From 529081f9ca41d8aa704ef044676c464978d5d62b Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Wed, 24 Jun 2020 00:14:18 +0200 Subject: [PATCH] Rewriting --- Data/RSDFT-CIPSI.org | 177 +++++-- Manuscript/h2o-dmc.pdf | Bin 0 -> 10750 bytes Manuscript/rsdft-cipsi-qmc.bib | 311 ++++++++++++ Manuscript/rsdft-cipsi-qmc.tex | 376 ++++++++++++++ {Manuscript => OLD}/All-cyclo.pdf | Bin {Manuscript => OLD}/Bib.bib | 0 {Manuscript => OLD}/DMC-RS-CI.pdf | Bin {Manuscript => OLD}/DMC-RSDFT.pdf | Bin {Manuscript => OLD}/Plot-Barrier.jpg | Bin {Manuscript => OLD}/Raw-data.pdf | Bin {Manuscript => OLD}/Reactant-Transition.jpg | Bin {Manuscript => OLD}/TODO.md | 0 {Manuscript => OLD}/biblio.bib | 0 {Manuscript => OLD}/frag_00278.jpg | Bin OLD/main.bbl | 519 ++++++++++++++++++++ {Manuscript => OLD}/main.tex | 0 OLD/mainNotes.bib | 2 + OLD/pass | 2 + 18 files changed, 1350 insertions(+), 37 deletions(-) create mode 100644 Manuscript/h2o-dmc.pdf create mode 100644 Manuscript/rsdft-cipsi-qmc.bib create mode 100644 Manuscript/rsdft-cipsi-qmc.tex rename {Manuscript => OLD}/All-cyclo.pdf (100%) rename {Manuscript => OLD}/Bib.bib (100%) rename {Manuscript => OLD}/DMC-RS-CI.pdf (100%) rename {Manuscript => OLD}/DMC-RSDFT.pdf (100%) rename {Manuscript => OLD}/Plot-Barrier.jpg (100%) rename {Manuscript => OLD}/Raw-data.pdf (100%) rename {Manuscript => OLD}/Reactant-Transition.jpg (100%) rename {Manuscript => OLD}/TODO.md (100%) rename {Manuscript => OLD}/biblio.bib (100%) rename {Manuscript => OLD}/frag_00278.jpg (100%) create mode 100644 OLD/main.bbl rename {Manuscript => OLD}/main.tex (100%) create mode 100644 OLD/mainNotes.bib create mode 100644 OLD/pass diff --git a/Data/RSDFT-CIPSI.org b/Data/RSDFT-CIPSI.org index 0c74bd1..a40a0fc 100644 --- a/Data/RSDFT-CIPSI.org +++ b/Data/RSDFT-CIPSI.org @@ -140,6 +140,7 @@ $$ [[file:h2o-dz-bfd.png]] *** QMC + #+TBLFM: $11=(2.*$6+$9-$3) :: $12=sqrt(2.*$7*$7+$10*$10+$4*$4) #+NAME:h2o-dz-bfd-qmc | | H_{2}O | | | | H | | | O | | | | | $\mu$ | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | E_atomization | | @@ -317,22 +318,22 @@ $$ *** QMC #+NAME:h2o-tz-bfd-qmc - | | H_{2}O | | | | H | | | O | | | | - | $\mu$ | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | E_atomization | | - |-------+--------+----------------+--------------+------+---------------+--------------+-------+----------------+--------------+--------------+--------------| - | 1.e-6 | 11 | -17.2567442368 | 0.0000674814 | 1 | -0.4999898051 | 0.0000065175 | 32 | -15.8930902787 | 0.0000578329 | 0.36367435 | 8.9349534e-5 | - | 0.20 | 23 | -17.2567293499 | 0.0000806448 | 1 | -0.5000210771 | 0.0000097491 | 18 | -15.8930200016 | 0.0000473829 | 0.36366719 | 9.4545295e-5 | - | 0.30 | 53 | -17.2537172349 | 0.0005108024 | 1 | -0.4999868887 | 0.0000149003 | 22 | -15.8928444962 | 0.0000484284 | 0.36089896 | 5.1352550e-4 | - | 0.50 | 1442 | -17.2577489156 | 0.0001590032 | 1 | -0.4999987389 | 0.0000138110 | 95 | -15.8933526777 | 0.0001305214 | 0.36439876 | 2.0663819e-4 | - | 0.75 | 3213 | -17.2583975528 | 0.0002828675 | 1 | -0.5000191125 | 0.0000109315 | 431 | -15.8962312871 | 0.0000796553 | 0.36212804 | 2.9427536e-4 | - | 1.00 | 6743 | -17.2609826285 | 0.0002158278 | 1 | -0.4999862089 | 0.0000119766 | 571 | -15.8985297947 | 0.0000845907 | 0.36248042 | 2.3243086e-4 | - | 1.75 | 54540 | -17.2635480054 | 0.0001864936 | 1 | -0.4999666515 | 0.0000131903 | 1064 | -15.9030314207 | 0.0000811325 | 0.36058328 | 2.0423103e-4 | - | 2.50 | 51691 | -17.2643468817 | 0.0003484087 | 1 | -0.4999801617 | 0.0000094324 | 1879 | -15.9044556541 | 0.0000797733 | 0.35993090 | 3.5767351e-4 | - | 3.80 | 103059 | -17.2636697373 | 0.0002950991 | 1 | -0.5000085861 | 0.0000085475 | 1846 | -15.9046408769 | 0.0000732723 | 0.35901169 | 3.0429990e-4 | - | 5.70 | 102599 | -17.2631675892 | 0.0003358884 | 1 | -0.5000055184 | 0.0000147803 | 1944 | -15.9042738928 | 0.0000731262 | 0.35888266 | 3.4439131e-4 | - | 8.50 | 101803 | -17.2632963114 | 0.0003799317 | 1 | -0.5000010336 | 0.0000123807 | 1943 | -15.9044520693 | 0.0000761691 | 0.35884217 | 3.8788709e-4 | - | inf | 200521 | -17.2638832283 | 0.0002702253 | 1 | -0.4999916320 | 0.0000135918 | 15699 | -15.9039263267 | 0.0000874602 | 0.35997364 | 2.8467609e-4 | - #+TBLFM: $11=(2.*$6+$9-$3) :: $12=sqrt(2.*$7*$7+$10*$10+$4*$4) + | | H_{2}O | | | | H | | | O | | | | + | $\mu$ | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | E_atomization | | + |-------+---------+----------------+--------------+------+---------------+--------------+-------+----------------+--------------+--------------+--------------| + | 1.e-6 | 23 | -17.2567442368 | 0.0000674814 | 1 | -0.4999898051 | 0.0000065175 | 32 | -15.8930902787 | 0.0000578329 | 0.36367435 | 8.9349534e-5 | + | 0.20 | 23 | -17.2567293499 | 0.0000806448 | 1 | -0.5000210771 | 0.0000097491 | 18 | -15.8930200016 | 0.0000473829 | 0.36366719 | 9.4545295e-5 | + | 0.30 | 219 | -17.2537172349 | 0.0005108024 | 1 | -0.4999868887 | 0.0000149003 | 22 | -15.8928444962 | 0.0000484284 | 0.36089896 | 5.1352550e-4 | + | 0.50 | 1699 | -17.2577489156 | 0.0001590032 | 1 | -0.4999987389 | 0.0000138110 | 95 | -15.8933526777 | 0.0001305214 | 0.36439876 | 2.0663819e-4 | + | 0.75 | 13362 | -17.2583975528 | 0.0002828675 | 1 | -0.5000191125 | 0.0000109315 | 431 | -15.8962312871 | 0.0000796553 | 0.36212804 | 2.9427536e-4 | + | 1.00 | 25673 | -17.2609826285 | 0.0002158278 | 1 | -0.4999862089 | 0.0000119766 | 571 | -15.8985297947 | 0.0000845907 | 0.36248042 | 2.3243086e-4 | + | 1.75 | 207475 | -17.2635480054 | 0.0001864936 | 1 | -0.4999666515 | 0.0000131903 | 1064 | -15.9030314207 | 0.0000811325 | 0.36058328 | 2.0423103e-4 | + | 2.50 | 858123 | -17.2643468817 | 0.0003484087 | 1 | -0.4999801617 | 0.0000094324 | 1879 | -15.9044556541 | 0.0000797733 | 0.35993090 | 3.5767351e-4 | + | 3.80 | 1621513 | -17.2636697373 | 0.0002950991 | 1 | -0.5000085861 | 0.0000085475 | 1846 | -15.9046408769 | 0.0000732723 | 0.35901169 | 3.0429990e-4 | + | 5.70 | 1629655 | -17.2631675892 | 0.0003358884 | 1 | -0.5000055184 | 0.0000147803 | 1944 | -15.9042738928 | 0.0000731262 | 0.35888266 | 3.4439131e-4 | + | 8.50 | 1643301 | -17.2632963114 | 0.0003799317 | 1 | -0.5000010336 | 0.0000123807 | 1943 | -15.9044520693 | 0.0000761691 | 0.35884217 | 3.8788709e-4 | + | inf | 1631982 | -17.2638832283 | 0.0002702253 | 1 | -0.4999916320 | 0.0000135918 | 15699 | -15.9039263267 | 0.0000874602 | 0.35997364 | 2.8467609e-4 | + #+TBLFM: |$11=(2.