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@ -4286,6 +4286,196 @@ plot data u 7:xtic(1) w l notitle, data u :7:8 w err notitle, 1 notitle, -1 noti
| | | | MAD | 1.01 |
#+TBLFM: @>$5='(org-sbe "madformula" (data @5$5..@>>$5) )
** Grossman
*** Data from the paper ([[doi:10.1063/1.1487829]])
| BeH | 43.0 | 0.2 | 46.90 | 0.01 | 0.06852566 | 3.18724e-4 | 0.074740778 | 1.59362e-5 |
| C2H2 | 390.0 | 0.4 | 386.9 | 0.2 | 0.6215118 | 6.37448e-4 | 0.61657158 | 3.18724e-4 |
| C2H4 | 533.5 | 0.4 | 531.9 | 0.1 | 0.85019627 | 6.37448e-4 | 0.84764648 | 1.59362e-4 |
| C2H6 | 669.3 | 0.4 | 666.3 | 0. | 1.0666099 | 6.37448e-4 | 1.0618290 | 0. |
| CH | 79.5 | 0.2 | 79.90 | 0.02 | 0.12669279 | 3.18724e-4 | 0.12733024 | 3.18724e-5 |
| CH2(1A1) | 169.7 | 0.4 | 170.6 | 0.4 | 0.27043731 | 6.37448e-4 | 0.27187157 | 6.37448e-4 |
| CH2(3B1) | 181.9 | 0.4 | 179.6 | 0.4 | 0.28987948 | 6.37448e-4 | 0.28621415 | 6.37448e-4 |
| CH3 | 290.9 | 0.2 | 289.3 | 0.2 | 0.46358406 | 3.18724e-4 | 0.46103427 | 3.18724e-4 |
| CH3Cl | 371.6 | 0.8 | 371.0 | 0. | 0.59218919 | 1.274896e-3 | 0.59123302 | 0. |
| CH4 | 395.0 | 0.2 | 392.5 | 0.1 | 0.6294799 | 3.18724e-4 | 0.62549585 | 1.59362e-4 |
| CN | 170.5 | 0.4 | 178 | 2.0 | 0.27171221 | 6.37448e-4 | 0.28366436 | 3.18724e-3 |
| CO | 253.2 | 0.3 | 256.2 | 0.2 | 0.40350458 | 4.78086e-4 | 0.40828544 | 3.18724e-4 |
| CO2 | 379.5 | 0.4 | 381.93 | 0.01 | 0.60477879 | 6.37448e-4 | 0.60865129 | 1.59362e-5 |
| CS | 165.4 | 0.5 | 169 | 6.0 | 0.26358475 | 7.9681e-4 | 0.26932178 | 9.56172e-3 |
| Cl2 | 54.3 | 0.2 | 57.18 | 0.01 | 0.086533566 | 3.18724e-4 | 0.091123192 | 1.59362e-5 |
| ClF | 53.7 | 0.6 | 59.1 | 0.1 | 0.085577394 | 9.56172e-4 | 0.094182942 | 1.59362e-4 |
| ClO | 55.4 | 0.4 | 63.42 | 0.02 | 0.088286548 | 6.37448e-4 | 0.10106738 | 3.18724e-5 |
| F2 | 32.0 | 0.8 | 36.9 | 0.1 | 0.05099584 | 1.274896e-3 | 0.058804578 | 1.59362e-4 |
| H2CO | 357.5 | 0.5 | 357.2 | 0.1 | 0.56971915 | 7.9681e-4 | 0.56924106 | 1.59362e-4 |
| H2O | 219.4 | 0.2 | 219.35 | 0.01 | 0.34964023 | 3.18724e-4 | 0.34956055 | 1.59362e-5 |
| H2O2 | 246.6 | 0.3 | 252.3 | 0. | 0.39298669 | 4.78086e-4 | 0.40207033 | 0. |
| H2S | 172.1 | 0.4 | 173.1 | 0.2 | 0.27426200 | 6.37448e-4 | 0.27585562 | 3.18724e-4 |
| H3COH | 483.8 | 0.5 | 480.8 | 0. | 0.77099336 | 7.9681e-4 | 0.76621250 | 0. |
| H3CSH | 446.0 | 0.4 | 445.1 | 0. | 0.71075452 | 6.37448e-4 | 0.70932026 | 0. |
| HCN | 302.0 | 0.8 | 301 | 2.0 | 0.48127324 | 1.274896e-3 | 0.47967962 | 3.18724e-3 |
| HCO | 269.8 | 0.4 | 270 | 2.0 | 0.42995868 | 6.37448e-4 | 0.4302774 | 3.18724e-3 |
| HCl | 103.4 | 0.4 | 102.2 | 0.5 | 0.16478031 | 6.37448e-4 | 0.16286796 | 7.9681e-4 |
| HF | 135.9 | 0.2 | 135.2 | 0.2 | 0.21657296 | 3.18724e-4 | 0.21545742 | 3.18724e-4 |
