├── Makefile ├── Pictures ├── heh.chk ├── q.png ├── fig.1.1.pdf ├── MPn-energy.pdf ├── heh2.gjf ├── wolniewicz.dat ├── heh.gjf ├── h2-exact-kolos.dat ├── W-heh.dat ├── MPn-energy.tex ├── h2-6-31gdp-fci.dat ├── h2-6-31gdp-mp2.dat ├── h2-6-31gdp-mp3.dat ├── h2-6-31gdp-uhf.dat ├── h2-sto-3g-fci.dat ├── h2-sto-3g-mp2.dat ├── h2-sto-3g-mp3.dat ├── h2-sto-3g-uhf.dat ├── heh-tot-ener.txt ├── heh_tot_ener.txt └── h2.dat ├── Chaps ├── preface.tex ├── AppendixD.tex ├── progess.tex ├── AppendixA.tex ├── AppendixC.tex └── Chap5.tex ├── NOTICE ├── .gitignore ├── .github └── workflows │ ├── texlive.profile │ ├── build.yml │ └── release.yml ├── README.md ├── bib.bib ├── main.tex ├── dev_guide.md ├── structure.tex ├── code ├── out └── AppendixBcode.f90 └── bookszabo.cls /Makefile: -------------------------------------------------------------------------------- 1 | 2 | main: 3 | latexmk -xelatex -halt-on-error -shell-escape main 4 | -------------------------------------------------------------------------------- /Pictures/heh.chk: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/NominHanggai/szaboqc/HEAD/Pictures/heh.chk -------------------------------------------------------------------------------- /Pictures/q.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/NominHanggai/szaboqc/HEAD/Pictures/q.png -------------------------------------------------------------------------------- /Chaps/preface.tex: -------------------------------------------------------------------------------- 1 | \chapter*{修订版前言} 2 | 修订版与第一版相比, 有三处变化较大. 一是增加了附录. 3 | \chapter*{第一版前言} 4 | -------------------------------------------------------------------------------- /Pictures/fig.1.1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/NominHanggai/szaboqc/HEAD/Pictures/fig.1.1.pdf -------------------------------------------------------------------------------- /Pictures/MPn-energy.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/NominHanggai/szaboqc/HEAD/Pictures/MPn-energy.pdf -------------------------------------------------------------------------------- /Pictures/heh2.gjf: -------------------------------------------------------------------------------- 1 | %mem=1gb 2 | %cpu=0-7 3 | % chk=heh.chk 4 | # pop=none scan hf/sto-3g units=au Geom=nocrowd 5 | 6 | we 7 | 8 | 1 1 9 | H 10 | He 1 R 11 | 12 | R 0.5 301 0.01 13 | -------------------------------------------------------------------------------- /Pictures/wolniewicz.dat: -------------------------------------------------------------------------------- 1 | 1.0 -.12453881 2 | 1.2 -.16493435 3 | 1.3 -.17234623 4 | 1.39 -.17445199 5 | 1.4 -.17447477 6 | 1.4011 -.17447498 7 | 1.41 -.17446041 8 | 1.5 -.17285408 9 | 1.6 -.16858212 10 | 1.8 -.15506752 11 | 2.0 -.13813155 12 | 2.2 -.12013035 13 | 2.4 -.10242011 14 | 2.6 -.08578740 15 | 2.8 -.07067758 16 | 3.0 -.05731738 17 | 3.2 -.04578647 18 | -------------------------------------------------------------------------------- /NOTICE: -------------------------------------------------------------------------------- 1 | Contributors 2 | 3 | @Mulliken: Most chapters 4 | Shirong Wang @hebrewsnabla: chap 4.5, various fixes, github actions 5 | @shenyi97: chap 3.6.2, various fixes 6 | @AllanChain: various fixes 7 | @PramSin: fixes in chap 1, improvement of autoref 8 | Mengyuan Wu @SeptemberMy: appendix A,B,C,D, various fixes 9 | @maki49: chap 3.8, 4.2, 5.2, various fixes 10 | @Usu171: various fixes 11 | @Bessgendre: fixes in chap 2 12 | YI Zeping @yizeyi18: revision and fixes in chap 1, 2 13 | 14 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # LaTeX temporary files 2 | *.aux 3 | *.log 4 | *.toc 5 | *.ent 6 | *.fls 7 | *.xdv 8 | 9 | # PDF output - usually a bad idea to keep this in Git 10 | *.pdf 11 | 12 | # Latexmk 13 | *.fdb_latexmk 14 | 15 | # SyncTeX 16 | *.synctex.gz 17 | *.synctex(busy) 18 | 19 | # LaTeX Beamer 20 | *.snm 21 | *.vrb 22 | *.nav 23 | *.out 24 | 25 | # BibTeX 26 | *.bbl 27 | *.blg 28 | 29 | # 'tikz/external' library files 30 | *.dpth 31 | *.dep 32 | *.log 33 | *.ent 34 | *.auxlock 35 | 36 | *.sh 37 | -------------------------------------------------------------------------------- /.github/workflows/texlive.profile: -------------------------------------------------------------------------------- 1 | # From latex3 2 | # https://github.com/latex3/latex3/blob/main/support/texlive.profile 3 | 4 | selected_scheme scheme-infraonly 5 | TEXDIR /tmp/texlive 6 | TEXMFSYSCONFIG /tmp/texlive/texmf-config 7 | TEXMFSYSVAR /tmp/texlive/texmf-var 8 | TEXMFLOCAL /tmp/texlive/texmf-local 9 | TEXMFHOME ~/texmf 10 | TEXMFCONFIG ~/.texlive/texmf-config 11 | TEXMFVAR ~/.texlive/texmf-var 12 | option_doc 0 13 | option_src 0 14 | tlpdbopt_autobackup 0 15 | 16 | -------------------------------------------------------------------------------- /Pictures/heh.gjf: -------------------------------------------------------------------------------- 1 | %mem=1gb 2 | %cpu=0-7 3 | % chk=heh.chk 4 | # pop=none scan hf/gen units=au Geom=nocrowd 5 | 6 | we 7 | 8 | 1 1 9 | H 10 | He 1 R 11 | 12 | R 0.5 301 0.01 13 | 14 | He 0 15 | S 3 1.24 16 | 6.36242139 0.15432897 17 | 1.15892300 0.53532814 18 | 0.31364979 0.44463454 19 | **** 20 | H 0 21 | S 3 1.0 22 | 3.42525091 0.15432897 23 | 0.62391373 0.53532814 24 | 0.16885540 0.44463454 25 | **** 26 | 27 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Modern Quantum Chemistry 现代量子化学 汉化版 2 | 3 | Attila Szabo & Neil Ostlund *Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory* 中文翻译。重新绘制了所有插图。 4 | 5 | pdf请到Release下载，最新版本：[0.2.7](https://github.com/Mulliken/szaboqc/releases/download/v0.2.7/szabo_zh-v0.2.7.pdf)。 6 | 7 | ## 手动编译 8 | 9 | Tex Live >=2021 下 XeLaTeX 编译。 10 | 11 | 如果编译后引用失效则可能需要使用xelatex再编译一次。 12 | 13 | 或者使用`make`. 14 | 15 | ## 关于贡献 16 | 17 | See [Contributing guide and tips](https://github.com/Mulliken/szaboqc/issues/11) and [dev guide](./dev_guide.md). 18 | -------------------------------------------------------------------------------- /Pictures/h2-exact-kolos.dat: -------------------------------------------------------------------------------- 1 | 0.9 -.083643176 2 | 1.0 -.124539660 3 | 1.1 -.150057312 4 | 1.2 -.164935191 5 | 1.3 -.172347100 6 | 1.35 -.173963679 7 | 1.4 -.174475668 8 | 1.45 -.174057026 9 | 1.5 -.172855034 10 | 1.6 -.168583330 11 | 1.7 -.162458684 12 | 1.8 -.155068695 13 | 2.0 -.138132913 14 | 2.2 -.120132069 15 | 2.4 -.102422553 16 | 2.6 -.085791176 17 | 2.8 -.070683159 18 | 3.0 -.057326175 19 | 3.2 -.045799543 20 | 3.4 -.036075238 21 | 3.6 -.028046110 22 | 3.8 -.021549547 23 | 4.0 -.016389951 24 | 4.2 -.012359608 25 | 4.4 -.009256107 26 | 4.6 -.006894750 27 | 4.8 -.005115473 28 | -------------------------------------------------------------------------------- /bib.bib: -------------------------------------------------------------------------------- 1 | @incollection{sutcliffe1975fundamentals, 2 | title={Fundamentals of computational quantum chemistry}, 3 | author={Sutcliffe, BT}, 4 | booktitle={Computational Techniques in Quantum Chemistry and Molecular Physics}, 5 | pages={1--105}, 6 | year={1975}, 7 | publisher={Springer} 8 | } 9 | 10 | 11 | 12 | 13 | 14 | @incollection{shavitt1963gaussian, 15 | title={The Gaussian function in calculations of statistical mechanics and quantum mechanics}, 16 | author={Shavitt, Isaiah}, 17 | booktitle={Methods of computational physics}, 18 | volume={2}, 19 | pages={1--45}, 20 | year={1963}, 21 | publisher={Academic Press}, 22 | editor={B. Alder, S. Fernbach, and M. Rotenberg}, 23 | adress={New York} 24 | } 25 | 26 | 27 | 28 | -------------------------------------------------------------------------------- /Pictures/W-heh.dat: -------------------------------------------------------------------------------- 1 | 1.1 1.24 0.7 2.9408985 8158.8 1.0515650 2 | 1.2 1.33 0.75 2.9620088 12792.0 1.0313685 3 | 1.3 1.42 0.80 2.9732408 15257.1 1.0164542 4 | 1.4 1.5168 0.8582 2.9779726 16295.6 1.0054215 5 | 1.42 1.5351 0.8674 2.9783531 16379.1 1.0035902 6 | 1.44 1.5534 0.8766 2.9785796 16428.8 1.0018681 7 | 1.46 1.5717 0.8858 2.9786667 