*$6+$9-$3) :: $12=sqrt(2.*$7*$7+$10*$10+$4*$4) #+BEGIN_SRC gnuplot :var data=h2o-tz-bfd-qmc :output file :file h2o-tz-bfd-qmc.png reset @@ -349,7 +350,7 @@ $$ [[file:h2o-tz-bfd-qmc.png]] *** ECMD - + #+NAME:h2o-tz-bfd-ecmd | | H_{2}O | | H | | O | | | | $\mu$ | N_{det} | E(CIPSI) | N_{det} | E(CIPSI) | N_{det} | E(CIPSI) | E_atomization | @@ -402,9 +403,6 @@ $$ } #+END_SRC - #+RESULTS: - [[file:./coefs-h2o.png]] - ** TODO cc-pVQZ-BFD #+begin_example HYDROGEN @@ -1986,7 +1984,7 @@ return result #+TBLFM: @>$7='(org-sbe "madformula" (data @5$7..@>>$7) ) :: @>$8='(org-sbe "maderrformula" (data @5$8..@>>$8) ) ***** MAD = 5.42 +/- 0.29 -*** TODO FCI +*** DONE FCI **** CIPSI ***** Table #+NAME: dz-cipsi-inf @@ -2065,28 +2063,85 @@ return result | | | | MAD | 23.62 | #+TBLFM: @>$5='(org-sbe "madformula" (data @5$5..@>>$5) ) ***** MAD = 23.62 -**** TODO QMC with Jastrow +**** QMC with Jastrow ***** Table #+NAME: dz-dmc-inf | | Total E | | Delta E | | Reference | Error | | | | Hartree | | Hartree | | Hartree | kcal/mol | | |---------+------------+----------+----------+----------+-----------+------------+----------| + | Be | -1.007625 | 0.000014 | 0.000000 | 0.000020 | | | | + | C | -5.431715 | 0.000065 | 0.000000 | 0.000092 | | | | + | Cl | -14.962307 | 0.000256 | 0.000000 | 0.000362 | | | | + | F | -24.193487 | 0.000257 | 0.000000 | 0.000363 | | | | + | H | -0.499996 | 0.000007 | 0.000000 | 0.000010 | | | | + | Li | -0.196329 | 0.000004 | 0.000000 | 0.000006 | | | | + | N | -9.798226 | 0.000105 | 0.000000 | 0.000149 | | | | + | Na | -0.182136 | 0.000008 | 0.000000 | 0.000011 | | | | + | O | -15.900621 | 0.000176 | 0.000000 | 0.000249 | | | | + | P | -6.472697 | 0.000089 | 0.000000 | 0.000126 | | | | + | S | -10.124922 | 0.000191 | 0.000000 | 0.000271 | | | | + | Si | -3.765137 | 0.000063 | 0.000000 | 0.000089 | | | | |---------+------------+----------+----------+----------+-----------+------------+----------| - | | | | | | | 5.38 | 0.19 | + | BeH | -1.579924 | 0.000058 | 0.072304 | 0.000060 | 0.079400 | -4.452718 | 0.037652 | + | C2H2 | -12.491177 | 0.001978 | 0.627755 | 0.001982 | 0.642400 | -9.189949 | 1.243661 | + | C2H4 | -13.747858 | 0.002165 | 0.884445 | 0.002169 | 0.899000 | -9.133250 | 1.360919 | + | C2H6 | -14.980890 | 0.002993 | 1.117486 | 0.002996 | 1.136900 | -12.182781 | 1.880250 | + | CH2_1A1 | -6.715937 | 0.000274 | 0.284230 | 0.000282 | 0.288900 | -2.930337 | 0.176733 | + | CH2_3B1 | -6.730926 | 0.000247 | 0.299220 | 0.000256 | 0.304100 | -3.062465 | 0.160633 | + | CH3 | -7.415930 | 0.000433 | 0.484228 | 0.000439 | 0.490800 | -4.124162 | 0.275260 | + | CH3Cl | -22.511728 | 0.003261 | 0.617719 | 0.003272 | 0.631000 | -8.333880 | 2.053169 | + | CH4 | -8.092352 | 0.001733 | 0.660654 | 0.001735 | 0.670300 | -6.052881 | 1.088669 | + | CH | -6.063309 | 0.000142 | 0.131598 | 0.000156 | 0.133900 | -1.444348 | 0.098110 | + | Cl2 | -30.006770 | 0.001417 | 0.082156 | 0.001507 | 0.094000 | -7.432073 | 0.945753 | + | ClF | -39.247103 | 0.001471 | 0.091309 | 0.001515 | 0.100100 | -5.516449 | 0.950520 | + | ClO | -30.948491 | 0.002849 | 0.085563 | 0.002866 | 0.104700 | -12.008590 | 1.798595 | + | CN | -15.511536 | 0.001716 | 0.281595 | 0.001721 | 0.288800 | -4.521091 | 1.079775 | + | CO | -21.740502 | 0.002046 | 0.408166 | 0.002055 | 0.413700 | -3.472682 | 1.289362 | + | CO2 | -37.838828 | 0.001777 | 0.605871 | 0.001812 | 0.621400 | -9.744893 | 1.137305 | + | CS | -15.809393 | 0.002280 | 0.252755 | 0.002289 | 0.274000 | -13.331407 | 1.436248 | + | F2 | -48.437580 | 0.002788 | 0.050606 | 0.002835 | 0.062200 | -7.275352 | 1.779096 | + | H2CO | -22.914591 | 0.002667 | 0.582263 | 0.002674 | 0.596700 | -9.059348 | 1.677865 | + | H2O | -17.257153 | 0.000554 | 0.356541 | 0.000582 | 0.371900 | -9.637832 | 0.365219 | + | H2O2 | -33.210861 | 0.001813 | 0.409628 | 0.001847 | 0.429400 | -12.406998 | 1.158935 | + | H2S | -11.410576 | 0.000694 | 0.285663 | 0.000720 | 0.292000 | -3.976713 | 0.451833 | + | H3COH | -24.133288 | 0.001467 | 0.800970 | 0.001480 | 0.818700 | -11.126046 | 0.928505 | + | H3CSH | -18.293307 | 0.003131 | 0.736687 | 0.003138 | 0.757000 | -12.746473 | 1.969176 | + | HCl | -15.630053 | 0.000503 | 0.167751 | 0.000564 | 0.171000 | -2.039062 | 0.354034 | + | HCN | -16.216313 | 0.002143 | 0.486376 | 0.002146 | 0.496900 | -6.603748 | 1.346757 | + | HCO | -22.262328 | 0.002545 | 0.429996 | 0.002552 | 0.444700 | -9.226696 | 1.601523 | + | HF | -24.908111 | 0.000426 | 0.214628 | 0.000498 | 0.226100 | -7.198764 | 0.312367 | + | HOCl | -31.608392 | 0.001374 | 0.245468 | 0.001409 | 0.264700 | -12.068155 | 0.883968 | + | Li2 | -0.431300 | 0.000009 | 0.038642 | 0.000012 | 0.038900 | -0.162060 | 0.007643 | + | LiF | -24.606552 | 0.000507 | 0.216736 | 0.000568 | 0.222000 | -3.303261 | 0.356414 | + | LiH | -0.788098 | 0.000017 | 0.091774 | 0.000019 | 0.092430 | -0.411737 | 0.011790 | + | N2 | -19.949099 | 0.001829 | 0.352648 | 0.001841 | 0.364600 | -7.499767 | 1.155305 | + | N2H4 | -22.254991 | 0.001486 | 0.658557 | 0.001501 | 0.699600 | -25.754667 | 0.941605 | + | Na2 | -0.390694 | 0.000021 | 0.026422 | 0.000027 | 0.026800 | -0.237078 | 0.016780 | + | NaCl | -15.300352 | 0.001292 | 0.155910 | 0.001317 | 0.157400 | -0.935284 | 0.826560 | + | NH | -10.425927 | 0.000232 | 0.127706 | 0.000255 | 0.133500 | -3.635724 | 0.159726 | + | NH2 | -11.079579 | 0.000445 | 0.281362 | 0.000457 | 0.290400 | -5.671417 | 0.286744 | + | NH3 | -11.758534 | 0.000858 | 0.460322 | 0.000865 | 0.475500 | -9.524365 | 0.542585 | + | NO | -25.925608 | 0.002478 | 0.226761 | 0.002487 | 0.244500 | -11.131121 | 1.560528 | + | O2 | -31.978613 | 0.002212 | 0.177370 | 0.002240 | 0.192400 | -9.431204 | 1.405783 | + | OH | -16.563942 | 0.000322 | 0.163325 | 0.000367 | 0.170200 | -4.313827 | 0.230286 | + | P2 | -13.109626 | 0.002003 | 0.164231 | 0.002011 | 0.186000 | -13.660387 | 1.262016 | + | PH2 | -7.712455 | 0.000403 | 0.239766 | 0.000413 | 0.244000 | -2.656750 | 0.259404 | + | PH3 | -8.350324 | 0.000878 | 0.377640 | 0.000883 | 0.389000 | -7.128757 | 0.553934 | + | S2 | -20.401043 | 0.002564 | 0.151199 | 0.002593 | 0.164000 | -8.032937 | 1.626850 | + | Si2 | -7.640979 | 0.000961 | 0.110705 | 0.000969 | 0.121000 | -6.460521 | 0.608056 | + | Si2H6 | -11.365871 | 0.002677 | 0.835623 | 0.002680 | 0.849000 | -8.394027 | 1.681992 | + | SiH2_1A1 | -5.006581 | 0.000257 | 0.241453 | 0.000265 | 0.243000 | -0.970756 | 0.166341 | + | SiH2_3B1 | -4.974331 | 0.000249 | 0.209203 | 0.000258 | 0.210000 | -0.499925 | 0.161639 | + | SiH3 | -5.624316 | 0.000625 | 0.359192 | 0.000629 | 0.363000 | -2.389674 | 0.394463 | + | SiH4 | -6.276132 | 0.001187 | 0.511013 | 0.001189 | 0.515000 | -2.502072 | 0.746197 | + | SiO | -19.954964 | 0.002217 | 0.289206 | 0.002225 | 0.306700 | -10.977712 | 1.396411 | + | SO2 | -42.290926 | 0.002157 | 0.364762 | 0.002194 | 0.414400 | -31.148522 | 1.377029 | + | SO | -26.212229 | 0.002411 | 0.186686 | 0.002425 | 0.200700 | -8.794218 | 1.521950 | + |---------+------------+----------+----------+----------+-----------+------------+----------| + | | | | | | | 7.38 | 1.08 | #+TBLFM: @>$7='(org-sbe "madformula" (data @5$7..