+------------------------------------+ ! Pb 11.06.99 CI/MBPT2 RCC ! +------------------------------------+ All-electron Dirac-Coulomb and RECP calculations of excitation energies for lead atom with combined CI/MBPT2 and RCC methods. The number of the grid points: II=491 The radius of the last point: R2=50.0 Point nucleus: AM=0.0 V**N-2 approximation (i.e. 6s^2 HF field) is used for MBPT The orbital energies for MBPT: Kval=1 (i.e assigned orbital energy is equal HFD energy of the last valence orbital of the same symmetry) Total valence energies for MBPT: E=-3.3 a.u. Kmax=9 Lmax=2 Nsx1=5 Nsx2=0 (Nav=40) Nmax=60 - all notations relay to sgc.exe and scrc.exe -------------- TeX file with the results of SCF cslculations ------------- -------------- with GRECP and RECP of Christiansen et al. ------------- %***cut--here************************************************************ \documentclass[12pt]{iopart} \begin{document} \begin{table} \caption{ Results of the SCF calculations of the low-lying states of the lead atom and its ions with 22 electron RECPs. All values are in a.u.$\times 10^{-5}$. } \begin{tabular}{lrrrrrr} %\br \cline{1-7} Conf. & &\multicolumn{2}{c}{Exact}& GRECP &Ionic GRECP& RECP $^{\rm b}$ \\ & &\multicolumn{2}{c}{DF } & & & \\ \cline{2-7} & & Transition & \multicolumn{4}{c}{Errors$^{\rm a}$}\\ & & energy & & & & \\ \cline{1-7} \multicolumn{7}{l}{States averaged over nonrelativistic configuration} \\ $6s^2 6p^2 $ & & 0 & & 0 & 0 & 0 \\ $6s^2 6p^1 6d^1 $ & & 16896 & & 2 & 20 & 5 \\ $6s^1 6p^1 6d^1 $ & & 81086 & & 21 & 148 & 165 \\ $6s^1 6p^2 6d^1 $ & & 47949 & & 25 & 145 & 137 \\ $6s^1 6p^2 5f^1 $ & & 50718 & & 25 & 143 & 135 \\ $6s^1 6p^2 $ & & 53850 & & 25 & 143 & 135 \\ \multicolumn{7}{l}{States averaged over relativistic configurations}\\ $6s_{1/2}^1 6p_{1/2}^2 $ & & 46484 & & 28 & 173 & 165 \\ $6s_{1/2}^1 6p_{3/2}^2 $ & & 58430 & & 28 & 131 & 105 \\ $6s_{1/2}^1 6p_{1/2}^1 6p_{3/2}^1$ & & 51281 & & 24 & 149 & 130 \\ \multicolumn{7}{l}{Terms }\\ $...$ $ (J=1/2)$ & & 55983 & & 45 & 163 & 140 \\ $...$ $ (J=3/2)$ & & 50706 & & 22 & 148 & 123 \\ $...$ $ (J=5/2)$ & & 50318 & & 14 & 144 & 121 \\ \cline{1-7} %\br \end{tabular} \noindent $^{\rm a}$In this table, errors were calculated as differences between the total energies from the GRECP and all-electron SCF calculations. The point nuclear model was employed.\\ \noindent $^{\rm b}$ RECP of Christiansen et al~\cite{Christ}. \end{table} \section*{References} \begin{thebibliography}{99} \bibitem{Christ} P.A.Christiansen 1999, private communication;\\ S.A.Wildman, G.A.DiLabio and P.A.Christiansen, J.\ Chem.