Working on MAD
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@ -582,6 +582,22 @@ return result
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#+RESULTS: sdformula
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: 0.0
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#+NAME: sderrformula
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#+BEGIN_SRC python :var data=[1.] :var data_err=[0.1] :exports none
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from math import sqrt
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n = len(data)
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average = sum(data)/n
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if n >1 : n = n-1
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variance_up = sum([(x+y - average)*(x+y - average) for (x,y) in zip(data,data_err)])/n
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variance_dn = sum([(x-y - average)*(x-y - average) for (x,y) in zip(data,data_err)])/n
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result = abs(sqrt(variance_up) - sqrt(variance_dn))
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print(result)
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return result
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#+END_SRC
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#+RESULTS: sderrformula
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: 1.1102230246251565e-16
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** pVDZ, BFD
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*** CCSD(T)
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**** Table
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@ -1144,11 +1160,12 @@ return result
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| SiH4 | -6.273878 | 0.000571 | 0.515119 | 0.000590 | 0.515000 | 0.074705 | 0.370052 | 95 |
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| SiO | -19.948315 | 0.000541 | 0.296918 | 0.000580 | 0.306700 | -6.138378 | 0.364006 | 23 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+------|
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| | 4.61 | 0.34 | -3.62 | 5.30 | | | | |
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| | 4.61 | 0.34 | -3.62 | 5.30 | 0.09 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 4.61 +/- 0.34
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*** $\mu=1/4$
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@ -1308,11 +1325,12 @@ return result
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| SiH4 | -6.274259 | 0.000352 | 0.514417 | 0.000375 | 0.515000 | -0.365936 | 0.235128 | 765 |
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| SiO | -19.950838 | 0.000744 | 0.298179 | 0.000772 | 0.306700 | -5.347322 | 0.484376 | 307 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+------|
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| | 4.04 | 0.37 | -3.13 | 4.88 | | | | |
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| | 4.04 | 0.37 | -3.13 | 4.88 | 0.10 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 4.04 +/- 0.37
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*** $\mu=1/2$
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@ -1548,11 +1566,12 @@ return result
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| SiH4 | -6.276686 | 0.000227 | 0.513844 | 0.000233 | 0.515000 | -0.725479 | 0.146503 | 10672 |
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| SiO | -19.954139 | 0.000398 | 0.297994 | 0.000406 | 0.306700 | -5.463401 | 0.254487 | 13428 |
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|---------+------------+----------+----------+----------+----------+------------+----------+--------|
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| | 3.74 | 0.35 | -3.53 | 4.03 | | | | |
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| | 3.74 | 0.35 | -3.53 | 4.03 | 0.23 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 3.74 +/- 0.35
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*** $\mu=1$
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@ -1712,11 +1731,12 @@ return result
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| SiH4 | -6.276709 | 0.000144 | 0.511981 | 0.000169 | 0.515000 | -1.894611 | 0.105885 | 400063 |
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| SiO | -19.958150 | 0.000200 | 0.297492 | 0.000209 | 0.306700 | -5.778347 | 0.131036 | 226347 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+----------|
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| | 5.42 | 0.29 | -5.14 | 4.55 | | | | |
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| | 5.42 | 0.29 | -5.14 | 4.55 | 0.03 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 5.42 +/- 0.29
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*** $\mu=2$
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@ -1876,11 +1896,12 @@ return result
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| SiH4 | -6.277594 | 0.000436 | 0.512572 | 0.000443 | 0.515000 | -1.523325 | 0.278234 | 1538523 |
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| SiO | -19.961945 | 0.000824 | 0.295735 | 0.000858 | 0.306700 | -6.880821 | 0.538188 | 1775195 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+----------|
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| | 5.98 | 0.83 | -5.91 | 4.79 | | | | |
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| | 5.98 | 0.83 | -5.91 | 4.79 | 0.71 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 5.98 +/- 0.83
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*** $\mu=5$
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@ -2040,11 +2061,12 @@ return result
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| SiH4 | -6.277334 | 0.000611 | 0.512261 | 0.000617 | 0.515000 | -1.718715 | 0.387297 | 3474506 |
