Push for manu
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@ -5488,7 +5488,7 @@ p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureUPYlpu.png]]
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[[file:/tmp/babel-eZHQur/figurebpHOZx.png]]
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#+begin_src R :results output :session *R* :exports both
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vdz
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@ -5507,14 +5507,14 @@ dev.off()
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1157 <NA> 0.6666667
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1224 <NA> 0.8333333
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1291 <NA> 1.0000000
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E E.lo E.hi Basis
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889 -48.41951 -48.41995 -48.41907 DZ
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956 -48.42197 -48.42239 -48.42155 DZ
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1023 -48.42486 -48.42568 -48.42403 DZ
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1090 -48.43245 -48.43273 -48.43216 DZ
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1157 -48.44106 -48.44179 -48.44033 DZ
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1224 -48.44532 -48.44737 -48.44327 DZ
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1291 -48.43758 -48.44037 -48.43479 DZ
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E E.lo E.hi Ndet Basis
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889 -48.41951 -48.41995 -48.41907 23 DZ
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956 -48.42197 -48.42239 -48.42155 8 DZ
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1023 -48.42486 -48.42568 -48.42403 1743 DZ
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1090 -48.43245 -48.43273 -48.43216 11952 DZ
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1157 -48.44106 -48.44179 -48.44033 829438 DZ
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1224 -48.44532 -48.44737 -48.44327 5326459 DZ
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1291 -48.43758 -48.44037 -48.43479 8302442 DZ
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Warning message:
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In format.data.frame(x, digits = digits, na.encode = FALSE) :
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corrupt data frame: columns will be truncated or padded with NAs
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@ -5717,6 +5717,72 @@ F2opt
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1.0000000
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#+end_example
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**** C2
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Parameters of the Jastrow:
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a r_{12} / (1 + b r_{12})
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p r_{12}^2 / (1 + p r_{12}^2)
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a = 1/2 b = 1.3820
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p_C = 0.4855 (energy minimization)
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#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
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p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
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fun.0 <- function(x) ((0.4855 * x) / (1 + 0.4855* x))**2
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fun.2 <- function(x) (0.5 * x) / (1 + 1.3820 * x)
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p <- p + stat_function(fun = fun.0) + xlim(0,5)
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p <- p + stat_function(fun = fun.2) + xlim(0,5)
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p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureS0c2r2.png]]
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#+begin_src R :results output :session *R* :exports both
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normalize <- function(x) {x / sqrt(sum(x^2))}
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FCI <- normalize(scan("c2_fci.dat"))
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dC2 = data.frame(FCI)
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dC2$mu_0 <- normalize(scan("c2_0.00.dat"))
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dC2$mu_025 <- normalize(scan("c2_0.25.dat"))
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dC2$mu_05 <- normalize(scan("c2_0.50.dat"))
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dC2$mu_75 <- normalize(scan("c2_0.75.dat"))
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dC2$mu_1 <- normalize(scan("c2_1.00.dat"))
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dC2$mu_2 <- normalize(scan("c2_2.00.dat"))
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dC2$mu_5 <- normalize(scan("c2_5.00.dat"))
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dC2$mu_inf<- normalize(scan("c2_inf.dat"))
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dC2$jast <- normalize(scan("c2_jast.dat"))
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overlap <- function(x) {x <- abs(crossprod(x,dC2$jast)[1]) }
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C2 <- apply(dC2,2,overlap)
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C2
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#+end_src
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#+RESULTS:
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#+begin_example
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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Read 200 items
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FCI mu_0 mu_025 mu_05 mu_75 mu_1 mu_2 mu_5
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0.9975648 0.8772789 0.9265083 0.9661503 0.9907511 0.9953970 0.9991335 0.9991130
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mu_inf jast
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0.9989082 1.0000000
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#+end_example
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**** Plot
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#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
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@ -5725,21 +5791,25 @@ X <- Mu/(Mu+1.)
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X[7] <- 1.