16447.9 1.0002507 8 | 1.4632 1.5779 0.8886 2.9786686 16448.3 1.0000019 9 | 1.4633 1.5779 0.8886 2.9786686 16448.3 0.9999940 10 | 1.48 1.5903 0.8953 2.9786250 16438.8 0.9987308 11 | 1.5 1.6089 0.9048 2.9784660 16403.9 0.9973042 12 | 1.6 1.70 0.95 2.9762475 15917.0 0.9914107 13 | 1.7 1.79 1.00 2.9723602 15063.8 0.9872358 14 | 1.8 1.88 1.05 2.9675157 14000.6 0.9843976 15 | -------------------------------------------------------------------------------- /main.tex: -------------------------------------------------------------------------------- 1 | %!TEX TS-program = xelatex 2 | %!TEX encoding = UTF-8 Unicode 3 | 4 | \documentclass[UTF8,scheme=chinese,heading]{ctexbook} 5 | 6 | %\linespread{1.2} 7 | 8 | \input{structure} 9 | 10 | \date{\today} 11 | 12 | 13 | \begin{document} 14 | %\include{./Chaps/preface} 15 | 16 | \include{./Chaps/progess} 17 | \tableofcontents % Print the table of contents itself 18 | \setcounter{page}{0} 19 | 20 | %注意替换\ts -> \mathscr{S}, \cs _> a^\dagger, \sch -> Schr\''odinger, \hs 为\mathscr{H}, \vs -> \mathscr{V}, 算子 -> 算符. \ht 21 | % 22 | \include{./Chaps/Chap1} 23 | \include{./Chaps/Chap2} 24 | \include{./Chaps/Chap3} 25 | \include{./Chaps/Chap4} 26 | \include{./Chaps/Chap5} 27 | \include{./Chaps/Chap6} 28 | \include{./Chaps/Chap7} 29 | 30 | \appendix 31 | \include{./Chaps/AppendixA} 32 | \include{./Chaps/AppendixB} 33 | \include{./Chaps/AppendixC} 34 | \include{./Chaps/AppendixD} 35 | 36 | \end{document} 37 | -------------------------------------------------------------------------------- /dev_guide.md: -------------------------------------------------------------------------------- 1 | 2 | ## mathscr 3 | 4 | 目前默认的`\mathscr`是`mathalpha`提供的 boondoxo 字体。要使用 rsfs 的话可以用 `\mathrsfs`. ([pr56](https://github.com/NominHanggai/szaboqc/pull/56)) 5 | 6 | ## autoref 7 | 8 | 尽量用 `\autoref` 来引用图表和公式，其引用章节、方程、图、表时自动写成“第……章（节）”、“式(x.y)”、“图x.y”、“表x.y”这样的样式。 9 | 详见[pr21](https://github.com/NominHanggai/szaboqc/pull/21). 10 | 11 | ## 公式编号 12 | 13 | (2.3a)(2.3b) 之类的编号用 `subequations` 实现。参见[pr17](https://github.com/NominHanggai/szaboqc/pull/17). 14 | 15 | 我们希望翻译版的公式编号和原书一一对应。 16 | 17 | ## 上波浪线 18 | 19 | 在书写带有上波浪线的符号时，使用 `\widetilde{}` $\widetilde{\Psi}$ 而不是 `\tilde{}` $\tilde{\Psi}$，以获得更好的显示效果。 20 | 21 | ## 希腊字母加粗 22 | 23 | **在数学环境下**，如果要加粗希腊字母，避免使用 `\boldsymbol{}`，应当使用 `\mathbf{}` 或者便捷方式 `\mbf{}`。由于字体原因，LaTeX 会自动将希腊字母的 `\boldsymbol{}` 转为 `\mathbf{}` 字体。 24 | 25 | ## 带有数学环境的章节名（书签） 26 | 27 | 如果章节名中有数学环境，为了在生成的 PDF 书签中正确显示章节名，需要使用 `\texorpdfstring{}{}`，其中 `` 是纯文本内容，不含数学环境。例如 H₂ 应写为 `\texorpdfstring{H$_2$}{H₂}`。 28 | -------------------------------------------------------------------------------- /Chaps/AppendixD.tex: -------------------------------------------------------------------------------- 1 | \chapter{$\hd$的分子积分作为键长的函数} 2 | \label{appendix:d} 3 | 所有的物理量都使用原子单位。这些积分是由正文\autoref{sec:3.5.2}中所述， 4 | 使用极小基组$STO-3G$在Slater指数$\zeta=1.24$下计算得到的。 5 | \begin{table}[h!] 6 | \centering 7 | \begin{tabular}{lllllll} 8 | \hline 9 | $\bm{R}$ & $\varepsilon_1 $& $\varepsilon_2 $ 10 | & $J_{11}$ & $J_{12}$& $J_{22}$& $K_{12}$ 11 | \\ \hline 12 | 0.6&$-0.7927$&1.3327&0.7469&0.7392&0.7817&0.1614 13 | \\0.8&$-0.7321$&1.1233&0.7330&0.7212&0.7607&0.1655 14 | \\1.0&$-0.6758$&0.9418&0.7144&0.7019&0.7388&0.1702 15 | \\1.2&$-0.6245$&0.7919&0.6947&0.6824&0.7176&0.1755 16 | \\1.4&$-0.5782$&0.6703&0.6746&0.6636&0.6975&0.1813 17 | \\1.6&$-0.5368$&0.5715&0.6545&0.6457&0.6786&0.1874 18 | \\1.8&$-0.4998$&0.4898&0.6349&0.6289&0.6608&0.1938 19 | \\2.0&$-0.4665$&0.4209&0.6162&0.6131&0.6439&0.2005 20 | \\2.5&$-0.3954$&0.2889&0.5751&0.5789&0.6057&0.2179 21 | \\3.0&$-0.3377$&0.1981&0.5432&0.5512&0.5734&0.2351 22 | \\4.0&$-0.2542$&0.0916&0.5026&0.5121&0.5259&0.2651 23 | \\5.0&$-0.2028$&0.0387&0.4808&0.4873&0.4947&0.2877 24 | \\7.5&$-0.1478$&$-0.0114$&0.4533&0.4540&0.4547&0.3206 25 | \\10.0&$-0.1293$&$-0.0292$&0.4373&0.4373&0.4373&0.3373 26 | \\20.0&$-0.1043$&$-0.0543$&0.4123&0.4123&0.4123&0.3623 27 | \\100.0&$-0.0843$&$-0.0743$&0.3923&0.3923&0.3923&0.3823 28 | \\$\infty$&$-0.0793$&$-0.0793$&0.3873&0.3873&0.3873&0.3873 29 | \\ \hline 30 | \end{tabular} 31 | \label{tD.1} 32 | \end{table} -------------------------------------------------------------------------------- /Pictures/MPn-energy.tex: -------------------------------------------------------------------------------- 1 | %!TEX TS-program = lualatex 2 | %!TEX encoding = UTF-8 Unicode 3 | \documentclass[]{book} 4 | \usepackage{amsmath,tikz,braket} 5 | \usepackage{pgfplots} 6 | \pgfplotsset{compat=newest} 7 | \begin{document} 8 | \begin{tikzpicture} 9 | \begin{axis}[ 10 | %standard, 11 | %ticks=none, 12 | axis y line=center, 13 | axis x line=middle, 14 | axis x line shift=-.55, 15 | every axis x label/.style={at={(axis description cs:1.02,.57)}}, 16 | xlabel={$M$}, 17 | every axis y label/.style={rotate=90,at={(axis description cs:-0.13,.54)}}, 18 | ylabel={$\displaystyle\sum_{n=2}^{M}E_0^{(n)}/\beta$}, 19 | % axis on top=true, 20 | xmin=1, 21 | xmax=19, 22 | xtick={10,16}, 23 | extra x ticks={2,4,6,8,12,18,14}, 24 | extra x tick style={ 25 | xticklabel style={yshift=0.5ex, anchor=south} 26 | }, 27 | ymin=1.45, 28 | ymax=2.45, 29 | ytick distance=.1, 30 | height=.66\textwidth, 31 | width=\textwidth, 32 | %restrict y to domain*=0:3, 33 | ] 34 | \addplot [domain=1:18 mark=none, dashed, thick, blue] table{gegenbauer.dat}; 35 | \addplot [only marks,blue,every mark/.append style={scale=1.3}]coordinates 36 | {(2,1.5) (3,2.25) (4,2.34375) (5,2.10938) (6,1.88672) (7,1.8457) (8,1.95154) (9,2.06708) (10,2.0912) (11,2.02801) (12,1.95462) (13,1.93831) (14,1.98141) (15,2.03325) (16,2.04521) (17,2.01342) (18,1.97432)}; 37 | %\node [left, blue] at (axis cs: 0,120) {$ \scr{E}_{\rm tot}(\{\bo{R}_A\})$}; 38 | %\node [above, blue] at (axis cs: .8,0) {$\{\bo{R}_A\}$}; 39 | \end{axis} 40 | \end{tikzpicture} 41 | %\tikzset{every mark/.append style={scale=2}} 42 | \end{document} 43 | -------------------------------------------------------------------------------- /.github/workflows/build.yml: -------------------------------------------------------------------------------- 1 | name: Build 2 | 3 | on: 4 | push: 5 | branches: 6 | - master 7 | 8 | pull_request: 9 | branches: 10 | - master 11 | 12 | env: 13 | CTAN_URL: https://mirrors.rit.edu/CTAN 14 | TL_PACKAGES: amscls amsmath anyfontsize blkarray booktabs braket caption chemformula ctex endnotes enumitem fancyhdr float graphics hyperref latexmk luatex85 l3packages mathtools metafont multirow needspace newfloat pgfplots scalerel siunitx stackengine threeparttable tools ulem xcolor xecjk xfrac 15 | TL_FONT_PCK: boondox fandol libertinus-fonts mathalpha psnfss collection-fontsrecommended 16 | 17 | jobs: 18 | build-on-ubuntu: 19 | runs-on: ubuntu-latest 20 | if: "!startsWith(github.ref, 'refs/tags/v')" 21 | env: 22 | SET_PATH: | 23 | export PATH=/tmp/texlive/bin/x86_64-linux:$PATH 24 | steps: 25 | - name: Set up Git repository 26 | uses: actions/checkout@v4 27 | 28 | - name: Install TeX Live 29 | run: | 30 | ${{ env.SET_PATH }} 31 | wget ${{ env.CTAN_URL }}/systems/texlive/tlnet/install-tl-unx.tar.gz 32 | tar -xzf install-tl-unx.tar.gz 33 | cd install-tl-20* && ./install-tl --profile ../.github/workflows/texlive.profile 34 | tlmgr install ${{ env.TL_PACKAGES }} ${{ env.TL_FONT_PCK }} 35 | tlmgr update --self --all --no-auto-install --repository=${{ env.CTAN_URL }}/systems/texlive/tlnet/ 36 | - name: Compile test file 37 | run: | 38 | ${{ env.SET_PATH }} 39 | make 40 | - name: Upload PDF 41 | uses: actions/upload-artifact@v4 42 | with: 43 | name: generated-pdf 44 | path: 45 | ./*.pdf 46 | -------------------------------------------------------------------------------- /.github/workflows/release.yml: -------------------------------------------------------------------------------- 1 | name: Release 2 | 3 | on: 4 | push: 5 | branches: 6 | - master 7 | 8 | tags: 9 | - v* 10 | 11 | env: 12 | CTAN_URL: https://mirrors.rit.edu/CTAN 13 | TL_PACKAGES: amscls amsmath anyfontsize blkarray booktabs braket caption chemformula ctex endnotes