\ Phys. 107 (1997) 9975 \end{thebibliography} \end{document} ************************************************************************** Basis for 4e-CI (optimized for 4e-CI) is derived from the numerical SCF calculations of the corresponding spinors for the following nonrelativistic averaged configurations (for positive ion states): 6p^2, [5d^9] 6d^1, [5d^9] 5f^1 [6p^1] 7s^1, [6p^1] 7p^1, [5d^7] 7d^1, [5d^7] 6f^1 [6s^1] 8s^1, [6s^1] 8p^1, [5d^6] 8d^1, [5d^8] 9p^1, [5d^9]10p^1, Then functions are "diagonalized" in 6s^2 HF field with an exception of 6p 7s and 7p ones. Those orbitals were taken "as is" from HFD calculations of neutral atom. Files with corresponding data - $HOME/atom/ci/res 20348 Jul 13 1998 10d 3001 Dec 11 1998 10p 30426 Jul 17 1998 11d 24361 Jul 16 1998 11p 22254 Jul 19 1998 12d 20259 Jul 17 1998 12p 20292 Jul 22 1998 13d 25349 Jul 20 1998 13p 10131 Jul 27 1998 14d 30 Jul 27 1998 14d.en 20253 Jul 23 1998 14p 30 Jul 23 1998 14p.en 20221 Jul 28 1998 15p 32 Jul 28 1998 15p.en 938 Nov 12 1997 5f 4340 Nov 30 1998 5g 3703 Oct 29 1998 6d 3670 Nov 29 1998 6f 3700 Oct 29 1998 7d 12084 Jul 6 1998 7f 20348 Jul 13 1998 10d 3001 Dec 11 1998 10p 30426 Jul 17 1998 11d 24361 Jul 16 1998 11p 22254 Jul 19 1998 12d 20259 Jul 17 1998 12p 20292 Jul 22 1998 13d 25349 Jul 20 1998 13p 10131 Jul 27 1998 14d 30 Jul 27 1998 14d.en 20253 Jul 23 1998 14p 30 Jul 23 1998 14p.en 20221 Jul 28 1998 15p 32 Jul 28 1998 15p.en 938 Nov 12 1997 5f 4340 Nov 30 1998 5g 3703 Oct 29 1998 6d 3670 Nov 29 1998 6f 3700 Oct 29 1998 7d 12084 Jul 6 1998 7f 4425 Nov 16 1997 8d 2256 Oct 29 1998 8p 3067 Oct 29 1998 8s 35916 Jul 10 1998 9d 2334 Oct 30 1998 9p Basis for MBPT (optimized for 36 correlated electrons): 6p^2, [6p^1] 7s^1, [5d^9] 6d^1, [6p^1] 7p^1, [6s^1] 8s^1, [5d^9] 5f^1, [5d^7] 7d^1, [6s^1] 8p^1, [5d^7] 9s^1, [5d^1] 5g^1, [5d^5] 6f^1, [5d^3] 8d^1, [5d^8] 9p^1, [5d^1] 10s^1, [5p^3] 6g^1, [5p^5] 7f^1, [5d^8] 9d^1, [5d^4] 10p^1, [4f^13] 11s^1, [4f^12] 7g^1, [4f^13]8f^1, [4f^8] 10d^1, [5p^3] 11p^1, [4f^8] 8g^1, [5d^1] 11d^1, where h and i functions are derived from the "non-diagonalized" 5g,6g,... ones by multipling their radial parts (large components in all-electron case) on radial variable r and r^2. Then functions are "diagonalized" in 6s^2 HF field. Files with corresponding data - $HOME/ci/res/mbpt 14910 Feb 22 11:10 10d.pt 11814 Feb 10 1999 10p.pt 12747 Feb 12 1999 10s.pt 16029 Feb 22 11:51 11d.pt 15887 Feb 16 1999 11p.pt 3967 Apr 6 1999 11p.pt.corr.last 10953 Feb 11 17:33 11s.pt 60 Feb 24 16:48 12d.pt 13133 Feb 18 1999 12s.pt 6447 Jan 30 1999 5f.pt 8810 Feb 3 1999 5g.pt 3746 Feb 20 1999 6f.pt 12523 Feb 9 1999 6g.pt 788 Apr 28 1999 6h.pt 14390 Feb 21 1999 7f.pt 5093 Feb 13 16:41 7g.pt.corr 721 Apr 28 1999 7h.pt 7874 Feb 1 1999 8d.pt 3869 Feb 22 10:33 8f.pt.corr 8484 Feb 24 13:03 8g.pt.corr 8683 Feb 10 1999 9d.pt 69 Feb 24 13:07 9f.pt 8098 Feb 2 1999 9s.pt 14910 Feb 22 11:10 10d.pt 11814 Feb 10 1999 10p.pt 12747 Feb 12 1999 10s.pt 16029 Feb 22 11:51 11d.pt 15887 