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H2O_FCI <- H2O["FCI"]
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F2_FCI <- F2["FCI"]
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C2_FCI <- C2["FCI"]
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H2O <- subset(H2O, names(H2O) != "jast" & names(H2O) != "FCI")
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F2 <- subset(F2 , names(F2 ) != "jast" & names(F2 ) != "FCI")
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C2 <- subset(C2 , names(C2 ) != "jast" & names(C2 ) != "FCI")
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F2opt <- subset(F2opt , names(F2opt) != "jast" & names(F2opt) != "FCI")
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ov = data.frame(Mu, X, H2O, F2, F2opt)
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ov = data.frame(Mu, X, H2O, F2, F2opt, C2)
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print(ov)
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Mu <- as.character(Mu)
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Mu[7] <- TeX("$\\infty$")
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H2O.spline_int <- as.data.frame(spline(ov$"X", ov$"H2O", n=101))
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F2.spline_int <- as.data.frame(spline(ov$"X", ov$"F2", n=101))
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C2.spline_int <- as.data.frame(spline(ov$"X", ov$"C2", n=101))
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F2opt.spline_int <- as.data.frame(spline(ov$"X", ov$"F2opt", n=101))
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p <- ggplot()
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p <- p + geom_point(aes(x=X,y=F2), color='red')
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p <- p + geom_point(aes(x=X,y=H2O), color='blue')
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p <- p + geom_point(aes(x=X,y=C2), color='green')
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#p <- p + geom_point(aes(x=X,y=F2opt), color='green')
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p <- p + geom_line(data = F2.spline_int, aes(x=x, y=y, colour='a'))
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p <- p + geom_line(data = H2O.spline_int, aes(x=x, y=y, colour='b'))
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@ -0,0 +1,200 @@
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||||
0.109846285060951E-01
|
||||
|
|
@ -0,0 +1,200 @@
|
|||
0.881982886736393
|
||||
0.0222983777287801
|
||||
-0.00700771381475242
|
||||
0.00463102267483265
|
||||
0.00946274313493148
|
||||
0.0186557730091422
|
||||
-0.00030778249637751
|
||||
0.00448972734710415
|
||||
-0.0124554019583896
|
||||
-0.00965794303209597
|
||||
-0.00396105042294517
|
||||
0.01283621718614
|
||||
-0.000170513410259279
|
||||
0.00911809534255359
|
||||
-0.00525322726637156
|
||||
-0.00034573639634314
|
||||
0.0078440493240993
|
||||
-0.337395510144737
|
||||
-0.0711043726559605
|
||||
0.0764944818198298
|
||||
-0.00993459305248185
|
||||
-0.0113924314571598
|
||||
-0.00430871944473571
|
||||
-0.00149301402263706
|
||||
-0.0269889377526652
|
||||
-0.0715197331959022
|
||||
-0.00463117146825351
|
||||
-0.00620355031944937
|
||||
0.00433122905704303
|
||||
-0.0495459958509328
|
||||
0.0818762208985481
|
||||
0.00689191152348026
|
||||
-0.0130071640555659
|
||||
0.0018095785271374
|
||||
-0.0236341672057347
|
||||
-0.0101455178979249
|
||||
0.0157421047982156
|
||||
0.00952271668966099
|
||||
-0.015841375558739
|
||||
-0.0148140597220675
|
||||
-0.0128743056629707
|
||||
0.00405846437030334
|
||||
0.00977276921176085
|
||||
-0.0985914023739978
|
||||
-0.0413548646156484
|
||||
-0.00528466869058297
|
||||
-0.00648883134061262
|
||||
0.0133522555116448
|
||||
-0.0756108952267605
|
||||
-0.102031433446374
|
||||
0.0466134283660787
|
||||
-0.0227691641648556
|
||||
-0.0137711480943095
|
||||
-0.0193389656582796
|
||||
0.0337903394274469
|
||||
-0.0116978054744949
|
||||
-0.0192533277597503
|
||||
0.00391062265769892
|
||||
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|
||||
0.0547042794826488
|
||||
-0.0262905014365859
|
||||
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|
||||
-0.00756333707818562
|
||||
0.00762076397072185
|
||||
-0.0163713707804138
|
||||
-0.0156141455899909
|
||||
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|
||||
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|
||||
0.0707807188400203
|
||||
-0.0495820650585793
|
||||
0.0131139797316869
|
||||
0.0108667208387176
|
||||
0.00989270553891231
