys.},
+ Pages = {223--230},
+ Title = {Calculation of 0--0 Excitation Energies of Organic Molecules by CIS(D) Quantum Chemical Methods},
+ Volume = 305,
+ Year = {2004}}
+
+@article{Gross_1988,
+ Author = {Gross, E. K. U. and Oliveira, L. N. and Kohn, W.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/PhysRevA.37.2809},
+ File = {/Users/loos/Zotero/storage/H33LDJJ6/Gross et al. - 1988 - Density-functional theory for ensembles of fractio.pdf},
+ Issn = {0556-2791},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {8},
+ Pages = {2809-2820},
+ Title = {Density-Functional Theory for Ensembles of Fractionally Occupied States. {{I}}. {{Basic}} Formalism},
+ Volume = {37},
+ Year = {1988},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2809}}
+
+@article{Gross_1988a,
+ Author = {Gross, E. K. U. and Oliveira, L. N. and Kohn, W.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/PhysRevA.37.2805},
+ File = {/Users/loos/Zotero/storage/24I8IUKS/Gross et al. - 1988 - Rayleigh-Ritz variational principle for ensembles .pdf},
+ Issn = {0556-2791},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {8},
+ Pages = {2805-2808},
+ Title = {Rayleigh-{{Ritz}} Variational Principle for Ensembles of Fractionally Occupied States},
+ Volume = {37},
+ Year = {1988},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2805}}
+
+@article{Grossman_2001,
+ Author = {Grossman, Jeffrey C. and Rohlfing, Michael and Mitas, Lubos and Louie, Steven G. and Cohen, Marvin L.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/PhysRevLett.86.472},
+ File = {/Users/loos/Zotero/storage/FGCU52SP/Grossman et al. - 2001 - High Accuracy Many-Body Calculational Approaches f.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = jan,
+ Number = {3},
+ Pages = {472-475},
+ Title = {High {{Accuracy Many}}-{{Body Calculational Approaches}} for {{Excitations}} in {{Molecules}}},
+ Volume = {86},
+ Year = {2001},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.86.472}}
+
+@article{Grossman_2002,
+ Author = {Grossman, Jeffrey C.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1487829},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Jul},
+ Number = {4},
+ Pages = {1434--1440},
+ Publisher = {AIP Publishing},
+ Title = {Benchmark quantum Monte Carlo calculations},
+ Url = {http://dx.doi.org/10.1063/1.1487829},
+ Volume = {117},
+ Year = {2002},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.1487829}}
+
+@article{Guareschi_2013,
+ Author = {Guareschi, Riccardo and Filippi, Claudia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct400876y},
+ File = {/Users/loos/Zotero/storage/HJKGZLDW/Guareschi and Filippi - 2013 - Ground- and Excited-State Geometry Optimization of.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = dec,
+ Number = {12},
+ Pages = {5513-5525},
+ Title = {Ground- and {{Excited}}-{{State Geometry Optimization}} of {{Small Organic Molecules}} with {{Quantum Monte Carlo}}},
+ Volume = {9},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct400876y}}
+
+@article{Guareschi_2014,
+ Author = {Guareschi, Riccardo and Floris, Franca Maria and Amovilli, Claudio and Filippi, Claudia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ct500723s},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = dec,
+ Number = {12},
+ Pages = {5528-5537},
+ Shorttitle = {Solvent {{Effects}} on {{Excited}}-{{State Structures}}},
+ Title = {Solvent {{Effects}} on {{Excited}}-{{State Structures}}: {{A Quantum Monte Carlo}} and {{Density Functional Study}}},
+ Volume = {10},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct500723s}}
+
+@article{Guareschi_2016a,
+ Author = {Guareschi, Riccardo and Zulfikri, Habiburrahman and Daday, Csaba and Floris, Franca Maria and Amovilli, Claudio and Mennucci, Benedetta and Filippi, Claudia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.jctc.6b00044},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Pages = {1674-1683},
+ Shorttitle = {Introducing {{QMC}}/{{MMpol}}},
+ Title = {Introducing {{QMC}}/{{MMpol}}: {{Quantum Monte Carlo}} in {{Polarizable Force Fields}} for {{Excited States}}},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00044}}
+
+@article{Gwaltney_1995,
+ Author = {Steven R. Gwaltney and Rodney J. Bartlett},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {https://doi.org/10.1016/0009-2614(95)00608-7},
+ Issn = {0009-2614},
+ Journal = {Chem. Phys. Lett.},
+ Number = {1},
+ Pages = {26--32},
+ Title = {An Application of the Equation-Of-Motion Coupled Cluster Method to the Excited States of Formaldehyde, Acetaldehyde, and Acetone},
+ Url = {http://www.sciencedirect.com/science/article/pii/0009261495006087},
+ Volume = {241},
+ Year = {1995},
+ Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0009261495006087},
+ Bdsk-Url-2 = {https://doi.org/10.1016/0009-2614(95)00608-7}}
+
+@article{Ha_1972,
+ Author = {Ha, Tae-Kyu},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/0022-2860(72)85029-4},
+ Issn = {00222860},
+ Journal = {J. Mol. Struct.},
+ Language = {en},
+ Month = may,
+ Number = {2},
+ Pages = {171-178},
+ Title = {Ab Initio Calculation of Cis-Trans Isomerization in Glyoxal},
+ Volume = {12},
+ Year = {1972},
+ Bdsk-Url-1 = {https://doi.org/10.1016/0022-2860(72)85029-4}}
+
+@article{Hadad_1993,
+ Author = {Hadad, Christopher M. and Foresman, James B. and Wiberg, Kenneth B.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/j100119a010},
+ Journal = {J. Phys. Chem.},
+ Number = {17},
+ Pages = {4293--4312},
+ Title = {Excited States of Carbonyl Compounds. 1. Formaldehyde and Acetaldehyde},
+ Url = {http://dx.doi.org/10.1021/j100119a010},
+ Volume = {97},
+ Year = {1993},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/j100119a010}}
+
+@article{Haghighi_Mood_2017,
+ Author = {Haghighi Mood, Kaveh and L{\"u}chow, Arne},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.jpca.7b05798},
+ Issn = {1520-5215},
+ Journal = {J. Phys. Chem. A},
+ Month = {Aug},
+ Number = {32},
+ Pages = {6165--6171},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Full Wave Function Optimization with Quantum Monte Carlo and Its Effect on the Dissociation Energy of FeS},
+ Url = {http://dx.doi.org/10.1021/acs.jpca.7b05798},
+ Volume = {121},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jpca.7b05798}}
+
+@article{Hald_2002,
+ Author = {Hald, Kasper and J\o{}rgensen, Poul and Christiansen, Ove and Koch, Henrik},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1457431},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = apr,
+ Number = {14},
+ Pages = {5963-5970},
+ Title = {Implementation of Electronic Ground States and Singlet and Triplet Excitation Energies in Coupled Cluster Theory with Approximate Triples Corrections},
+ Volume = {116},
+ Year = {2002},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1457431}}
+
+@article{Hammond_1987,
+ Author = {Brian L. Hammond and Peter J. Reynolds and William A. Lester},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.453345},
+ Journal = {J. Chem. Phys.},
+ Month = {jul},
+ Number = {2},
+ Pages = {1130--1136},
+ Publisher = {{AIP} Publishing},
+ Title = {Valence quantum Monte Carlo with ab initio effective core potentials},
+ Url = {https://doi.org/10.1063%2F1.453345},
+ Volume = {87},
+ Year = 1987,
+ Bdsk-Url-1 = {https://doi.org/10.1063%2F1.453345},
+ Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.453345}}
+
+@article{Handy_1999,
+ Author = {Handy, Nicholas C. and Tozer, David J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/(SICI)1096-987X(19990115)20:1<106::AID-JCC11>3.0.CO;2-P},
+ Issn = {0192-8651, 1096-987X},
+ Journal = {J. Comput. Chem.},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {106-113},
+ Title = {Excitation Energies of Benzene from {{Kohn}}-{{Sham}} Theory},
+ Volume = {20},
+ Year = {1999},
+ Bdsk-Url-1 = {https://doi.org/10.1002/(SICI)1096-987X(19990115)20:1%3C106::AID-JCC11%3E3.0.CO;2-P}}
+
+@article{Harbach_2014,
+ Author = {Harbach, Philipp H. P. and Wormit, Michael and Dreuw, Andreas},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4892418},
+ File = {/Users/loos/Zotero/storage/GP5QMR6N/Harbach et al. - 2014 - The third-order algebraic diagrammatic constructio.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {6},
+ Pages = {064113},
+ Shorttitle = {The Third-Order Algebraic Diagrammatic Construction Method ({{ADC}}(3)) for the Polarization Propagator for Closed-Shell Molecules},
+ Title = {The Third-Order Algebraic Diagrammatic Construction Method ({{ADC}}(3)) for the Polarization Propagator for Closed-Shell Molecules: {{Efficient}} Implementation and Benchmarking},
+ Volume = {141},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4892418}}
+
+@article{Hashimoto_1996,
+ Author = {Hashimoto, T. and Nakano, H. and Hirao, K.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.471286},
+ File = {/Users/loos/Zotero/storage/M979MTK8/Hashimoto et al. - 1996 - Theoretical study of the valence π→π excited stat.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = apr,
+ Number = {16},
+ Pages = {6244-6258},
+ Shorttitle = {Theoretical Study of the Valence $\Pi\rightarrow\pi$* Excited States of Polyacenes},
+ Title = {Theoretical Study of the Valence $\Pi\rightarrow\pi$* Excited States of Polyacenes: {{Benzene}} and Naphthalene},
+ Volume = {104},
+ Year = {1996},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.471286}}
+
+@article{Hashimoto_1996a,
+ Author = {Hashimoto, T. and Nakano, H. and Hirao, K.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.471286},
+ File = {/Users/loos/Zotero/storage/M979MTK8/Hashimoto et al. - 1996 - Theoretical study of the valence π→π excited stat.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = apr,
+ Number = {16},
+ Pages = {6244-6258},
+ Shorttitle = {Theoretical Study of the Valence $\Pi\rightarrow\pi$* Excited States of Polyacenes},
+ Title = {Theoretical Study of the Valence $\Pi\rightarrow\pi$* Excited States of Polyacenes: {{Benzene}} and Naphthalene},
+ Volume = {104},
+ Year = {1996},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.471286}}
+
+@article{Hattig_2000,
+ Author = {H\"attig, C. and Weigend, F.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {J. Chem. Phys.},
+ Pages = {5154--5161},
+ Title = {CC2 Excitation Energy Calculations on Large Molecules Using the Resolution of the Identity Approximation},
+ Volume = 113,
+ Year = 2000}
+
+@article{Hattig12,
+ Author = {C. Hattig and W. Klopper and A. Kohn and D. P. Tew},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Journal = {Chem. Rev.},
+ Pages = {4},
+ Volume = {112},
+ Year = {2012}}
+
+@article{Hay_1974,
+ Author = {Hay, P. Jeffrey and Shavitt, Isaiah},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Volume = {60},
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+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1681456}}
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+@article{Head-Gordon_1994,
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+ Journal = {Chem. Phys. Lett.},
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+ Year = {1994}}
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+@article{Head-Gordon_1995,
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Publisher = {John Wiley \& Sons, Inc.},
+ Timestamp = {2014.11.24},
+ Title = {Molecular Electronic-Structure Theory},
+ Year = {2013}}
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+@article{Hirao_1983,
+ Author = {Hirao, K.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ Issn = {0021-9606, 1089-7690},
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+@article{Hirata_2000,
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Doi = {10.1016/S0009-2614(00)00772-7},
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+ Title = {High-Order Determinantal Equation-of-Motion Coupled-Cluster Calculations for Electronic Excited States},
+ Volume = {326},
+ Year = {2000},
+ Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(00)00772-7}}
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+@article{Hollauer_1991,
+ Author = {Hollauer, Eduardo and Nascimento, Marco Antonio Chaer},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+@article{Holmes_2016,
+ Author = {Holmes, Adam A. and Tubman, Norm M. and Umrigar, C. J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ Issn = {1549-9626},
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+ Pages = {3674--3680},
+ Publisher = {American Chemical Society (ACS)},
+ Title = {Heat-Bath Configuration Interaction: An Efficient Selected Configuration Interaction Algorithm Inspired by Heat-Bath Sampling},
+ Url = {http://dx.doi.org/10.1021/acs.jctc.6b00407},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jctc.6b00407}}
+
+@article{Holmes_2017,
+ Author = {Holmes, Adam A. and Umrigar, C. J. and Sharma, Sandeep},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4998614},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Oct},
+ Number = {16},
+ Pages = {164111},
+ Publisher = {AIP Publishing},
+ Title = {Excited states using semistochastic heat-bath configuration interaction},
+ Url = {http://dx.doi.org/10.1063/1.4998614},
+ Volume = {147},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4998614}}
+
+@article{Hoyer_2016,
+ Author = {Hoyer, Chad E. and Ghosh, Soumen and Truhlar, Donald G. and Gagliardi, Laura},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Journal = {J. Phys. Chem. Lett.},
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+ Pages = {586-591},
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+ Volume = {7},
+ Year = {2016},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1021/acs.jpclett.5b02773}}
