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Working on MAD

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This commit is contained in:
Anthony Scemama 2020-08-02 19:15:57 +02:00
parent def2087319
commit da7f0809f9
2 changed files with 203 additions and 87 deletions
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@ -582,6 +582,22 @@ return result
#+RESULTS: sdformula #+RESULTS: sdformula
: 0.0 : 0.0
#+NAME: sderrformula
#+BEGIN_SRC python :var data=[1.] :var data_err=[0.1] :exports none
from math import sqrt
n = len(data)
average = sum(data)/n
if n >1 : n = n-1
variance_up = sum([(x+y - average)*(x+y - average) for (x,y) in zip(data,data_err)])/n
variance_dn = sum([(x-y - average)*(x-y - average) for (x,y) in zip(data,data_err)])/n
result = abs(sqrt(variance_up) - sqrt(variance_dn))
print(result)
return result
#+END_SRC
#+RESULTS: sderrformula
: 1.1102230246251565e-16
** pVDZ, BFD ** pVDZ, BFD
*** CCSD(T) *** CCSD(T)
**** Table **** Table
@ -1144,11 +1160,12 @@ return result
| SiH4 | -6.273878 | 0.000571 | 0.515119 | 0.000590 | 0.515000 | 0.074705 | 0.370052 | 95 | | SiH4 | -6.273878 | 0.000571 | 0.515119 | 0.000590 | 0.515000 | 0.074705 | 0.370052 | 95 |
| SiO | -19.948315 | 0.000541 | 0.296918 | 0.000580 | 0.306700 | -6.138378 | 0.364006 | 23 | | SiO | -19.948315 | 0.000541 | 0.296918 | 0.000580 | 0.306700 | -6.138378 | 0.364006 | 23 |
|---------+------------+----------+----------+----------+-----------+------------+----------+------| |---------+------------+----------+----------+----------+-----------+------------+----------+------|
| | 4.61 | 0.34 | -3.62 | 5.30 | | | | | | | 4.61 | 0.34 | -3.62 | 5.30 | 0.09 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 4.61 +/- 0.34 ***** MAD = 4.61 +/- 0.34
*** $\mu=1/4$ *** $\mu=1/4$
@ -1308,11 +1325,12 @@ return result
| SiH4 | -6.274259 | 0.000352 | 0.514417 | 0.000375 | 0.515000 | -0.365936 | 0.235128 | 765 | | SiH4 | -6.274259 | 0.000352 | 0.514417 | 0.000375 | 0.515000 | -0.365936 | 0.235128 | 765 |
| SiO | -19.950838 | 0.000744 | 0.298179 | 0.000772 | 0.306700 | -5.347322 | 0.484376 | 307 | | SiO | -19.950838 | 0.000744 | 0.298179 | 0.000772 | 0.306700 | -5.347322 | 0.484376 | 307 |
|---------+------------+----------+----------+----------+-----------+------------+----------+------| |---------+------------+----------+----------+----------+-----------+------------+----------+------|
| | 4.04 | 0.37 | -3.13 | 4.88 | | | | | | | 4.04 | 0.37 | -3.13 | 4.88 | 0.10 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 4.04 +/- 0.37 ***** MAD = 4.04 +/- 0.37
*** $\mu=1/2$ *** $\mu=1/2$
@ -1548,11 +1566,12 @@ return result
| SiH4 | -6.276686 | 0.000227 | 0.513844 | 0.000233 | 0.515000 | -0.725479 | 0.146503 | 10672 | | SiH4 | -6.276686 | 0.000227 | 0.513844 | 0.000233 | 0.515000 | -0.725479 | 0.146503 | 10672 |
| SiO | -19.954139 | 0.000398 | 0.297994 | 0.000406 | 0.306700 | -5.463401 | 0.254487 | 13428 | | SiO | -19.954139 | 0.000398 | 0.297994 | 0.000406 | 0.306700 | -5.463401 | 0.254487 | 13428 |
|---------+------------+----------+----------+----------+----------+------------+----------+--------| |---------+------------+----------+----------+----------+----------+------------+----------+--------|
| | 3.74 | 0.35 | -3.53 | 4.03 | | | | | | | 3.74 | 0.35 | -3.53 | 4.03 | 0.23 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 3.74 +/- 0.35 ***** MAD = 3.74 +/- 0.35
*** $\mu=1$ *** $\mu=1$
@ -1712,11 +1731,12 @@ return result