|
||||
0.0231113396345966
|
||||
-0.00855695322967444
|
||||
0.0725669523773959
|
||||
0.0514236718258817
|
||||
-0.0925841808975707
|
||||
0.0194380306421912
|
||||
0.000730570515309862
|
||||
-0.0108278437152388
|
||||
0.0182042881298797
|
||||
-0.00545133695512132
|
||||
-0.00947885525198547
|
||||
0.00882365233005454
|
||||
-0.0326498326483643
|
||||
0.0559669852166831
|
||||
0.0124041286095178
|
||||
0.00649946004214022
|
||||
-0.00650614986234504
|
||||
0.00407706248542536
|
||||
0.00651530391776104
|
||||
0.013447118349483
|
||||
-0.0134939060495744
|
||||
0.00565109159485104
|
||||
0.0294640045855332
|
||||
-0.00955622274933703
|
||||
0.0222204159377911
|
||||
0.0133466129749798
|
||||
-0.00721633610288384
|
||||
-0.0086040283137278
|
||||
0.00340371293203504
|
||||
-0.012844093543297
|
||||
0.0121257065338001
|
||||
0.00268855738749708
|
||||
0.00118279275124873
|
||||
-0.0153429047422775
|
||||
-0.015616547785309
|
||||
0.0129986755216289
|
||||
-0.0107568537084456
|
||||
-0.0130470312315803
|
||||
0.0104556444029914
|
||||
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|
||||
-0.00719032006182835
|
||||
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|
||||
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|
||||
-0.000336751895784083
|
||||
-0.00330660433831328
|
||||
0.00140165273739631
|
||||
0.0107159290429912
|
||||
0.00552948813933582
|
||||
-0.0137448910012184
|
||||
0.00749751479758479
|
||||
0.00887272196500938
|
||||
-0.0124943703336521
|
||||
-0.0172269856835151
|
||||
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|
||||
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|
||||
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|
||||
-0.00406570845243238
|
||||
0.00048435760011601
|
||||
0.0105345039142465
|
||||
-0.0201850286749165
|
||||
0.00546754742568234
|
||||
0.00564722258456261
|
||||
-0.0179937361721281
|
||||
-0.0120951853991906
|
||||
-0.00617003473303931
|
||||
-0.00408525582586736
|
||||
-0.00151657586731412
|
||||
-0.00458159624101839
|
||||
-0.00627763345850877
|
||||
-0.010456268102569
|
||||
0.00261965704162741
|
||||
0.0105821294864581
|
||||
0.0131796611662535
|
||||
-0.0056904594035664
|
||||
-0.00637383224732294
|
||||
-0.0104380912708849
|
||||
-0.00719740299000661
|
||||
-0.00569934816846962
|
||||
0.00324336844641772
|
||||
-0.00855906610749913
|
||||
-0.00583583049252428
|
||||
-0.00842972795121058
|
||||
-0.012444201853323
|
||||
-0.0133488041521598
|
||||
-0.00245733952585063
|
||||
-0.00161441536671584
|
||||
-0.00935765492929235
|
||||
-0.00395995663912395
|
||||
0.012306797131557
|
||||
-0.00955343910342383
|
||||
-0.00483645751117614
|
||||
-0.0106509881800608
|
||||
-0.0006152347489508
|
||||
0.0036263283613646
|
||||
-0.0039261472903939
|
||||
0.0103190288685123
|
||||
0.00668053510370978
|
||||
-0.00151422542417983
|
||||
-0.00283255673740532
|
||||
0.00717357508078575
|
||||
-0.00556968959239721
|
||||
-0.00756156148745145
|
||||
0.000657155382757425
|
||||
0.00545993387358546
|
||||
-0.00295271942532878
|
||||
-0.0102939079358432
|
||||
0.00045686000680757
|
||||
-0.00725595564991065
|
||||
0.00249965944852971
|
||||
-0.0140745119884327
|
||||
0.000858668335540681
|
||||
0.00328573254558175
|
||||
0.00213602634438687
|
||||
0.00957752547434445
|
||||
-0.00529675687686024
|
||||
-0.00989365284374646
|
||||
0.0128937799030866
|
||||
-0.00634117428048179
|
||||
0.00714977162777464
|
||||
-0.000571554334043656
|
||||
-0.00777046441209764
|
||||
0.00743298767791391
|
||||
0.0166481329287521
|
||||
-0.00472787660262257
|
||||
-0.0128645827089702
|
||||
-0.0163106234477167
|
||||
0.0136246272924602
|
||||
|
|
@ -716,4 +716,16 @@ note={Gaussian Inc. Wallingford CT}
|
|||
issn = {0021-9606},
|
||||
publisher = {American Institute of Physics},
|
||||
doi = {10.1063/1.1757439}
|
||||
}
|
||||
|
||||
|
||||
@article{Applencourt_2018,
|
||||
author = {Applencourt, Thomas and Gasperich, Kevin and
|
||||
Scemama, Anthony},
|
||||
title = {{Spin adaptation with determinant-based selected configuration interaction}},
|
||||
journal = {arXiv},
|
||||
year = {2018},
|
||||
month = {Dec},
|
||||
eprint = {1812.06902},
|
||||
url = {https://arxiv.org/abs/1812.06902v1}
|
||||
}
|
||||
|
|
@ -77,7 +77,7 @@ of selected configuration interaction (sCI)
|
|||
methods\cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
|
||||
the sizes of the systems that could be computed at the FCI level, but
|
||||
the scaling remains exponential unless some bias is introduced leading
|
||||
to a loss of size extensivity.