@>>$7) ) :: @>$8='(org-sbe "maderrformula" (data @5$8..@>>$8) ) -***** MAD = 8.05 +/- 0.42 -***** Figure - #+BEGIN_SRC gnuplot :var data=data-g2-qmc-fci-jast :output file :file data-g2-qmc-dz-fci-jast.png -reset -set xtics rotate -plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 notitle, \ - data2 u 7:xtic(1) w l notitle, data2 u :7:8 w err notitle, 1 notitle, -1 notitle - #+END_SRC - - #+RESULTS: - [[file:data-g2-qmc-dz-1.0-jast.png]] - +***** MAD = 7.38 +/- 1.08 ** pVTZ, BFD *** DONE CCSD(T) **** Table @@ -4210,8 +4265,6 @@ for name, basis, method, mu in data: #+RESULTS: [[file:rsdft.csv]] -** RUNNING - - DZ FCI DMC ** TODO Temps CPU pour DMC @@ -4305,3 +4358,53 @@ for name, basis, method, mu in data: - [ ] DMC TZ FCI (tres cher...) - [ ] DMC TZ Hartree-Fock + + +* Work +** Figure 1 + #+NAME:tab1 + | $\mu$ | | N_{det} | E(DMC) | Error | N_{det} | E(DMC) | Error | + |------+------------+--------+----------------+--------------+---------+----------------+--------------| + | 0.00 | 0. | 11 | -17.2535927658 | 0.0000616883 | 23 | -17.2567442368 | 0.0000674814 | + | 0.20 | 0.16666667 | 23 | -17.2537299553 | 0.0000673868 | 23 | -17.2567293499 | 0.0000806448 | + | 0.50 | 0.33333333 | 1442 | -17.2539120512 | 0.0001475010 | 1699 | -17.2577489156 | 0.0001590032 | + | 0.75 | 0.42857143 | 3213 | -17.2551351593 | 0.0001757492 | 13362 | -17.2583975528 | 0.0002828675 | + | 1.00 | 0.5 | 6743 | -17.2566320269 | 0.0001881680 | 25673 | -17.2609826285 | 0.0002158278 | + | 1.75 | 0.63636364 | 54540 | -17.2595366309 | 0.0003193539 | 207475 | -17.2635480054 | 0.0001864936 | + | 2.50 | 0.71428571 | 51691 | -17.2593693798 | 0.0002564347 | 858123 | -17.2643468817 | 0.0003484087 | + | 3.80 | 0.79166667 | 103059 | -17.2586706893 | 0.0002222800 | 1621513 | -17.2636697373 | 0.0002950991 | + | 5.70 | 0.85074627 | 102599 | -17.2577349871 | 0.0002927319 | 1629655 | -17.2631675892 | 0.0003358884 | + | 8.50 | 0.89473684 | 101803 | -17.2572986405 | 0.0002283964 | 1643301 | -17.2632963114 | 0.0003799317 | + | inf | 1 | 200521 | -17.2567650693 | 0.0005779960 | 1631982 | -17.2638832283 | 0.0002702253 | + #+TBLFM: $2=$1/($1+1) + + #+BEGIN_SRC gnuplot :var data=tab1 :output file :file ../Manuscript/h2o-dmc.pdf +set termoption font "Times" +set termoption fontscale 0.6 +reset +set grid +set xrange [.0:1.00] +set key top right + +set xlabel "{/Symbol m}" +set ylabel "E_{DMC}" +set format y "%.3f" + +set ytics 0.001 +set xtics ("DFT" 0, "0.2" 0.2/1.2, "0.5" 0.5/1.5, "0.75" 0.75/1.75, "1.0" 0.5, \ + "1.75" 1.75/2.75, "2.5" 2.5/3.5, "3.8" 3.8/4.8, "5.7" 5.7/6.7, "8.5" 8.5/9.5, "FCI" 1) + +set style fill transparent solid 0.25 # partial transparency +set style fill noborder # no separate top/bottom lines + +plot data u 2:($4+$5):($4-$5) w filledcurves notitle lc "blue", \ + data u 2:4 w l lc "blue" title "VDZ-BFD",\ + data u 2:($7+$8):($7-$8) w filledcurves notitle lc "red", \ + data u 2:7 w l lc "red" title "VTZ-BFD" + + #+END_SRC + + #+RESULTS: + [[file:../Manuscript/h2o-dmc.pdf]] + + diff --git a/Manuscript/h2o-dmc.pdf b/Manuscript/h2o-dmc.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f25a0e7130220f3bfc88545c08a09935143d105b GIT binary patch literal 10750 zcmeHtbzGEN)UJX_NJ-aFQUWsr3?a=>BArSOFmy9?3JTJVG)R|}G>FozG>A0PAt^}S 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Levy}, + journal = {Proc. Natl. Acad. Sci. U.S.A.}, + pages = {6062}, + year = {1979}, + volume = {76} +} + +@article{Lie-IJQC-83, + author = {E. H. Lieb}, + title = {}, + journal = {Int. J. Quantum Chem.}, + pages = {24}, + year = {1983}, + volume = {{24}} +} + +@incollection{Sav-INC-96, + author = {A. Savin}, + title = {On Degeneracy, Near Degeneracy and Density Functional Theory}, + booktitle = {Recent Developments of Modern Density Functional Theory}, + publisher = {Elsevier}, + address = {Amsterdam}, + editor = {J. M. Seminario}, + pages = {327-357}, + year = {1996} +} + +@article{TouColSav-PRA-04, + author = {J. Toulouse and F. Colonna and A. Savin}, + title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory}, + publisher = {APS}, + year = {2004}, + journal = {Phys. Rev. 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Phys.}, + volume = {149}, + number = {3}, + pages = {034108}, + year = {2018}, + month = {Jul}, + issn = {0021-9606}, + publisher = {American Institute of Physics}, + doi = {10.1063/1.5041327} +} + +@article{Giner_2015, + author = {Giner, Emmanuel and Scemama, Anthony and + Caffarel, Michel}, + title = {{Fixed-node diffusion Monte Carlo potential energy curve of the fluorine molecule + F2 using selected configuration interaction trial wavefunctions}}, + journal = {J. Chem. Phys.}, + volume = {142}, + number = {4}, + pages = {044115}, + year = {2015}, + month = {Jan}, + issn = {0021-9606}, + publisher = {American Institute of Physics}, + doi = {10.1063/1.4905528} +} + +@article{Coldwell_1977, + author = {Coldwell, Robert Lynn}, + title = {{Zero Monte Carlo error or quantum mechanics is easier}}, + journal = {Int. J. Quantum Chem.}, + volume = {12}, + number = {S11}, + pages = {215--222}, + year = {1977}, + month = {Jan}, + issn = {0020-7608}, + publisher = {John Wiley {\&} Sons, Ltd}, + doi = {10.1002/qua.560120826} +} + +@article{Umrigar_2005, + author = {Umrigar, C. J. and Filippi, Claudia}, + title = {{Energy and Variance Optimization of Many-Body Wave Functions}}, + journal = {Phys. Rev. Lett.