| HOCl | 152.8 | 0.4 | 156.3 | 0.5 | 0.24350514 | 6.37448e-4 | 0.24908281 | 7.9681e-4 |
| Li2 | 23.5 | 0.2 | 23.9 | 0.7 | 0.03745007 | 3.18724e-4 | 0.038087518 | 1.115534e-3 |
| LiF | 145.1 | 0.4 | 138 | 2.0 | 0.23123426 | 6.37448e-4 | 0.21991956 | 3.18724e-3 |
| LiH | 55.3 | 0.2 | 56.00 | 0.01 | 0.088127186 | 3.18724e-4 | 0.08924272 | 1.59362e-5 |
| N2 | 221.0 | 0.8 | 225.1 | 0.4 | 0.35219002 | 1.274896e-3 | 0.35872386 | 6.37448e-4 |
| N2H4 | 406.8 | 0.9 | 405.4 | 0. | 0.64828462 | 1.434258e-3 | 0.64605355 | 0. |
| NH | 78.2 | 0.4 | 79.0 | 0.4 | 0.12462108 | 6.37448e-4 | 0.12589598 | 6.37448e-4 |
| NH2 | 169.2 | 0.4 | 170.0 | 0.3 | 0.26964050 | 6.37448e-4 | 0.2709154 | 4.78086e-4 |
| NH3 | 276.5 | 0.2 | 276.7 | 0.1 | 0.44063593 | 3.18724e-4 | 0.44095465 | 1.59362e-4 |
| NO | 142.9 | 0.4 | 150.06 | 0.04 | 0.22772830 | 6.37448e-4 | 0.23913862 | 6.37448e-5 |
| Na2 | 17.3 | 0.2 | 16.8 | 0.3 | 0.027569626 | 3.18724e-4 | 0.026772816 | 4.78086e-4 |
| NaCl | 98.8 | 0.3 | 97.3 | 0.5 | 0.15744966 | 4.78086e-4 | 0.15505923 | 7.9681e-4 |
| O2 | 111.7 | 0.5 | 117.96 | 0.02 | 0.17800735 | 7.9681e-4 | 0.18798342 | 3.18724e-5 |
| OH | 101.2 | 0.3 | 101.4 | 0.3 | 0.16127434 | 4.78086e-4 | 0.16159307 | 4.78086e-4 |
| P2 | 107.9 | 0.2 | 116.1 | 0.5 | 0.17195160 | 3.18724e-4 | 0.18501928 | 7.9681e-4 |
| PH2 | 143.7 | 0.2 | 144.7 | 0.6 | 0.22900319 | 3.18724e-4 | 0.23059681 | 9.56172e-4 |
| PH3 | 224.8 | 0.2 | 228.6 | 0.4 | 0.35824578 | 3.18724e-4 | 0.36430153 | 6.37448e-4 |
| S2 | 98.3 | 0.3 | 100.66 | 0.07 | 0.15665285 | 4.78086e-4 | 0.16041379 | 1.115534e-4 |
| SO | 117.6 | 0.6 | 123.4 | 0.3 | 0.18740971 | 9.56172e-4 | 0.19665271 | 4.78086e-4 |
| SO2 | 240.0 | 0.8 | 254.0 | 0.2 | 0.3824688 | 1.274896e-3 | 0.40477948 | 3.18724e-4 |
| Si2 | 73.3 | 0.2 | 74.0 | 0. | 0.11681235 | 3.18724e-4 | 0.11792788 | 0. |
| Si2H6 | 505.8 | 0.4 | 500.1 | 0. | 0.80605300 | 6.37448e-4 | 0.79696936 | 0. |
| SiH2(1A1) | 145.5 | 0.2 | 144.4 | 0.2 | 0.23187171 | 3.18724e-4 | 0.23011873 | 3.18724e-4 |
| SiH2(3B1) | 125.8 | 0.2 | 123.4 | 0.2 | 0.20047740 | 3.18724e-4 | 0.19665271 | 3.18724e-4 |
| SiH3 | 215.1 | 0.2 | 214 | 1.0 | 0.34278766 | 3.18724e-4 | 0.34103468 | 1.59362e-3 |
| SiH4 | 305.8 | 0.2 | 302.6 | 0.5 | 0.48732900 | 3.18724e-4 | 0.48222941 | 7.9681e-4 |
| SiO | 186.7 | 0.2 | 190 | 2.0 | 0.29752885 | 3.18724e-4 | 0.3027878 | 3.18724e-3 |
#+TBLFM: $6=$2*0.00159362 :: $7=$3*0.00159362 :: $8=$4*0.00159362 :: $9=$5*0.00159362
*** Data
#+NAME: grossman-data
| | ae_cal | Error ae_cal | ae_nr | ae_diff |
| | Hartree | Hartree | Hartree | kcal/mol |
|---------+-------------+--------------+----------+--------------------|
| BeH | 0.06852566 | 3.18724e-4 | 0.079400 | -6.8236719 |