| SiH4 | -6.276709 | 0.000144 | 0.511981 | 0.000169 | 0.515000 | -1.894611 | 0.105885 | 400063 | | SiH4 | -6.276709 | 0.000144 | 0.511981 | 0.000169 | 0.515000 | -1.894611 | 0.105885 | 400063 |
| SiO | -19.958150 | 0.000200 | 0.297492 | 0.000209 | 0.306700 | -5.778347 | 0.131036 | 226347 | | SiO | -19.958150 | 0.000200 | 0.297492 | 0.000209 | 0.306700 | -5.778347 | 0.131036 | 226347 |
|---------+------------+----------+----------+----------+-----------+------------+----------+----------| |---------+------------+----------+----------+----------+-----------+------------+----------+----------|
| | 5.42 | 0.29 | -5.14 | 4.55 | | | | | | | 5.42 | 0.29 | -5.14 | 4.55 | 0.03 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 5.42 +/- 0.29 ***** MAD = 5.42 +/- 0.29
*** $\mu=2$ *** $\mu=2$
@ -1876,11 +1896,12 @@ return result
| SiH4 | -6.277594 | 0.000436 | 0.512572 | 0.000443 | 0.515000 | -1.523325 | 0.278234 | 1538523 | | SiH4 | -6.277594 | 0.000436 | 0.512572 | 0.000443 | 0.515000 | -1.523325 | 0.278234 | 1538523 |
| SiO | -19.961945 | 0.000824 | 0.295735 | 0.000858 | 0.306700 | -6.880821 | 0.538188 | 1775195 | | SiO | -19.961945 | 0.000824 | 0.295735 | 0.000858 | 0.306700 | -6.880821 | 0.538188 | 1775195 |
|---------+------------+----------+----------+----------+-----------+------------+----------+----------| |---------+------------+----------+----------+----------+-----------+------------+----------+----------|
| | 5.98 | 0.83 | -5.91 | 4.79 | | | | | | | 5.98 | 0.83 | -5.91 | 4.79 | 0.71 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 5.98 +/- 0.83 ***** MAD = 5.98 +/- 0.83
*** $\mu=5$ *** $\mu=5$
@ -2040,11 +2061,12 @@ return result
| SiH4 | -6.277334 | 0.000611 | 0.512261 | 0.000617 | 0.515000 | -1.718715 | 0.387297 | 3474506 | | SiH4 | -6.277334 | 0.000611 | 0.512261 | 0.000617 | 0.515000 | -1.718715 | 0.387297 | 3474506 |
| SiO | -19.963125 | 0.001051 | 0.297073 | 0.001093 | 0.306700 | -6.040980 | 0.685650 | 4210107 | | SiO | -19.963125 | 0.001051 | 0.297073 | 0.001093 | 0.306700 | -6.040980 | 0.685650 | 4210107 |
|---------+------------+----------+----------+----------+-----------+------------+----------+----------| |---------+------------+----------+----------+----------+-----------+------------+----------+----------|
| | 6.18 | 0.84 | -6.13 | 4.87 | | | | | | | 6.18 | 0.84 | -6.13 | 4.87 | 0.55 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 6.18 +/- 0.84 ***** MAD = 6.18 +/- 0.84
*** FCI *** FCI
@ -2203,11 +2225,12 @@ return result
| SiH4 | -6.276132 | 0.001187 | 0.511013 | 0.001189 | 0.515000 | -2.502072 | 0.746197 | 2954175 | | SiH4 | -6.276132 | 0.001187 | 0.511013 | 0.001189 | 0.515000 | -2.502072 | 0.746197 | 2954175 |
| SiO | -19.954964 | 0.002217 | 0.289206 | 0.002225 | 0.306700 | -10.977712 | 1.396411 | 4457124 | | SiO | -19.954964 | 0.002217 | 0.289206 | 0.002225 | 0.306700 | -10.977712 | 1.396411 | 4457124 |
|---------+------------+----------+----------+----------+-----------+------------+----------+----------| |---------+------------+----------+----------+----------+-----------+------------+----------+----------|
| | 7.38 | 1.08 | -7.38 | 5.67 | | | | | | | 7.38 | 1.08 | -7.38 | 5.67 | 0.68 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 7.38 +/- 1.08 ***** MAD = 7.38 +/- 1.08
*** Optimal mu *** Optimal mu
@ -2286,11 +2309,12 @@ return result
| SiH4 | -6.277692 | 0.000996 | 0.512431 | 0.001014 | 0.515000 | -1.612261 | 0.636414 | | SiH4 | -6.277692 | 0.000996 | 0.512431 | 0.001014 | 0.515000 | -1.612261 | 0.636414 |