}, + volume = {94}, + number = {15}, + pages = {150201}, + year = {2005}, + month = {Apr}, + issn = {1079-7114}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.94.150201} +} + diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex new file mode 100644 index 0000000..6026758 --- /dev/null +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -0,0 +1,376 @@ +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace,subfigure} + +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} + +\usepackage{txfonts} + +\usepackage[ + colorlinks=true, + citecolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{HTML}{009900} +\usepackage[normalem]{ulem} + +\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France} +\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States} +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France} + + +\begin{document} + +\title{Enabling high accuracy diffusion Monte Carlo calculations with + range-separated density functional theory and selected configuration interaction} + +\author{Anthony Scemama} +\affiliation{\LCPQ} +\author{Emmanuel Giner} +\email{emmanuel.giner@lct.jussieu.fr} +\affiliation{\LCT} +\author{Anouar Benali} +\email{benali@anl.gov} +\affiliation{\ANL} +\author{Pierre-Fran\c{c}ois Loos} +\email{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} + + +\begin{abstract} +\end{abstract} + +\maketitle + + +\section{Introduction} +\label{sec:intro} + + +\section{Range-separated DFT in a nutshell} +\label{sec:rsdft} + +\subsection{Exact equations} +\label{sec:exact} + +The exact ground-state energy of a $N$-electron system with +nuclei-electron potential $v_\mathrm{ne}(\textbf{r})$ can be expressed +by the following minimization over $N$-representable densities +$n$~\cite{Lev-PNAS-79,Lie-IJQC-83} +\begin{equation} + E_0 = \min_n \left\{ \mathcal{F}[n] + \int v_\mathrm{ne}(\textbf{r}) n(\textbf{r}) \mathrm{d} \textbf{r} \right\}, + \label{E0} +\end{equation} +with the standard constrained-search universal density functional +\begin{equation} + \mathcal{F}[n] = \min_{\Psi\rightarrow n} \left \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}|\Psi \right \rangle, + \label{Fn} +\end{equation} +where $\hat{T}$ and $\hat{W}_\mathrm{ee}$ are the kinetic-energy and +Coulomb electron-electron interaction operators, respectively. The +minimizing multideterminant wave function in Eq.~\eqref{Fn} will be +denoted by $\Psi[n]$. In RS-DFT, the universal density functional is +decomposed as~\cite{Sav-INC-96,TouColSav-PRA-04} +\begin{equation} + \mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n], + \label{Fdecomp} +\end{equation} +where $\mathcal{F}^{\mathrm{lr},\mu}[n]$ is a long-range (lr) +universal density functional +\begin{equation} +\label{lr_univ_fonc} +\mathcal{F}^{\mathrm{lr},\mu}[n]= \min_{\Psi\rightarrow n} \left + \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu}|\Psi +\right \rangle, +\end{equation} +and $\bar{E}_{\mathrm{Hxc}}^{\,\mathrm{sr,}\mu}[n]$ is the +complementary short-range (sr) Hartree-exchange-correlation (Hxc) +density functional. In Eq.~(\ref{lr_univ_fonc}), +$\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ is the long-range +electron-electron interaction defined as +\begin{equation} + \hat{W}_\mathrm{ee}^{\mathrm{lr},\mu} = \frac{1}{2} \iint w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12}) \hat{n}_2(\textbf{r}_1,\textbf{r}_2) + \mathrm{d} \textbf{r}_1 \mathrm{d} \textbf{r}_2, +\end{equation} +with the error-function potential +$w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12})=\mathrm{erf}(\mu\, r_{12} +)/r_{12}$ (expressed with the interelectronic distance $r_{12} = +||\textbf{r}_1-\textbf{r}_2||$) and the pair-density operator +$\hat{n}_2(\textbf{r}_1, \textbf{r}_2)=\hat{n}(\textbf{r}_1) +\hat{n}(\textbf{r}_2) - \delta(\textbf{r}_1-\textbf{r}_2) +\hat{n}(\textbf{r}_1)$ where $\hat{n}(\textbf{r})$ is the density +operator. The range-separation parameter $\mu$ corresponds to an +inverse distance controlling the range of the separation and the +strength of the interaction at $r_{12} = 0$. For a given density, we +will denote by $\Psi^\mu[n]$ the minimizing multideterminant wave +function in Eq.~\eqref{lr_univ_fonc}. Inserting the decomposition of +Eq.~\eqref{Fdecomp} into Eq.~\eqref{E0}, and recomposing the two-step +minimization into a single one, leads to the following expression for +the exact ground-state electronic energy +\begin{equation} + \label{min_rsdft} E_0= \min_{\Psi} \left\{ +\left + \langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\right +\rangle ++ \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\right\}, +\end{equation} +where the minimization is done over normalized $N$-electron +multideterminant wave functions, $\hat{V}_{\mathrm{ne}} = \int +v_{\mathrm{ne}} (\textbf{r}) \hat{n}(\textbf{r}) \mathrm{d} +\textbf{r}$, and $n_\Psi$ refers to the density of $\Psi$, i.e. +$n_\Psi(\textbf{r})=\left \langle\Psi|\hat{n}(\textbf{r})|\Psi\right +\rangle$. +The minimizing multideterminant wave function $\Psi^\mu$ in +Eq.~\eqref{min_rsdft} can be determined by the self-consistent +eigenvalue equation +\begin{equation} + \label{rs-dft-eigen-equation} + \hat{H}^\mu[n_{\Psi^{\mu}}] \ket{\Psi^{\mu}}= \mathcal{E}^{\mu} \ket{\Psi^{\mu}}, +\end{equation} +with the long-range interacting Hamiltonian +\begin{equation} + \label{H_mu} + \hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}+ \hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}], +\end{equation} +where +$\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n]=\int \delta \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n]/\delta n(\textbf{r}) \, \hat{n}(\textbf{r}) \mathrm{d} \textbf{r}$ +is the complementary short-range Hartree-exchange-correlation +potential operator. Note that $\Psi^{\mu}$ is not the exact physical +ground-state wave function but only an effective wave +function. However, its density $n_{\Psi^{\mu}}$ is the exact physical +ground-state density. Once $\Psi^{\mu}$ has been calculated, the exact +electronic ground-state energy is obtained by +\begin{equation} + \label{E-rsdft} + E_0= \braket{\Psi^{\mu}|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi^{\mu}}+\bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_{\Psi^\mu}]. +\end{equation} +Note that, for $\mu=0$, the long-range interaction vanishes, +$w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus RS-DFT +reduces to standard KS-DFT. For $\mu\to\infty$, the long-range +interaction becomes