| C2H2 | 0.6215118 | 6.37448e-4 | 0.642400 | -13.107391 |
| C2H4 | 0.85019627 | 6.37448e-4 | 0.899000 | -30.624446 |
| C2H6 | 1.0666099 | 6.37448e-4 | 1.136900 | -44.107190 |
| CH | 0.12669279 | 3.18724e-4 | 0.133900 | -4.5225399 |
| CH2_1A1 | 0.27043731 | 6.37448e-4 | 0.288900 | -11.585378 |
| CH2_3B1 | 0.28987948 | 6.37448e-4 | 0.304100 | -8.9234071 |
| CH3 | 0.46358406 | 3.18724e-4 | 0.490800 | -17.078061 |
| CH3Cl | 0.59218919 | 1.274896e-3 | 0.631000 | -24.353867 |
| CH4 | 0.6294799 | 3.18724e-4 | 0.670300 | -25.614701 |
| CN | 0.27171221 | 6.37448e-4 | 0.288800 | -10.722625 |
| CO | 0.40350458 | 4.78086e-4 | 0.413700 | -6.3976481 |
| CO2 | 0.60477879 | 6.37448e-4 | 0.621400 | -10.429845 |
| CS | 0.26358475 | 7.9681e-4 | 0.274000 | -6.5355919 |
| Cl2 | 0.086533566 | 3.18724e-4 | 0.094000 | -4.6852035 |
| ClF | 0.085577394 | 9.56172e-4 | 0.100100 | -9.1129667 |
| ClO | 0.088286548 | 6.37448e-4 | 0.104700 | -10.299477 |
| F2 | 0.05099584 | 1.274896e-3 | 0.062200 | -7.0306347 |
| H2CO | 0.56971915 | 7.9681e-4 | 0.596700 | -16.930542 |
| H2O | 0.34964023 | 3.18724e-4 | 0.371900 | -13.968054 |
| H2O2 | 0.39298669 | 4.78086e-4 | 0.429400 | -22.849431 |
| H2S | 0.27426200 | 6.37448e-4 | 0.292000 | -11.130633 |
| H3COH | 0.77099336 | 7.9681e-4 | 0.818700 | -29.936020 |
| H3CSH | 0.71075452 | 6.37448e-4 | 0.757000 | -29.019139 |
| HCN | 0.48127324 | 1.274896e-3 | 0.496900 | -9.8058257 |
| HCO | 0.42995868 | 6.37448e-4 | 0.444700 | -9.2502102 |
| HCl | 0.16478031 | 6.37448e-4 | 0.171000 | -3.9028689 |
| HF | 0.21657296 | 3.18724e-4 | 0.226100 | -5.9782382 |
| HOCl | 0.24350514 | 6.37448e-4 | 0.264700 | -13.299821 |
| Li2 | 0.03745007 | 3.18724e-4 | 0.038900 | -0.90983421 |
| LiF | 0.23123426 | 6.37448e-4 | 0.222000 | 5.7945181 |
| LiH | 0.088127186 | 3.18724e-4 | 0.092430 | -2.7000251 |
| N2 | 0.35219002 | 1.274896e-3 | 0.364600 | -7.7872893 |
| N2H4 | 0.64828462 | 1.434258e-3 | 0.699600 | -32.200512 |
| NH | 0.12462108 | 6.37448e-4 | 0.133500 | -5.5715415 |
| NH2 | 0.26964050 | 6.37448e-4 | 0.290400 | -13.026631 |
| NH3 | 0.44063593 | 3.18724e-4 | 0.475500 | -21.877279 |
| NO | 0.22772830 | 6.37448e-4 | 0.244500 | -10.524278 |
| Na2 | 0.027569626 | 3.18724e-4 | 0.026800 | 0.48294198 |
| NaCl | 0.15744966 | 4.78086e-4 | 0.157400 | 0.031161758 |
| O2 | 0.17800735 | 7.9681e-4 | 0.192400 | -9.0314190 |
| OH | 0.16127434 | 4.78086e-4 | 0.170200 | -5.6008710 |
| P2 | 0.17195160 | 3.18724e-4 | 0.186000 | -8.8154014 |
| PH2 | 0.22900319 | 3.18724e-4 | 0.244000 | -9.4105307 |
| PH3 | 0.35824578 | 3.18724e-4 | 0.389000 | -19.298340 |
| S2 | 0.15665285 | 4.78086e-4 | 0.164000 | -4.6103525 |
| SO | 0.18740971 | 9.56172e-4 | 0.200700 | -8.3396857 |