|
||||
to a loss of size consistency.
|
||||
|
||||
The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
|
||||
the exact solution of the Schrödinger equation with an additional
|
||||
|
|
@ -330,6 +330,9 @@ RS-Hamiltonian is performed iteratively with the updated density.
|
|||
The convergence of the algorithm was further improved
|
||||
by introducing a direct inversion in the iterative subspace (DIIS)
|
||||
step to extrapolate the density both in the outer and inner loops.
|
||||
Note that any range-separated post-Hartree-Fock method can be
|
||||
implemented using this scheme by just replacing the CIPSI step by the
|
||||
post-HF method of interest.
|
||||
|
||||
\section{Computational details}
|
||||
\label{sec:comp-details}
|
||||
|
|
@ -350,6 +353,8 @@ Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
|
|||
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
|
||||
The convergence criterion for stopping the CIPSI calculations
|
||||
was $\EPT < 1$~m\hartree{} $\vee \Ndet > 10^7$.
|
||||
All the wave functions are eigenfunctions of the $S^2$ operator, as
|
||||
described in ref~\onlinecite{Applencourt_2018}.
|
||||
|
||||
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
|
||||
in the determinant localization approximation (DLA),\cite{Zen_2019}
|
||||
|
|
@ -357,7 +362,8 @@ where 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, in the DLA the fixed-node
|
||||
energy is independent of the Jatrow factor, as in all-electron
|
||||
calculations.
|
||||
calculations. Simple Jastrow factors were used to reduce the
|
||||
fluctuations of the local energy.
|
||||
|
||||
|
||||
|
||||
|
|
@ -508,30 +514,42 @@ 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.
|
||||
|
||||
\subsection{Size-consistence}
|
||||
\subsection{Size-consistency}
|
||||
|
||||
An extremely important feature required to get accurate
|
||||
atomization energies is size-extensivity, since the numbers of
|
||||
correlated electron pairs in the isolated atoms are different from
|
||||
those of the molecules.
|
||||
atomization energies is size-consistency (or strict separability),
|
||||
since the numbers of correlated electron pairs in the isolated atoms
|
||||
are different from those of the molecules.
|
||||
The energy computed within density functional theory is size-consistent, and
|
||||
as it is a mean-field method the convergence to the complete basis set
|
||||
(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to
|
||||
the calculation of atomization energies, especially with small basis
|
||||
sets. But going to the CBS limit will converge to biased atomization
|
||||
energies because of the use of approximate density functionals.
|
||||
|
||||
On the other hand, FCI is also size-consistent, but the convergence of
|
||||
the FCI energies to the CBS limit is much slower because of the
|
||||
description of short-range electron correlation using atom-centered
|
||||
functions. But ultimately the exact energy will be reached.
|
||||
|
||||
In the context of selected CI calculations, when the variational energy is
|
||||
extrapolated to the FCI energy\cite{Holmes_2017} there is no
|
||||
size-consistence error. But when the selected wave function is used
|
||||
as a reference for post-Hartree-Fock methods (sCI+PT2 for instance)
|
||||
or for QMC calculations, there is a residual size-consistence error
|
||||
originating from the truncation of the determinant space.
|
||||
size-consistency error. But when the truncated sCI wave function is used
|
||||
as a reference for post-Hartree-Fock methods such as sCI+PT2
|
||||
or for QMC calculations, there is a residual size-consistency error
|
||||
originating from the truncation of the wave function.