enumitem fancyhdr float graphics hyperref latexmk luatex85 l3packages mathtools metafont multirow needspace newfloat pgfplots scalerel siunitx stackengine threeparttable tools ulem xcolor xecjk xfrac 14 | TL_FONT_PCK: boondox fandol libertinus-fonts mathalpha psnfss collection-fontsrecommended 15 | 16 | jobs: 17 | release: 18 | runs-on: ubuntu-latest 19 | if: startsWith(github.ref, 'refs/tags/v') 20 | env: 21 | SET_PATH: | 22 | export PATH=/tmp/texlive/bin/x86_64-linux:$PATH 23 | steps: 24 | - name: Set up Git repository 25 | uses: actions/checkout@v4 26 | 27 | - name: Set Version 28 | run: echo "VERSION=${GITHUB_REF##*/}" >> $GITHUB_ENV 29 | 30 | - name: Install TeX Live 31 | run: | 32 | ${{ env.SET_PATH }} 33 | wget ${{ env.CTAN_URL }}/systems/texlive/tlnet/install-tl-unx.tar.gz 34 | tar -xzf install-tl-unx.tar.gz 35 | cd install-tl-20* && ./install-tl --profile ../.github/workflows/texlive.profile 36 | tlmgr install ${{ env.TL_PACKAGES }} ${{ env.TL_FONT_PCK }} 37 | tlmgr update --self --all --no-auto-install --repository=${{ env.CTAN_URL }}/systems/texlive/tlnet/ 38 | - name: Compile test file 39 | run: | 40 | ${{ env.SET_PATH }} 41 | make 42 | mv main.pdf szabo_zh-$VERSION.pdf 43 | - name: Create release 44 | uses: "marvinpinto/action-automatic-releases@latest" 45 | with: 46 | repo_token: "${{ secrets.GITHUB_TOKEN }}" 47 | prerelease: false 48 | files: | 49 | szabo_zh-v*.pdf 50 | -------------------------------------------------------------------------------- /Chaps/progess.tex: -------------------------------------------------------------------------------- 1 | \chapter*{进度表} 2 | 3 | \makeatletter 4 | \@date 5 | \makeatother 6 | 7 | \begin{itemize} 8 | \item[\CheckedBox] 第一章 数学预备 9 | \item[\CheckedBox] 第二章 多电子波函数与多电子算符 10 | \item[\DSquare] 第三章 Hatree-Fock近似 11 | \begin{itemize} 12 | \item[\CheckedBox] 3.1-3.5 13 | \item[\CheckedBox] 3.6 14 | % \begin{itemize} 15 | % \item[\CheckedBox] 3.6.1,3.6.2 16 | % \item[\Square] 3.6.3, 3.6.4 17 | % \end{itemize} 18 | \item[\DSquare] 3.7 19 | \begin{itemize} 20 | \item [\DSquare] 3.7.1-3.7.2 21 | \item [\Square] 3.7.3-3.7.4 22 | \end{itemize} 23 | \item[\DSquare] 3.8 24 | \begin{itemize} 25 | \item[\CheckedBox] 3.8.1-3.8.5 26 | \item[\Square] 3.8.6 27 | \item[\DSquare] 3.8.7 28 | \end{itemize} 29 | \end{itemize} 30 | \item[\DSquare] 31 | 第四章 组态相互作用 32 | \begin{itemize} 33 | \item[\DSquare] 4.1 34 | \item[\CheckedBox] 4.2 35 | \item[\Square] 4.3 36 | \item[\CheckedBox] 4.4 37 | \item[\DSquare] 4.5 38 | \item[\Square] 4.6 39 | \end{itemize} 40 | \item[\DSquare] 第五章 对理论与耦合对理论 41 | \begin{itemize} 42 | \item[\DSquare] 5.1 43 | \item[\DSquare] 5.2 44 | \begin{itemize} 45 | \item [\CheckedBox] 5.2.1 46 | \item [\CheckedBox] 5.2.2 47 | \item [\Square] 5.2.3-5.2.4 48 | \end{itemize} 49 | \item[\Square] 5.3 50 | \end{itemize} 51 | \item[\CheckedBox] 第六章 多体微扰论 52 | % \begin{itemize} 53 | % \item[\CheckedBox] 6.1-6.8 54 | % \item[\CheckedBox] 6.8 55 | % \end{itemize} 56 | \item[\DSquare] 单粒子多体格林函数 57 | \begin{itemize} 58 | \item[\CheckedBox] 7.1-7.2 59 | \item[\DSquare] 7.3 60 | \item[\Square] 7.4-7.5 61 | \end{itemize} 62 | \item[\DSquare] 附录 \begin{itemize} 63 | \item[\CheckedBox] 附录A 64 | \item[\CheckedBox] 附录B 65 | \item[\DSquare] 附录C 66 | \begin{itemize} 67 | \item[\CheckedBox] C.1-C.4 68 | \item[\Square] C.5-C.7 69 | \end{itemize} 70 | \item[\CheckedBox] 附录D 71 | \end{itemize} 72 | \end{itemize} 73 | -------------------------------------------------------------------------------- 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-------------------------------------------------------------------------------- 1 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2 | % The Szabo Book 3 | % Structural Definitions File 4 | % Version 0.1 (5/4/20) 5 | % 6 | \RequirePackage{luatex85,shellesc} 7 | \usepackage[top=1.7cm,bottom=1.5cm,left=1.7cm,right=1.7cm,headsep=18pt,papersize={13.8176cm,21.6cm} 8 | ]{geometry} 9 | \usepackage{indentfirst} 10 | \usepackage{graphicx} % Required for including pictures 11 | \graphicspath{{Pictures/}} 12 | \usepackage{multirow} 13 | \usepackage{tikz,pgfplots} % Required for drawing custom shapes 14 | %\pgfplotsset{compat=newest} 15 | \usepackage{booktabs,threeparttable 16 | } % Booktabs required for nicer horizontal rules in tables 17 | \usepackage[shortlabels]{enumitem} 18 | \setenumerate{leftmargin=*} 19 | \usepackage{xcolor} % Required for specifying colors by name 20 | %\definecolor{ocre}{RGB}{0,0,0} % Define the favorite color used for highlighting throughout the book 21 | \bibliographystyle{alpha} 22 | 23 | 24 | %------------------------------------------------------------------- 25 | % Chinese Settings 26 | %------------------------------------------------------------------- 27 | \usepackage[format=hang,labelfont=bf]{caption} 28 | %\captionsetup{font={scriptsize}} 29 | \captionsetup{figurename={图}, tablename={表}} 30 | %\addto{\captionsenglish}{\renewcommand{\contentsname}{目\quad录}} 31 | \def\chntoday{\the\year~年~\the\month~月~\the\day~日} 32 | % adjust the width of tables to adapt it to the textwidth 33 | \usepackage{tabularx} 34 | \newcolumntype{Y}{>{\centering\arraybackslash}X} % New flag to centering the width-adapted columns 35 | 36 | \usepackage{amsmath} 37 | \usepackage{anyfontsize} 38 | \usepackage{amssymb} 39 | \usepackage{bm} 40 | 41 | \usepackage{mathtools} % For *rcases* environment 42 | \usepackage[scr=boondoxo, scrscaled=1]{mathalfa} 43 | \DeclareMathAlphabet{\mathrsfs}{U}{rsfs}{m}{n} 44 | \DeclareFontFamily{U}{rsfs}{\skewchar\font127 } 45 | \DeclareFontShape{U}{rsfs}{m}{n}{% 46 | <-6> rsfs5 47 | <6-8> rsfs7 48 | <8-> rsfs10 49 | }{} 50 | %\usepackage{mathrsfs} 51 | %\usepackage{mathptmx} 52 | \usepackage{mathpazo} 53 | %\usepackage{old-arrows} % 54 | \usepackage[T1]{fontenc} 55 | %\usepackage{newpxtext,newpxmath} 56 | %\usepackage{newtxtext} 57 | %\usepackage{newtxmath,} %A systematic solution to Roman fonts, including math fonts. 58 | \usepackage{braket,ulem} 59 | \renewcommand{\braket}[1]{\Braket{#1}} 60 | \renewcommand{\ket}[1]{\Ket{#1}} 61 | \renewcommand{\bra}[1]{\Bra{#1}} 62 | \ctexset{ 63 | chapter={ 64 | name = {第,章}, 65 | format = \Large\bfseries\raggedright, 66 | aftername = \par\bigskip}, 67 | section/format = \fontsize{8.5pt}{\baselineskip}\selectfont\bfseries\raggedright, 68 | subsection/format = \fontsize{8pt}{\baselineskip}\selectfont\bfseries\raggedright 69 | } 70 | 71 | %----------------------------------------------------------------- 72 | % Page Headers 73 | %----------------------------------------------------------------- 74 | 75 | \usepackage{fancyhdr} % Required for header and footer configuration 76 | \renewcommand{\chaptermark}[1]{\markboth{\sffamily\normalsize\CTEXthechapter\ #1}{}} % Chapter text font settings 77 | \renewcommand{\sectionmark}[1]{\markright{\normalfont\kaishu\normalsize\CTEXthesection\hspace{5pt}#1}{}} 78 | 79 | %---------------------------------------------------------------------------------------- 80 | \let\bar\undifined 81 | \newcommand{\bar}[1]{\overline{#1}} 82 | \newcommand{\scr}[1]{\mathscr{#1}} 83 | \newcommand{\bo}[1]{\mathbf{#1}} 84 | \renewcommand{\epsilon}{\varepsilon} 85 | \newcommand{\sch}{Schr\"odinger} 86 | \newcommand{\ha}{Hamiltonian} 87 | %\newcommand{\dd}{\text{d}} 88 | \usepackage{xparse} 89 | \newcommand{\dd}[1]{\mathrm{d}\mathbf{#1}} 90 | \let\dd\undefined 91 | \NewDocumentCommand{\dd}{ g }{% 92 | \IfValueTF{#1} 93 | {% https://tex.stackexchange.com/q/53068/5764 94 | \if\relax\detokenize{#1}\relax 95 | \mathrm{d} % 96 | \else 97 | \mathrm{d}\mathbf{#1}% 98 | \fi} 99 | {\mathrm{d}}% 100 | } 101 | \newcommand{\ddx}{\dd\mathbf{x}} 102 | 103 | \newcommand{\db}[1]{\dd\bo{#1}} 104 | \newcommand{\hs}{\mathscr{H}} 105 | \newcommand{\vs}{\mathscr{V}} 106 | \newcommand{\es}{\mathscr{E}} 107 | \renewcommand{\emph}[1]{\textbf{#1}} 108 | 109 | 110 | \usepackage{calc} 111 | %use \settowidth, \widthof 112 | 113 | \usepackage{endnotes} 114 | \renewcommand{\makeenmark}{\hbox{$^\theenmark$}} 115 | \renewcommand{\notesname}{\bf 注释} 116 | \makeatletter 117 | \@addtoreset{endnote}{chapter} 118 | \makeatother 119 | 120 | %----------------------------------------------------------------- 121 | % the command to generate two lines between which are excerise content, using amsthm to number the exercise. 