| SO2 | 0.3824688 | 1.274896e-3 | 0.414400 | -20.036897 |
| Si2 | 0.11681235 | 3.18724e-4 | 0.121000 | -2.6277594 |
| Si2H6 | 0.80605300 | 6.37448e-4 | 0.849000 | -26.949335 |
| SiH2_1A1 | 0.23187171 | 3.18724e-4 | 0.243000 | -6.9830261 |
| SiH2_3B1 | 0.20047740 | 3.18724e-4 | 0.210000 | -5.9754521 |
| SiH3 | 0.34278766 | 3.18724e-4 | 0.363000 | -12.683287 |
| SiH4 | 0.48732900 | 3.18724e-4 | 0.515000 | -17.363612 |
| SiO | 0.29752885 | 3.18724e-4 | 0.306700 | -5.7549165 |
|---------+-------------+--------------+----------+--------------------|
| | | | MAD | 12.499646457509431 |
#+TBLFM: $5=($2-$4)/0.00159362
#+TBLFM: @>$5='(org-sbe "madformula" (data @5$5..@>>$5) )
*** Grossman's reference values
#+NAME: grossman-ref
| | ae_cal | error ae_cal | ae_nr | ae_diff |
| | Hartree | Hartree | Hartree | kcal/mol |
|---------+-------------+-------------+----------+-------------------|
| BeH | 0.074740778 | 1.59362e-5 | 0.079400 | -2.9236719 |
| C2H2 | 0.61657158 | 3.18724e-4 | 0.642400 | -16.207389 |
| C2H4 | 0.84764648 | 1.59362e-4 | 0.899000 | -32.224445 |
| C2H6 | 1.0618290 | 0. | 1.136900 | -47.107215 |
| CH | 0.12733024 | 3.18724e-5 | 0.133900 | -4.1225386 |
| CH2_1A1 | 0.27187157 | 6.37448e-4 | 0.288900 | -10.685377 |
| CH2_3B1 | 0.28621415 | 6.37448e-4 | 0.304100 | -11.223410 |
| CH3 | 0.46103427 | 3.18724e-4 | 0.490800 | -18.678060 |
| CH3Cl | 0.59123302 | 0. | 0.631000 | -24.953866 |
| CH4 | 0.62549585 | 1.59362e-4 | 0.670300 | -28.114701 |
| CN | 0.28366436 | 3.18724e-3 | 0.288800 | -3.2226252 |
| CO | 0.40828544 | 3.18724e-4 | 0.413700 | -3.3976481 |
| CO2 | 0.60865129 | 1.59362e-5 | 0.621400 | -7.9998431 |
| CS | 0.26932178 | 9.56172e-3 | 0.274000 | -2.9355932 |
| Cl2 | 0.091123192 | 1.59362e-5 | 0.094000 | -1.8052032 |
| ClF | 0.094182942 | 1.59362e-4 | 0.100100 | -3.7129667 |
| ClO | 0.10106738 | 3.18724e-5 | 0.104700 | -2.2794769 |
| F2 | 0.058804578 | 1.59362e-4 | 0.062200 | -2.1306347 |
| H2CO | 0.56924106 | 1.59362e-4 | 0.596700 | -17.230544 |
| H2O | 0.34956055 | 1.59362e-5 | 0.371900 | -14.018053 |
| H2O2 | 0.40207033 | 0. | 0.429400 | -17.149427 |
| H2S | 0.27585562 | 3.18724e-4 | 0.292000 | -10.130633 |
| H3COH | 0.76621250 | 0. | 0.818700 | -32.936020 |
| H3CSH | 0.70932026 | 0. | 0.757000 | -29.919140 |
| HCN | 0.47967962 | 3.18724e-3 | 0.496900 | -10.805826 |
| HCO | 0.4302774 | 3.18724e-3 | 0.444700 | -9.0502127 |
| HCl | 0.16286796 | 7.9681e-4 | 0.171000 | -5.1028727 |
| HF | 0.21545742 | 3.18724e-4 | 0.226100 | -6.6782420 |
| HOCl | 0.24908281 | 7.9681e-4 | 0.264700 | -9.7998205 |
| Li2 | 0.038087518 | 1.115534e-3 | 0.038900 | -0.50983421 |
| LiF | 0.21991956 | 3.18724e-3 | 0.222000 | -1.3054806 |
| LiH | 0.08924272 | 1.59362e-5 | 0.092430 | -2.0000251 |