| SiO | -19.963415 | 0.001961 | 0.296567 | 0.002038 | 0.306700 | -6.358308 | 1.278744 | | SiO | -19.963415 | 0.001961 | 0.296567 | 0.002038 | 0.306700 | -6.358308 | 1.278744 |
|---------+------------+----------+----------+----------+-----------+------------+----------| |---------+------------+----------+----------+----------+-----------+------------+----------|
| | 5.85 | 1.75 | -5.63 | 4.79 | | | | | | 5.85 | 1.75 | -5.63 | 4.79 | 1.11 | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 5.85 +/- 1.75 ***** MAD = 5.85 +/- 1.75
** pVTZ, BFD ** pVTZ, BFD
@ -2857,11 +2881,12 @@ return result
| SiH4 | -6.277757 | 0.000145 | 0.519191 | 0.000151 | 0.515000 | 2.629616 | 0.094560 | 95 | | SiH4 | -6.277757 | 0.000145 | 0.519191 | 0.000151 | 0.515000 | 2.629616 | 0.094560 | 95 |
| SiO | -19.955719 | 0.000131 | 0.303829 | 0.000205 | 0.306700 | -1.801746 | 0.128826 | 47 | | SiO | -19.955719 | 0.000131 | 0.303829 | 0.000205 | 0.306700 | -1.801746 | 0.128826 | 47 |
|---------+------------+----------+----------+----------+-----------+------------+----------+------| |---------+------------+----------+----------+----------+-----------+------------+----------+------|
| | 3.52 | 0.19 | -1.03 | 4.39 | | 3.52 | 0.19 | | | | 3.52 | 0.19 | -1.03 | 4.39 | 0.04 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 3.52 +/- 0.19 ***** MAD = 3.52 +/- 0.19
*** $\mu=1/4$ *** $\mu=1/4$
@ -3021,11 +3046,12 @@ return result
| SiH4 | -6.279817 | 0.000685 | 0.519634 | 0.000691 | 0.515000 | 2.908075 | 0.433856 | 2019 | | SiH4 | -6.279817 | 0.000685 | 0.519634 | 0.000691 | 0.515000 | 2.908075 | 0.433856 | 2019 |
| SiO | -19.958859 | 0.000768 | 0.305851 | 0.000780 | 0.306700 | -0.532719 | 0.489741 | 845 | | SiO | -19.958859 | 0.000768 | 0.305851 | 0.000780 | 0.306700 | -0.532719 | 0.489741 | 845 |
|---------+------------+----------+----------+----------+-----------+------------+----------+-------| |---------+------------+----------+----------+----------+-----------+------------+----------+-------|
| | 3.39 | 0.77 | -0.59 | 4.44 | | 3.39 | 0.77 | | | | 3.39 | 0.77 | -0.59 | 4.44 | 0.34 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 3.39 +/- 0.77 ***** MAD = 3.39 +/- 0.77
*** $\mu=1/2$ *** $\mu=1/2$
@ -3260,11 +3286,12 @@ return result
| SiH4 | -6.281142 | 0.000138 | 0.517298 | 0.000148 | 0.515000 | 1.442121 | 0.092620 | 41778 | | SiH4 | -6.281142 | 0.000138 | 0.517298 | 0.000148 | 0.515000 | 1.442121 | 0.092620 | 41778 |
| SiO | -19.963472 | 0.000273 | 0.304821 | 0.000282 | 0.306700 | -1.178797 | 0.177217 | 26149 | | SiO | -19.963472 | 0.000273 | 0.304821 | 0.000282 | 0.306700 | -1.178797 | 0.177217 | 26149 |
|---------+------------+----------+----------+----------+-----------+------------+----------+---------| |---------+------------+----------+----------+----------+-----------+------------+----------+---------|
| | 2.46 | 0.18 | -1.72 | 3.02 | | | | | | | 2.46 | 0.18 | -1.72 | 3.02 | 0.06 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 2.46 +/- 0.18 ***** MAD = 2.46 +/- 0.18
*** $\mu=1$ *** $\mu=1$
@ -3424,11 +3451,12 @@ return result
| SiH4 | -6.282083 | 0.000683 | 0.515237 | 0.000709 | 0.515000 | 0.148971 | 0.445190 | 1594422 | | SiH4 | -6.282083 | 0.000683 | 0.515237 | 0.000709 | 0.515000 | 0.148971 | 0.445190 | 1594422 |
| SiO | -19.967377 | 0.002598 | 0.303241 | 0.002611 | 0.306700 | -2.170380 | 1.638215 | 1660400 | | SiO | -19.967377 | 0.002598 | 0.303241 | 0.002611 | 0.306700 | -2.170380 | 1.638215 | 1660400 |