Feb 16 1999 11p.pt 3967 Apr 6 1999 11p.pt.corr.last 10953 Feb 11 17:33 11s.pt 60 Feb 24 16:48 12d.pt 13133 Feb 18 1999 12s.pt 6447 Jan 30 1999 5f.pt 8810 Feb 3 1999 5g.pt 3746 Feb 20 1999 6f.pt 12523 Feb 9 1999 6g.pt 788 Apr 28 1999 6h.pt 14390 Feb 21 1999 7f.pt 5093 Feb 13 16:41 7g.pt.corr 721 Apr 28 1999 7h.pt 7874 Feb 1 1999 8d.pt 3869 Feb 22 10:33 8f.pt.corr 8484 Feb 24 13:03 8g.pt.corr 8683 Feb 10 1999 9d.pt 69 Feb 24 13:07 9f.pt 8098 Feb 2 1999 9s.pt Transitions between the following states are considered: State Leading conf. (J,parity) 0g 6s^2 6p^2/P^3 1g 6s^2 6p^2/P^3 2g 6s^2 6p^2/P^3 2g 6s^2 6p^2/D^1 0g 6s^2 6p^2/S^1 Total energies in a.u. with orbitals from all-electron Dirac-Coulomb calculations (point nuclei model) spd-basis spdf-basis spdfg-bas spdfgh-bas 4e-CI -20919.71073404 -20919.71428843 [3,5,3,2] -20919.67959669 -20919.68233896 -20919.66440575 -20919.66918744 -20919.61767300 -20919.62429876 -20919.58086526 -20919.58710595 22e-CI/MBPT2 -20919.78012775 -20919.78272751 [3,5,3,2]/ -20919.74669349 -20919.74920007 [4,5,4,2,2,2] -20919.73340011 -20919.73588426 -20919.68589738 -20919.68827331 -20919.64707797 -20919.64939176 22e-CI/MBPT2 -20919.68983675 [3,5,3]/ -20919.65869888 [3,5,3] -20919.64318133 -20919.59577983 -20919.55695411 14e-CI/MBPT2 -20919.70530667 -20919.79266959 [3,5,3,2]/ -20919.67414334 -20919.75948018 [7,6,6,4,4] -20919.65864340 -20919.74609221 -20919.61123921 -20919.69876846 -20919.57231985 -20919.65964019 22e-CI/MBPT2 -20919.69979797 -20919.77510058 -20919.79307308 [3,5,3,2]/ -20919.66852654 -20919.74243942 -20919.75823543 [7,6,6,2,4] -20919.65302613 -20919.72892233 -20919.74470804 -20919.60550678 -20919.68144425 -20919.69537475 -20919.56664888 -20919.64222589 -20919.65707696 14e-CI/MBPT2 -20919.78611380 [3,5,3,2]/ -20919.75311642 [4,5,4,2,2,2] -20919.73974601 -20919.69263530 -20919.65355134 36e-CI/MBPT2 -20919.79345794 [3,5,3,2]/ -20919.75949211 [7,6,6,4,4] -20919.74621382 -20919.69752325 -20919.65878845 Transition energies in 1/cm from all-electron Dirac-Coulomb calculations (point nucleu) |------------------------------------| | Experimental transition energies: | | in 1/cm | | 0 | | 7819 | | 10650 | | 21457 | | 29466 | |------------------------------------| sp-basis spd-basis spdf-basis spdfg-basis spdfgh-basis 4e-CI 0 0 | 0 | [3,5,3,2] 7157 6833 | 7012 | 10649 10167 | 9898 | 21154 20424 | 19750 | 28174 28502 | 27913 | | | 22e-CI/MBPT2 | | 0 0 [3,5,3,2]/ | | 7338 7358 [4,5,4,2,2,2] | | 10255 10280 | | 20681 20730 | | 29201 29263 | | 22e-CI/MBPT2 0 | | [3,5,3]/ 6834 | | [3,5,3] 10239 | | 20643 | | 29164 | | | | 14e-CI/MBPT2 0 | 0 | 0 [3,5,3,2]/ 6839 | 7248 | 7284 [7,6,6,4,4] 10241 | 10250 | 10222 20645 | 20567 | 20608 29187 | 29138 | 29196 | | | | 22e-CI/MBPT2 0 | 0 | 0 [3,5,3,2]/ 6863 | 7168 | 7646 [7,6,6,2,4] 