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| SiO | -19.963125 | 0.001051 | 0.297073 | 0.001093 | 0.306700 | -6.040980 | 0.685650 | 4210107 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+----------|
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| | 6.18 | 0.84 | -6.13 | 4.87 | | | | |
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| | 6.18 | 0.84 | -6.13 | 4.87 | 0.55 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 6.18 +/- 0.84
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*** FCI
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@ -2203,11 +2225,12 @@ return result
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| SiH4 | -6.276132 | 0.001187 | 0.511013 | 0.001189 | 0.515000 | -2.502072 | 0.746197 | 2954175 |
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| SiO | -19.954964 | 0.002217 | 0.289206 | 0.002225 | 0.306700 | -10.977712 | 1.396411 | 4457124 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+----------|
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| | 7.38 | 1.08 | -7.38 | 5.67 | | | | |
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| | 7.38 | 1.08 | -7.38 | 5.67 | 0.68 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 7.38 +/- 1.08
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*** Optimal mu
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@ -2286,11 +2309,12 @@ return result
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| SiH4 | -6.277692 | 0.000996 | 0.512431 | 0.001014 | 0.515000 | -1.612261 | 0.636414 |
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| SiO | -19.963415 | 0.001961 | 0.296567 | 0.002038 | 0.306700 | -6.358308 | 1.278744 |
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|---------+------------+----------+----------+----------+-----------+------------+----------|
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| | 5.85 | 1.75 | -5.63 | 4.79 | | | |
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| | 5.85 | 1.75 | -5.63 | 4.79 | 1.11 | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 5.85 +/- 1.75
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** pVTZ, BFD
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@ -2857,11 +2881,12 @@ return result
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| SiH4 | -6.277757 | 0.000145 | 0.519191 | 0.000151 | 0.515000 | 2.629616 | 0.094560 | 95 |
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| SiO | -19.955719 | 0.000131 | 0.303829 | 0.000205 | 0.306700 | -1.801746 | 0.128826 | 47 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+------|
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| | 3.52 | 0.19 | -1.03 | 4.39 | | 3.52 | 0.19 | |
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| | 3.52 | 0.19 | -1.03 | 4.39 | 0.04 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 3.52 +/- 0.19
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*** $\mu=1/4$
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@ -3021,11 +3046,12 @@ return result
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| SiH4 | -6.279817 | 0.000685 | 0.519634 | 0.000691 | 0.515000 | 2.908075 | 0.433856 | 2019 |
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| SiO | -19.958859 | 0.000768 | 0.305851 | 0.000780 | 0.306700 | -0.532719 | 0.489741 | 845 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+-------|
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| | 3.39 | 0.77 | -0.59 | 4.44 | | 3.39 | 0.77 | |
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| | 3.39 | 0.77 | -0.59 | 4.44 | 0.34 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 3.39 +/- 0.77
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*** $\mu=1/2$
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@ -3260,11 +3286,12 @@ return result
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| SiH4 | -6.281142 | 0.000138 | 0.517298 | 0.000148 | 0.515000 | 1.442121 | 0.092620 | 41778 |
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| SiO | -19.963472 | 0.000273 | 0.304821 | 0.000282 | 0.306700 | -1.178797 | 0.177217 | 26149 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+---------|
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| | 2.46 | 0.18 | -1.72 | 3.02 | | | | |
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| | 2.46 | 0.18 | -1.72 | 3.02 | 0.06 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 2.46 +/- 0.18
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*** $\mu=1$
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@ -3424,11 +3451,12 @@ return result
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| SiH4 | -6.282083 | 0.000683 | 0.515237 | 0.000709 | 0.515000 | 0.148971 | 0.445190 | 1594422 |
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| SiO | -19.967377 | 0.002598 | 0.303241 | 0.002611 | 0.306700 | -2.170380 | 1.638215 | 1660400 |
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|---------+------------+----------+----------+----------+-----------+------------+----------+----------|
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| | 4.38 | 0.94 | -4.24 | 5.11 | | | | |
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| | 4.38 | 0.94 | -4.24 | 5.11 | 0.31 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
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***** MAD = 4.38 +/- 0.94
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***** Figure
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@ -3441,6 +3469,20 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
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#+RESULTS:
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[[file:data-g2-qmc-tz-1.0-jast.png]]
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*** FCI