| N2 | 0.35872386 | 6.37448e-4 | 0.364600 | -3.6872906 |
| N2H4 | 0.64605355 | 0. | 0.699600 | -33.600513 |
| NH | 0.12589598 | 6.37448e-4 | 0.133500 | -4.7715390 |
| NH2 | 0.2709154 | 4.78086e-4 | 0.290400 | -12.226629 |
| NH3 | 0.44095465 | 1.59362e-4 | 0.475500 | -21.677282 |
| NO | 0.23913862 | 6.37448e-5 | 0.244500 | -3.3642776 |
| Na2 | 0.026772816 | 4.78086e-4 | 0.026800 | -0.017058019 |
| NaCl | 0.15505923 | 7.9681e-4 | 0.157400 | -1.4688382 |
| O2 | 0.18798342 | 3.18724e-5 | 0.192400 | -2.7714135 |
| OH | 0.16159307 | 4.78086e-4 | 0.170200 | -5.4008672 |
| P2 | 0.18501928 | 7.9681e-4 | 0.186000 | -0.61540392 |
| PH2 | 0.23059681 | 9.56172e-4 | 0.244000 | -8.4105307 |
| PH3 | 0.36430153 | 6.37448e-4 | 0.389000 | -15.498343 |
| S2 | 0.16041379 | 1.115534e-4 | 0.164000 | -2.2503545 |
| SO | 0.19665271 | 4.78086e-4 | 0.200700 | -2.5396832 |
| SO2 | 0.40477948 | 3.18724e-4 | 0.414400 | -6.0368971 |
| Si2 | 0.11792788 | 0. | 0.121000 | -1.9277620 |
| Si2H6 | 0.79696936 | 0. | 0.849000 | -32.649339 |
| SiH2_1A1 | 0.23011873 | 3.18724e-4 | 0.243000 | -8.0830248 |
| SiH2_3B1 | 0.19665271 | 3.18724e-4 | 0.210000 | -8.3754534 |
| SiH3 | 0.34103468 | 1.59362e-3 | 0.363000 | -13.783286 |
| SiH4 | 0.48222941 | 7.9681e-4 | 0.515000 | -20.563616 |
| SiO | 0.3027878 | 3.18724e-3 | 0.306700 | -2.4549140 |
|---------+-------------+-------------+----------+-------------------|
| | | | MAD | 11.00762358960377 |
#+TBLFM: $5=($2-$4)/0.00159362
#+TBLFM: @>$5='(org-sbe "madformula" (data @5$5..@>>$5) )
** Data export
#+begin_src python :results output :session :exports code

17
Manuscript/Makefile Executable file
View File

@ -0,0 +1,17 @@
TARGET=rsdft-cipsi-qmc.pdf
SRC=rsdft-cipsi-qmc.tex
.PHONY: all
all: $(TARGET)
$(TARGET): $(SRC)
latexmk -pdf $(SRC)
.PHONY: clean
clean:
-rm -f $(TARGET)
follow:
latexmk -pdf -view=pdf -bibtex -pvc -gg -interaction=nonstopmode $(SRC)

30
Manuscript/rsdft-cipsi-qmc.bib
View File

@ -597,3 +597,33 @@ note={Gaussian Inc. Wallingford CT}
publisher = {American Chemical Society},
doi = {10.1021/acs.jpca.9b01933}
}
@article{Dubecky_2014,
author = {Dubecký, Matúš and Derian, René and
Jurečka, Petr and Mitas, Lubos and Hobza, Pavel and Otyepka, Michal},
title = {{Quantum Monte Carlo for noncovalent interactions: an efficient protocol attaining
benchmark accuracy}},
journal = {Phys. Chem. Chem. Phys.},
volume = {16},
number = {38},
pages = {20915--20923},
year = {2014},
month = {Sep},
issn = {1463-9084},
publisher = {The Royal Society of Chemistry},
doi = {10.1039/C4CP02093F}
}
@article{Grossman_2002,
author = {Grossman, Jeffrey C.},
title = {{Benchmark quantum Monte Carlo calculations}},
journal = {J. Chem. Phys.},