|---------+------------+----------+----------+----------+-----------+------------+----------+----------| |---------+------------+----------+----------+----------+-----------+------------+----------+----------|
| | 4.38 | 0.94 | -4.24 | 5.11 | | | | | | | 4.38 | 0.94 | -4.24 | 5.11 | 0.31 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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) );%0.2f
#+TBLFM: @>$6='(org-sbe "sderrformula" (data @5$7..@>>$7) (data_err @5$8..@>>$8) );%0.2f
***** MAD = 4.38 +/- 0.94 ***** MAD = 4.38 +/- 0.94
***** Figure ***** Figure
@ -3441,6 +3469,20 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
#+RESULTS: #+RESULTS:
[[file:data-g2-qmc-tz-1.0-jast.png]] [[file:data-g2-qmc-tz-1.0-jast.png]]
*** 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 =
** pVQZ, BFD ** pVQZ, BFD
*** CCSD(T) *** CCSD(T)
**** Table **** Table
@ -4004,11 +4046,12 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
| SiH4 | -6.279824 | 0.000455 | 0.521153 | 0.000465 | 0.515000 | 3.861238 | 0.291920 | 95 | | SiH4 | -6.279824 | 0.000455 | 0.521153 | 0.000465 | 0.515000 | 3.861238 | 0.291920 | 95 |
| SiO | -19.956758 | 0.000627 | 0.305065 | 0.000650 | 0.306700 | -1.025872 | 0.407693 | 95 | | SiO | -19.956758 | 0.000627 | 0.305065 | 0.000650 | 0.306700 | -1.025872 | 0.407693 | 95 |
|---------+------------+----------+----------+----------+-----------+-----------+----------+------| |---------+------------+----------+----------+----------+-----------+-----------+----------+------|
| | 3.16 | 0.26 | -0.12 | 4.12 | | | | | | | 3.16 | 0.26 | -0.12 | 4.119 | 0.025 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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 = 3.16 +/- 0.26 ***** MAD = 3.16 +/- 0.26
*** $\mu=1/4$ *** $\mu=1/4$
@ -4168,11 +4211,12 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
| SiH4 | -6.281099 | 0.000478 | 0.520692 | 0.000479 | 0.515000 | 3.571494 | 0.300315 | 2251 | | SiH4 | -6.281099 | 0.000478 | 0.520692 | 0.000479 | 0.515000 | 3.571494 | 0.300315 | 2251 |
| SiO | -19.961401 | 0.000239 | 0.308118 | 0.000246 | 0.306700 | 0.889539 | 0.154490 | 2155 | | SiO | -19.961401 | 0.000239 | 0.308118 | 0.000246 | 0.306700 | 0.889539 | 0.154490 | 2155 |
|---------+------------+----------+----------+----------+-----------+-----------+----------+-------| |---------+------------+----------+----------+----------+-----------+-----------+----------+-------|
| | 2.90 | 0.25 | 0.39 | 3.74 | | | | | | | 2.90 | 0.25 | 0.39 | 3.745 | 0.005 | | | |
#+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f #+TBLFM: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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 ***** 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 | | 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 | | 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: @>$2='(org-sbe "madformula" (data @5$7..@>>$7) );%0.2f
#+TBLFM: @>$3='(org-sbe "maderrformula" (data @5$8..@>>$8) );%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: @>$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 ***** 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 ** SHCI CBS extrpolation
#+NAME: cbs-shci #+NAME: cbs-shci
| | e_cal | ae_cal | ae_nr | ae_diff | | | e_cal | ae_cal | ae_nr | ae_diff |

195
Manuscript/rsdft-cipsi-qmc.tex
View File

@ -681,13 +681,7 @@ noticeable with $\mu=5$~bohr$^{-1}$.