the standard Coulomb interaction, +$w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and +thus RS-DFT reduces to standard wave-function theory (WFT). + +\subsection{Approximations} +\label{sec:approx} + +Provided that the exact short-range complementary functional is known +and that the wave function $\Psi^{\mu}$ is developed in a complete +basis set, the theory is exact and therefore provides the exact +density and ground state energy whatever value of $\mu$ is chosen to +split the universal Levy-Lieb functional (see Eqs. \eqref{Fdecomp} and +\eqref{lr_univ_fonc}). Of course in practice, two kind of +approximations must be made: i) the approximation of the wave function +by using a finite basis set, ii) the approximation on the exact +complementary functionals. As long as approximations are made, the +theory is not exact anymore and might depend on $\mu$. + +Despite its $\mu$ dependence, approximated RS-DFT schemes provide +potentially appealing features as: i) the approximated wave function +$\Psi^{\mu}$ is an eigenvector of a non-divergent operator +$\hat{H}^\mu[n_{\Psi^{\mu}}]$ (see Eqs. \eqref{H_mu} and +\eqref{rs-dft-eigen-equation}) and therefore converges more rapidly +with respect to the basis set~\cite{FraMusLupTou-JCP-15}, and also +produces more compact wave function~\cite{GinPraFerAssSavTou-JCP-18} +as it will be illustrated here, ii) the semi-local approximations for +RS-DFT complementary functionals are usually better suited to describe +the remaining short-range part of the electron-electron correlation. + +In this work, we use the short-range version of the +Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange and +correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} +(see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}) which +takes the form +\begin{eqnarray} + \bar{E}^{\mathrm{sr},\mu,\textsc{pbe}}_{\mathrm{x/c}}[n] = \int \bar{e}_\mathrm{{x/c}}^\mathrm{sr,\mu,\textsc{pbe}}(n(\textbf{r}),\nabla n(\textbf{r})) \, \mathrm{d}\textbf{r}. +\end{eqnarray} + + + + + +\section{Computational details} +\label{sec:comp-details} + +All the calculations were made using BFD +pseudopotentials\cite{Burkatzki_2008} with the associated double, triple +and quadruple zeta basis sets. +CCSD(T) and DFT calculations were made with +\emph{Gaussian09},\cite{g09} using an unrestricted Hartree-Fock +determinant as a reference for open-shell systems. All the CIPSI +calculations and range-separated CIPSI calculations were made with +\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020} +Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013} +in the determinant localization approximation.\cite{Zen_2019} + +In the determinant localization approximation, only the determinantal +component of the trial wave function is present in the expression of +the wave function on which the pseudopotential is localized. Hence, +the pseudopotential operator does not depend on the Jastrow factor, as +it is the case in all-electron calculations. This improves the +reproducibility of the results, as they depend only on parameters +optimized in a deterministic framework. + + +\section{Influence of the range-separation parameter on the fixed-node + error} +\label{sec:mu-dmc} +\begin{table}[h] + \caption{Fixed-node energies of the water molecule.} + \label{tab:h2o-dmc} + \centering + \begin{tabular}{crlrl} + \hline + & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ}\\ + $\mu$ & $N_{\text{det}}$ & E(DMC) & $N_{\text{det}}$ & E(DMC)\\ + \hline + $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$\\ + $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$\\ + $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$\\ + $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$\\ + $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$\\ + $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$\\ + $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$\\ + $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$\\ + $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$\\ + $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$\\ + $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$\\ + $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$\\ + \hline + \end{tabular} +\end{table} + +\begin{figure}[h] + \centering + \includegraphics[width=\columnwidth]{h2o-dmc.pdf} + \caption{Fixed-node energies of the water molecule with variable + values of $\mu$.} + \label{fig:h2o-dmc} +\end{figure} +The water molecule was taken at the equilibrium +geometry,\cite{Caffarel_2016} and RSDFT-CIPSI wave functions were +generated with BFD pseudopotentials and the corresponding double-zeta +basis set using multiple values of the range-separation parameter +$\mu$. The convergence criterion for stopping the CIPSI calculation +was set to 1~m$E_h$ on the PT2 correction. Then, these wave functions +were used as trial wave functions for FN-DMC calculations, and the +corresponding energies are shown in table~\ref{tab:h2o-dmc} and +figure~\ref{fig:h2o-dmc}. + + +\section{Atomization energy benchmarks} +\label{sec:atomization} + +Atomization energies are challenging for post-Hartree-Fock methods +because their calculation requires a perfect balance in the +description of atoms and molecules. Basis sets used in molecular +calculations are atom-centered, so they are always better adapted to +atoms than molecules and atomization energies usually tend to be +underestimated. +In the context of FN-DMC calculations, the nodal surface is imposed by +the trial wavefunction which is expanded on an atom-centered basis +set. So we expect the fixed-node error to be also related to the basis +set incompleteness error. +Increasing the size of the basis set improves the description of +the density and of electron correlation, but also reduces the +imbalance in the quality of the description of the atoms and the +molecule, leading to more accurate atomization energies. + +Another important feature required to get accurate atomization +energies is size-extensivity, since the numbers of correlated electrons +in the isolated atoms are different from the number of correlated +electrons in the molecule. +In the context of selected CI calculations, when the variational energy is +extrapolated to the FCI energy\cite{Holmes_2017} there is obviously no +size-consistence error. But when the selected wave function is used +for as a reference for post-Hartree-Fock methods or QMC calculations, +there is a residual size-consistence error originating from the +truncation of the determinant space. + +QMC calculations can be made size-consistent by extrapolating the +FN-DMC energy to estimate the energy obtained with the FCI as a trial +wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the +size-consistence error can be reduced by choosing the number of +selected determinants such that the sum of the PT2 corrections on the +atoms is equal to the PT2 correction of the molecule, enforcing that +the variational dissociation potential energy surface (PES) is +parallel to the perturbatively corrected PES, which is an accurate +estimate of the FCI PES.