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2016/08/19 v6685 L3 File and I/O operations +\l_iow_line_count_int=\count127 +\l__iow_target_count_int=\count128 +\l__iow_current_line_int=\count129 +\l__iow_current_word_int=\count130 +\l__iow_current_indentation_int=\count131 +L3 Module: l3skip 2016/08/19 v6692 L3 Dimensions and skips \c_zero_dim=\dimen124 \c_max_dim=\dimen125 \l_tmpa_dim=\dimen126 @@ -422,110 +441,56 @@ Package: expl3 2018/03/05 L3 programming layer (code) \l_tmpb_muskip=\muskip15 \g_tmpa_muskip=\muskip16 \g_tmpb_muskip=\muskip17 -\l_keys_choice_int=\count133 -\c__fp_leading_shift_int=\count134 -\c__fp_middle_shift_int=\count135 -\c__fp_trailing_shift_int=\count136 -\c__fp_big_leading_shift_int=\count137 -\c__fp_big_middle_shift_int=\count138 -\c__fp_big_trailing_shift_int=\count139 -\c__fp_Bigg_leading_shift_int=\count140 -\c__fp_Bigg_middle_shift_int=\count141 -\c__fp_Bigg_trailing_shift_int=\count142 -\c__fp_rand_size_int=\count143 -\c__fp_rand_four_int=\count144 -\c__fp_rand_eight_int=\count145 -\l__sort_length_int=\count146 -\l__sort_min_int=\count147 -\l__sort_top_int=\count148 -\l__sort_max_int=\count149 -\l__sort_true_max_int=\count150 -\l__sort_block_int=\count151 -\l__sort_begin_int=\count152 -\l__sort_end_int=\count153 -\l__sort_A_int=\count154 -\l__sort_B_int=\count155 -\l__sort_C_int=\count156 -\l__tl_build_start_index_int=\count157 -\l__tl_build_index_int=\count158 -\l__tl_analysis_normal_int=\count159 -\l__tl_analysis_index_int=\count160 -\l__tl_analysis_nesting_int=\count161 -\l__tl_analysis_type_int=\count162 -\l__regex_internal_a_int=\count163 -\l__regex_internal_b_int=\count164 -\l__regex_internal_c_int=\count165 -\l__regex_balance_int=\count166 -\l__regex_group_level_int=\count167 -\l__regex_mode_int=\count168 -\c__regex_cs_in_class_mode_int=\count169 -\c__regex_cs_mode_int=\count170 -\l__regex_catcodes_int=\count171 -\l__regex_default_catcodes_int=\count172 -\c__regex_catcode_D_int=\count173 -\c__regex_catcode_S_int=\count174 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-\l__coffin_display_offset_dim=\dimen145 -\l__coffin_display_x_dim=\dimen146 -\l__coffin_display_y_dim=\dimen147 +L3 Module: l3keys 2016/09/21 v6711 L3 Key-value interfaces +\l_keys_choice_int=\count132 +L3 Module: l3fp 2016/11/21 v6761 L3 Floating points +\c__fp_leading_shift_int=\count133 +\c__fp_middle_shift_int=\count134 +\c__fp_trailing_shift_int=\count135 +\c__fp_big_leading_shift_int=\count136 +\c__fp_big_middle_shift_int=\count137 +\c__fp_big_trailing_shift_int=\count138 +\c__fp_Bigg_leading_shift_int=\count139 +\c__fp_Bigg_middle_shift_int=\count140 +\c__fp_Bigg_trailing_shift_int=\count141 +L3 Module: l3box 2016/08/19 v6685 L3 Experimental boxes +\c_empty_box=\box43 +\l_tmpa_box=\box44 +\l_tmpb_box=\box45 +\g_tmpa_box=\box46 +\g_tmpb_box=\box47 +L3 Module: l3coffins 2016/08/19 v6685 L3 Coffin code layer +\l__coffin_internal_box=\box48 +\l__coffin_internal_dim=\dimen130 +\l__coffin_offset_x_dim=\dimen131 +\l__coffin_offset_y_dim=\dimen132 +\l__coffin_x_dim=\dimen133 +\l__coffin_y_dim=\dimen134 +\l__coffin_x_prime_dim=\dimen135 +\l__coffin_y_prime_dim=\dimen136 +\c_empty_coffin=\box49 +\l__coffin_aligned_coffin=\box50 +\l__coffin_aligned_internal_coffin=\box51 +\l_tmpa_coffin=\box52 +\l_tmpb_coffin=\box53 +\l__coffin_display_coffin=\box54 +\l__coffin_display_coord_coffin=\box55 +\l__coffin_display_pole_coffin=\box56 +\l__coffin_display_offset_dim=\dimen137 +\l__coffin_display_x_dim=\dimen138 +\l__coffin_display_y_dim=\dimen139 +L3 Module: l3color 2016/06/13 v6596 L3 Experimental color support +L3 Module: l3sys 2015/09/25 v6087 L3 Experimental system/runtime functions +L3 Module: l3candidates 2016/08/19 v6688 L3 Experimental additions to l3kernel +\l__box_top_dim=\dimen140 +\l__box_bottom_dim=\dimen141 +\l__box_left_dim=\dimen142 +\l__box_right_dim=\dimen143 +\l__box_top_new_dim=\dimen144 +\l__box_bottom_new_dim=\dimen145 +\l__box_left_new_dim=\dimen146 +\l__box_right_new_dim=\dimen147 +\l__box_internal_box=\box57 \l__coffin_bounding_shift_dim=\dimen148 \l__coffin_left_corner_dim=\dimen149 \l__coffin_right_corner_dim=\dimen150 @@ -533,17 +498,19 @@ Package: expl3 2018/03/05 L3 programming layer (code) \l__coffin_top_corner_dim=\dimen152 \l__coffin_scaled_total_height_dim=\dimen153 \l__coffin_scaled_width_dim=\dimen154 +L3 Module: l3luatex 2016/08/18 v6679 L3 Experimental LuaTeX-specific functions ) -(/usr/local/texlive/2018/texmf-dist/tex/latex/l3kernel/l3pdfmode.def -File: l3pdfmode.def 2017/03/18 v L3 Experimental driver: PDF mode -\l__driver_color_stack_int=\count270 +(/usr/share/texlive/texmf-dist/tex/latex/l3kernel/l3pdfmode.def +File: l3pdfmode.def 2016/08/18 v6679 L3 Experimental driver: PDF mode +\l__driver_color_stack_int=\count142 +\l__driver_tmp_box=\box58 )) -Package: xparse 2018/02/21 L3 Experimental document command parser -\l__xparse_current_arg_int=\count271 -\g__xparse_grabber_int=\count272 -\l__xparse_m_args_int=\count273 -\l__xparse_mandatory_args_int=\count274 -\l__xparse_v_nesting_int=\count275 +Package: xparse 2016/11/21 v6760 L3 Experimental document command parser +\l__xparse_current_arg_int=\count143 +\l__xparse_m_args_int=\count144 +\l__xparse_mandatory_args_int=\count145 +\l__xparse_processor_int=\count146 +\l__xparse_v_nesting_int=\count147 ) ................................................. . LaTeX info: "xparse/define-command" @@ -581,31 +548,6 @@ Package: xparse 2018/02/21 L3 Experimental document command parser . Defining command \@quantity with sig. ' t\big t\Big t\bigg t\Bigg g o d() . d|| ' on line 62. ................................................. - -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\big ' for the command '\@quantity' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\Big ' for the command '\@quantity' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\bigg ' for the command '\@quantity' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\Bigg ' for the command '\@quantity' should be a single -* character. -************************************************* ................................................. . LaTeX info: "xparse/define-command" . @@ -738,60 +680,12 @@ Package: xparse 2018/02/21 L3 Experimental document command parser . Defining command \@braces with sig. ' s t\big t\Big t\bigg t\Bigg m m m ' on . line 156. ................................................. -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\big ' for the command '\@braces' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\Big ' for the command '\@braces' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\bigg ' for the command '\@braces' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\Bigg ' for the command '\@braces' should be a single -* character. -************************************************* ................................................. . LaTeX info: "xparse/define-command" . . Defining command \fbraces with sig. ' s t\big t\Big t\bigg t\Bigg m m m m ' . on line 174. ................................................. -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\big ' for the command '\fbraces' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\Big ' for the command '\fbraces' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\bigg ' for the command '\fbraces' should be a single -* character. -************************************************* -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\Bigg ' for the command '\fbraces' should be a single -* character. -************************************************* ................................................. . LaTeX info: "xparse/define-command" . @@ -1323,12 +1217,6 @@ Package: xparse 2018/02/21 L3 Experimental document command parser . . Defining command \bra with sig. ' s m t\ket s g ' on line 526. ................................................. -************************************************* -* LaTeX warning: "xparse/not-single-char" -* -* Argument delimiter '\ket ' for the command '\bra' should be a single -* character. -************************************************* ................................................. . LaTeX info: "xparse/define-command" . @@ -1501,29 +1389,91 @@ Package: xparse 2018/02/21 L3 Experimental document command parser . . Defining command \homework with sig. '' on line 764. ................................................. -) (/usr/local/texlive/2018/texmf-dist/tex/latex/mhchem/mhchem.sty -Package: mhchem 2017/07/24 v4.07 for typesetting chemical formulae +) +(/usr/share/texlive/texmf-dist/tex/latex/mhchem/mhchem.sty +Package: mhchem 2017/01/16 v4.06 for typesetting chemical formulae -(/usr/local/texlive/2018/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty -Package: l3keys2e 2018/02/21 LaTeX2e option processing using LaTeX3 keys -) (/usr/local/texlive/2018/texmf-dist/tex/latex/tools/calc.sty -Package: calc 2017/05/25 v4.3 Infix arithmetic (KKT,FJ) -\calc@Acount=\count276 -\calc@Bcount=\count277 +(/usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty +Package: l3keys2e 2016/11/21 v6760 LaTeX2e option processing using LaTeX3 keys +) +(/usr/share/texlive/texmf-dist/tex/latex/l3experimental/l3str/l3regex.sty +Package: l3regex 2016/11/21 v6760 L3 Experimental regular expressions + +(/usr/share/texlive/texmf-dist/tex/latex/l3experimental/l3str/l3tl-build.sty +Package: l3tl-build 2016/11/21 v6760 L3 Experimental token list construction +\l__tl_build_start_index_int=\count148 +\l__tl_build_index_int=\count149 +) +(/usr/share/texlive/texmf-dist/tex/latex/l3experimental/l3str/l3tl-analysis.sty +Package: l3tl-analysis 2016/06/13 v6589 L3 Experimental token lists analysis +\l__tl_analysis_normal_int=\count150 +\l__tl_analysis_index_int=\count151 +\l__tl_analysis_nesting_int=\count152 +\l__tl_analysis_type_int=\count153 +) (/usr/share/texlive/texmf-dist/tex/latex/l3experimental/l3str/l3flag.sty +Package: l3flag 2016/11/21 v6760 L3 Experimental flags +) +(/usr/share/texlive/texmf-dist/tex/latex/l3experimental/l3str/l3str-convert.sty +Package: l3str-convert 2016/09/07 v6699 L3 Experimental string encoding convers +ions +\l__str_internal_int=\count154 +\c__str_replacement_char_int=\count155 +) +\l__regex_internal_a_int=\count156 +\l__regex_internal_b_int=\count157 +\l__regex_internal_c_int=\count158 +\l__regex_balance_int=\count159 +\l__regex_group_level_int=\count160 +\l__regex_mode_int=\count161 +\l__regex_catcodes_int=\count162 +\l__regex_default_catcodes_int=\count163 +\c__regex_catcode_D_int=\count164 +\c__regex_catcode_S_int=\count165 +\c__regex_catcode_L_int=\count166 +\c__regex_catcode_O_int=\count167 +\c__regex_catcode_A_int=\count168 +\c__regex_all_catcodes_int=\count169 +\l__regex_show_lines_int=\count170 +\l__regex_min_state_int=\count171 +\l__regex_max_state_int=\count172 +\l__regex_left_state_int=\count173 +\l__regex_right_state_int=\count174 +\l__regex_capturing_group_int=\count175 +\l__regex_min_pos_int=\count176 +\l__regex_max_pos_int=\count177 +\l__regex_current_pos_int=\count178 +\l__regex_start_pos_int=\count179 +\l__regex_success_pos_int=\count180 +\l__regex_current_char_int=\count181 +\l__regex_current_catcode_int=\count182 +\l__regex_last_char_int=\count183 +\l__regex_case_changed_char_int=\count184 +\l__regex_current_state_int=\count185 +\l__regex_step_int=\count186 +\l__regex_min_active_int=\count187 +\l__regex_max_active_int=\count188 +\l__regex_replacement_csnames_int=\count189 +\l__regex_match_count_int=\count190 +\l__regex_submatch_int=\count191 +\l__regex_zeroth_submatch_int=\count192 +) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +Package: calc 2014/10/28 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count193 +\calc@Bcount=\count194 \calc@Adimen=\dimen155 \calc@Bdimen=\dimen156 \calc@Askip=\skip61 \calc@Bskip=\skip62 LaTeX Info: Redefining \setlength on input line 80. LaTeX Info: Redefining \addtolength on input line 81. -\calc@Ccount=\count278 +\calc@Ccount=\count195 \calc@Cskip=\skip63 ) -(/usr/local/texlive/2018/texmf-dist/tex/latex/chemgreek/chemgreek.sty +(/usr/share/texlive/texmf-dist/tex/latex/chemgreek/chemgreek.sty Package: chemgreek 2016/12/20 v1.1 interfaceforuprightgreeklettersforuseinchemi stry (CN) -\l__chemgreek_tmpa_int=\count279 -\g__chemgreek_tmpa_int=\count280 +\l__chemgreek_tmpa_int=\count196 +\g__chemgreek_tmpa_int=\count197 ................................................. . LaTeX info: "xparse/define-command" . @@ -1575,7 +1525,7 @@ stry (CN) . Defining command \selectchemgreekmapping with sig. 'm' on line 491. ................................................. ) -\l__mhchem_ce_distanceFromLastComma_int=\count281 +\l__mhchem_ce_distanceFromLastComma_int=\count198 \__mhchem_arrow_extPgfLength_skip=\skip64 \mhchem@minispace@tmp=\skip65 \mhchem@minibackspace@tmp=\skip66 @@ -1586,7 +1536,7 @@ stry (CN) \mhchem@smallbondwidth=\skip71 \mhchem@bondtmp@minussidebearingleft=\skip72 \mhchem@bondtmp@minussidebearingright=\skip73 -\l__mhchem_option_version_int=\count282 +\l__mhchem_option_version_int=\count199 \mhchem@option@minusmathsidebearingleft=\dimen157 \mhchem@option@minusmathsidebearingright=\dimen158 \mhchem@option@minustextsidebearingleft=\dimen159 @@ -1600,27 +1550,27 @@ Package mhchem Warning: You did not specify a 'version' option for the mhchem (mhchem) different output with the same input in future versions . -) (/usr/local/texlive/2018/texmf-dist/tex/latex/psnfss/mathpazo.sty +) (/usr/share/texlive/texmf-dist/tex/latex/psnfss/mathpazo.sty Package: mathpazo 2005/04/12 PSNFSS-v9.2a Palatino w/ Pazo Math (D.Puga, WaS) \symupright=\mathgroup9 ) -(/usr/local/texlive/2018/texmf-dist/tex/latex/libertine/libertine.sty -Package: libertine 2018/01/03 (Bob Tennent) Supports Libertine and Biolinum fon +(/usr/share/texlive/texmf-dist/tex/latex/libertine/libertine.sty +Package: libertine 2014/11/25 (Bob Tennent) Supports Libertine and Biolinum fon ts for all LaTeX engines. -(/usr/local/texlive/2018/texmf-dist/tex/latex/xkeyval/xkeyval.sty +(/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty Package: xkeyval 2014/12/03 v2.7a package option processing (HA) -(/usr/local/texlive/2018/texmf-dist/tex/generic/xkeyval/xkeyval.tex -(/usr/local/texlive/2018/texmf-dist/tex/generic/xkeyval/xkvutils.tex +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +(/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex \XKV@toks=\toks25 \XKV@tempa@toks=\toks26 ) -\XKV@depth=\count283 +\XKV@depth=\count266 File: xkeyval.tex 2014/12/03 v2.7a key=value parser (HA) )) -(/usr/local/texlive/2018/texmf-dist/tex/latex/base/textcomp.sty -Package: textcomp 2017/04/05 v2.0i Standard LaTeX package +(/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +Package: textcomp 2017/02/22 v2.0g Standard LaTeX package Package textcomp Info: Sub-encoding information: (textcomp) 5 = only ISO-Adobe without \textcurrency (textcomp) 4 = 5 + \texteuro @@ -1634,84 +1584,9 @@ Package textcomp Info: Sub-encoding information: (textcomp) See the documentation for details. 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PDF statistics: - 328 PDF objects out of 1000 (max. 8388607) - 302 compressed objects within 4 object streams + 327 PDF objects out of 1000 (max. 8388607) + 301 compressed objects within 4 object streams 98 named destinations out of 1000 (max. 500000) 45585 words of extra memory for PDF output out of 51595 (max. 10000000) diff --git a/Manuscript/CI-F12.pdf b/Manuscript/CI-F12.pdf index 609a062..b963f7b 100644 Binary files a/Manuscript/CI-F12.pdf and b/Manuscript/CI-F12.pdf differ diff --git a/Manuscript/CI-F12.synctex.gz b/Manuscript/CI-F12.synctex.gz deleted file mode 100644 index bacdf3e..0000000 Binary files a/Manuscript/CI-F12.synctex.gz and /dev/null differ diff --git a/Manuscript/CI-F12.tex b/Manuscript/CI-F12.tex index 9edae17..d5abaac 100644 --- a/Manuscript/CI-F12.tex +++ b/Manuscript/CI-F12.tex @@ -49,9 +49,9 @@ % addresses \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} -\begin{document} +\begin{document} -\title{Dressing the configuration interaction matrix with explicit correlation} +\title{Dressing the configuration interaction matrix with explicit correlation} \author{Anthony Scemama} \affiliation{\LCPQ} @@ -98,36 +98,36 @@ Atomic units are used throughout. Inspired by a number of previous research \cite{Shiozaki11}, our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part \begin{equation} \label{eq:D} - \kD = \sum_{I} c_I \kI + \kD = \sum_{I} c_I \kI \end{equation} composed by a linear combination of determinants $\kI$ of coefficients $c_I$ and an ``explicitly-correlated'' part \begin{equation} \label{eq:WF-F12-CIPSI} - \kF = \sum_{I} t_I \hQ f \kI + \kF = \sum_{I} t_I \hQ f \kI \end{equation} with coefficients $t_I$. The projector \begin{equation} - \hQ = \hI - \sum_{I} \dyad{I} + \hQ = \hI - \sum_{I} \dyad{I} \end{equation} ensures the orthogonality between $\kD$ and $\kF$, and \begin{equation} \label{eq:Ja} - f = \sum_{i < j} \gamma_{ij} f_{ij} + f = \sum_{i < j} \gamma_{ij} f_{ij} \end{equation} is a correlation factor, and \begin{equation} - \gamma_{ij} = - \begin{cases} - 1/2, & \text{for opposite-spin electrons}, - \\ - 1/4, & \text{for same-spin electrons}. - \end{cases} + \gamma_{ij} = + \begin{cases} + 1/2, & \text{for opposite-spin electrons}, + \\ + 1/4, & \text{for same-spin electrons}. + \end{cases} \end{equation} As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as \begin{equation} - f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}. + f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}. \end{equation} %---------------------------------------------------------------- @@ -136,23 +136,23 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various Our primary goal is to introduce the explicit correlation between electrons at low computational cost. Therefore, assuming that $\hH \ket{\Psi} = E \Psi$, one can write, by projection over $\bra{I}$, \begin{equation} - c_I \qty[ H_{II} + c_I^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0. + c_I \qty[ H_{II} + c_I^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0. \end{equation} where $H_{IJ} = \mel{I}{\hH}{J}$. Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian: \begin{equation} \label{eq:DrH} - \oH_{IJ} = - \begin{cases} - H_{II} + c_I^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$}, - \\ - H_{IJ}, & \text{otherwise}, - \end{cases} + \oH_{IJ} = + \begin{cases} + H_{II} + c_I^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$}, + \\ + H_{IJ}, & \text{otherwise}, + \end{cases} \end{equation} with \begin{equation} \label{eq:IHF} - \mel{I}{\hH}{F} = \sum_J t_J \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ], + \mel{I}{\hH}{F} = \sum_J t_J \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ], \end{equation} and $f_{IJ} = \mel{I}{f}{J}$. It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis. @@ -164,7 +164,7 @@ First, as one may have realized, the coefficients $t_I$ are unknown. \alert{This yields the following linear system of equations \begin{equation} \label{eq:tI} - \sum_J (\delta_{IJ} + f_{IJ}) t_J = c_I, + \sum_J (\delta_{IJ} + f_{IJ}) t_J = c_I, \end{equation} which can be easily solved using standard linear algebra packages.} @@ -175,13 +175,13 @@ For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}. %%% FIG 1 %%% %\begin{figure} -% \includegraphics[width=\linewidth]{fig1} -% \caption{ -% \label{fig:CBS} -% Schematic representation of the various orbital spaces and their notation. -% The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).} +% \includegraphics[width=\linewidth]{fig1} +% \caption{ +% \label{fig:CBS} +% Schematic representation of the various orbital spaces and their notation. +% The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).} %\end{figure} -%%% %%% +%%% %%% %---------------------------------------------------------------- \textit{Matrix elements.---} @@ -203,17 +203,17 @@ This CBS is built as the union of the orbital basis set (OBS) $\qty{p}$ (divided In the CBS, one can write \begin{equation} \label{eq:RI} - \hI = \sum_{A \in \mA} \dyad{A}{A} + \hI = \sum_{A \in \mA} \dyad{A}{A} \end{equation} where $\mA$ is the set of all the determinants $\kA$ corresponding to electronic excitations from occupied orbitals $\qty{i}$ to the extended virtual orbital space $\qty{a} \cup \qty{\alpha}$. Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eqref{eq:IHF}, one gets \begin{equation} \label{eq:IHF-RI} \begin{split} - \mel{I}{\hH}{F} - & = \sum_J t_J \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ] - \\ - & = \sum_J t_J \sum_{A \in \mC} H_{IA} f_{AJ}, + \mel{I}{\hH}{F} + & = \sum_J t_J \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ] + \\ + & = \sum_J t_J \sum_{A \in \mC} H_{IA} f_{AJ}, \end{split} \end{equation} where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$. @@ -231,35 +231,35 @@ Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero \cite{Kutzelnigg91}. %\begin{gather} -% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j} -% \\ -% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta} -% \\ -% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta} +% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j} +% \\ +% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta} +% \\ +% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta} %\end{gather} % %\begin{gather} -% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j} -% \\ -% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0 -% \\ -% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} = +% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j} +% \\ +% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0 +% \\ +% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} = %\end{gather} % %\begin{gather} -% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} = -% \\ -% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} = -% \\ -% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} = +% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} = +% \\ +% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} = +% \\ +% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} = %\end{gather} % %\begin{gather} -% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} = -% \\ -% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} = -% \\ -% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} = +% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} = +% \\ +% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} = +% \\ +% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} = %\end{gather} %---------------------------------------------------------------- @@ -267,81 +267,86 @@ Here, we will eschew the generalized Brillouin condition (GBC) which set these t %---------------------------------------------------------------- In all the calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a} \begin{equation} - f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la}, + f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la}, \end{equation} which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}, i.e. \begin{equation} - \exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ). + \exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ). \end{equation} The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study. All the calculations have been performed with Quantum Package \cite{QP}. -%%% TABLE 1 %%% +%%% TABLE 1 %%% \begin{table} \caption{ \label{tab:atoms} FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.} \begin{ruledtabular} -\begin{tabular}{lcccc} -Atom & cc-pVXZ & FCI-F12 & CIPSI & FCI \\ +\begin{tabular}{lcccd} + Atom & $N$ & FCI-F12 & CIPSI & \text{FCI} \\ \hline -\ce{He} & D & & & $-2.887\,595$\footnotemark[1] \\ - & T & & & $-2.900\,232$\footnotemark[1] \\ - & Q & & & $-2.902\,411$\footnotemark[1] \\ - & 5 & & & $-2.903\,152$\footnotemark[1] \\ - & 6 & & & $-2.903\,432$\footnotemark[1] \\ - & $\infty$ & & & $-2.903\,724$\footnotemark[2] \\ -\hline -\ce{Li} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-7.478\,060$\footnotemark[3] \\ -\hline -\ce{Be} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-14.667\,356$\footnotemark[4] \\ -\hline -\ce{B} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-24.653\,866$\footnotemark[5] \\ + \ce{He} & D & & & -2.887\,595 \footnotemark[1] \\ + (cc-pV$N$Z) & T & & & -2.900\,232 \footnotemark[1] \\ + & Q & & & -2.902\,411 \footnotemark[1] \\ + & 5 & & & -2.903\,152 \footnotemark[1] \\ + & 6 & & & -2.903\,432 \footnotemark[1] \\ + & $\infty$ & & & -2.903\,724 \footnotemark[2] \\ \hline -\ce{C} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-37.840\,129$\footnotemark[6] \\ + \ce{Li} & D & & & -7.466\,025 (FCI) \\ + (cc-pCV$N$Z) & T & & & -7.474\,251 (FCI) \\ + & Q & & & -7.476\,373 (CIPSI) \\ + & $\infty$ & & & -7.478\,060 \footnotemark[3] \\ \hline -\ce{N} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $ 54.588917$\footnotemark[7] \\ -\hline -\ce{O} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-75.066892$\footnotemark[7] \\ + \ce{Be} & D & & & -14.651\,833 (FCI) \\ + (cc-pCV$N$Z) & T & & & -14.662\,369 (CIPSI) \\ + & Q & & & -14.665\,566 (CIPSI) \\ + & $\infty$ & & & -14.667\,356 \footnotemark[4] \\ + & $\infty$ & & & -14.667\,39 (TOTO) \\ \hline -\ce{F} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-99.733\,424$\footnotemark[7] \\ + \ce{B} & D & & & -24.619\,099 (CIPSI) \\ + (cc-pwCV$N$Z) & T & & & -24.643\,222 (CIPSI) \\ + & Q & & & -24.650\,331 (CIPSI) \\ + & 5 & & & -24.652\,309 (CIPSI) \\ + & $\infty$ & & & -24.653\,866 \footnotemark[5] \\ + & $\infty$ & & & -24.653\,90 (TOTO) \\ \hline -\ce{Ne} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & $-128.937\,274$\footnotemark[7] \\ + \ce{C} & D & & & -37.792\,466 (CIPSI) \\ + (cc-pwCV$N$Z) & T & & & -37.829\,847 (CIPSI) \\ + & Q & & & -37.839\,816 (CIPSI) \\ + & 5 & & & -37.842\,731 (CIPSI) \\ + & $\infty$ & & & -37.840\,129 6 \\ + & $\infty$ & & & -37.845\,0 (TOTO) \\ +\hline + \ce{N} & D & & & \\ + (cc-pwCV$N$Z) & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & -54.588\,917 \footnotemark[7] \\ + & $\infty$ & & & -54.589\,3 (TOTO) \\ +\hline + \ce{O} & D & & & \\ + (cc-pwCV$N$Z) & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & -75.066\,892 \footnotemark[7] \\ + & $\infty$ & & & -75.067\,4 (TOTO) \\ +\hline + \ce{F} & D & & & \\ + (cc-pwCV$N$Z) & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & -99.733\,424 \footnotemark[7] \\ + & $\infty$ & & & -99.734\,1 (TOTO) \\ +\hline + \ce{Ne} & D & & & \\ + (cc-pwCV$N$Z) & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & -128.937\,274 \footnotemark[7] \\ + & $\infty$ & & & -128.938\,3 (TOTO) \\ \end{tabular} -\end{ruledtabular} +\end{ruledtabular} \footnotetext[1]{Reference \onlinecite{Kong12}} \footnotetext[2]{Reference \onlinecite{Nakashima07}} \footnotetext[3]{Reference \onlinecite{Puchalski09}} @@ -360,23 +365,23 @@ CIPSI, FCI-F12 i-FCIQMC and FCI total ground-state energy of the \ce{H2}, \ce{F2 The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.} \begin{ruledtabular} \begin{tabular}{lccccc} -Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & \mcc{i-FCIQMC} & \mcc{FCI} \\ +Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & \mcc{i-FCIQMC} & \mcc{FCI} \\ \hline -\ce{H2} & D & & & \\ - & T & & & \\ - & Q & & & \\ - & 5 & & & \\ - & $\infty$ & & & & $-1.174\,476$\footnotemark[1] \\ +\ce{H2} & D & & & \\ + & T & & & \\ + & Q & & & \\ + & 5 & & & \\ + & $\infty$ & & & & $-1.174\,476$\footnotemark[1] \\ \hline -\ce{F2} & D & $-199.099\,28$\footnotemark[2] & & $-199.099\,41(9)$\footnotemark[3] \\ - & T & $-199.296\,5$\footnotemark[2] & & $-199.297\,7(1)$\footnotemark[3] \\ +\ce{F2} & D & $-199.099\,28$\footnotemark[2] & & $-199.099\,41(9)$\footnotemark[3] \\ + & T & $-199.296\,5$\footnotemark[2] & & $-199.297\,7(1)$\footnotemark[3] \\ \hline -\ce{H2O} & D & $-76.282\,136$\footnotemark[4] & & & $-76.282\,865$\footnotemark[5] \\ - & T & $-76.388\,287$\footnotemark[4] & & & $-76.390\,158$\footnotemark[5] \\ - & Q & $-76.419\,324$\footnotemark[4] & & & $-76.421\,148$\footnotemark[5] \\ - & 5 & $-76.428\,550$\footnotemark[4] & & & $-76.431\,105$\footnotemark[5] \\ +\ce{H2O} & D & $-76.282\,136$\footnotemark[4] & & & $-76.282\,865$\footnotemark[5] \\ + & T & $-76.388\,287$\footnotemark[4] & & & $-76.390\,158$\footnotemark[5] \\ + & Q & $-76.419\,324$\footnotemark[4] & & & $-76.421\,148$\footnotemark[5] \\ + & 5 & $-76.428\,550$\footnotemark[4] & & & $-76.431\,105$\footnotemark[5] \\ \end{tabular} -\end{ruledtabular} +\end{ruledtabular} \footnotetext[1]{Reference \onlinecite{Pachucki10}} \footnotetext[2]{Reference \onlinecite{Giner15}} \footnotetext[3]{Reference \onlinecite{Cleland12}}