10265 | 10135 | 10614 20694 | 20555 | 21442 29222 | 29162 | 29847 | | 22e-CI/MBPT2 | 0 | 0 [3,5,3,2]/ | 7339 | 7426 [7,6,6,4,4] | 10336 | 10356 | 20749 | 20884 | 29287 | 29433 | | | | 14e-CI/MBPT2 | | 0 [3,5,3,2]/ | | 7242 [4,5,4,2,2,2] | | 10176 | | 20516 | | 29094 | | 36e-CI/MBPT2 | | 0 [3,5,3,2]/ | | 7454 [7,6,6,4,4] | | 10368 | | 21055 | | 29556 |--------------------------------------------------------------------------| | RCC | |--------------------------------------------------------------------------| Numerical HFD basis set was taken - in the same way as in CI/MBPT2 Total energies - in 1/cm - only for ground state spd-bas spdf-bas spdfg-bas spdfgh-bas 4eRCC -20919.70758946 -20919.71162396 [3,5,3,2] 4eRCC -20919.71117175 -20919.717752771 [7,6,6,4,4] 22eRCC -20919.88266650 -20920.359200681 -20920.36316162 [7,6,6,4,4,2] 22eRCC -20919.80044265 [3,5,3] 14eRCC -20919.80866317 -20920.07220877 -20920.10646296 [7,6,6,4,4] 36eRCC [7,6,6,4,4,2] -20920.44880326 -20920.91589645 -20920.91988336 Transition energies in 1/cm - first number in each column-ionization potential |------------------------------------| | Experimental transition energies: | | in 1/cm | | 0 | | 7819 | | 10650 | | 21457 | | 29466 | |------------------------------------| sp-bas spd-bas | spdf-bas | spdfg-bas spdfgh-bas | | 4eRCC 53796 55911 | 56367 | [3,5,3,2] 6945 6532 | 6658 | 10421 9719 | 9421 | 20834 19687 | 19194 | 28264 28491 | 28104 | | | 4eRCC 56343 | 56896 *+2p | 57053 [7,6,6,4,4] 6569 | 6735 | 6772 9790 | 9540 | 9557 19839 | 19415 | 19459 28753 | 28370 | 28368 | | 22eRCC 57741 | | 59648 59778 [7,6,6,4,4,2] 6912 | | 7369 7397 10272 | | 10286 10299 20729 | | 20839 20867 29845 | | 29665 29672 | | 22eRCC 57290 | | [3,5,3] 6832 | | 10165 | | 20504 | | 29522 | | | | 14eRCC 57845 | 58695 | 59490 [7,6,6,4,4] 6855 | 6880 | 7189 10214 | 9892 | 10111 20623 | 20063 | 20494 29779 | 29222 | 29396 | | 36eRCC | 58773 | 59697 59828 [7,6,6,4,4,2] | 7035 | 7405 7434 | 10040 | 10330 10343 | 20348 | 20927 20956 | 29419 | 29757 29763 Absolute errors (in comparison with the experimental data) in 1/cm of all-electron Dirac-Coulomb calculations (point nucleus) E_exper-E_calc spd-basis | spdf-basis | spdfg-basis spdfgh-basis | | 4el-CI 0 | 0 | [3,5,3,2] 986 | 807 | 483 | 752 | 1033 | 1707 | 964 | 1553 | | | 22el-CI/MBPT2 | | 0 0 [3,5,3,2]/ | | 481 461 [4,5,4,2,2,2] | | 395 370 | | 776 727 | | 265 203 | | 22el-CI/MBPT2 0 | | [3,5,3]/ 985 | | [3,5,3] 411 | | 814 | | 302 | | | | 14e-CI/MBPT2 0 | | [3,5,3]/ 980 | | [7,6,6] 409 | | 812 | | 279 | | | | 22e-CI/MBPT2 0 | 0 | 0 [3,5,3,2]/ 956 | 651 | 173 [7,6,6,2,4] 385 | 515 | 36 763 | 902 | 15 244 | 304 | -381 | | | | 14e-CI/MBPT2 | | 0 [3,5,3,2]/ | | 577 [4,5,4,2,2,2] | | 474 | | 941 | | 372 | | 36e-CI/MBPT2 | | 0 [3,5,3,2]/ | | 365 [7,6,6,4,4] | | 282 | | 