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+@article{Hsu_2001,
+ Author = {Hsu, Chao-Ping and Hirata, So and {Head-Gordon}, Martin},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Doi = {10.1021/jp0024367},
+ File = {/Users/loos/Zotero/storage/I6XQ5V25/Hsu et al. - 2001 - Excitation Energies from Time-Dependent Density Fu.pdf},
+ Issn = {1089-5639, 1520-5215},
+ Journal = {J. Phys. Chem. A},
+ Language = {en},
+ Month = jan,
+ Number = {2},
+ Pages = {451-458},
+ Shorttitle = {Excitation {{Energies}} from {{Time}}-{{Dependent Density Functional Theory}} for {{Linear Polyene Oligomers}}},
+ Title = {Excitation {{Energies}} from {{Time}}-{{Dependent Density Functional Theory}} for {{Linear Polyene Oligomers}}: {{Butadiene}} to {{Decapentaene}}},
+ Volume = {105},
+ Year = {2001},
+ Bdsk-Url-1 = {https://doi.org/10.1021/jp0024367}}
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+@article{Hubner_1998,
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {0009-2614},
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+ Doi = {10.1016/j.chemphys.2011.03.019},
+ File = {/Users/loos/Zotero/storage/A4JUV4M4/Huix-Rotllant et al. - 2011 - Assessment of dressed time-dependent density-funct.pdf},
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+ Journal = {Chem. Phys.},
+ Language = {en},
+ Month = nov,
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+ Year = {2011},
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+ Volume = {89},
+ Year = {1988},
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+@article{Innes_1988,
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+@article{Kannengiesser_2014,
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {1089-7690},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ File = {/Users/loos/Zotero/storage/Z4NLZBMD/Kazaryan et al. - 2008 - Excitation Energies from Spin-Restricted Ensemble-.pdf},
+ Issn = {1089-5639, 1520-5215},
+ Journal = {J. Phys. Chem. A},
+ Language = {en},
+ Month = dec,
+ Number = {50},
+ Pages = {12980-12988},
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+ Author = {Maitra, Neepa T.},
+ Booktitle = {Fundamentals of Time-Dependent Density Functional Theory},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1007/978-3-642-23518-4_8},
+ Editor = {Marques, Miguel A.L. and Maitra, Neepa T. and Nogueira, Fernando M.S. and Gross, E.K.U. and Rubio, Angel},
+ File = {/Users/loos/Zotero/storage/MAFNZHIQ/Maitra - 2012 - Memory History , Initial-State Dependence , and D.pdf},
+ Isbn = {978-3-642-23517-7 978-3-642-23518-4},
+ Pages = {167-184},
+ Publisher = {Springer Berlin Heidelberg},
+ Title = {Memory: History , Initial-State Dependence , and Double-Excitations},
+ Volume = {837},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1007/978-3-642-23518-4_8}}
+
+@article{Manathunga_2016,
+ Author = {Manathunga, Madushanka and Yang, Xuchun and Luk, Hoi Ling and Gozem, Samer and Frutos, Luis Manuel and Valentini, Alessio and Ferr\`e, Nicolas and Olivucci, Massimo},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.jctc.5b00945},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = feb,
+ Number = {2},
+ Pages = {839-850},
+ Title = {Probing the {{Photodynamics}} of {{Rhodopsins}} with {{Reduced Retinal Chromophores}}},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b00945}}
+
+@article{Manten_2001,
+ Author = {Manten, Sebastian and L{\"u}chow, Arne},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1394757},
+ Issn = {1089-7690},
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+ Month = {Sep},
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+ Pages = {5362--5366},
+ Publisher = {AIP Publishing},
+ Title = {On the accuracy of the fixed-node diffusion quantum Monte Carlo method},
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+ Volume = {115},
+ Year = {2001},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.1394757}}
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+@article{Marquart_1963,
+ Author = {Marquart, J. R. and Berkowitz, J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {1089-7690},
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+ Pages = {283--285},
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+@article{Mason_1959,
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Pages = {1263-1268},
+ Title = {248. {{The}} Electronic Spectra of {{N}}-Heteroaromatic Systems. {{Part IV}}. {{The}} Vibrational Structure of the N$\rightarrow\pi$ Band of Sym-Tetrazine},
+ Volume = {0},
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+@article{Matos_1987,
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+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ Issn = {0021-9606, 1089-7690},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {1089-7690},
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+ Publisher = {AIP Publishing},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/jcc.21102},
+ File = {/Users/loos/Zotero/storage/6CVASLJL/Mazur and W{\l}odarczyk - 2009 - Application of the dressed time-dependent density .pdf},
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+ Journal = {J. Comput. Chem.},
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+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {0021-9606, 1089-7690},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Year = {2009},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Doi = {10.1021/jp908032x},
+ Issn = {1089-5639, 1520-5215},
+ Journal = {J. Phys. Chem. A},
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+ Month = nov,
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+ Pages = {12749-12753},
+ Shorttitle = {Optimizing {{Conical Intersections}} by {{Spin}}-{{Flip Density Functional Theory}}},
+ Title = {Optimizing {{Conical Intersections}} by {{Spin}}-{{Flip Density Functional Theory}}: {{Application}} to {{Ethylene}}},
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+ Year = {2009},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ File = {/Users/loos/Zotero/storage/8RF2NYW3/Mitas - 2006 - Structure of Fermion Nodes and Nodal Cells.pdf},
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+ Journal = {Phys. Rev. Lett.},
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Isbn = {978-0-08-045519-8},
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+ Year = {2011}}
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+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ File = {/Users/loos/Zotero/storage/IC928IFM/Senjean et al. - 2015 - Linear interpolation method in ensemble Kohn-Sham .pdf},
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+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.92.012518}}
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+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
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+ File = {/Users/loos/Zotero/storage/B4S5PIVV/Senjean et al. - 2016 - Combining linear interpolation with extrapolation .pdf},
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+ Language = {en},
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+ Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2015.1119902}}
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+ Volume = 63,
+ Year = 1954}
+
+@article{Tawada_2004,
+ Author = {Tawada, Yoshihiro and Tsuneda, Takao and Yanagisawa, Susumu and Yanai, Takeshi and Hirao, Kimihiko},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.1688752},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {18},
+ Pages = {8425-8433},
+ Title = {A Long-Range-Corrected Time-Dependent Density Functional Theory},
+ Volume = {120},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1688752}}
+
+@article{Theophilou_1979,
+ Abstract = {In this paper it is shown that the density can be used as the basic variable for calculating the properties of excited states. The correspondence is not between an eigenstate and its density, as is the case with the ground state, but between the subspace spanned by the number of lowest-energy eigenstates and the sum of their densities. An extension of the Hohenberg-Kohn-Sham theory for excited states has also been developed. The equations derived are similar in form to those for the ground-state density but the interpretation is different. The lowest-order approximation of the present theory coincides with Slater's `transition-state' theory.},
+ Author = {Theophilou, A K},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1088/0022-3719/12/24/013},
+ File = {/Users/loos/Zotero/storage/BKC3FYW9/Theophilou - 1979 - The energy density functional formalism for excite.pdf},
+ Issn = {0022-3719},
+ Journal = {J. Phys. C},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Pages = {5419-5430},
+ Title = {The Energy Density Functional Formalism for Excited States},
+ Volume = {12},
+ Year = {1979},
+ Bdsk-Url-1 = {https://doi.org/10.1088/0022-3719/12/24/013}}
+
+@article{Toulouse_2007,
+ Author = {Toulouse, Julien and Umrigar, C. J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.2437215},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Feb},
+ Number = {8},
+ Pages = {084102},
+ Publisher = {AIP Publishing},
+ Title = {Optimization of quantum Monte Carlo wave functions by energy minimization},
+ Url = {http://dx.doi.org/10.1063/1.2437215},
+ Volume = {126},
+ Year = {2007},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2437215}}
+
+@article{Toulouse_2008,
+ Author = {Toulouse, Julien and Umrigar, C. J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.2908237},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {17},
+ Pages = {174101},
+ Title = {Full Optimization of {{Jastrow}}\textendash{{Slater}} Wave Functions with Application to the First-Row Atoms and Homonuclear Diatomic Molecules},
+ Volume = {128},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.2908237}}
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+@article{Tozer_1998,
+ Author = {Tozer, David J. and Handy, Nicholas C.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.477711},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {23},
+ Pages = {10180-10189},
+ Shorttitle = {Improving Virtual {{Kohn}}\textendash{{Sham}} Orbitals and Eigenvalues},
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+ Volume = {109},
+ Year = {1998},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.477711}}
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+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-12-11 14:08:40 +0100},
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+ Issn = {0026-8976, 1362-3028},
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+ Month = oct,
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+ Volume = {97},
+ Year = {1999},
+ Bdsk-Url-1 = {https://doi.org/10.1080/00268979909482888}}
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+ Author = {Tozer, David J. and Handy, Nicholas C.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1039/a910321j},
+ File = {/Users/loos/Zotero/storage/TFJP3V8Z/Tozer and Handy - 2000 - On the determination of excitation energies using .pdf},
+ Issn = {14639076, 14639084},
+ Journal = {Phys. Chem. Chem. Phys.},
+ Language = {en},
+ Number = {10},
+ Pages = {2117-2121},
+ Title = {On the Determination of Excitation Energies Using Density Functional Theory},
+ Volume = {2},
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+ Bdsk-Url-1 = {https://doi.org/10.1039/a910321j}}
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+ Author = {Tozer, David J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
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+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
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+ Pages = {12697-12699},
+ Title = {Relationship between Long-Range Charge-Transfer Excitation Energy Error and Integer Discontinuity in {{Kohn}}\textendash{{Sham}} Theory},
+ Volume = {119},
+ Year = {2003},
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+ Volume = {11},
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+ Author = {C. J. Umrigar},
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+ Editor = {M. P. Nightingale and C. J. Umrigar},
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+ Volume = {30},
+ Year = {1975},
+ Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(75)80039-X}}
+
+@article{Williamson_2002,
+ Author = {Williamson, Andrew J. and Grossman, Jeffrey C. and Hood, Randolph Q. and Puzder, Aaron and Galli, Giulia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/PhysRevLett.89.196803},
+ File = {/Users/loos/Zotero/storage/7PV4DECC/Williamson et al. - 2002 - Quantum Monte Carlo Calculations of Nanostructure .pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = oct,
+ Number = {19},
+ Pages = {196803},
+ Shorttitle = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}},
+ Title = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}: {{Application}} to {{Silicon Quantum Dots}}},
+ Volume = {89},
+ Year = {2002},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.89.196803}}
+
+@article{Williamson_2002a,
+ Author = {Williamson, Andrew J. and Grossman, Jeffrey C. and Hood, Randolph Q. and Puzder, Aaron and Galli, Giulia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1103/PhysRevLett.89.196803},
+ File = {/Users/loos/Zotero/storage/XMBEEF79/Williamson et al. - 2002 - Quantum Monte Carlo Calculations of Nanostructure .pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = oct,
+ Number = {19},
+ Pages = {196803},
+ Shorttitle = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}},
+ Title = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}: {{Application}} to {{Silicon Quantum Dots}}},