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**** CIPSI
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***** Table
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#+NAME: tz-cipsi-inf
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| | e_cal | ae_cal | ae_nr | ae_diff | Ndet |
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| | Hartree | Hartree | Hartree | kcal/mol | |
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|---------+------------+----------+----------+------------+----------|
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|---------+------------+----------+----------+------------+----------|
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|---------+------------+----------+----------+------------+----------|
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| | | | | | |
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#+TBLFM: @>$3='(org-sbe "madformula" (data @5$5..@>>$5) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$5..@>>$5) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$5..@>>$5) );%0.2f
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***** MAD =
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** pVQZ, BFD
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*** CCSD(T)
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**** Table
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@ -4004,11 +4046,12 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
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| SiH4 | -6.279824 | 0.000455 | 0.521153 | 0.000465 | 0.515000 | 3.861238 | 0.291920 | 95 |
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| SiO | -19.956758 | 0.000627 | 0.305065 | 0.000650 | 0.306700 | -1.025872 | 0.407693 | 95 |
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|---------+------------+----------+----------+----------+-----------+-----------+----------+------|
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| | 3.16 | 0.26 | -0.12 | 4.12 | | | | |
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| | 3.16 | 0.26 | -0.12 | 4.119 | 0.025 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.3f
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#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.3f
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***** MAD = 3.16 +/- 0.26
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*** $\mu=1/4$
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@ -4168,11 +4211,12 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
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| SiH4 | -6.281099 | 0.000478 | 0.520692 | 0.000479 | 0.515000 | 3.571494 | 0.300315 | 2251 |
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| SiO | -19.961401 | 0.000239 | 0.308118 | 0.000246 | 0.306700 | 0.889539 | 0.154490 | 2155 |
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|---------+------------+----------+----------+----------+-----------+-----------+----------+-------|
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| | 2.90 | 0.25 | 0.39 | 3.74 | | | | |
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| | 2.90 | 0.25 | 0.39 | 3.745 | 0.005 | | | |
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#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
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#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
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#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
|
||||
#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
|
||||
#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.3f
|
||||
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.3f
|
||||
|
||||
***** MAD = 2.90 +/- 0.25
|
||||
|
||||
@ -4333,12 +4377,27 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
|
||||
| SiH4 | -6.284045 | 0.000114 | 0.519529 | 0.000144 | 0.515000 | 2.841690 | 0.090184 | 95042 |
|
||||
| SiO | -19.967776 | 0.000424 | 0.308334 | 0.000463 | 0.306700 | 1.025334 | 0.290557 | 127437 |
|
||||
|---------+------------+----------+----------+----------+-----------+-----------+----------+---------|
|
||||
| | 2.06 | 0.35 | -0.44 | 2.74 | | | | |
|
||||
| | 2.06 | 0.35 | -0.44 | 2.740 | 0.125 | | | |
|
||||
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
|
||||
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%0.2f
|
||||
#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$7..@>>$7) );%0.2f
|
||||
#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.2f
|
||||
#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$7..@>>$7) );%0.3f
|
||||
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.3f
|
||||
***** MAD = 2.06 +/- 0.35
|
||||
*** FCI
|
||||
**** CIPSI
|
||||
***** Table
|
||||
#+NAME: tz-cipsi-inf
|
||||
| | e_cal | ae_cal | ae_nr | ae_diff | Ndet |
|
||||
| | Hartree | Hartree | Hartree | kcal/mol | |
|
||||
|---------+------------+----------+----------+------------+----------|
|
||||
|---------+------------+----------+----------+------------+----------|
|
||||
|---------+------------+----------+----------+------------+----------|
|
||||
| | | | | | |
|
||||
#+TBLFM: @>$3='(org-sbe "madformula" (data @5$5..@>>$5) );%0.2f
|
||||
#+TBLFM: @>$4='(org-sbe "mudformula" (data @5$5..@>>$5) );%0.2f
|
||||
#+TBLFM: @>$5='(org-sbe "sdformula" (data @5$5..@>>$5) );%0.2f
|
||||
***** MAD =
|
||||
** SHCI CBS extrpolation
|
||||
#+NAME: cbs-shci
|
||||
| | e_cal | ae_cal | ae_nr | ae_diff |
|
||||
|
@ -681,13 +681,7 @@ noticeable with $\mu=5$~bohr$^{-1}$.
|
||||
|
||||
\subsection{Benchmark}
|
||||
|
||||
The 55 molecules of the benchmark for the Gaussian-1
|
||||
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
|
||||
of the RS-DFT-CIPSI trial wave functions for energy differences.