402 sp-basis | | -90 4elRCC 0 | 0 | [3,5,3,2] 1287 | | 931 | | 1770 | | 975 | | | | | | 4elRCC 0 | | 0 [7,6,6,4,4] 1250 | | 1047 860 | | 1093 1618 | | 1998 713 | | 1098 | | 22elRCC 0 | | 0 0 [7,6,6,4,4,2] 907 | | 450 422 378 | | 364 351 728 | | 618 590 -379 | | -199 -206 | | 22elRCC 0 | | [3,5,3] 987 | | 485 | | 953 | | -56 | | | | 14elRCC 0 | 0 | 0 [7,6,6,4,4] 964 | 939 | 630 436 | 758 | 539 834 | 1394 | 963 -313 | 244 | 70 | | 36eRCC | 0 | 0 0 [7,6,6,4,4,2] | 784 | 414 385 | 610 | 320 307 | 1109 | 530 501 | 47 | -291 -297 Differences in total state energies in 1/cm for 4 el CI/MBPT2 and RCC: First number - ground state: E_total_CI - E_total_RCC 4e-CI/PT - 4e-RCC [3,5] [3,5,3] [3,5,3,2] -334 -690 -586 -120 -389 -232 -104 -242 -109 -11 47 -30 -412 -680 -777 ---------------------------------------------------------------- | GRECP with CI/PT and RCC | ---------------------------------------------------------------- Basises for RECP calculations are to be obtained by optimization in each case separately! A few first functions in each symmetry are taken from Hartree-Fock procedure with frozen core 6s^2 and different ionic states. Data for Pb2+ are in files - $HOME/atom/ionres 3379 Apr 3 16:22 10p.christ 7367 Apr 2 17:02 5f.christ 10866 Mar 11 14:54 5f.grecp 9250 Apr 1 16:20 6d.christ 9146 Mar 9 16:14 6d.grecp 1547 Apr 3 12:58 6f.christ 1555 Mar 11 16:30 6f.grecp 2569 Mar 31 13:27 6p.christ 1950 Mar 9 12:42 6p.grecp 1560 Apr 1 18:43 7d.christ 2589 Mar 10 12:40 7d.grecp 9171 Apr 3 16:24 7p.christ 5524 Mar 9 14:34 7p.grecp 7278 Mar 31 14:52 7s.christ 4273 Mar 9 13:52 7s.grecp 2610 Apr 2 17:59 8d.christ 2614 Mar 11 15:39 8d.grecp 2171 Apr 1 17:13 8p.christ 6882 Mar 9 18:26 8p.grecp 5898 Mar 31 17:40 8s.christ 5709 Mar 9 15:13 8s.grecp 3828 Apr 1 19:58 9p.christ 4561 Mar 10 18:04 9p.grecp *.grecp - orbitals taken from HFJ calculations with GRECP *.christ - ||--||--||--|| RECP by Christiansen et al. Then basis is extended by multyplying radial parts by r to the size [15,14,12,8] in case of Pb atom: for all-electron calculations - the same basis set as above. for GRECP - initial basis from 4 electron CI optimization. for RECP by Christiansen et al - initial basis from 4 electron CI optim. then extension to [15,14,12,8]. 22el numerical GRECP and RECP by Christiansen et al. for Pb are used Transition energies for CI/MBPT2 with GRECP spd basis spdf basis spdfg basis spdfgh basis 22el-CI/MBPT2 0 0 [3,5,3,2]/ 7396 7418 [4,5,4,2,2,2] 10321 10348 20762 20814 29262 29327 4e-CI 0 [3,5,3,2] 7038 9934 19796 27970 14e RCC 59385 [4,5,4,2,2] 7211 10117 20525 29392 Differences in transition energies for CI/PT2 and GRECP/CI/PT2: E_grecp_ci/pt - E_ci/pt in 1/cm: 0 71 58 126 69 sp spd spdf spdfg spdfgh 14el RCC 0 0 0 0 [4,5,4,2,2] 2 6 47 91 -2 -2 26 71 -3 4 60 151 -31 -41 35 