+ Volume = {89},
+ Year = {2002},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.89.196803}}
+
+@article{Witek_2003,
+ Author = {Witek, Henryk A. and Nakano, Haruyuki and Hirao, Kimihiko},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/jcc.10311},
+ Issn = {0192-8651, 1096-987X},
+ Journal = {J. Comput. Chem.},
+ Language = {en},
+ Month = sep,
+ Number = {12},
+ Pages = {1390-1400},
+ Title = {Multireference Perturbation Theory with Optimized Partitioning. {{II}}. {{Applications}} to Molecular Systems},
+ Volume = {24},
+ Year = {2003},
+ Bdsk-Url-1 = {https://doi.org/10.1002/jcc.10311}}
+
+@article{Woodcock_2002,
+ Author = {Woodcock, H. Lee and Schaefer, Henry F. and Schreiner, Peter R.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/jp0212895},
+ File = {/Users/loos/Zotero/storage/T47SGG9V/Woodcock et al. - 2002 - Problematic Energy Differences between Cumulenes a.pdf},
+ Issn = {1089-5639, 1520-5215},
+ Journal = {J. Phys. Chem. A},
+ Language = {en},
+ Month = dec,
+ Number = {49},
+ Pages = {11923-11931},
+ Shorttitle = {Problematic {{Energy Differences}} between {{Cumulenes}} and {{Poly}}-Ynes},
+ Title = {Problematic {{Energy Differences}} between {{Cumulenes}} and {{Poly}}-Ynes: {{Does This Point}} to a {{Systematic Improvement}} of {{Density Functional Theory}}?},
+ Volume = {106},
+ Year = {2002},
+ Bdsk-Url-1 = {https://doi.org/10.1021/jp0212895}}
+
+@article{Wouters_2014,
+ Author = {Wouters, Sebastian and Poelmans, Ward and Ayers, Paul W. and Van Neck, Dimitri},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/j.cpc.2014.01.019},
+ Issn = {00104655},
+ Journal = {Computer Physics Communications},
+ Language = {en},
+ Month = jun,
+ Number = {6},
+ Pages = {1501-1514},
+ Shorttitle = {{{CheMPS2}}},
+ Title = {{{CheMPS2}}: {{A}} Free Open-Source Spin-Adapted Implementation of the Density Matrix Renormalization Group for Ab Initio Quantum Chemistry},
+ Volume = {185},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.cpc.2014.01.019}}
+
+@article{Wu_2007,
+ Author = {Wu, Z. J. and Wang, M. Y. and Su, Z. M.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/jcc.20603},
+ Issn = {1096-987X},
+ Journal = {J. Comput. Chem.},
+ Month = {Feb},
+ Number = {3},
+ Pages = {703--714},
+ Publisher = {Wiley-Blackwell},
+ Title = {Electronic structures and chemical bonding in diatomic ScX to ZnX (X = S, Se, Te)},
+ Url = {http://dx.doi.org/10.1002/jcc.20603},
+ Volume = {28},
+ Year = {2007},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.20603}}
+
+@article{Xiao_2016,
+ Author = {Xiao, Suo and Li, Xiaopeng and Sun, Weiwei and Guan, Baoqin and Wang, Yong},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/j.cej.2016.05.068},
+ Issn = {1385-8947},
+ Journal = {Chem. Eng. J.},
+ Month = {Dec},
+ Pages = {251--259},
+ Publisher = {Elsevier BV},
+ Title = {General and facile synthesis of metal sulfide nanostructures: In situ microwave synthesis and application as binder-free cathode for Li-ion batteries},
+ Url = {http://dx.doi.org/10.1016/j.cej.2016.05.068},
+ Volume = {306},
+ Year = {2016},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cej.2016.05.068}}
+
+@article{Xu_2013,
+ Abstract = {We present a new approach to calculating potential energy surfaces for photochemical reactions by combining self-consistent-field calculations for single-reference ground and excited states with symmetrycorrected spin-flip Tamm-Dancoff approximation calculations for multireference electronic states. The method is illustrated by an application with the M05-2X exchange-correlation functional to cis-trans isomerization of the penta-2,4-dieniminium cation, which is a model (with three conjugated double bonds) of the protonated Schiff base of retinal. We find good agreement with multireference configuration interaction-plus-quadruples (MRCISD+Q) wave function calculations along three key paths in the strong-interaction region of the ground and first excited singlet states.},
+ Author = {Xu, Xuefei and Gozem, Samer and Olivucci, Massimo and Truhlar, Donald G.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/jz301935x},
+ File = {/Users/loos/Zotero/storage/W3NS8BIA/Xu et al. - 2013 - Combined Self-Consistent-Field and Spin-Flip Tamm--.pdf},
+ Issn = {1948-7185},
+ Journal = {J. Phys. Chem. Lett.},
+ Language = {en},
+ Month = jan,
+ Number = {2},
+ Pages = {253-258},
+ Title = {Combined {{Self}}-{{Consistent}}-{{Field}} and {{Spin}}-{{Flip Tamm}}\textendash{{Dancoff Density Functional Approach}} to {{Potential Energy Surfaces}} for {{Photochemistry}}},
+ Volume = {4},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1021/jz301935x}}
+
+@article{Yanai_2004,
+ Author = {Yanai, Takeshi and Tew, David P and Handy, Nicholas C},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1016/j.cplett.2004.06.011},
+ Issn = {00092614},
+ Journal = {Chem. Phys. Lett.},
+ Language = {en},
+ Month = jul,
+ Number = {1-3},
+ Pages = {51-57},
+ Title = {A New Hybrid Exchange\textendash{}Correlation Functional Using the {{Coulomb}}-Attenuating Method ({{CAM}}-{{B3LYP}})},
+ Volume = {393},
+ Year = {2004},
+ Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2004.06.011}}
+
+@article{Yang_2013a,
+ Author = {Yang, Weitao and {Mori-S\'anchez}, Paula and Cohen, Aron J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4817183},
+ File = {/Users/loos/Zotero/storage/FXN44KXZ/Yang et al. - 2013 - Extension of many-body theory and approximate dens.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = sep,
+ Number = {10},
+ Pages = {104114},
+ Title = {Extension of Many-Body Theory and Approximate Density Functionals to Fractional Charges and Fractional Spins},
+ Volume = {139},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4817183}}
+
+@article{Yang_2014,
+ Author = {Yang, Zeng-Hui and Trail, John R. and {Pribram-Jones}, Aurora and Burke, Kieron and Needs, Richard J. and Ullrich, Carsten A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-12-11 14:04:22 +0100},
+ Doi = {10.1103/PhysRevA.90.042501},
+ File = {/Users/loos/Zotero/storage/432NXITC/Yang et al. - 2014 - Exact and approximate Kohn-Sham potentials in ense.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = oct,
+ Number = {4},
+ Pages = {042501},
+ Title = {Exact and Approximate {{Kohn}}-{{Sham}} Potentials in Ensemble Density-Functional Theory},
+ Volume = {90},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.90.042501}}
+
+@article{Yang_2017,
+ Author = {Yang, Zeng-Hui and {Pribram-Jones}, Aurora and Burke, Kieron and Ullrich, Carsten A.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-12-11 14:03:56 +0100},
+ Doi = {10.1103/PhysRevLett.119.033003},
+ File = {/Users/loos/Zotero/storage/6UU7W786/Yang et al. - 2017 - Direct Extraction of Excitation Energies from Ense.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = jul,
+ Number = {3},
+ Pages = {033003},
+ Title = {Direct {{Extraction}} of {{Excitation Energies}} from {{Ensemble Density}}-{{Functional Theory}}},
+ Volume = {119},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.119.033003}}
+
+@article{Zhang_1996,
+ Author = {Zhang, Nan and Hayase, Takasuke and Kawamata, Hiroshi and Nakao, Kojiro and Nakajima, Atsushi and Kaya, Koji},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.471048},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Mar},
+ Number = {10},
+ Pages = {3413--3419},
+ Publisher = {AIP Publishing},
+ Title = {Photoelectron spectroscopy of iron--sulfur cluster anions},
+ Url = {http://dx.doi.org/10.1063/1.471048},
+ Volume = {104},
+ Year = {1996},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.471048}}
+
+@article{Zhao_2016,
+ Author = {Zhao, Luning and Neuscamman, Eric},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.jctc.6b00480},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {3719-3726},
+ Title = {Equation of {{Motion Theory}} for {{Excited States}} in {{Variational Monte Carlo}} and the {{Jastrow Antisymmetric Geminal Power}} in {{Hilbert Space}}},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00480}}
+
+@article{Zhou_2017,
+ Author = {Zhou, Xiaojun and Wang, Fan},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1002/jcc.24750},
+ Issn = {1096-987X},
+ Journal = {J. Comput. Chem.},
+ Keywords = {FN-DMC ⋅ barrier heights ⋅ H-transfer reactions ⋅ CCSD(T) ⋅ density functional theory},
+ Number = {11},
+ Pages = {798--806},
+ Title = {Barrier heights of hydrogen-transfer reactions with diffusion quantum monte carlo method},
+ Url = {http://dx.doi.org/10.1002/jcc.24750},
+ Volume = {38},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.24750}}
+
+@article{Zimmerman_2009,
+ Author = {Zimmerman, Paul M. and Toulouse, Julien and Zhang, Zhiyong and Musgrave, Charles B. and Umrigar, C. J.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.3220671},
+ File = {/Users/loos/Zotero/storage/RAGHIL93/Zimmerman et al. - 2009 - Excited states of methylene from quantum Monte Car.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = sep,
+ Number = {12},
+ Pages = {124103},
+ Title = {Excited States of Methylene from Quantum {{Monte Carlo}}},
+ Volume = {131},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.3220671}}
+
+@article{Zimmerman_2011,
+ Author = {Zimmerman, Paul M. and Bell, Franziska and Casanova, David and {Head-Gordon}, Martin},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ja208431r},
+ Issn = {0002-7863, 1520-5126},
+ Journal = {J. Am. Chem. Soc.},
+ Language = {en},
+ Month = dec,
+ Number = {49},
+ Pages = {19944-19952},
+ Shorttitle = {Mechanism for {{Singlet Fission}} in {{Pentacene}} and {{Tetracene}}},
+ Title = {Mechanism for {{Singlet Fission}} in {{Pentacene}} and {{Tetracene}}: {{From Single Exciton}} to {{Two Triplets}}},
+ Volume = {133},
+ Year = {2011},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ja208431r}}
+
+@article{Zimmerman_2013,
+ Author = {Zimmerman, Paul M. and Musgrave, Charles B. and {Head-Gordon}, Martin},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/ar3001734},
+ Issn = {0001-4842, 1520-4898},
+ Journal = {Acc. Chem. Res.},
+ Language = {en},
+ Month = jun,
+ Number = {6},
+ Pages = {1339-1347},
+ Title = {A {{Correlated Electron View}} of {{Singlet Fission}}},
+ Volume = {46},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ar3001734}}
+
+@article{Zimmerman_2017,
+ Author = {Zimmerman, Paul M.},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1063/1.4977727},
+ Issn = {1089-7690},
+ Journal = {J. Chem. Phys.},
+ Month = {Mar},
+ Number = {10},
+ Pages = {104102},
+ Publisher = {AIP Publishing},
+ Title = {Incremental full configuration interaction},
+ Url = {http://dx.doi.org/10.1063/1.4977727},
+ Volume = {146},
+ Year = {2017},
+ Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4977727}}
+
+@article{Zulfikri_2016,
+ Author = {Zulfikri, Habiburrahman and Amovilli, Claudio and Filippi, Claudia},
+ Date-Added = {2018-10-24 22:38:52 +0200},
+ Date-Modified = {2018-10-24 22:38:52 +0200},
+ Doi = {10.1021/acs.jctc.5b01077},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = mar,
+ Number = {3},
+ Pages = {1157-1168},
+ Title = {Multiple-{{Resonance Local Wave Functions}} for {{Accurate Excited States}} in {{Quantum Monte Carlo}}},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01077}}
+
+@article{Hohenberg_1964,
+ Annote = {Hohenberg-Kohn theorem},
+ Author = {P. Hohenberg and W. Kohn},
+ Date-Added = {2018-10-24 22:32:22 +0200},
+ Date-Modified = {2018-12-11 14:09:26 +0100},
+ Doi = {10.1103/PhysRev.136.B864},
+ Journal = {Phys. Rev.},
+ Pages = {B864--B871},
+ Title = {Inhomogeneous electron gas},
+ Volume = {136},
+ Year = {1964},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.136.B864}}
+
+@article{Kohn_1965,
+ Author = {W. Kohn and L. J. Sham},
+ Date-Added = {2018-10-24 22:32:22 +0200},
+ Date-Modified = {2018-12-11 14:09:45 +0100},
+ Doi = {10.1103/PhysRev.140.A1133},
+ Journal = {Phys. Rev.},
+ Pages = {A1133--A1138},
+ Title = {Self-consistent equations including exchange and correlation effects},
+ Volume = {140},
+ Year = {1965},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.140.A1133}}
+
+@article{Ceperley_1980,
+ Author = {D. M. Ceperley and B. J. Alder},
+ Date-Added = {2018-10-24 22:31:35 +0200},
+ Date-Modified = {2018-10-24 22:31:41 +0200},
+ Journal = {Phys. Rev. Lett.},
+ Keywords = {dft, qmech; jellium},
+ Pages = {566--569},
+ Title = {Ground State of the Electron Gas by a Stochastic Method},
+ Volume = {45},
+ Year = {1980}}
+
+@article{Cohen_2012,
+ Author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula and Yang, Weitao},
+ Doi = {10.1021/cr200107z},
+ File = {/Users/loos/Zotero/storage/TKMU8F5B/Cohen et al. - 2012 - Challenges for Density Functional Theory.pdf},
+ Issn = {0009-2665, 1520-6890},
+ Journal = {Chem. Rev.},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {289-320},
+ Title = {Challenges for {{Density Functional Theory}}},
+ Volume = {112},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1021/cr200107z}}
+
+@article{Cohen_2008,
+ Author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula and Yang, Weitao},
+ Doi = {10.1103/PhysRevB.77.115123},
+ File = {/Users/loos/Zotero/storage/QYTAN7UQ/Cohen et al. - 2008 - Fractional charge perspective on the band gap in d.pdf},
+ Issn = {1098-0121, 1550-235X},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = mar,
+ Number = {11},
+ Title = {Fractional Charge Perspective on the Band Gap in Density-Functional Theory},
+ Volume = {77},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.77.115123}}
+
+@article{Mori-Sanchez_2008,
+ Author = {{Mori-S\'anchez}, Paula and Cohen, Aron J. and Yang, Weitao},