122 | \let\openbox\undefined 123 | \usepackage{amsthm} 124 | \usepackage{needspace} 125 | \newtheoremstyle{nx}{}{}{\normalfont}{}{\bfseries}{}{1em}{} 126 | \theoremstyle{nx} 127 | \newtheorem{Xercise}{$\quad$练习}[chapter] 128 | \newenvironment{xercise}{\Needspace{2\baselineskip}\par\vspace{\baselineskip}\noindent\hrule\vspace{0\baselineskip}\Xercise\label{ex:\theXercise} 129 | } 130 | {\vspace{.5\baselineskip}\noindent\hrule\par\endXercise} 131 | 132 | \newcommand\Next{\endXercise\hrule\Xercise\label{ex:\theXercise}} 133 | \newcommand{\exercise}[1]{\begin{xercise}#1\end{xercise}} 134 | %---------------------------------------------------------------------------------------- 135 | 136 | % 引用章节、方程、图、表时自动写成“第……章（节）”、“式(x.y)”、“图x.y”、“表x.y”这样的型式： 137 | \def\chapterautorefname~#1\null{第#1章\null} 138 | \def\sectionautorefname~#1\null{小节#1\null} 139 | \def\subsectionautorefname~#1\null{小节#1\null} 140 | \def\subsubsectionautorefname~#1\null{小节#1\null} 141 | \def\equationautorefname~#1\null{式(#1)\null} 142 | \def\tableautorefname~#1\null{表#1\null} 143 | \def\figureautorefname~#1\null{图#1\null} 144 | \def\appendixautorefname~#1\null{附录#1\null} 145 | \def\Xerciseautorefname~#1\null{练习#1\null} 146 | 147 | %======================================================== 148 | 149 | \newcommand{\tu}{\begin{figure}[h] 150 | \includegraphics[width=.5\textwidth]{q.png} 151 | \end{figure}} 152 | % Feynman Diagram Setting 153 | \usetikzlibrary{arrows.meta} 154 | \newcommand{\midarrow}{\tikz \draw[-triangle 45] (0,0) -- +(.05,0);} 155 | \tikzset{> = latex, color=blue} 156 | \usetikzlibrary{decorations.markings,decorations.pathreplacing} 157 | \tikzset{ 158 | % style to apply some styles to each segment of a path 159 | on each segment/.style={draw=blue, 160 | decorate, 161 | decoration={ 162 | show path construction, 163 | moveto code={}, 164 | lineto code={ 165 | \path [#1] 166 | (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast); 167 | }, 168 | curveto code={ 169 | \path [#1] (\tikzinputsegmentfirst) 170 | .. controls 171 | (\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb) 172 | .. 173 | (\tikzinputsegmentlast); 174 | }, 175 | closepath code={ 176 | \path [#1] 177 | (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast); 178 | }, 179 | }, 180 | }, 181 | % style to add an arrow in the middle of a path 182 | mid arrow/.style={postaction={decorate,decoration={ 183 | markings, 184 | mark=at position #1 with {\draw[arrows = {-Latex[width=0pt 7, length=5pt]}] (0pt,.0pt) -- (1.8pt,0pt);} 185 | }}}, 186 | mid arrow/.default={.55}, 187 | mid arrow seg/.style={postaction={on each segment={decorate,decoration={ 188 | markings, 189 | mark=at position #1 with {\draw[arrows = {-Latex[width=0pt 7, length=5pt]}] (0pt,.0pt) -- (1.8pt,0pt);}} 190 | }}}, 191 | mid arrow seg/.default={.55}, 192 | % style to add an reverse arrow in the middle of a path 193 | mid reverse arrow/.style={postaction={decorate,decoration={ 194 | markings, 195 | mark=at position .45 with {\draw[arrows = {-Latex[width=0pt 7, length=5pt]}] (1.8pt,.0pt) -- (0pt,0pt);} 196 | }}}, 197 | mid reverse arrow seg/.style={postaction={on each segment={decorate,decoration={ 198 | markings, 199 | mark=at position #1 with {\draw[arrows = {-Latex[width=0pt 7, length=5pt]}] (1.8pt,.0pt) -- (0pt,0pt);}} 200 | }}}, 201 | above arrow/.style={to path={-- ++(0, .2) -| (\tikztotarget)},arrows = {-Latex[width=0pt 7, length=5pt]}}, 202 | below arrow/.style={to path={-- ++(0,-.2) -| (\tikztotarget)}} 203 | } 204 | \newlength{\wletter} 205 | \newlength{\hletter} 206 | \newcommand{\tikzmark}[1]{ 207 | \settowidth{\wletter}{\text{$\mathsurround=0pt i$}} 208 | \settoheight{\hletter}{\text{$\mathsurround=0pt i$}} 209 | \tikz[overlay,remember picture,inner sep=0,minimum