112 22el-CI/MBPT2 0 0 [3,5,3,2]/ 58 60 [4,5,4,2,2,2] 66 68 81 84 61 64 0 4e-CI 26 [3,5,3,2] 36 46 57 14e-CI/MBPT2 0 [3,5,3,2]/ 63 [4,5,4,2,2,2] 67 82 49 ---------------------------------------------------------------------------- | Comparation of calculations with and without screening | ---------------------------------------------------------------------------- sigma means inclusion one-electron diagrams in PT2 screening - inclusion two-electron ones Important: in CI/PT2 core-core correlations weren't involved 4e-CI | 22e-CI/PT with sigma only| 22e-CI/PT sigma+screening | [3,5,3,2] | [3,5,3,2]/[7,6,6,4,4] | [3,5,3,2]/[7,6,6,4,4] | --------------------------------------------------------------------------- | Ground state total energies (a.u) | --------------------------------------------------------------------------- | -20919.71428843 | -20919.80356404 | -20919.79307308 | --------------------------------------------------------------------------- | Transition energies | --------------------------------------------------------------------------- | 0 | 0 | 0 | | 7012 | 7892 | 7646 | | 9898 | 10917 | 10614 | | 19750 | 21920 | 21442 | | 27913 | 30338 | 29847 | --------------------------------------------------------------------------- | Absolute errors E_exper - E_calc | |-------------------------------------------------------------------------- | 0 | 0 | 0 | | 807 | -73 | 173 | | 752 | -267 | 36 | | 1707 | -463 | 15 | | 1553 | -872 | -381 | --------------------------------------------------------------------------- Screening influence basis [3,5,3,2]/[7,6,6,4,4] --------------------------------------------------------------------------------------------------- Electrons in CI/PT | 4 | 22 | 22 | 22 | 22 | 36 | 36 | 36 | --------------------------------------------------------------------------------------------------- Electrons involved in one-el | | | | | | | | | diagrams(sigma) "over" 4e in CI | 0 | 18 | 18 | 18 | 18 | 32 | 32 | 32 | --------------------------------------------------------------------------------------------------- Electrons involved in two-el | | | 5d | 5dp | 5dps | | | | diag. (screening)"over" 4e in CI| 0 | 0 | 10 | 16 | 18 | 0 | 10 | 32 | --------------------------------------------------------------------------------------------------- | Absolute errors E_exper - E_calc 1/cm | --------------------------------------------------------------------------------------------------- | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 807 | -73 | 205 | 177 | 173 | -177 | 99 | 365 | | 752 | -267 | 62 | 39 | 36 | -382 | -56 | 282 | | 1707 | -463 | 72 | 22 | 15 | -693 | -165 | 402 | | 1553 | -872 | -328 | -375 | -381 | -1113 | -575 | -90 | ---------------------------------------------------------------------------------------------------