+ Doi = {10.1103/PhysRevLett.100.146401},
+ File = {/Users/loos/Zotero/storage/Y4U67Q4C/Mori-S{\'a}nchez et al. - 2008 - Localization and Delocalization Errors in Density .pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = apr,
+ Number = {14},
+ Title = {Localization and {{Delocalization Errors}} in {{Density Functional Theory}} and {{Implications}} for {{Band}}-{{Gap Prediction}}},
+ Volume = {100},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.100.146401}}
+
+@article{Cohen_2008a,
+ Author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula and Yang, Weitao},
+ Doi = {10.1063/1.2987202},
+ File = {/Users/loos/Zotero/storage/GG72PJT7/Cohen et al. - 2008 - Fractional spins and static correlation error in d.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = sep,
+ Number = {12},
+ Pages = {121104},
+ Title = {Fractional Spins and Static Correlation Error in Density Functional Theory},
+ Volume = {129},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.2987202}}
+
+@article{Gross_1988b,
+ Author = {Gross, E. K. U. and Oliveira, L. N. and Kohn, W.},
+ Doi = {10.1103/PhysRevA.37.2809},
+ File = {/Users/loos/Zotero/storage/V9XVGNHV/Gross et al. - 1988 - Density-functional theory for ensembles of fractio.pdf},
+ Issn = {0556-2791},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {8},
+ Pages = {2809-2820},
+ Title = {Density-Functional Theory for Ensembles of Fractionally Occupied States. {{I}}. {{Basic}} Formalism},
+ Volume = {37},
+ Year = {1988},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2809}}
+
+@article{Oliveira_1988a,
+ Author = {Oliveira, L. N. and Gross, E. K. U. and Kohn, W.},
+ Doi = {10.1103/PhysRevA.37.2821},
+ File = {/Users/loos/Zotero/storage/FCL4XIJQ/Oliveira et al. - 1988 - Density-functional theory for ensembles of fractio.pdf},
+ Issn = {0556-2791},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {8},
+ Pages = {2821-2833},
+ Title = {Density-Functional Theory for Ensembles of Fractionally Occupied States. {{II}}. {{Application}} to the {{He}} Atom},
+ Volume = {37},
+ Year = {1988},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2821}}
+
+@article{Gross_1988c,
+ Author = {Gross, E. K. U. and Oliveira, L. N. and Kohn, W.},
+ Doi = {10.1103/PhysRevA.37.2805},
+ File = {/Users/loos/Zotero/storage/3T7YFYJX/Gross et al. - 1988 - Rayleigh-Ritz variational principle for ensembles .pdf},
+ Issn = {0556-2791},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {8},
+ Pages = {2805-2808},
+ Title = {Rayleigh-{{Ritz}} Variational Principle for Ensembles of Fractionally Occupied States},
+ Volume = {37},
+ Year = {1988},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2805}}
+
+@article{Mirtschink_2013,
+ Author = {Mirtschink, Andr\'e and Seidl, Michael and {Gori-Giorgi}, Paola},
+ Doi = {10.1103/PhysRevLett.111.126402},
+ File = {/Users/loos/Zotero/storage/MWU5VCE9/Mirtschink et al. - 2013 - Derivative Discontinuity in the Strong-Interaction.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = sep,
+ Number = {12},
+ Title = {Derivative {{Discontinuity}} in the {{Strong}}-{{Interaction Limit}} of {{Density}}-{{Functional Theory}}},
+ Volume = {111},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.111.126402}}
+
+@article{Levy_1995,
+ Author = {Levy, Mel},
+ Doi = {10.1103/PhysRevA.52.R4313},
+ File = {/Users/loos/Zotero/storage/W7CIW7AW/Levy - 1995 - Excitation energies from density-functional orbita.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = dec,
+ Number = {6},
+ Pages = {R4313-R4315},
+ Title = {Excitation Energies from Density-Functional Orbital Energies},
+ Volume = {52},
+ Year = {1995},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.52.R4313}}
+
+@article{Theophilou_1979a,
+ Abstract = {In this paper it is shown that the density can be used as the basic variable for calculating the properties of excited states. The correspondence is not between an eigenstate and its density, as is the case with the ground state, but between the subspace spanned by the number of lowest-energy eigenstates and the sum of their densities. An extension of the Hohenberg-Kohn-Sham theory for excited states has also been developed. The equations derived are similar in form to those for the ground-state density but the interpretation is different. The lowest-order approximation of the present theory coincides with Slater's `transition-state' theory.},
+ Author = {Theophilou, A K},
+ Doi = {10.1088/0022-3719/12/24/013},
+ File = {/Users/loos/Zotero/storage/E5Y224GW/Theophilou - 1979 - The energy density functional formalism for excite.pdf},
+ Issn = {0022-3719},
+ Journal = {J. Phys. C Solid State Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Pages = {5419-5430},
+ Title = {The Energy Density Functional Formalism for Excited States},
+ Volume = {12},
+ Year = {1979},
+ Bdsk-Url-1 = {https://doi.org/10.1088/0022-3719/12/24/013}}
+
+@article{Valone_1980,
+ Author = {Valone, Steven M.},
+ Doi = {10.1063/1.440656},
+ File = {/Users/loos/Zotero/storage/8IEEMICU/Valone - 1980 - A one‐to‐one mapping between one‐particle densitie.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = nov,
+ Number = {9},
+ Pages = {4653-4655},
+ Title = {A One-to-one Mapping between One-particle Densities and Some {\emph{n}} -particle Ensembles},
+ Volume = {73},
+ Year = {1980},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.440656}}
+
+@incollection{Filatov_2015d,
+ Abstract = {Ensemble density functional theory (DFT) is a novel time-independent formalism for obtaining excitation energies of many-body fermionic systems. A considerable advantage of ensemble DFT over the more common Kohn\textendash{}Sham (KS) DFT and time-dependent DFT formalisms is that it enables one to account for strong non-dynamic electron correlation in the ground and excited states of molecular systems in a transparent and accurate fashion. Despite its positive aspects, ensemble DFT has not so far found its way into the repertoire of methods of modern computational chemistry, probably because of the perceived lack of practically affordable implementations of the theory. The spin-restricted ensemble-referenced KS (REKS) method is perhaps the first computationally feasible implementation of the ideas behind ensemble DFT which enables one to describe accurately electronic transitions in a wide class of molecular systems, including strongly correlated molecules (biradicals, molecules undergoing bond breaking/formation), extended $\pi$-conjugated systems, donor\textendash{}acceptor charge transfer adducts, etc.},
+ Address = {Cham},
+ Author = {Filatov, Michael},
+ Booktitle = {Density-{{Functional Methods}} for {{Excited States}}},
+ Doi = {10.1007/128_2015_630},
+ Editor = {Ferr\'e, Nicolas and Filatov, Michael and {Huix-Rotllant}, Miquel},
+ File = {/Users/loos/Zotero/storage/7MGLS9WA/Filatov - 2015 - Ensemble DFT Approach to Excited States of Strongl.pdf},
+ Isbn = {978-3-319-22080-2 978-3-319-22081-9},
+ Language = {en},
+ Pages = {97-124},
+ Publisher = {{Springer International Publishing}},
+ Title = {Ensemble {{DFT Approach}} to {{Excited States}} of {{Strongly Correlated Molecular Systems}}},
+ Volume = {368},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1007/128_2015_630}}
+
+@article{Filatov_2015e,
+ Author = {Filatov, Michael},
+ Doi = {10.1002/wcms.1209},
+ File = {/Users/loos/Zotero/storage/BEWPPY67/Filatov - 2015 - Spin-restricted ensemble-referenced Kohn-Sham meth.pdf},
+ Issn = {17590876},
+ Journal = {Wiley Interdiscip. Rev. Comput. Mol. Sci.},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Pages = {146-167},
+ Shorttitle = {Spin-Restricted Ensemble-Referenced {{Kohn}}-{{Sham}} Method},
+ Title = {Spin-Restricted Ensemble-Referenced {{Kohn}}-{{Sham}} Method: Basic Principles and Application to Strongly Correlated Ground and Excited States of Molecules},
+ Volume = {5},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1002/wcms.1209}}
+
+@article{Kazaryan_2008a,
+ Author = {Kazaryan, Andranik and Heuver, Jeroen and Filatov, Michael},
+ Doi = {10.1021/jp8033837},
+ File = {/Users/loos/Zotero/storage/QWBJKBS3/Kazaryan et al. - 2008 - Excitation Energies from Spin-Restricted Ensemble-.pdf},
+ Issn = {1089-5639, 1520-5215},
+ Journal = {J. Phys. Chem. A},
+ Language = {en},
+ Month = dec,
+ Number = {50},
+ Pages = {12980-12988},
+ Shorttitle = {Excitation {{Energies}} from {{Spin}}-{{Restricted Ensemble}}-{{Referenced Kohn}}-{{Sham Method}}},
+ Title = {Excitation {{Energies}} from {{Spin}}-{{Restricted Ensemble}}-{{Referenced Kohn}}-{{Sham Method}}: {{A State}}-{{Average Approach}} {\textsuperscript{\textdagger}}},
+ Volume = {112},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1021/jp8033837}}
+
+@article{Filatov_1999,
+ Abstract = {The energy and density of situations with strong non-dynamic correlation are formulated as weighted sums {\v Z}ensembles. of energies and densities of symmetry-adapted reference KS determinants. A computational scheme termed the spin-restricted ensemble-referenced Kohn\textendash{}Sham {\v Z}REKS. method is devised for these cases. An optimal set of orthonormal one-electron orbitals and their optimal occupation numbers are obtained from minimization of the ground state energy with respect to the density. The REKS method is applied to several model problems, rotation in C2 H 4, dissociation of H 2, and the singlet-triplet energy gaps in substituted trimethylene diradicals. q 1999 Elsevier Science B.V. All rights reserved.},
+ Author = {Filatov, Michael and Shaik, Sason},
+ Doi = {10.1016/S0009-2614(99)00336-X},
+ File = {/Users/loos/Zotero/storage/URSNLFXD/Filatov and Shaik - 1999 - A spin-restricted ensemble-referenced Kohn--Sham me.pdf},
+ Issn = {00092614},
+ Journal = {Chem. Phys. Lett.},
+ Language = {en},
+ Month = may,
+ Number = {5-6},
+ Pages = {429-437},
+ Title = {A Spin-Restricted Ensemble-Referenced {{Kohn}}\textendash{{Sham}} Method and Its Application to Diradicaloid Situations},
+ Volume = {304},
+ Year = {1999},
+ Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(99)00336-X}}
+
+@article{Steinmann_2013,
+ Author = {Steinmann, Stephan N. and Yang, Weitao},
+ Doi = {10.1063/1.4817849},
+ File = {/Users/loos/Zotero/storage/TTITC4WH/Steinmann and Yang - 2013 - Wave function methods for fractional electrons.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {7},
+ Pages = {074107},
+ Title = {Wave Function Methods for Fractional Electrons},
+ Volume = {139},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4817849}}
+
+@article{Mori-Sanchez_2009,
+ Author = {{Mori-S\'anchez}, Paula and Cohen, Aron J. and Yang, Weitao},
+ Doi = {10.1103/PhysRevLett.102.066403},
+ File = {/Users/loos/Zotero/storage/Y55S3LBQ/Mori-S{\'a}nchez et al. - 2009 - Discontinuous Nature of the Exchange-Correlation F.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = feb,
+ Number = {6},
+ Title = {Discontinuous {{Nature}} of the {{Exchange}}-{{Correlation Functional}} in {{Strongly Correlated Systems}}},
+ Volume = {102},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.102.066403}}
+
+@article{Cohen_2009,
+ Abstract = {In this work the behavior of MP2 for fractional occupations is investigated. The consideration of fractional charge behavior gives a simple derivation of an expression for the chemical potential (or the derivative of energy with respect to the number of electrons) of MP2. A generalized optimized effective potential formalism (OEP) has been developed in which the OEP is a nonlocal potential, which can be applied to explicit functionals of the orbitals and eigenvalues and also facilitates the evaluation of the chemical potential. The MP2 derivative improves upon the corresponding Koopmans' theorem in Hartree-Fock theory for the ionization energy and also gives a good estimate of the electron affinity. In strongly correlated systems with degeneracies and fractional spins, MP2 diverges, and another corrected second-order perturbative method ameliorates this failure for the energy but still does not recapture the correct behavior for the energy derivatives that yield the gap. Overall we present a view of wave function based methods and their behavior for fractional charges and spins that offers insight into the application of these methods to challenging chemical problems.},
+ Author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula and Yang, Weitao},
+ Doi = {10.1021/ct8005419},
+ File = {/Users/loos/Zotero/storage/KE5WMCA8/Cohen et al. - 2009 - Second-Order Perturbation Theory with Fractional C.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Pages = {786-792},
+ Title = {Second-{{Order Perturbation Theory}} with {{Fractional Charges}} and {{Fractional Spins}}},
+ Volume = {5},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1021/ct8005419}}
+
+@article{Mori-Sanchez_2012,
+ Author = {{Mori-S\'anchez}, Paula and Cohen, Aron J. and Yang, Weitao},
+ Doi = {10.1103/PhysRevA.85.042507},
+ File = {/Users/loos/Zotero/storage/E9F5G9CV/Mori-S{\'a}nchez et al. - 2012 - Failure of the random-phase-approximation correlat.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Title = {Failure of the Random-Phase-Approximation Correlation Energy},
+ Volume = {85},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.85.042507}}
+
+@article{Mori-Sanchez_2014,
+ Author = {{Mori-S\'anchez}, Paula and Cohen, Aron J.},
+ Doi = {10.1039/C4CP01170H},
+ File = {/Users/loos/Zotero/storage/EIX2IAFR/Mori-S{\'a}nchez and Cohen - 2014 - The derivative discontinuity of the exchange--corre.pdf},
+ Issn = {1463-9076, 1463-9084},
+ Journal = {Phys Chem Chem Phys},
+ Language = {en},