volume = {117},
number = {4},
pages = {1434--1440},
year = {2002},
month = {Jul},
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.1487829}
}

93
Manuscript/rsdft-cipsi-qmc.tex
View File

@ -50,19 +50,21 @@
\section{Introduction}
\label{sec:intro}
The full configuration interaction (FCI) method \eg{within an incomplete basis set}
leads to the exact solution of the Schrödinger equation with an approximate Hamiltonian
\eg{which consists in the exact one projected onto } \sout{expressed in} a finite basis of Slater determinants.
The FCI wave function can be interpreted as the exact solution of the
true Hamiltonian obtained with the additional constraint that it
can only span the space provided by the basis. At the complete
basis set (CBS) limit, the constraint vanishes and the exact solution
is obtained.
The full configuration interaction (FCI) method within a finite atomic
basis set leads to an approximate solution of the Schrödinger
equation.
This solution is the eigenpair of an approximate Hamiltonian, which is
the projection of the exact Hamiltonian onto the finite basis of all
possible Slater determinants.
The FCI wave function can be interpreted as the constrained solution of the
true Hamiltonian, where the solution is forced to span the space
provided by the basis.
At the complete basis set (CBS) limit, the constraint vanishes and the
exact solution is obtained.
Hence the FCI method enables a systematic improvement of the
calculations by \sout{increasing the size} \eg{improving the quality} of the basis set. Nevertheless,
its exponential scaling with the number of electrons and with the size
of the basis is prohibitive to treat large systems.
calculations by improving the quality of the basis set.
Nevertheless, its exponential scaling with the number of electrons and
with the size of the basis is prohibitive for large systems.
In recent years, the introduction of new algorithms\cite{Booth_2009}
and the
revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
@ -75,32 +77,42 @@ to a loss of size extensivity.
The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
the exact solution of the Schrödinger equation with an additional
constraint, imposing the solution to have the same nodal hypersurface
as a given trial wave function. This approximation, known as the
\emph{fixed-node} approximation, \eg{is variational with respect to the nodes of the trial wave function: the DMC energy obtained with a given trial wave function is an upper bound to the exact energy, and the latter is recovered only }
when the nodes of the trial wave function coincide with the nodes of the exact wave function\sout{, the exact energy and wave function are obtained}.