\subsection{Benchmark} \subsection{Benchmark}
The 55 molecules of the benchmark for the Gaussian-1 \begin{squeezetable}
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{table*} \begin{table*}
\caption{Mean absolute error (MAE), mean signed errors (MSE) and \caption{Mean absolute error (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with the different methods 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} \label{tab:mad}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{ll rrr rrr rrr} \begin{tabular}{ll rrr rrr rrr}
Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VQZ-BFD & \phantom{} \\ 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 \\ \phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
\hline \hline
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\ 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 \\ 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 \\ 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 \\ B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
\hline \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 \hline
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\ 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{} & 1/4 & 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{} & 1/2 & 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{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\ \phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\ \phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
\phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\ \phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
\hline \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 \\ 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(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{} & 1/4 & 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(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{} & 1/2 & 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(29) & -5.14(\phantom{0.}29) & 4.55 & 4.38(94) & -4.24(94) & 5.11 & --- & --- & --- \\ \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(83) & -5.91(\phantom{0.}83) & 4.79 & --- & --- & --- & --- & --- & --- \\ \phantom{} & 2 & 5.98(\phantom{0.}83) & -5.91(\phantom{0.}83) & 4.79(\phantom{0.}71) & --- & --- & --- & --- & --- & --- \\
\phantom{} & 5 & 6.18(84) & -6.13(\phantom{0.}84) & 4.87 & --- & --- & --- & --- & --- & --- \\ \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{} & \(\infty\) & 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 & --- & --- & --- & --- & --- & --- \\ \phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & --- & --- & --- & --- & --- & --- \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \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} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{g2-dmc.pdf} \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 trial wave functions. Each dot corresponds to an atomization
energy. energy.
The boxes contain the data between first and third quartiles, and 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} \label{fig:g2-dmc}
\end{figure} \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} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{g2-ndet.pdf} \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} \label{fig:g2-ndet}
\end{figure} \end{figure}
The number of determinants in the wave functions are shown in
figure~\ref{fig:n2-ndet}. For all the calculations, the stopping The number of determinants in the trial wave functions are shown in
criterion of the CIPSI algorithm was $\EPT < 1$~m\hartree{} or $\Ndet > figure~\ref{fig:g2-ndet}. As expected, the number of determinants
10^7$. is smaller when $\mu$ is small and larger when $\mu$ is large.
For FCI, we have given extrapolated values at $\EPT\rightarrow 0$.
At $\mu=0$ the number of determinants is not equal to one because 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. 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 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 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 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 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 have converged to the Kohn-Sham orbitals with $\mu=0$, and the
been the KS single determinant. 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} \section{Conclusion}
We have seen that introducing short-range correation via We have seen that introducing short-range correation via
a range-separated Hamiltonian in a full CI expansion yields improved a range-separated Hamiltonian in a full CI expansion yields improved
nodes, especially with small basis sets. The effect is similar to the nodal surfaces, especially with small basis sets. The effect of sr-DFT
effect of re-optimizing the CI coefficients in the presence of a on the determinant expansion is similar to the effect of re-optimizing
Jastrow factor, but without the burden of performing a stochastic the CI coefficients in the presence of a Jastrow factor, but without
optimization. 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 Varying the range-separation parameter $\mu$ and aproaching the
parameter $\mu$. The size-consistency error is controlled, as well as the RS-DFT-FCI with CIPSI provides a way to adapt the number of
invariance with respect to the spin projection $m_s$. determinants in the trial wave function, leading to size-consistent
Finding the optimal value of $\mu$ gives the lowest FN-DMC energies FN-DMC energies.
within basis set. However, if one wants to compute an energy We propose two methods. The first one is for the computation of
difference, one should not minimize the accurate total energies by a one-parameter optimization of the FN-DMC
FN-DMC energies of the reactants independently. It is preferable to energy via the variation of the parameter $\mu$.
choose a value of $\mu$ for which the fixed-node errors are well The second method is for the computation of energy differences, where
balanced, leading to a good cncellation of errors. We found that a the target is not the lowest possible FN-DMC energies but the best
value of $\mu=0.5$~bohr${^-1}$ is the value where the errors are the possible cancellation of errors. Using a fixed value of $\mu$
smallest. Moreover, such a small value of $\mu$ gives extermely 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 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.
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\begin{acknowledgments} \begin{acknowledgments}
An award of computer time was provided by the Innovative and Novel This work was performed using HPC resources from GENCI-TGCC (Grand
Computational Impact on Theory and Experiment (INCITE) program. This Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
research has used resources of the Argonne Leadership Computing 2019-0510.
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.
\end{acknowledgments} \end{acknowledgments}