\cite{Giner_2015} + +Another source of size-consistence error in QMC calculation may originate +from the Jastrow factor. Usually, the Jastrow factor contains +one-electron, two-electron and one-nucleus-two-electron terms. +The problematic part is the two-electron term, whose simplest form can +be expressed as +\begin{equation} + J_\text{ee} = \sum_i \sum_{j +\bibitem [{\citenamefont {Levy}(1979)}]{Lev-PNAS-79}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Levy}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Proc. + Natl. Acad. Sci. U.S.A.}\ }\textbf {\bibinfo {volume} {76}},\ \bibinfo + {pages} {6062} (\bibinfo {year} {1979})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Lieb}(1983)}]{Lie-IJQC-83}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~H.}\ \bibnamefont + {Lieb}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Int. J. + Quantum Chem.}\ }\textbf {\bibinfo {volume} {{24}}},\ \bibinfo {pages} {24} + (\bibinfo {year} {1983})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Savin}(1996)}]{Sav-INC-96}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Savin}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Recent + Developments of Modern Density Functional Theory}}},\ \bibinfo {editor} + {edited by\ \bibinfo {editor} {\bibfnamefont {J.~M.}\ \bibnamefont + {Seminario}}}\ (\bibinfo {publisher} {Elsevier},\ \bibinfo {address} + {Amsterdam},\ \bibinfo {year} {1996})\ pp.\ \bibinfo {pages} + {327--357}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Toulouse}\ \emph {et~al.}(2004)\citenamefont + {Toulouse}, \citenamefont {Colonna},\ and\ \citenamefont + {Savin}}]{TouColSav-PRA-04}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Toulouse}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Colonna}}, \ + and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}},\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ + }\textbf {\bibinfo {volume} {70}},\ \bibinfo {pages} {062505} (\bibinfo + {year} {2004})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Franck}\ \emph {et~al.}(2015)\citenamefont {Franck}, + \citenamefont {Mussard}, \citenamefont {Luppi},\ and\ \citenamefont + {Toulouse}}]{FraMusLupTou-JCP-15}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont + {Franck}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Mussard}}, + \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Luppi}}, \ and\ \bibinfo + {author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href@noop {} + {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf + {\bibinfo {volume} {142}},\ \bibinfo {pages} {074107} (\bibinfo {year} + {2015})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Perdew}\ \emph + {et~al.}(1996{\natexlab{a}})\citenamefont {Perdew}, \citenamefont {Burke},\ + and\ \citenamefont {Ernzerhof}}]{PerBurErn-PRL-96}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont + {Perdew}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}}, \ + and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Ernzerhof}},\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ + }\textbf {\bibinfo {volume} {77}},\ \bibinfo {pages} {3865} (\bibinfo {year} + {1996}{\natexlab{a}})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Goll}\ \emph {et~al.}(2006)\citenamefont {Goll}, + \citenamefont {Werner}, \citenamefont {Stoll}, \citenamefont {Leininger}, + \citenamefont {Gori-Giorgi},\ and\ \citenamefont + {Savin}}]{GolWerStoLeiGorSav-CP-06}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Goll}}, \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Werner}}, + \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Stoll}}, \bibinfo + {author} {\bibfnamefont {T.}~\bibnamefont {Leininger}}, \bibinfo {author} + {\bibfnamefont {P.}~\bibnamefont {Gori-Giorgi}}, \ and\ \bibinfo {author} + {\bibfnamefont {A.}~\bibnamefont {Savin}},\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {Chem. Phys.}\ }\textbf {\bibinfo {volume} + {329}},\ \bibinfo {pages} {276} (\bibinfo {year} {2006})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Toulouse}\ \emph {et~al.}(2005)\citenamefont + {Toulouse}, \citenamefont {Colonna},\ and\ \citenamefont + {Savin}}]{TouColSav-JCP-05}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Toulouse}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Colonna}}, \ + and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}},\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ + }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages} {014110} (\bibinfo + {year} {2005})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Goll}\ \emph {et~al.}(2005)\citenamefont {Goll}, + \citenamefont {Werner},\ and\ \citenamefont {Stoll}}]{GolWerSto-PCCP-05}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Goll}}, \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Werner}}, \ + and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Stoll}},\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem. + Phys.}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {3917} (\bibinfo + {year} {2005})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hohenberg}\ and\ \citenamefont + {Kohn}(1964)}]{Hohenberg1964}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont + {Hohenberg}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont + {Kohn}},\ }\href {\doibase 10.1103/PhysRevB.7.1912} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {136}},\ + \bibinfo {pages} {B864} (\bibinfo {year} {1964})},\ \Eprint + {http://arxiv.org/abs/1108.5632} {arXiv:1108.5632} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Kohn}\ and\ \citenamefont + {Sham}(1965{\natexlab{a}})}]{kohn1965physrev}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont + {Kohn}}\ and\ \bibinfo {author} {\bibfnamefont {L.