|
||||
|
||||
|
||||
|
||||
%\begin{squeezetable}
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{Mean absolute error (MAE), mean signed errors (MSE) and
|
||||
standard deviations (RMSD) obtained with the different methods and
|
||||
@ -695,41 +689,94 @@ of the RS-DFT-CIPSI trial wave functions for energy differences.
|
||||
\label{tab:mad}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ll rrr rrr rrr}
|
||||
Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VQZ-BFD & \phantom{} \\
|
||||
\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
|
||||
Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VQZ-BFD & \phantom{} \\
|
||||
\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
|
||||
\hline
|
||||
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
|
||||
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
|
||||
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
|
||||
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
|
||||
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
|
||||
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
|
||||
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
|
||||
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
|
||||
\hline
|
||||
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
|
||||
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
|
||||
\hline
|
||||
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
|
||||
\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
|
||||
\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
|
||||
\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
|
||||
\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
|
||||
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
|
||||
\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
|
||||
\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
|
||||
\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
|
||||
\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
|
||||
\hline
|
||||
DMC@RS-DFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30 & 3.52(19) & -1.03(19) & 4.39 & 3.16(26) & -0.12(26) & 4.12 \\
|
||||
\phantom{} & 1/4 & 4.04(37) & -3.13(\phantom{0.}37) & 4.88 & 3.39(77) & -0.59(77) & 4.44 & 2.90(25) & 0.25(25) & 3.74 \\
|
||||
\phantom{} & 1/2 & 3.74(35) & -3.53(\phantom{0.}35) & 4.03 & 2.46(18) & -1.72(18) & 3.02 & 2.06(35) & -0.44(35) & 2.74 \\
|
||||
\phantom{} & 1 & 5.42(29) & -5.14(\phantom{0.}29) & 4.55 & 4.38(94) & -4.24(94) & 5.11 & --- & --- & --- \\
|
||||
\phantom{} & 2 & 5.98(83) & -5.91(\phantom{0.}83) & 4.79 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & 5 & 6.18(84) & -6.13(\phantom{0.}84) & 4.87 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79 & --- & --- & --- & --- & --- & --- \\
|
||||
DMC@RS-DFT-CIPSI & 0 & 4.61(\phantom{0.}34) & -3.62(\phantom{0.}34) & 5.30(\phantom{0.}09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
|
||||
\phantom{} & 1/4 & 4.04(\phantom{0.}37) & -3.13(\phantom{0.}37) & 4.88(\phantom{0.}10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
|
||||
\phantom{} & 1/2 & 3.74(\phantom{0.}35) & -3.53(\phantom{0.}35) & 4.03(\phantom{0.}23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
|
||||
\phantom{} & 1 & 5.42(\phantom{0.}29) & -5.14(\phantom{0.}29) & 4.55(\phantom{0.}03) & 4.38(94) & -4.24(94) & 5.11(31) & --- & --- & --- \\
|
||||
\phantom{} & 2 & 5.98(\phantom{0.}83) & -5.91(\phantom{0.}83) & 4.79(\phantom{0.}71) & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & 5 & 6.18(\phantom{0.}84) & -6.13(\phantom{0.}84) & 4.87(\phantom{0.}55) & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(\phantom{0.}68) & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & --- & --- & --- & --- & --- & --- \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
%\end{squeezetable}
|
||||
\end{squeezetable}
|
||||
|
||||
The 55 molecules of the benchmark for the Gaussian-1
|
||||
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the
|
||||
performance of the RS-DFT-CIPSI trial wave functions in the context of
|
||||
energy differences. Calculations were made in the double-, triple-
|
||||
and quadruple-zeta basis sets with different values of $\mu$, and using
|
||||
natural orbitals of a preliminary CIPSI calculation.
|
||||
For comparison, we have computed the energies of all the atoms and
|
||||
molecules at the DFT level with different density functionals, and at
|
||||
the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean
|
||||
absolute errors (MAE), mean signed errors (MSE) and standard
|
||||
deviations (RMSD). For FCI (RS-DFT-CIPSI, $\mu=\infty$), we have
|
||||
given extrapolated values at $\EPT\rightarrow 0$.