+ Number = {28},
+ Pages = {14378-14387},
+ Title = {The Derivative Discontinuity of the Exchange\textendash{}Correlation Functional},
+ Volume = {16},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1039/C4CP01170H}}
+
+@article{Cohen_2007,
+ Author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula and Yang, Weitao},
+ Doi = {10.1063/1.2741248},
+ File = {/Users/loos/Zotero/storage/LV8BEL9G/Cohen et al. - 2007 - Development of exchange-correlation functionals wi.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {19},
+ Pages = {191109},
+ Title = {Development of Exchange-Correlation Functionals with Minimal Many-Electron Self-Interaction Error},
+ Volume = {126},
+ Year = {2007},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.2741248}}
+
+@article{Franck_2014,
+ Author = {Franck, Odile and Fromager, Emmanuel},
+ Date-Modified = {2018-12-09 14:18:03 +0100},
+ Doi = {10.1080/00268976.2013.858191},
+ File = {/Users/loos/Zotero/storage/MH52WCNZ/Franck and Fromager - 2014 - Generalised adiabatic connection in ensemble densi.pdf},
+ Issn = {0026-8976, 1362-3028},
+ Journal = {Mol. Phys.},
+ Language = {en},
+ Month = jun,
+ Number = {12},
+ Pages = {1684-1701},
+ Shorttitle = {Generalised Adiabatic Connection in Ensemble Density-Functional Theory for Excited States},
+ Title = {Generalised Adiabatic Connection in Ensemble Density-Functional Theory for Excited States: Example of the {{H}} {\textsubscript{2}} Molecule},
+ Volume = {112},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2013.858191}}
+
+@article{Gidopoulos_2002,
+ Author = {Gidopoulos, N. I. and Papaconstantinou, P. G. and Gross, E. K. U.},
+ Doi = {10.1103/PhysRevLett.88.033003},
+ File = {/Users/loos/Zotero/storage/RRB3BXVQ/Gidopoulos et al. - 2002 - Spurious Interactions, and Their Correction, in th.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = jan,
+ Number = {3},
+ Title = {Spurious {{Interactions}}, and {{Their Correction}}, in the {{Ensemble}}-{{Kohn}}-{{Sham Scheme}} for {{Excited States}}},
+ Volume = {88},
+ Year = {2002},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.88.033003}}
+
+@article{Pribram-Jones_2014,
+ Author = {{Pribram-Jones}, Aurora and Yang, Zeng-hui and Trail, John R. and Burke, Kieron and Needs, Richard J. and Ullrich, Carsten A.},
+ Doi = {10.1063/1.4872255},
+ File = {/Users/loos/Zotero/storage/D8SVJKXC/Pribram-Jones et al. - 2014 - Excitations and benchmark ensemble density functio.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {18},
+ Pages = {18A541},
+ Title = {Excitations and Benchmark Ensemble Density Functional Theory for Two Electrons},
+ Volume = {140},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4872255}}
+
+@article{Carrascal_2018,
+ Abstract = {The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequencydependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequencydependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the groundstate exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. Many exact results, limits, and expansions about those limits are given in the Appendices.},
+ Author = {Carrascal, Diego J. and Ferrer, Jaime and Maitra, Neepa and Burke, Kieron},
+ Doi = {10.1140/epjb/e2018-90114-9},
+ File = {/Users/loos/Zotero/storage/YFNPCZLK/Carrascal et al. - 2018 - Linear response time-dependent density functional .pdf},
+ Issn = {1434-6028, 1434-6036},
+ Journal = {Eur. Phys. J. B},
+ Language = {en},
+ Month = jul,
+ Number = {7},
+ Title = {Linear Response Time-Dependent Density Functional Theory of the {{Hubbard}} Dimer},
+ Volume = {91},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1140/epjb/e2018-90114-9}}
+
+@article{Carrascal_2015,
+ Abstract = {This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy\textendash{}Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the `failure' of the Kohn\textendash{}Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree\textendash{}Fock and Bethe ansatz local density approximation) to show their successes and limitations. We also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for density functional energy components, several limits of the exact functional (weak- and strong-coupling, symmetric and asymmetric), various adiabatic connection results, proofs of exact conditions for this model, and the origin of the Hubbard model from a minimal basis model for stretched H2.},
+ Author = {Carrascal, D J and Ferrer, J and Smith, J C and Burke, K},
+ Doi = {10.1088/0953-8984/27/39/393001},
+ File = {/Users/loos/Zotero/storage/LRMWNYEQ/Carrascal et al. - 2015 - The Hubbard dimer a density functional case study.pdf},
+ Issn = {0953-8984, 1361-648X},
+ Journal = {J. Phys. Condens. Matter},
+ Language = {en},
+ Month = oct,
+ Number = {39},
+ Pages = {393001},
+ Shorttitle = {The {{Hubbard}} Dimer},
+ Title = {The {{Hubbard}} Dimer: A Density Functional Case Study of a Many-Body Problem},
+ Volume = {27},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1088/0953-8984/27/39/393001}}
+
+@article{Levy_1999,
+ Author = {Levy, Mel and Nagy, \'Agnes},
+ Doi = {10.1103/PhysRevLett.83.4361},
+ File = {/Users/loos/Zotero/storage/TAQZN9B5/Levy and Nagy - 1999 - Variational Density-Functional Theory for an Indiv.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = nov,
+ Number = {21},
+ Pages = {4361-4364},
+ Title = {Variational {{Density}}-{{Functional Theory}} for an {{Individual Excited State}}},
+ Volume = {83},
+ Year = {1999},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.83.4361}}
+
+@article{Perdew_1985,
+ Author = {Perdew, John P. and Levy, Mel},
+ Doi = {10.1103/PhysRevB.31.6264},
+ File = {/Users/loos/Zotero/storage/4MC7Q9WE/Perdew and Levy - 1985 - Extrema of the density functional for the energy .pdf},
+ Issn = {0163-1829},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = may,
+ Number = {10},
+ Pages = {6264-6272},
+ Shorttitle = {Extrema of the Density Functional for the Energy},
+ Title = {Extrema of the Density Functional for the Energy: {{Excited}} States from the Ground-State Theory},
+ Volume = {31},
+ Year = {1985},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.31.6264}}
+
+@article{Borgoo_2015,
+ Abstract = {This expression gives the difference between an excitation energy E1 - E0 and the corresponding Kohn\textendash{}Sham orbital energy difference $\epsilon$1 - $\epsilon$0 as a partial derivative of the exchange-correlation energy of an ensemble of states Exc,w[$\rho$]. Through Lieb maximisation, on input full-CI density functions, the exchange-correlation energy is evaluated accurately and the partial derivative is evaluated numerically using finite difference. The equality is studied numerically for different geometries of the H2 molecule and different ensemble weights. We explore the adiabatic connection for the ensemble exchange-correlation energy. The latter may prove useful when modelling the unknown weight dependence of the exchange-correlation energy.},
+ Address = {Athens, Greece},
+ Author = {Borgoo, Alex and Teale, Andy M. and Helgaker, Trygve},
+ Date-Modified = {2018-12-11 14:07:51 +0100},
+ Doi = {10.1063/1.4938857},
+ File = {/Users/loos/Zotero/storage/HRM8DPU6/Borgoo et al. - 2015 - Excitation energies from ensemble DFT.pdf},
+ Journal = {AIP Conf. Proc.},
+ Language = {en},
+ Pages = {090049},
+ Title = {Excitation Energies from Ensemble {{DFT}}},
+ Volume = {1702},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4938857}}
+
+@article{Levy_2014,
+ Author = {Levy, Mel and Zahariev, Federico},
+ Doi = {10.1103/PhysRevLett.113.113002},
+ File = {/Users/loos/Zotero/storage/4C87D4GM/Levy and Zahariev - 2014 - Ground-State Energy as a Simple Sum of Orbital Ene.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = sep,
+ Number = {11},
+ Shorttitle = {Ground-{{State Energy}} as a {{Simple Sum}} of {{Orbital Energies}} in {{Kohn}}-{{Sham Theory}}},
+ Title = {Ground-{{State Energy}} as a {{Simple Sum}} of {{Orbital Energies}} in {{Kohn}}-{{Sham Theory}}: {{A Shift}} in {{Perspective}} through a {{Shift}} in {{Potential}}},
+ Volume = {113},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.113.113002}}
+
+@article{Kohn_1986,
+ Author = {Kohn, Walter},
+ Doi = {10.1103/PhysRevA.34.737},
+ File = {/Users/loos/Zotero/storage/5LC78TCX/Kohn - 1986 - Density-functional theory for excited states in a .pdf},
+ Issn = {0556-2791},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = aug,
+ Number = {2},
+ Pages = {737-741},
+ Title = {Density-Functional Theory for Excited States in a Quasi-Local-Density Approximation},
+ Volume = {34},
+ Year = {1986},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.34.737}}
+
+@article{Nagy_2001,
+ Abstract = {Recently an optimized potential method (OPM) has been derived for ensembles of excited states. Here an alternative OPM is proposed. The ensemble Kohn\textendash{}Sham potential in the generalized version of the Krieger\textendash{}Li\textendash{}Iafrate approximation to the OPM method is obtained.},
+ Author = {Nagy, \'A.},
+ Date-Modified = {2018-12-11 14:00:17 +0100},
+ Doi = {10.1088/0953-4075/34/12/305},
+ File = {/Users/loos/Zotero/storage/N7CH5INL/Nagy - 2001 - An alternative optimized potential method for ense.pdf},
+ Issn = {0953-4075, 1361-6455},
+ Journal = {J. Phys. B At. Mol. Opt. Phys.},
+ Language = {en},
+ Month = jun,
+ Number = {12},
+ Pages = {2363-2370},
+ Title = {An Alternative Optimized Potential Method for Ensembles of Excited States},
+ Volume = {34},
+ Year = {2001},
+ Bdsk-Url-1 = {https://doi.org/10.1088/0953-4075/34/12/305}}
+
+@article{Nagy_1998,
+ Author = {Nagy, \'A},
+ Date-Modified = {2018-12-11 14:00:30 +0100},
+ Doi = {10.1002/(SICI)1097-461X(1998)69:3<247::AID-QUA4>3.0.CO;2-V},
+ File = {/Users/loos/Zotero/storage/28AU5L4W/Nagy - 1998 - Optimized potential method for ensembles of excite.pdf},
+ Issn = {0020-7608, 1097-461X},
+ Journal = {Int. J. Quantum Chem.},
+ Language = {en},
+ Number = {3},
+ Pages = {247-254},
+ Title = {Optimized Potential Method for Ensembles of Excited States},
+ Volume = {69},
+ Year = {1998},
+ Bdsk-Url-1 = {https://doi.org/10.1002/(SICI)1097-461X(1998)69:3%3C247::AID-QUA4%3E3.0.CO;2-V}}
+
+@article{Kraisler_2013,
+ Author = {Kraisler, Eli and Kronik, Leeor},
+ Date-Modified = {2018-12-11 14:05:18 +0100},
+ Doi = {10.1103/PhysRevLett.110.126403},
+ File = {/Users/loos/Zotero/storage/ELUWJQ66/Kraisler and Kronik - 2013 - Piecewise Linearity of Approximate Density Functio.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = mar,
+ Number = {12},
+ Pages = {126403},
+ Shorttitle = {Piecewise {{Linearity}} of {{Approximate Density Functionals Revisited}}},
+ Title = {Piecewise {{Linearity}} of {{Approximate Density Functionals Revisited}}: {{Implications}} for {{Frontier Orbital Energies}}},
+ Volume = {110},
+ Year = {2013},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.110.126403}}
+
+@article{Filatov_2015f,
+ Author = {Filatov, Michael and {Huix-Rotllant}, Miquel and Burghardt, Irene},
+ Doi = {10.1063/1.4919773},
+ File = {/Users/loos/Zotero/storage/HEEF3N7Y/Filatov et al. - 2015 - Ensemble density functional theory method correctl.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {18},
+ Pages = {184104},
+ Title = {Ensemble Density Functional Theory Method Correctly Describes Bond Dissociation, Excited State Electron Transfer, and Double Excitations},
+ Volume = {142},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4919773}}
+
+@article{Kraisler_2014,
+ Author = {Kraisler, Eli and Kronik, Leeor},
+ Doi = {10.1063/1.4871462},
+ File = {/Users/loos/Zotero/storage/SLYF4MBS/Kraisler and Kronik - 2014 - Fundamental gaps with approximate density function.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = may,
+ Number = {18},
+ Pages = {18A540},
+ Shorttitle = {Fundamental Gaps with Approximate Density Functionals},
+ Title = {Fundamental Gaps with Approximate Density Functionals: {{The}} Derivative Discontinuity Revealed from Ensemble Considerations},
+ Volume = {140},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4871462}}
+
+@article{Gorling_2015,
+ Author = {G\"orling, Andreas},
+ Doi = {10.1103/PhysRevB.91.245120},
+ File = {/Users/loos/Zotero/storage/DNV6TU95/G{\"o}rling - 2015 - Exchange-correlation potentials with proper discon.pdf},
+ Issn = {1098-0121, 1550-235X},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = jun,
+ Number = {24},
+ Title = {Exchange-Correlation Potentials with Proper Discontinuities for Physically Meaningful {{Kohn}}-{{Sham}} Eigenvalues and Band Structures},
+ Volume = {91},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.91.245120}}
+
+@article{Pittalis_2006b,
+ Author = {Pittalis, S. and Kurth, S. and Gross, E. K. U.},
+ Doi = {10.1063/1.2338038},
+ File = {/Users/loos/Zotero/storage/GFM8R2WS/Pittalis et al. - 2006 - On the degeneracy of atomic states within exact-ex.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {084105},
+ Title = {On the Degeneracy of Atomic States within Exact-Exchange (Spin-) Density Functional Theory},
+ Volume = {125},
+ Year = {2006},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.2338038}}
+
+@article{Nagy_2005,
+ Author = {Nagy, \'A. and Liu, S. and Bartolloti, L.},
+ Date-Modified = {2018-12-11 13:59:45 +0100},
+ Doi = {10.1063/1.1871933},
+ File = {/Users/loos/Zotero/storage/J4ZHPWPL/Nagy et al. - 2005 - Generalized density functional theory for degenera.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = apr,
+ Number = {13},
+ Pages = {134107},
+ Title = {Generalized Density Functional Theory for Degenerate States},
+ Volume = {122},
+ Year = {2005},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.1871933}}
+