~J.}\ \bibnamefont + {Sham}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physical + Review}\ }\textbf {\bibinfo {volume} {140}},\ \bibinfo {pages} {A1133} + (\bibinfo {year} {1965}{\natexlab{a}})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Sun}\ \emph {et~al.}(2017)\citenamefont {Sun}, + \citenamefont {Berkelbach}, \citenamefont {Blunt}, \citenamefont {Booth}, + \citenamefont {Guo}, \citenamefont {Li}, \citenamefont {Liu}, \citenamefont + {McClain}, \citenamefont {Sayfutyarova}, \citenamefont {Sharma}, + \citenamefont {Wouters},\ and\ \citenamefont {Chan}}]{PYSCF}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont + {Sun}}, \bibinfo {author} {\bibfnamefont {T.~C.}\ \bibnamefont {Berkelbach}}, + \bibinfo {author} {\bibfnamefont {N.~S.}\ \bibnamefont {Blunt}}, \bibinfo + {author} {\bibfnamefont {G.~H.}\ \bibnamefont {Booth}}, \bibinfo {author} + {\bibfnamefont {S.}~\bibnamefont {Guo}}, \bibinfo {author} {\bibfnamefont + {Z.}~\bibnamefont {Li}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Liu}}, \bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont {McClain}}, + \bibinfo {author} {\bibfnamefont {E.~R.}\ \bibnamefont {Sayfutyarova}}, + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Sharma}}, \bibinfo + {author} {\bibfnamefont {S.}~\bibnamefont {Wouters}}, \ and\ \bibinfo + {author} {\bibfnamefont {G.~K.}\ \bibnamefont {Chan}},\ }\href {\doibase + 10.1002/wcms.1340} {\enquote {\bibinfo {title} {Pyscf: the python‐based + simulations of chemistry framework},}\ } (\bibinfo {year} {2017}),\ \Eprint + {http://arxiv.org/abs/https://onlinelibrary.wiley.com/doi/pdf/10.1002/wcms.1340} + {https://onlinelibrary.wiley.com/doi/pdf/10.1002/wcms.1340} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Kohn}\ and\ \citenamefont + {Sham}(1965{\natexlab{b}})}]{Kohn1965}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont + {Kohn}}\ and\ \bibinfo {author} {\bibfnamefont {L.~J.}\ \bibnamefont + {Sham}},\ }\href {\doibase 10.1103/PhysRev.140.A1133} {\bibfield {journal} + {\bibinfo {journal} {Physical Review}\ }\textbf {\bibinfo {volume} {140}},\ + \bibinfo {pages} {A1133} (\bibinfo {year} {1965}{\natexlab{b}})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Perdew}\ \emph + {et~al.}(1996{\natexlab{b}})\citenamefont {Perdew}, \citenamefont {Burke},\ + and\ \citenamefont {Ernzerhof}}]{PBE}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont + {Perdew}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}}, \ + and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Ernzerhof}},\ }\href + {\doibase 10.1103/PhysRevLett.77.3865} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {77}},\ \bibinfo + {pages} {3865} (\bibinfo {year} {1996}{\natexlab{b}})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Becke}(1993)}]{B3LYP1}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~D.}\ \bibnamefont + {Becke}},\ }\href {\doibase 10.1063/1.464913} {\bibfield {journal} {\bibinfo + {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo {volume} + {98}},\ \bibinfo {pages} {5648} (\bibinfo {year} {1993})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1063/1.464913} + {https://doi.org/10.1063/1.464913} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Lee}\ \emph {et~al.}(1988)\citenamefont {Lee}, + \citenamefont {Yang},\ and\ \citenamefont {Parr}}]{B3LYP2}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Lee}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yang}}, \ and\ + \bibinfo {author} {\bibfnamefont {R.~G.}\ \bibnamefont {Parr}},\ }\href + {\doibase 10.1103/PhysRevB.37.785} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {37}},\ \bibinfo {pages} {785} + (\bibinfo {year} {1988})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Vosko}\ \emph {et~al.}(1980)\citenamefont {Vosko}, + \citenamefont {Wilk},\ and\ \citenamefont {Nusair}}]{B3LYP3}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~H.}\ \bibnamefont + {Vosko}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Wilk}}, \ and\ + \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Nusair}},\ }\href + {\doibase 10.1139/p80-159} {\bibfield {journal} {\bibinfo {journal} + {Canadian Journal of Physics}\ }\textbf {\bibinfo {volume} {58}},\ \bibinfo + {pages} {1200} (\bibinfo {year} {1980})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1139/p80-159} + {https://doi.org/10.1139/p80-159} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Stephens}\ \emph {et~al.}(1994)\citenamefont + {Stephens}, \citenamefont {Devlin}, \citenamefont {Chabalowski},\ and\ + \citenamefont {Frisch}}]{B3LYP4}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~J.}\ \bibnamefont + {Stephens}}, \bibinfo {author} {\bibfnamefont {F.~J.}\ \bibnamefont + {Devlin}}, \bibinfo {author} {\bibfnamefont {C.~F.}\ \bibnamefont + {Chabalowski}}, \ and\ \bibinfo {author} {\bibfnamefont {M.~J.}\ \bibnamefont + {Frisch}},\ }\href {\doibase 10.1021/j100096a001} {\bibfield {journal} + {\bibinfo {journal} {The Journal of Physical Chemistry}\ }\textbf {\bibinfo + {volume} {98}},\ \bibinfo {pages} {11623} (\bibinfo {year} {1994})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1021/j100096a001} + {https://doi.org/10.1021/j100096a001} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Adamo}\ and\ \citenamefont + {Barone}(1999)}]{adamo1999jchemphys}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Adamo}}\ and\ \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Barone}},\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {The Journal of + Chemical Physics}\ }\textbf {\bibinfo {volume} {110}},\ \bibinfo {pages} + {6158} (\bibinfo {year} {1999})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Zhang}\ and\ \citenamefont + {Yang}(1998)}]{PhysRevLett.80.890}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont + {Zhang}}\ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Yang}},\ + }\href {\doibase 10.1103/PhysRevLett.80.890} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {80}},\ \bibinfo + {pages} {890} (\bibinfo {year} {1998})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Zhao}\ and\ \citenamefont {Truhlar}(2006)}]{M06-L}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont + {Zhao}}\ and\ \bibinfo {author} {\bibfnamefont {D.