|
||||
|
||||
In this benchmark, the great majority of the systems are well
|
||||
described by a single determinant. Therefore, the atomization energies
|
||||
calculated at the DFT level are relatively accurate, even when
|
||||
the basis set is small. The introduction of exact exchange (B3LYP and
|
||||
PBE0) make the results more sensitive to the basis set, and reduce the
|
||||
accuracy. Thanks to the single-reference character of these systems,
|
||||
the CCSD(T) energy is an excellent estimate of the FCI energy, as
|
||||
shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)
|
||||
and FCI energies.
|
||||
The imbalance of the quality of description of molecules compared
|
||||
to atoms is exhibited by a very negative value of the MSE for
|
||||
CCSD(T) and FCI/VDZ-BFD, which is reduced by a factor of two
|
||||
when going to the triple-zeta basis, and again by a factor of two when
|
||||
going to the quadruple-zeta basis.
|
||||
|
||||
This large imbalance at the double-zeta level affects the nodal
|
||||
surfaces, because although the FN-DMC energies obtained with near-FCI
|
||||
trial wave functions are much lower than the single-determinant FN-DMC
|
||||
energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is
|
||||
larger than the single-determinant MAE (4.61~$\pm$ 0.34 kcal/mol).
|
||||
Using the FCI trial wave function the MSE is equal to the
|
||||
negative MAE which confirms that all the atomization energies are
|
||||
underestimated. This confirms that some of the basis-set
|
||||
incompleteness error is transferred in the fixed-node error.
|
||||
|
||||
Within the double-zeta basis set, the calculations could be done for the
|
||||
whole range of values of $\mu$, and the optimal value of $\mu$ was
|
||||
estimated for each system by searching for the minimum of the spline
|
||||
interpolation curve of the FN-DMC energy as a function of $\mu$.
|
||||
This corresponds the the line labelled by the \emph{Opt} value of $\mu$
|
||||
in the table. Using the optimal value of $\mu$ clearly improves the
|
||||
MAE, the MSE an the RMSD compared the the FCI wave function. This
|
||||
result is in line with the common knowledge that re-optimizing
|
||||
the determinantal component of the trial wave function in the presence
|
||||
of electron correlation reduces the errors due to the basis set incompleteness.
|
||||
These calculations were done only for the smallest basis set
|
||||
because of the expensive computational cost of the QMC calculations
|
||||
when the trial wave function is expanded on more than a few million
|
||||
determinants.
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{g2-dmc.pdf}
|
||||
\caption{Errors in the DMC atomization energies with the different
|
||||
\caption{Errors in the FN-DMC atomization energies with the different
|
||||
trial wave functions. Each dot corresponds to an atomization
|
||||
energy.
|
||||
The boxes contain the data between first and third quartiles, and
|
||||
@ -738,6 +785,18 @@ DMC@RS-DFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.3
|
||||
\label{fig:g2-dmc}
|
||||
\end{figure}
|
||||
|
||||
Searching for the optimal value of $\mu$ may be too costly, so we have
|
||||
computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
|
||||
are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
|
||||
in table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
|
||||
0.5~bohr$^{-1}$ for all three basis sets. Is is the value for which
|
||||
the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
|
||||
3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values
|
||||
are even lower than those obtained with the optimal value of
|
||||
$\mu$. Although the FN-DMC energies are higher, the numbers show that
|
||||
they are more consistent from one system to another, giving improved
|
||||
cancellations of errors.
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{g2-ndet.pdf}
|
||||
@ -749,61 +808,59 @@ DMC@RS-DFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.3
|
||||
\label{fig:g2-ndet}
|
||||
\end{figure}
|
||||
|
||||
The number of determinants in the wave functions are shown in
|
||||
figure~\ref{fig:n2-ndet}. For all the calculations, the stopping
|
||||
criterion of the CIPSI algorithm was $\EPT < 1$~m\hartree{} or $\Ndet >
|
||||
10^7$.
|
||||
For FCI, we have given extrapolated values at $\EPT\rightarrow 0$.
|
||||
At $\mu=0$ the number of determinants is not equal to one because
|
||||
we have used the natural orbitals of a first CIPSI calculation.
|
||||
|
||||
The number of determinants in the trial wave functions are shown in
|
||||
figure~\ref{fig:g2-ndet}. As expected, the number of determinants
|
||||
is smaller when $\mu$ is small and larger when $\mu$ is large.