+@article{Mori-Sanchez_2018,
+ Abstract = {A stochastic minimization method for a real-space wave function, $\Psi$(r1, r2...rn), constrained to a chosen density, $\rho$(r), is developed. It enables the explicit calculation of the Levy constrained search, F[$\rho$] = min$\Psi\rightarrow\rho$\langle$\Psi$|T\^ + V\^ ee|$\Psi$\rangle, which gives the exact functional of density functional theory. This general method is illustrated in the evaluation of F[$\rho$] for densities in one dimension with a soft-Coulomb interaction. Additionally, procedures are given to determine the first and second functional derivatives, $\delta$F/$\delta\rho$(r) and $\delta$2F/[$\delta\rho$(r)$\delta\rho$(r${'}$)]. For a chosen external potential, v(r), the functional and its derivatives are used in minimizations over densities to give the exact energy, Ev, without needing to solve the Schr\"odinger equation.},
+ Author = {{Mori-S\'anchez}, Paula and Cohen, Aron J.},
+ Doi = {10.1021/acs.jpclett.8b02332},
+ File = {/Users/loos/Zotero/storage/YCB2R94I/Mori-S{\'a}nchez and Cohen - 2018 - Exact Density Functional Obtained via the Levy Con.pdf},
+ Issn = {1948-7185},
+ Journal = {J. Phys. Chem. Lett.},
+ Language = {en},
+ Month = sep,
+ Number = {17},
+ Pages = {4910-4914},
+ Title = {Exact {{Density Functional Obtained}} via the {{Levy Constrained Search}}},
+ Volume = {9},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.8b02332}}
+
+@article{Cohen_2016,
+ Author = {Cohen, Aron J. and {Mori-S\'anchez}, Paula},
+ Doi = {10.1103/PhysRevA.93.042511},
+ File = {/Users/loos/Zotero/storage/T3UVHSB9/Cohen and Mori-S{\'a}nchez - 2016 - Landscape of an exact energy functional.pdf},
+ Issn = {2469-9926, 2469-9934},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Title = {Landscape of an Exact Energy Functional},
+ Volume = {93},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.93.042511}}
+
+@article{Grossi_2017,
+ Abstract = {Exact pieces of information on the adiabatic connection integrand, W$\lambda$[$\rho$], which allows evaluation of the exchange-correlation energy of Kohn-Sham density functional theory, can be extracted from the leading terms in the strong coupling limit ($\lambda$ $\rightarrow$ $\infty$, where $\lambda$ is the strength of the electron-electron interaction). In this work, we first compare the theoretical prediction for the two leading terms in the strong coupling limit with data obtained via numerical implementation of the exact Levy functional in the simple case of two electrons confined in one dimension, confirming the asymptotic exactness of these two terms. We then carry out a first study on the incorporation of the Fermionic statistics at large coupling $\lambda$, both numerical and theoretical, confirming that spin effects enter at orders $\sim$e-$\surd\lambda$.},
+ Author = {Grossi, Juri and Kooi, Derk P. and Giesbertz, Klaas J. H. and Seidl, Michael and Cohen, Aron J. and {Mori-S\'anchez}, Paula and {Gori-Giorgi}, Paola},
+ Doi = {10.1021/acs.jctc.7b00998},
+ File = {/Users/loos/Zotero/storage/N7476DG2/Grossi et al. - 2017 - Fermionic Statistics in the Strongly Correlated Li.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = dec,
+ Number = {12},
+ Pages = {6089-6100},
+ Title = {Fermionic {{Statistics}} in the {{Strongly Correlated Limit}} of {{Density Functional Theory}}},
+ Volume = {13},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00998}}
+
+@article{Requist_2008,
+ Author = {Requist, Ryan and Pankratov, Oleg},
+ Doi = {10.1103/PhysRevB.77.235121},
+ File = {/Users/loos/Zotero/storage/V99YCARD/Requist and Pankratov - 2008 - Generalized Kohn-Sham system in one-matrix functio.pdf},
+ Issn = {1098-0121, 1550-235X},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = jun,
+ Number = {23},
+ Title = {Generalized {{Kohn}}-{{Sham}} System in One-Matrix Functional Theory},
+ Volume = {77},
+ Year = {2008},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.77.235121}}
+
+@article{Requist_2010,
+ Author = {Requist, Ryan and Pankratov, Oleg},
+ Doi = {10.1103/PhysRevA.81.042519},
+ File = {/Users/loos/Zotero/storage/9AXX6QF4/Requist and Pankratov - 2010 - Adiabatic approximation in time-dependent reduced-.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = apr,
+ Number = {4},
+ Title = {Adiabatic Approximation in Time-Dependent Reduced-Density-Matrix Functional Theory},
+ Volume = {81},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.042519}}
+
+@article{Schonhammer_1995,
+ Author = {Sch\"onhammer, K. and Gunnarsson, O. and Noack, R. M.},
+ Doi = {10.1103/PhysRevB.52.2504},
+ File = {/Users/loos/Zotero/storage/68DMD75P/Sch{\"o}nhammer et al. - 1995 - Density-functional theory on a lattice Comparison.pdf},
+ Issn = {0163-1829, 1095-3795},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = jul,
+ Number = {4},
+ Pages = {2504-2510},
+ Shorttitle = {Density-Functional Theory on a Lattice},
+ Title = {Density-Functional Theory on a Lattice: {{Comparison}} with Exact Numerical Results for a Model with Strongly Correlated Electrons},
+ Volume = {52},
+ Year = {1995},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.52.2504}}
+
+@article{Schonhammer_1988,
+ Author = {Schonhammer, K. and Gunnarsson, O.},
+ Doi = {10.1103/PhysRevB.37.3128},
+ File = {/Users/loos/Zotero/storage/MW5GAI97/Schonhammer and Gunnarsson - 1988 - Difference between the quasiparticle and the Kohn-.pdf},
+ Issn = {0163-1829},
+ Journal = {Phys. Rev. B},
+ Language = {en},
+ Month = feb,
+ Number = {6},
+ Pages = {3128-3129},
+ Title = {Difference between the Quasiparticle and the {{Kohn}}-{{Sham}}-{{Fermi}} Surfaces},
+ Volume = {37},
+ Year = {1988},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.37.3128}}
+
+@article{Giarrusso_2018,
+ Abstract = {Using the formalism of the conditional amplitude, we study the response part of the exchange-correlation potential in the strong-coupling limit of density functional theory, analyzing its peculiar features and comparing it with the response potential averaged over the coupling constant for small atoms and for the hydrogen molecule. We also use a simple one-dimensional model of a stretched heteronuclear molecule to derive exact properties of the response potential in the strong-coupling limit. The simplicity of the model allows us to unveil relevant features also of the exact Kohn-Sham potential and its different components, namely the appearance of a second peak in the correlation kinetic potential on the side of the most electronegative atom.},
+ Author = {Giarrusso, Sara and Vuckovic, Stefan and {Gori-Giorgi}, Paola},
+ Doi = {10.1021/acs.jctc.8b00386},
+ File = {/Users/loos/Zotero/storage/WIEX8B37/Giarrusso et al. - 2018 - Response Potential in the Strong-Interaction Limit.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {4151-4167},
+ Shorttitle = {Response {{Potential}} in the {{Strong}}-{{Interaction Limit}} of {{Density Functional Theory}}},
+ Title = {Response {{Potential}} in the {{Strong}}-{{Interaction Limit}} of {{Density Functional Theory}}: {{Analysis}} and {{Comparison}} with the {{Coupling}}-{{Constant Average}}},
+ Volume = {14},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00386}}
+
+@article{Vuckovic_2018,
+ Abstract = {Approximate exchange-correlation functionals built by modeling in a nonlinear way the adiabatic connection (AC) integrand of density functional theory have many attractive features, being virtually parameter-free and satisfying different exact properties, but they also have a fundamental flaw: they violate the size-consistency condition, crucial to evaluate interaction energies of molecular systems. We show that size consistency in the AC-based functionals can be restored in a very simple way at no extra computational cost. Results on a large set of benchmark molecular interaction energies show that functionals based on the interaction strength interpolation approximations are significantly more accurate than second-order perturbation theory.},
+ Author = {Vuckovic, Stefan and {Gori-Giorgi}, Paola and Della Sala, Fabio and Fabiano, Eduardo},
+ Doi = {10.1021/acs.jpclett.8b01054},
+ File = {/Users/loos/Zotero/storage/7FQTH2WL/Vuckovic et al. - 2018 - Restoring Size Consistency of Approximate Function.pdf},
+ Issn = {1948-7185},
+ Journal = {J. Phys. Chem. Lett.},
+ Language = {en},
+ Month = jun,
+ Number = {11},
+ Pages = {3137-3142},
+ Title = {Restoring {{Size Consistency}} of {{Approximate Functionals Constructed}} from the {{Adiabatic Connection}}},
+ Volume = {9},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.8b01054}}
+
+@article{Vuckovic_2017,
+ Abstract = {From a simplified version of the mathematical structure of the strong coupling limit of the exact exchange-correlation functional, we construct an approximation for the electronic repulsion energy at physical coupling strength, which is fully nonlocal. This functional is self-interaction free and yields energy densities within the definition of the electrostatic potential of the exchange-correlation hole that are locally accurate and have the correct asymptotic behavior. The model is able to capture strong correlation effects that arise from chemical bond dissociation, without relying on error cancellation. These features, which are usually missed by standard density functional theory (DFT) functionals, are captured by the highly nonlocal structure, which goes beyond the ``Jacob's ladder'' framework for functional construction, by using integrals of the density as the key ingredient. Possible routes for obtaining the full exchange-correlation functional by recovering the missing kinetic component of the correlation energy are also implemented and discussed.},
+ Author = {Vuckovic, Stefan and {Gori-Giorgi}, Paola},
+ Doi = {10.1021/acs.jpclett.7b01113},
+ File = {/Users/loos/Zotero/storage/YJUN4JS9/Vuckovic and Gori-Giorgi - 2017 - Simple Fully Nonlocal Density Functionals for Elec.pdf},
+ Issn = {1948-7185},
+ Journal = {J. Phys. Chem. Lett.},
+ Language = {en},
+ Month = jul,
+ Number = {13},
+ Pages = {2799-2805},
+ Title = {Simple {{Fully Nonlocal Density Functionals}} for {{Electronic Repulsion Energy}}},
+ Volume = {8},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.7b01113}}
+
+@article{Vuckovic_2017a,
+ Author = {Vuckovic, Stefan and Levy, Mel and {Gori-Giorgi}, Paola},
+ Doi = {10.1063/1.4997311},
+ File = {/Users/loos/Zotero/storage/4VRT9538/Vuckovic et al. - 2017 - Augmented potential, energy densities, and virial .pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = dec,
+ Number = {21},
+ Pages = {214107},
+ Title = {Augmented Potential, Energy Densities, and Virial Relations in the Weak- and Strong-Interaction Limits of {{DFT}}},
+ Volume = {147},
+ Year = {2017},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.4997311}}
+
+@article{Vuckovic_2016,
+ Abstract = {The construction of density-functional approximations is explored by modeling the adiabatic connection locally, using energy densities defined in terms of the electrostatic potential of the exchange-correlation hole. These local models are more amenable to the construction of size-consistent approximations than their global counterparts. In this work we use accurate input local ingredients to assess the accuracy of a range of local interpolation models against accurate exchange-correlation energy densities. The importance of the strictly correlated electrons (SCE) functional describing the strong coupling limit is emphasized, enabling the corresponding interpolated functionals to treat strong correlation effects. In addition to exploring the performance of such models numerically for the helium and beryllium isoelectronic series and the dissociation of the hydrogen molecule, an approximate analytic model is presented for the initial slope of the local adiabatic connection. Comparisons are made with approaches based on global models, and prospects for future approximations based on the local adiabatic connection are discussed.},
+ Author = {Vuckovic, Stefan and Irons, Tom J. P. and Savin, Andreas and Teale, Andrew M. and {Gori-Giorgi}, Paola},
+ Doi = {10.1021/acs.jctc.6b00177},
+ File = {/Users/loos/Zotero/storage/AB9FFH6Y/Vuckovic et al. - 2016 - Exchange--Correlation Functionals via Local Interpo.pdf},
+ Issn = {1549-9618, 1549-9626},
+ Journal = {J. Chem. Theory Comput.},
+ Language = {en},
+ Month = jun,
+ Number = {6},
+ Pages = {2598-2610},
+ Title = {Exchange\textendash{{Correlation Functionals}} via {{Local Interpolation}} along the {{Adiabatic Connection}}},
+ Volume = {12},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00177}}
+
+@article{Lani_2016,
+ Author = {Lani, Giovanna and Di Marino, Simone and Gerolin, Augusto and {van Leeuwen}, Robert and {Gori-Giorgi}, Paola},
+ Doi = {10.1039/C6CP00339G},
+ File = {/Users/loos/Zotero/storage/J8LH8GAY/Lani et al. - 2016 - The adiabatic strictly-correlated-electrons functi.pdf},
+ Issn = {1463-9076, 1463-9084},
+ Journal = {Phys. Chem. Chem. Phys.},
+ Language = {en},
+ Number = {31},
+ Pages = {21092-21101},
+ Shorttitle = {The Adiabatic Strictly-Correlated-Electrons Functional},
+ Title = {The Adiabatic Strictly-Correlated-Electrons Functional: Kernel and Exact Properties},
+ Volume = {18},
+ Year = {2016},
+ Bdsk-Url-1 = {https://doi.org/10.1039/C6CP00339G}}
+
+@article{Malet_2012,
+ Author = {Malet, Francesc and {Gori-Giorgi}, Paola},
+ Doi = {10.1103/PhysRevLett.109.246402},
+ File = {/Users/loos/Zotero/storage/8XH47Q89/Malet and Gori-Giorgi - 2012 - Strong Correlation in Kohn-Sham Density Functional.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = dec,
+ Number = {24},