|
||||
|
||||
QMC energies 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
|
||||
size-consistency 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
|
||||
fragments is equal to the PT2 correction of the molecule, enforcing that
|
||||
the variational potential energy surface (PES) is
|
||||
parallel to the perturbatively corrected PES, which is an accurate
|
||||
estimate of the FCI PES.\cite{Giner_2015}
|
||||
parallel to the perturbatively corrected PES, which is a relatively
|
||||
accurate estimate of the FCI PES.\cite{Giner_2015}
|
||||
|
||||
Another source of size-consistence error in QMC calculations originates
|
||||
Another source of size-consistency error in QMC calculations originates
|
||||
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
|
||||
|
|
@ -545,9 +563,9 @@ or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
|
|||
One can easily see that this parameterization of the two-body
|
||||
interation is not size-consistent: the dissociation of a
|
||||
diatomic molecule $AB$ with a parameter $b_{AB}$
|
||||
will lead to two different two-body Jastrow factors on each atom, each
|
||||
will lead to two different two-body Jastrow factors, each
|
||||
with its own optimal value $b_A$ and $b_B$. To remove the
|
||||
size-consistence error on a PES using this ansätz for $J_\text{ee}$,
|
||||
size-consistency error on a PES using this ansätz for $J_\text{ee}$,
|
||||
one needs to impose that the parameters of $J_\text{ee}$ are fixed:
|
||||
$b_A = b_B = b_{AB}$.
|
||||
|
||||
|
|
@ -560,25 +578,39 @@ localization approximation,\cite{Zen_2019} the Jastrow factor is
|
|||
removed from the wave function on which the pseudopotential is localized.
|
||||
The great advantage of this approximation is that the FN-DMC energy
|
||||
only depends on the parameters of the determinantal component. Using a
|
||||
size-inconsistent Jastrow factor, or a non-optimal Jastrow factor will
|
||||
not introduce an additional size-consistence error in FN-DMC
|
||||
calculations, although it will reduce the statistical errors by
|
||||
reducing the variance of the local energy. Moreover, the
|
||||
integrals involved in the pseudo-potential are computed analytically
|
||||
and the computational cost of the pseudo-potential is dramatically
|
||||
reduced (for more detail, see Ref.~\onlinecite{Scemama_2015}).
|
||||
non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will
|
||||
not introduce an additional error in FN-DMC calculations, although it
|
||||
will reduce the statistical errors by reducing the variance of the
|
||||
local energy. Moreover, the integrals involved in the pseudo-potential
|
||||
are computed analytically and the computational cost of the
|
||||
pseudo-potential is dramatically reduced (for more detail, see
|
||||
Ref.~\onlinecite{Scemama_2015}).
|
||||
|
||||
In this section, we make a numerical verification that the produced
|
||||
wave functions are size-consistent. We have computed the energy of the
|
||||
dissocited fluorine dimer, where the two atoms are at a distance of 50~\AA.
|
||||
We expect that the energy of this system is equal to twice the energy
|
||||
of the fluorine atom.
|
||||
|
||||
The energy computed within density functional theory is extensive, and
|
||||
as it is a mean-field method the convergence to the complete basis set
|
||||
(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to
|
||||
the calculation of atomization energies, especially with small basis
|
||||
sets, but going to the CBS limit will converge to biased atomization
|
||||
energies because of the use of approximate density functionals.
|
||||
On the other hand, the convergence of the FCI energies to the CBS
|
||||
limit will be slower because of the description of short-range electron
|
||||
correlation with atom-centered functions, but ultimately the exact
|
||||
energy will be reached.
|
||||
%\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{FN-DMC Energies of the fluorine atom and the dissociated fluorine
|
||||
dimer, and size-consistency error.}
|
||||
\label{tab:size-cons}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{clll}
|
||||
$\mu$ & F & Dissociated F$_2$ & Size-consistency error \\
|
||||
\hline
|
||||
0.00 & & & \\
|
||||
0.25 & & & \\
|
||||
0.50 & & & \\
|
||||
1.00 & & & \\
|
||||
2.00 & & & \\
|
||||
5.00 & & & \\
|
||||
$\infty$ & & & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
|
||||
\subsection{Spin-invariance}
|
||||
|
|
@ -700,7 +732,7 @@ Jastrow factor, but without the burden of performing a stochastic
|
|||
optimization.
|
||||
The proposed procedure provides a method to optimize the
|
||||
FN-DMC energy via a single parameter, namely the range-separation
|
||||
parameter $\mu$. The size-consistence error is controlled, as well as the
|
||||
parameter $\mu$. The size-consistency error is controlled, as well as the
|
||||
invariance with respect to the spin projection $m_s$.
|
||||
Finding the optimal value of $\mu$ gives the lowest FN-DMC energies
|
||||
within basis set. However, if one wants to compute an energy
|
||||
|
|
|
|||
Loading…
Reference in New Issue