The DMC method is attractive because its scaling is polynomial with
the number of electrons and with the size of the trial wave
function. Moreover, the total energies obtained are usually below
those obtained with FCI \eg{in computationally tractable basis sets} because the fixed-node approximation imposes
less constraints on the solution than the finite-basis approximation.
as a given trial wave function.
Within this so-called \emph{fixed-node} approximation,
the DMC energy associated with a given trial wave function is an upper
bound to the exact energy, and the latter is recovered only when the
nodes of the trial wave function coincide with the nodes of the exact
wave function.
The polynomial scaling with the number of electrons and with the size
of the trial wave function makes the DMC method attractive.
In addition, the total energies obtained are usually far below
those obtained with the FCI method in computationally tractable basis
sets because the constraints imposed by the fixed-node approximation
are less severe than the constraints imposed by the finite-basis
approximation.
\eg{The qualitative picture of the electronic structure of weakly correlated systems, such as organic molecule near their equilibrium geometry, }\sout{In many cases, the systems under study} are well described by a single
Slater determinant. \sout{Single-determinant} DMC \eg{with a single-determinant trial wave function} can be used as a post-Hatree-Fock
single-reference method with an accuracy comparable to coupled cluster\eg{mettre la ref}.
The qualitative picture of the electronic structure of weakly
correlated systems, such as organic molecules near their equilibrium
geometry, is usually well represented with a single Slater
determinant.
DMC with a single-determinant trial wave function can be used as a
single-reference post-Hatree-Fock method, with an accuracy comparable
to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
The favorable scaling of QMC and its adequation with massively
parallel architectures makes it an attractive alternative for large
parallel architectures makes it a favourable alternative for large
systems.
The Slater determinant is determined by the nature of the molecular
orbitals. Three main options are commonly used: Hartree-Fock (HF),
Kohn-Sham (KS) or natural orbitals (NO) of a correlated wave function.
The nodal surfaces obtained with the KS determinant are in general
better than those obtained with the HF determinant,\cite{Per_2012} and
of comparable quality to those obtained with a Slater determinant
built with NOs.\cite{Wang_2019}
\eg{The choice of the Slater determinant entirely depends on the type of orbitals used to build it, for which three main options are available: the Kohn-Sham\ref{} (KS) scheme, the HF scheme or the natural orbitals (NO) of a correlated wave function. }
As it has been shown by many studies\cite{Per_2012}, the nodal surfaces obtained with the
KS determinant are in general better than those obtained with
the HF determinant, and of comparable quality
to those obtained with a Slater determinant built with NO.\cite{Wang_2019}
However, the fixed-node approximation is much more difficult to
control than the finite-basis approximation, as it is not possible
to minimize directly the DMC energy with respect to the variational
parameters of the trial wave function.
As it is not possible to minimize directly the DMC energy with respect
to the variational parameters of the trial wave function, the
fixed-node approximation is much more difficult to control than the
finite-basis approximation.
The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph{Jastrow factor}, taking
account of the electron-electron cusp and the short-range correlation
@ -117,13 +129,12 @@ When the basis set is increased, the trial wave function tends to the
exact wave function, so the nodal surface can be systematically
improved.\cite{Caffarel_2016}
This technique has the advantage that using FCI nodes in a given basis
set is well defined and has a unique solution. The optimization of the
wave function is deterministic, so the calculations are reproducible
and don't require the expertise of a QMC expert. However,
this technique can't be applied to large systems because of the
exponential scaling of the size of the wave function. Extrapolation
techniques have been applied to estimate the DMC energy of a FCI
wave function in a large basis set,\cite{Scemama_2018} and other
set is well defined, so the calculations are reproducible in a
black-box way without needing any expertise in QMC.
But this technique can't be applied to large systems because of the
exponential scaling of the size of the wave function.
Extrapolation techniques have been used to estimate the DMC energy of
FCI wave functions in a large basis sets,\cite{Scemama_2018} and other
authors have used a combination of the two approaches where CIPSI
trial wave functions are re-optimized under the presence of a Jastrow
factor.\cite{Giner_2016,Dash_2018,Dash_2019}