+ Title = {Strong {{Correlation}} in {{Kohn}}-{{Sham Density Functional Theory}}},
+ Volume = {109},
+ Year = {2012},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.109.246402}}
+
+@article{Seidl_2010,
+ Author = {Seidl, Michael and {Gori-Giorgi}, Paola},
+ Doi = {10.1103/PhysRevA.81.012508},
+ File = {/Users/loos/Zotero/storage/D2GW9MHB/Seidl and Gori-Giorgi - 2010 - Adiabatic connection at negative coupling strength.pdf},
+ Issn = {1050-2947, 1094-1622},
+ Journal = {Phys. Rev. A},
+ Language = {en},
+ Month = jan,
+ Number = {1},
+ Title = {Adiabatic Connection at Negative Coupling Strengths},
+ Volume = {81},
+ Year = {2010},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.012508}}
+
+@article{Gori-Giorgi_2009,
+ Author = {{Gori-Giorgi}, Paola and Seidl, Michael and Vignale, G.},
+ Doi = {10.1103/PhysRevLett.103.166402},
+ File = {/Users/loos/Zotero/storage/N7UAPX2V/Gori-Giorgi et al. - 2009 - Density-Functional Theory for Strongly Interacting.pdf},
+ Issn = {0031-9007, 1079-7114},
+ Journal = {Phys. Rev. Lett.},
+ Language = {en},
+ Month = oct,
+ Number = {16},
+ Title = {Density-{{Functional Theory}} for {{Strongly Interacting Electrons}}},
+ Volume = {103},
+ Year = {2009},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.166402}}
+
+@article{Giarrusso_2018a,
+ Abstract = {The response part of the exchange-correlation potential of Kohn\textendash{}Sham density functional theory plays a very important role, for example for the calculation of accurate band gaps and excitation energies. Here we analyze this part of the potential in the limit of infinite interaction in density functional theory, showing that in the one-dimensional case it satisfies a very simple sum rule.},
+ Author = {Giarrusso, Sara and {Gori-Giorgi}, Paola and Giesbertz, Klaas J. H.},
+ Doi = {10.1140/epjb/e2018-90301-8},
+ File = {/Users/loos/Zotero/storage/6FTETEYK/Giarrusso et al. - 2018 - Sum-rules of the response potential in the strongl.pdf},
+ Issn = {1434-6028, 1434-6036},
+ Journal = {Eur. Phys. J. B},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Title = {Sum-Rules of the Response Potential in the Strongly-Interacting Limit of {{DFT}}},
+ Volume = {91},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1140/epjb/e2018-90301-8}}
+
+@article{Perarnau-Llobet_2015,
+ Author = {{Perarnau-Llobet}, Mart\'i and Hovhannisyan, Karen V. and Huber, Marcus and Skrzypczyk, Paul and Brunner, Nicolas and Ac\'in, Antonio},
+ Doi = {10.1103/PhysRevX.5.041011},
+ File = {/Users/loos/Zotero/storage/QBAJ3LUY/Perarnau-Llobet et al. - 2015 - Extractable Work from Correlations.pdf},
+ Issn = {2160-3308},
+ Journal = {Phys. Rev. X},
+ Language = {en},
+ Month = oct,
+ Number = {4},
+ Title = {Extractable {{Work}} from {{Correlations}}},
+ Volume = {5},
+ Year = {2015},
+ Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevX.5.041011}}
+
+@article{Theophilou_2018,
+ Author = {Theophilou, Andreas K.},
+ Doi = {10.1063/1.5038262},
+ File = {/Users/loos/Zotero/storage/3IRI7ZW8/Theophilou - 2018 - A novel density functional theory for atoms, molec.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {7},
+ Pages = {074104},
+ Title = {A Novel Density Functional Theory for Atoms, Molecules, and Solids},
+ Volume = {149},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.5038262}}
+
+@article{Shea_2018,
+ Author = {Shea, Jacqueline A. R. and Neuscamman, Eric},
+ Doi = {10.1063/1.5045056},
+ File = {/Users/loos/Zotero/storage/7H7W9SRJ/Shea and Neuscamman - 2018 - Communication A mean field platform for excited s.pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Pages = {081101},
+ Shorttitle = {Communication},
+ Title = {Communication: {{A}} Mean Field Platform for Excited State Quantum Chemistry},
+ Volume = {149},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.5045056}}
+
+@article{Bredas_2014,
+ Author = {Bredas, Jean-Luc},
+ Doi = {10.1039/C3MH00098B},
+ File = {/Users/loos/Zotero/storage/QZ92N6TD/Bredas - 2014 - Mind the gap!.pdf},
+ Issn = {2051-6347, 2051-6355},
+ Journal = {Mater Horiz},
+ Language = {en},
+ Number = {1},
+ Pages = {17-19},
+ Title = {Mind the Gap!},
+ Volume = {1},
+ Year = {2014},
+ Bdsk-Url-1 = {https://doi.org/10.1039/C3MH00098B}}
+
+@article{Vanzini_2018,
+ Abstract = {The spectral potential is the dynamical generalization of the Kohn\textendash{}Sham potential. It targets, in principle exactly, the spectral function in addition to the electronic density. Here we examine the spectral potential in one of the simplest solvable models exhibiting a non-trivial interplay between electron-electron interaction and inhomogeneity, namely the asymmetric Hubbard dimer. We discuss a general strategy to introduce approximations, which consists in calculating the spectral potential in the homogeneous limit (here represented by the symmetric Hubbard dimer) and importing it in the real inhomogeneous system through a suitable ``connector''. The comparison of different levels of approximation to the spectral potential with the exact solution of the asymmetric Hubbard dimer gives insights about the advantages and the difficulties of this connector strategy for applications in real materials.},
+ Author = {Vanzini, Marco and Reining, Lucia and Gatti, Matteo},
+ Doi = {10.1140/epjb/e2018-90277-3},
+ File = {/Users/loos/Zotero/storage/98HZJ25E/Vanzini et al. - 2018 - Spectroscopy of the Hubbard dimer the spectral po.pdf},
+ Issn = {1434-6028, 1434-6036},
+ Journal = {Eur. Phys. J. B},
+ Language = {en},
+ Month = aug,
+ Number = {8},
+ Shorttitle = {Spectroscopy of the {{Hubbard}} Dimer},
+ Title = {Spectroscopy of the {{Hubbard}} Dimer: The Spectral Potential},
+ Volume = {91},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1140/epjb/e2018-90277-3}}
+
+@article{Sagredo_2018,
+ Author = {Sagredo, Francisca and Burke, Kieron},
+ Date-Modified = {2018-10-24 22:49:56 +0200},
+ Doi = {10.1063/1.5043411},
+ File = {/Users/loos/Zotero/storage/DBPV69BW/Sagredo and Burke - 2018 - Accurate double excitations from ensemble density .pdf},
+ Issn = {0021-9606, 1089-7690},
+ Journal = {J. Chem. Phys.},
+ Language = {en},
+ Month = oct,
+ Number = {13},
+ Pages = {134103},
+ Title = {Accurate Double Excitations from Ensemble Density Functional Calculations},
+ Volume = {149},
+ Year = {2018},
+ Bdsk-Url-1 = {https://doi.org/10.1063/1.5043411}}
diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
new file mode 100644
index 0000000..406cefb
--- /dev/null
+++ b/Manuscript/eDFT.tex
@@ -0,0 +1,469 @@
+\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
+
+\usepackage{mathpazo,libertine}
+\newcommand{\alert}[1]{\textcolor{red}{#1}}
+\definecolor{darkgreen}{RGB}{0, 180, 0}
+
+\usepackage{hyperref}
+\hypersetup{
+ colorlinks=true,
+ linkcolor=blue,
+ filecolor=blue,
+ urlcolor=blue,
+ citecolor=blue
+}
+
+%useful stuff
+\newcommand{\cdash}{\multicolumn{1}{c}{---}}
+\newcommand{\mc}{\multicolumn}
+\newcommand{\mr}{\multirow}
+\newcommand{\fnm}{\footnotemark}
+\newcommand{\fnt}{\footnotetext}
+\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
+\newcommand{\la}{\lambda}
+\newcommand{\si}{\sigma}
+
+% functionals, potentials, densities, etc
+\newcommand{\eps}{\epsilon}
+\newcommand{\e}[2]{\eps_\text{#1}^{#2}}
+\renewcommand{\v}[2]{v_\text{#1}^{#2}}
+\newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}}
+\newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}}
+\newcommand{\n}[1]{n^{#1}}
+\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
+\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
+
+
+% energies
+\newcommand{\EHF}{E_\text{HF}}
+\newcommand{\Ec}{E_\text{c}}
+\newcommand{\Ecat}{E_\text{cat}}
+\newcommand{\Eneu}{E_\text{neu}}
+\newcommand{\Eani}{E_\text{ani}}
+\newcommand{\EPT}{E_\text{PT2}}
+\newcommand{\EFCI}{E_\text{FCI}}
+
+% matrices
+\newcommand{\br}{\bm{r}}
+\newcommand{\bw}{\bm{w}}
+\newcommand{\bG}{\bm{G}}
+\newcommand{\bS}{\bm{S}}
+\newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}}
+\newcommand{\bHc}{\bm{H}^\text{c}}
+\newcommand{\bF}[1]{\bm{F}^{#1}}
+\newcommand{\Ex}[1]{\Omega^{#1}}
+\newcommand{\E}[1]{E^{#1}}
+
+% elements
+\newcommand{\ew}[1]{w_{#1}}
+\newcommand{\eG}[1]{G_{#1}}
+\newcommand{\eS}[1]{S_{#1}}
+\newcommand{\eGamma}[2]{\Gamma_{#1}^{#2}}
+\newcommand{\eHc}[1]{H_{#1}^\text{c}}
+\newcommand{\eF}[2]{F_{#1}^{#2}}
+
+% Numbers
+\newcommand{\Nel}{N}
+\newcommand{\Nbas}{K}
+
+% Ao and MO basis
+\newcommand{\MO}[2]{\phi_{#1}^{#2}}
+\newcommand{\cMO}[2]{c_{#1}^{#2}}
+\newcommand{\AO}[1]{\chi_{#1}}
+
+
+% units
+\newcommand{\IneV}[1]{#1~eV}
+\newcommand{\InAU}[1]{#1~a.u.}
+\newcommand{\InAA}[1]{#1~\AA}
+
+\newcommand{\SI}{\textcolor{blue}{supplementary material}}
+
+\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
+\newcommand{\LCQ}{Laboratoire de Chimie Quantique, Institut de Chimie, CNRS, Universit\'e de Strasbourg, Strasbourg, France}
+
+%%% added by Manu %%%
+
+\newcommand{\manu}[1]{{\textcolor{blue}{ Manu: #1 }} }
+\newcommand{\beq}{\begin{eqnarray}}
+\newcommand{\eeq}{\nonumber\end{eqnarray}}
+%%%%
+
+\begin{document}
+
+\title{Weight-dependent local density-functional approximations for ensembles}
+
+\author{Pierre-Fran\c{c}ois Loos}
+\email{loos@irsamc.ups-tlse.fr}
+\affiliation{\LCPQ}
+\author{Emmanuel Fromager}
+\email{fromagere@unistra.fr}
+\affiliation{\LCQ}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{abstract}
+We report a first generation of local, weight-dependent correlation density-functional approximations (DFAs) that incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
+These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
+The resulting eDFAs, based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence.
+Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
+\end{abstract}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\maketitle
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Introduction.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials \cite{ParrBook}.
+The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}(\br)$, the latter being a much simpler quantity than the electronic wave function.
+The complexity of the many-body problem is then transferred to the xc functional.
+Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} suffers, in practice, from various deficiencies \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}.
+The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge.
+Another issue, which is partly connected to the previous one, is the description of electronically-excited states.
+
+The standard approach for modeling excited states in DFT is linear response time-dependent DFT (TDDFT) \cite{Casida}.
+In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong.
+Moreover, in exact TDDFT, the xc functional is time dependent.
+The simplest and most widespread approximation in state-of-the-art electronic structure programs where TDDFT is implemented consists in neglecting memory effects \cite{Casida}.
+In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time.
+As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT \cite{Maitra_2004,Cave_2004,Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}.
+
+When affordable (i.e.~for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
+The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e.,~with the same set of orbitals.
+Interestingly, a similar approach exists in DFT.
+Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK) \cite{Gross_1988, Gross_1988a, Oliveira_1988}, and is a generalization of Theophilou's variational principle for equi-ensembles \cite{Theophilou_1979}.
+In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
+This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies.
+It actually accounts for the infamous derivative discontinuity contribution to energy gaps.
+\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
+Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018a,Gould_2018b,Sagredo_2018,Ayers_2018,Deur_2018a,Deur_2018b,Kraisler_2013, Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Senjean_2018,Smith_2016}.
+The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature.
+Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018a,Deur_2018b,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed \cite{Yang_2014,Yang_2017}.
+In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open.
+A first step towards this goal is presented in this Letter with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
+\alert{Mention WIDFA?}
+
+In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}.
+Moreover, the present method relies on exact Hartree-Fock (HF) exchange, eschewing the so-called ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}.
+Atomic units are used throughout.\\
+
+\manu{I added some key equations in the following. Will polish the all
+thing later on.\\
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Generalized KS-eDFT}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+}
+
+In eDFT, the ensemble energy
+\begin{equation}
+ \E{\bw}=(1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
+\end{equation}
+is obtained variationally as follows,
+In analogy with ground-state generalized KS-DFT, we consider the
+following partitioning of
+
+the ensemble Levy-Lieb functional
+\begin{equation}
+ F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
+\end{equation}
+
+\begin{equation}
+F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
+\end{equation}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{eKS for excited states.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Here, we explain how to perform a self-consistent KS calculation for ensembles (eKS) in the context of excited states.