|
||||
|
||||
Note that when $\mu=0$ the number of determinants is not equal to one because
|
||||
we have used the natural orbitals of a first CIPSI calculation, and
|
||||
not the sr-PBE orbitals.
|
||||
So the Kohn-Sham determinant is expressed as a linear combination of
|
||||
determinants built with natural orbitals. Note that it is possible to add
|
||||
determinants built with natural orbitals. It is possible to add
|
||||
an extra step to the algorithm to compute the natural orbitals from the
|
||||
RS-DFT/CIPSI wave function, and re-do the RS-DFT/CIPSI calculation with
|
||||
these orbitals to get an even more compact expansion. In that case, we would
|
||||
have obtained the Kohn-Sham orbitals with $\mu=0$, and the solution would have
|
||||
been the KS single determinant.
|
||||
have converged to the Kohn-Sham orbitals with $\mu=0$, and the
|
||||
solution would have been the PBE single determinant.
|
||||
|
||||
We could have obtained
|
||||
single-determinant wave functions by using the natural orbitals of a first
|
||||
|
||||
|
||||
\section{Conclusion}
|
||||
|
||||
We have seen that introducing short-range correation via
|
||||
a range-separated Hamiltonian in a full CI expansion yields improved
|
||||
nodes, especially with small basis sets. The effect is similar to the
|
||||
effect of re-optimizing the CI coefficients in the presence of a
|
||||
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-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
|
||||
difference, one should not minimize the
|
||||
FN-DMC energies of the reactants independently. It is preferable to
|
||||
choose a value of $\mu$ for which the fixed-node errors are well
|
||||
balanced, leading to a good cncellation of errors. We found that a
|
||||
value of $\mu=0.5$~bohr${^-1}$ is the value where the errors are the
|
||||
smallest. Moreover, such a small value of $\mu$ gives extermely
|
||||
nodal surfaces, especially with small basis sets. The effect of sr-DFT
|
||||
on the determinant expansion is similar to the effect of re-optimizing
|
||||
the CI coefficients in the presence of a Jastrow factor, but without
|
||||
the burden of performing a stochastic optimization.
|
||||
|
||||
Varying the range-separation parameter $\mu$ and aproaching the
|
||||
RS-DFT-FCI with CIPSI provides a way to adapt the number of
|
||||
determinants in the trial wave function, leading to size-consistent
|
||||
FN-DMC energies.
|
||||
We propose two methods. The first one is for the computation of
|
||||
accurate total energies by a one-parameter optimization of the FN-DMC
|
||||
energy via the variation of the parameter $\mu$.
|
||||
The second method is for the computation of energy differences, where
|
||||
the target is not the lowest possible FN-DMC energies but the best
|
||||
possible cancellation of errors. Using a fixed value of $\mu$
|
||||
increases the consistency of the trial wave functions, and we have found
|
||||
that $\mu=0.5$~bohr${^-1}$ is the value where the cancellation of
|
||||
errors is the most effective.
|
||||
Moreover, such a small value of $\mu$ gives extermely
|
||||
compact wave functions, making this recipe a good candidate for
|
||||
accurate calcultions of large systems with a multi-reference character.
|
||||
the accurate description of the whole potential energy surfaces of
|
||||
large systems.
|
||||
|
||||
|
||||
|
||||
|
||||
%%---------------------------------------
|
||||
\begin{acknowledgments}
|
||||
An award of computer time was provided by the Innovative and Novel
|
||||
Computational Impact on Theory and Experiment (INCITE) program. This
|
||||
research has used resources of the Argonne Leadership Computing
|
||||
Facility, which is a DOE Office of Science User Facility supported
|
||||
under Contract DE-AC02-06CH11357. AB, was supported by the
|
||||
U.S. Department of Energy, Office of Science, Basic Energy Sciences,
|
||||
Materials Sciences and Engineering Division, as part of the
|
||||
Computational Materials Sciences Program and Center for Predictive
|
||||
Simulation of Functional Materials.
|
||||
This work was performed using HPC resources from GENCI-TGCC (Grand
|
||||
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
|
||||
2019-0510.
|
||||
\end{acknowledgments}
|
||||
|
||||
|
||||
|
Loading…
Reference in New Issue
Block a user