~G.}\ \bibnamefont + {Truhlar}},\ }\href {\doibase 10.1063/1.2370993} {\bibfield {journal} + {\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo + {volume} {125}},\ \bibinfo {pages} {194101} (\bibinfo {year} {2006})},\ + \Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.2370993} + {https://doi.org/10.1063/1.2370993} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bender}\ and\ \citenamefont + {Davidson}(1969)}]{Bender69}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~F.}\ \bibnamefont + {Bender}}\ and\ \bibinfo {author} {\bibfnamefont {E.~R.}\ \bibnamefont + {Davidson}},\ }\href {\doibase 10.1103/PhysRev.183.23} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {183}},\ + \bibinfo {pages} {23} (\bibinfo {year} {1969})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Whitten}\ and\ \citenamefont + {Hackmeyer}(1969)}]{Whitten69}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~L.}\ \bibnamefont + {Whitten}}\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Hackmeyer}},\ }\href {\doibase 10.1063/1.1671985} {\bibfield {journal} + {\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo + {volume} {51}},\ \bibinfo {pages} {5584} (\bibinfo {year} {1969})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1063/1.1671985} + {https://doi.org/10.1063/1.1671985} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Huron}\ \emph {et~al.}(1973)\citenamefont {Huron}, + \citenamefont {Malrieu},\ and\ \citenamefont {Rancurel}}]{Huron73}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont + {Huron}}, \bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont {Malrieu}}, + \ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Rancurel}},\ + }\href {\doibase 10.1063/1.1679199} {\bibfield {journal} {\bibinfo + {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo {volume} + {58}},\ \bibinfo {pages} {5745} (\bibinfo {year} {1973})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1063/1.1679199} + {https://doi.org/10.1063/1.1679199} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Buenker}\ \emph {et~al.}(1978)\citenamefont + {Buenker}, \citenamefont {Peyerimhoff},\ and\ \citenamefont + {Butscher}}]{Buenker78}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~J.}\ \bibnamefont + {Buenker}}, \bibinfo {author} {\bibfnamefont {S.~D.}\ \bibnamefont + {Peyerimhoff}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont + {Butscher}},\ }\href {\doibase 10.1080/00268977800100581} {\bibfield + {journal} {\bibinfo {journal} {Molecular Physics}\ }\textbf {\bibinfo + {volume} {35}},\ \bibinfo {pages} {771} (\bibinfo {year} {1978})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1080/00268977800100581} + {https://doi.org/10.1080/00268977800100581} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Evangelisti}\ \emph {et~al.}(1983)\citenamefont + {Evangelisti}, \citenamefont {Daudey},\ and\ \citenamefont + {Malrieu}}]{EVANGELISTI1983}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {Evangelisti}}, \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont + {Daudey}}, \ and\ \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont + {Malrieu}},\ }\href {\doibase https://doi.org/10.1016/0301-0104(83)85011-3} + {\bibfield {journal} {\bibinfo {journal} {Chemical Physics}\ }\textbf + {\bibinfo {volume} {75}},\ \bibinfo {pages} {91 } (\bibinfo {year} + {1983})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Cimiraglia}(1985)}]{Cimiraglia1985}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Cimiraglia}},\ }\href {\doibase 10.1063/1.449362} {\bibfield {journal} + {\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo + {volume} {83}},\ \bibinfo {pages} {1746} (\bibinfo {year} {1985})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1063/1.449362} + {https://doi.org/10.1063/1.449362} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Chirlian}\ and\ \citenamefont + {Francl}(1987)}]{Chirlian1987}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {L.~E.}\ \bibnamefont + {Chirlian}}\ and\ \bibinfo {author} {\bibfnamefont {M.~M.}\ \bibnamefont + {Francl}},\ }\href {\doibase 10.1002/jcc.540080616} {\bibfield {journal} + {\bibinfo {journal} {Journal of Computational Chemistry}\ }\textbf {\bibinfo + {volume} {8}},\ \bibinfo {pages} {894} (\bibinfo {year} {1987})},\ \Eprint + {http://arxiv.org/abs/https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.540080616} + {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.540080616} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Giner}\ \emph {et~al.}(2013)\citenamefont {Giner}, + \citenamefont {Scemama},\ and\ \citenamefont {Caffarel}}]{Giner2013}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Giner}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \ + and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Caffarel}},\ }\href + {\doibase 10.1139/cjc-2013-0017} {\bibfield {journal} {\bibinfo {journal} + {Canadian Journal of Chemistry}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo + {pages} {879} (\bibinfo {year} {2013})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1139/cjc-2013-0017} + {https://doi.org/10.1139/cjc-2013-0017} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Caffarel}\ \emph {et~al.}(2014)\citenamefont + {Caffarel}, \citenamefont {Giner}, \citenamefont {Scemama},\ and\ + \citenamefont {Ramírez-Solís}}]{Caffarel2014}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Caffarel}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Giner}}, + \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Scemama}}, \ and\ + \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Ramírez-Solís}},\ + }\href {\doibase 10.1021/ct5004252} {\bibfield {journal} {\bibinfo + {journal} {Journal of Chemical Theory and Computation}\ }\textbf {\bibinfo + {volume} {10}},\ \bibinfo {pages} {5286} (\bibinfo {year} {2014})},\ \bibinfo + {note} {pMID: 26583212},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1021/ct5004252} + {https://doi.org/10.1021/ct5004252} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Caffarel}\ \emph + {et~al.}(2016{\natexlab{a}})\citenamefont {Caffarel}, \citenamefont + {Applencourt}, \citenamefont {Giner},\ and\ \citenamefont + {Scemama}}]{Caffarel16}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont 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