+In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
+i) the ground state ($I=0$), ii) the first singly-excited state ($I=1$), and iii) the first doubly-excited state ($I=2$).
+Generalization to a larger number of states is straightforward and is left for future work.
+By definition, the ensemble energy is
+\begin{equation}
+ \E{\bw} = (1 - \ew{1} - \ew{2}) \E{(0)} + \ew{1} \E{(1)} + \ew{2} \E{(2)}.
+\end{equation}
+$\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the to the single excitation and double excitation, respectively.
+\alert{To ensure the GOK variational principle \cite{Gross_1988a}, the weights must fulfil the following conditions: $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$. T2: I don't understand the asymmetry of the weights in this equation.}
+Note that, in order to extract individual energies from a single eKS calculation (see below), the weights must remain independent.
+By construction, the excitation energies are
+\begin{equation}
+\label{eq:Ex}
+ \Ex{(I)} = \pdv{\E{(I)}}{\ew{I}} = \E{(I)} - \E{(0)}.
+\end{equation}
+In the following, the orbitals $\MO{p}{\bw}(\br)$ are defined as linear combination of basis functions $\AO{\mu}(\br)$, such as
+\begin{equation}
+ \MO{p}{\bw}(\br) = \sum_{\mu=1}^{\Nbas} \cMO{\mu p}{\bw} \, \AO{\mu}(\br).
+\end{equation}
+
+Within the self-consistent eKS process, one is looking for the following weight-dependent density matrix:
+\begin{equation}
+\label{eq:Gamma}
+ \bGamma{\bw} = (1-\ew{1}-\ew{2}) \bGamma{(0)} - \ew{1} \bGamma{(1)} - \ew{2} \bGamma{(2)},
+\end{equation}
+where $\bw = (\ew{1},\ew{2})$ and $\bGamma{(I)}$ is the $I$th-state density matrix with elements
+\begin{equation}
+\label{eq:eGamma}
+ \eGamma{\mu\nu}{(I)} = \sum_{i=1}^{\Nel-I} \cMO{\mu i}{\bw} \cMO{\nu i}{\bw} + \sum_{a=\Nel+1}^{\Nel+I} \cMO{\mu a}{\bw} \cMO{\nu a}{\bw}.
+\end{equation}
+The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamma{\bw}$ in Eq.~\eqref{eq:Gamma} are obtained by diagonalizing the following Fock matrix
+\begin{multline}
+\label{eq:F}
+ \eF{\mu\nu}{\bw}
+ = \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si}
+ \\
+ + \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \AO{\mu}(\br) \AO{\nu}(\br) d\br,
+\end{multline}
+which itself depends on $\bGamma{\bw}$.
+In Eq.~\eqref{eq:F}, $\bHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$,
+\begin{equation}
+ (\mu\nu|\la\si) = \iint \frac{\AO\mu(\br_1) \AO\nu(\br_1) \AO\la(\br_2) \AO\si(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
+\end{equation}
+are two-electron repulsion integrals,
+$\v{c}{\bw}[\n{}(\br)] = \n{}(\br) \e{c}{\bw}[\n{}(\br)]$ and $\e{c}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study.
+The one-electron ensemble density is
+\begin{equation}
+ \n{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
+\end{equation}
+with a similar expression for $\n{(I)}(\br)$, while the ensemble energy reads
+\begin{multline}
+\label{eq:Ew}
+ \E{\bw}
+ = \Tr(\bGamma{\bw} \, \bHc)
+ + \frac{1}{2} \Tr(\bGamma{\bw} \, \bG \, \bGamma{\bw})
+ \\
+ + \int \e{c}{\bw}[\n{\bw}(\br)] \n{\bw}(\br) d\br.
+\end{multline}
+The self-consistent process described above is carried on until $\max \abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}} < \tau$, where $\tau$ is a user-defined threshold and $\eS{\mu\nu} = \braket{\AO{\mu}}{\AO{\nu}}$ are elements of the overlap matrix $\bS$.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Extracting individual energies.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Following Deur and Fromager \cite{Deur_2018b}, it is possible to extract individual energies, $\E{(I)}$, from the ensemble energy [Eq.~\eqref{eq:Ew}] as follows:
+\begin{multline}
+ \E{(I)} = \Tr(\bGamma{(I)} \, \bHc) + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)})
+ \\
+ + \int \left. \fdv{\v{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{(I)}(\br) d\br
+ + \LZ{c}{} + \DD{c}{(I)}.
+\end{multline}
+Note that a \emph{single} eKS calculation is required to extract the three individual energies.
+The correlation part of the (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by
+\begin{align}
+ \LZ{c}{} & = - \int \left. \fdv{\e{c}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
+ \\
+ \DD{c}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int \left. \pdv{\e{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br.
+\end{align}
+Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
+The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\e{c}{\bw}(\n{})$.
+\alert{Mention LIM?}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Uniform electron gases.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas model (also known as jellium) \cite{ParrBook, Loos_2016}.
+One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}.
+From this point of view, using finite uniform electron gases \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs is much more relevant \cite{Loos_2014a, Loos_2014b, Loos_2017a}.
+Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
+Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires \cite{Meyer_2009, Deshpande_2010}.
+
+As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e.~a circle) \cite{Loos_2012, Loos_2013a, Loos_2014b}.
+The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform.
+As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
+This is a necessary condition for being able to model derivative discontinuities.
+
+%%% TABLE 1 %%%
+\begin{table*}
+ \caption{
+ \label{tab:OG_func}
+ Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
+% \begin{ruledtabular}
+ \begin{tabular}{lcddd}
+ \hline\hline
+ State & $I$ & \tabc{$a^{(I)}$} & \tabc{$b^{(I)}$} & \tabc{$c^{(I)}$} \\
+ \hline
+ Ground state & $0$ & -0.0137078 & 0.0538982 & 0.0751740 \\
+ Singly-excited state & $1$ & -0.0238184 & 0.00413142 & 0.0568648 \\
+ Doubly-excited state & $2$ & -0.00935749 & -0.0261936 & 0.0336645 \\
+ \hline\hline
+ \end{tabular}
+% \end{ruledtabular}
+\end{table*}
+%%% %%% %%% %%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Density-functional approximations for ensembles.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
+As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
+All these states have the same (uniform) density $\n{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
+We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
+
+Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e.~per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
+\begin{equation}
+\label{eq:ec}
+ \e{c}{(I)}(\n{}) = \frac{a^{(I)}\,\n{}}{\n{} + b^{(I)} \sqrt{\n{}} + c^{(I)}},
+\end{equation}
+where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
+The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
+Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
+Combining these, one can build a three-state weight-dependent correlation eDFA:
+\begin{equation}
+\label{eq:ecw}
+ \e{c}{\bw}(\n{}) = (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}) + \ew{1} \e{c}{(1)}(\n{}) + \ew{2} \e{c}{(2)}(\n{}).
+\end{equation}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{LDA-centered functional.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
+\begin{equation}
+\label{eq:becw}
+ \be{c}{\bw}(\n{}) = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}) + \ew{1} \be{c}{(1)}(\n{}) + \ew{2} \be{c}{(2)}(\n{}),
+\end{equation}
+where
+\begin{equation}
+ \be{c}{(I)}(\n{}) = \e{c}{(I)}(\n{}) + \e{c}{\text{LDA}}(\n{}) - \e{c}{(0)}(\n{}).
+\end{equation}
+The local-density approximation (LDA) correlation functional,
+\begin{equation}
+ \e{c}{\text{LDA}}(\n{}) = a^\text{LDA} \, F\qty[1,\frac{3}{2},c^\text{LDA}, \frac{a^\text{LDA}(1-c^\text{LDA})}{b^\text{LDA}} {\n{}}^{-1}],
+\end{equation}
+specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function \cite{NISTbook}, and
+\begin{align}
+ a^\text{LDA} & = - \frac{\pi^2}{360},
+ &
+ b^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
+ &
+ c^\text{LDA} & = 2.408779.
+\end{align}
+Equation \eqref{eq:becw} can be recast
+\begin{equation}
+\label{eq:eLDA}
+\begin{split}
+ \be{c}{\bw}(\n{})
+ & = \e{c}{\text{LDA}}(\n{})
+ \\
+ & + \ew{1} \qty[\e{c}{(1)}(\n{})-\e{c}{(0)}(\n{})] + \ew{2} \qty[\e{c}{(2)}(\n{})-\e{c}{(0)}(\n{})],
+\end{split}
+\end{equation}
+which nicely highlights the centrality of the LDA in the present eDFA.
+In particular, $\be{c}{(0,0)}(\n{}) = \e{c}{\text{LDA}}(\n{})$.
+Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
+Finally, we note that, by construction,
+\begin{equation}
+ \left. \pdv{\be{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} = \be{c}{(J)}[\n{\bw}(\br)] - \be{c}{(0)}[\n{\bw}(\br)].
+\end{equation}
+\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors \cite{Gould_2018b}.}
+%%% FIG 1 %%%
+\begin{figure}
+ \includegraphics[width=\linewidth]{EvsL_5}
+ \caption{
+ \label{fig:EvsL}
+ Error with respect to FCI in single and double excitation energies for 5-boxium for various methods and box length $L$.
+ Graphs for additional values of $\Nel$ can be found as {\SI}.
+ \alert{T2: we could combine Figs. 1 and 2 into a single figure.}
+ }
+\end{figure}
+%%% %%% %%%
+
+%%% FIG 2 %%%
+\begin{figure}
+ \includegraphics[width=\linewidth]{EvsN_1}
+ \caption{
+ \label{fig:EvsN}
+ Error with respect to FCI in single and double excitation energies for $\Nel$-boxium for various methods and number of electrons $\Nel$ at $L=\pi$.
+ Graphs for additional values of $L$ can be found as {\SI}.
+ }
+\end{figure}
+%%% %%% %%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Computational details.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
+Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium.
+In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \Nel \le 7$.
+\alert{Comment on the quality of these density: density- and functional-driven errors?}
+
+These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
+For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime \cite{Rogers_2017,Rogers_2016}.
+We use as basis functions the (orthonormal) orbitals of the one-electron system, i.e.
+\begin{equation}
+ \AO{\mu}(x) =
+ \begin{cases}
+ \sqrt{2/L} \cos(\mu \pi x/L), & \mu \text{ is odd,}
+ \\
+ \sqrt{2/L} \sin(\mu \pi x/L), & \mu \text{ is even,}
+ \end{cases}
+\end{equation}
+with $ \mu = 1,\ldots,\Nbas$ and $\Nbas = 30$ for all calculations.
+For the self-consistent calculations (such as HF, KS or eKS), the convergence threshold has been set to $\tau = 10^{-7}$.
+For KS and eKS calculations, a Gauss-Legendre quadrature is employed to compute numerical integrals.
+
+In order to test the present eLDA functional we have performed various sets of calculations.
+To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
+For the single excitations, we have also performed time-dependent HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations \cite{Dreuw_2005}.
+For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also tested.
+
+Concerning the eKS calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Results and discussion.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e.~$\Nel = 5$).
+Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.
+In the weakly correlated regime (i.e.~small $L$), all methods provide accurate estimates of the excitation energies.
+When the box gets larger, they start to deviate.
+For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
+TDLDA yields larger errors at large $L$ by underestimating the excitation energies.
+TDA-TDLDA slightly corrects this trend thanks to error compensation.
+Concerning the eLDA functional, our results clearly evidences that the equi-weights [i.e. $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [i.e. $\bw = (0,0)$].
+This is especially true for the single excitation which is significantly improved by using state-averaged weights.
+The effect on the double excitation is less pronounced.
+Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.
+This conclusion is verified for smaller and larger number of electrons (see {\SI}).
+\alert{Shall I test the one-electron system for self-interaction?}
+
+Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies, for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
+The graphs associated with other $L$ values are reported as {\SI}.
+Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations.
+As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDHF or TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$, an error of the same order as CIS or TDA-TDLDA.
+Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only a few percent.
+
+\alert{Need further discussion on DD and LZ shift. Linearity of energy wrt weights?}
+
+\alert{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.
+However, when the box gets larger, there is a strong mixing between different degree of excitations.
+In particular, the single and double excitations strongly mix.
+This is clearly evidenced if one looks at the weights of the different configurations in the FCI wave function.
+In one hand, if one does construct a eDFA with a single state (either single or double), one clearly sees that the results quickly deteriorates when the box gets larger.
+On the other hand, building a functional which does mix singles and doubles corrects this by allowing configuration mixing.}
+
+\alert{It might be useful to add eHF results where one switch off the correlation part.
+For both zero weight and state-averaged weights?
+It would highlight the contribution of the derivative discontinuity.}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Concluding remarks.---}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In the present Letter, we have constructed a weight-dependent three-state DFA in the context of ensemble DFT.
+This eDFA delivers accurate excitation energies for both single and double excitations.
+Generalization to a larger number of states is straightforward and will be investigated in future work.
+Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.
+
+Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap.
+This can be done by constructing DFAs for the one- and three-electron ground state systems, and combining them with the two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and \eqref{eq:ecw}.
+However, as shown by Senjean and Fromager \cite{Senjean_2018}, one must modify the weights accordingly in order to maintain a constant density.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\textit{Acknowledgements.---}
+E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\bibliography{eDFT}
+
+\end{document}
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