height=\hletter] \node[shift={(-.4\wletter,.45\hletter)}] (#1) {};} 210 | \newcommand{\dian}{\tikz{\filldraw[blue](0,0)circle(1pt);}} 211 | 212 | \usetikzlibrary{calc} 213 | %\usepackage[compat=1.1.0]{tikz-feynman} 214 | \usepackage{float} 215 | \renewcommand{\thefootnote}{\roman{footnote}} % Change the marks of footnotes to numbers 216 | \usepackage{blkarray} %分块矩阵 217 | \definecolor{darkred}{HTML}{990000} 218 | \usepackage[bookmarksnumbered=true, colorlinks=true, linkcolor=darkred, urlcolor=darkred]{hyperref} 219 | \usepackage{chemformula} 220 | 221 | 222 | \usepackage{wasysym} 223 | %For the symbol \CheckedBox 224 | \usepackage{stackengine,scalerel} 225 | \newcommand\DSquare{\ThisStyle{\ensurestackMath{% 226 | \stackinset{c}{}{c}{}{\scalebox{.5}{$\SavedStyle\blacksquare$}} 227 | {\SavedStyle\square}}}} 228 | %For a filled box. 229 | %CR: https://tex.stackexchange.com/questions/553834/is-there-a-intermediate-checkedbox-symbol-in-latex 230 | 231 | %----------------------------------------------------------------- 232 | % Convinent abbreviations 233 | %----------------------------------------------------------------- 234 | \renewcommand{\normalsize}{\fontsize{8pt}{9.6pt}\selectfont} 235 | \newcommand{\diagsize}{\fontsize{5pt}{6.6pt}\selectfont} 236 | \newcommand{\cs}{a^\dagger} 237 | \newcommand{\hd}{\mathrm{H}_2} 238 | \newcommand{\hft}{Hartree-Fock} 239 | \newcommand{\twoe}{r_{12}^{-1}} 240 | \newcommand{\tp}{\tilde{\Phi}} 241 | %\usepackage[symbol]{footmisc} 242 | \newcommand{\bfr}{\mathbf{r}} 243 | \newcommand{\heh}{\mathrm{HeH}^+} 244 | \newcommand{\au}{\,\mathrm{a.u.}} 245 | %\newcommand{\ts}{\mathscr{S}} %总自旋算符 246 | \newcommand{\ts}{\mathrsfs{S}} 247 | \usepackage{xeCJKfntef} % the package containing the command \CJKuderline*{} for the \mci command 248 | \newcommand{\mci}[1]{\CJKunderline{#1}} 249 | \newcommand{\phrase}[1]{\CJKunderline{#1}} 250 | %%为方便而定义的简单代替, 最后要全部换回来. 251 | \newcommand{\jt}{\ket{\Phi_0}} 252 | \newcommand{\hjt}{\ket{\Psi_0}} 253 | \newcommand{\jtn}{\Phi_0} 254 | \newcommand{\hjtn}{\Psi_0} 255 | 256 | %Convenience commands defined by Hao 257 | \newcommand{\mrm}{\mathrm} 258 | \newcommand{\mbf}{\mathbf} 259 | \newcommand{\mcr}{\mathscr} 260 | \newcommand{\op}{\mathscr} 261 | \newcommand{\bs}[2]{\ensuremath\left\{ \vec{#1}_{#2} \right\}} 262 | \newcommand{\kbs}[1]{\ensuremath\left\{ \ket{#1} \right\}} 263 | \newcommand{\bbs}[1]{\ensuremath\left\{ \bra{#1} \right\}} 264 | \newcommand{\madj}[1]{\ensuremath{\mbf #1}^\dagger} 265 | \newcommand{\oadj}[1]{\ensuremath{\op #1}^\dagger} 266 | \newcommand{\tr}[1]{\ensuremath{\mrm{tr}}\,#1} 267 | %\newcommand{\ket}[1]{\ensuremath\left\vert #1\right\rangle} 268 | %\newcommand{\bra}[1]{\ensuremath\left\langle #1\right\vert} 269 | \newcommand{\Langle}{\ensuremath\Big\langle} 270 | \newcommand{\Rangle}{\ensuremath\Big\rangle} 271 | \newcommand{\Lvert}{\ensuremath\Big\vert} 272 | \newcommand{\olp}[2]{\ensuremath\langle #1 \vert #2 \rangle} 273 | \newcommand{\Olp}[2]{\ensuremath\left\langle #1 \middle\vert #2 \right\rangle} 274 | \newcommand{\oup}[2]{\ensuremath\left\vert #1 \right\rangle \left\langle #2 \right\vert} 275 | \newcommand{\mele}[3]{\ensuremath\langle #1\vert \op{#2} \vert #3\rangle} 276 | \newcommand{\Mele}[3]{\ensuremath\left\langle #1\middle\vert \op{#2} \middle\vert #3\right\rangle} 277 | \newcommand\bigzero{\makebox(0,0){\text{\huge0}}} 278 | %=============END=================== 279 | 280 | -------------------------------------------------------------------------------- /code/out: -------------------------------------------------------------------------------- 1 | STO-3G FOR ATOMIC NUMBERS 2.00 AND 1.00 2 | 3 | 4 | R ZETA1 ZETA2 S12 T11 5 | 6 | 1.463200 2.092500 1.240000 0.450770 2.164313 7 | 8 | 9 | T12 T22 V11A V12A V22A 10 | 11 | 0.167013 0.760033 -4.139827 -1.102912 -1.265246 12 | 13 | 14 | V11B V12B V22B V1111 V2111 15 | 16 | -0.677230 -0.411305 -1.226615 1.307152 0.437279 17 | 18 | 19 | V2121 V2211 V2221 V2222 20 | 21 | 0.177267 0.605703 0.311795 0.774608 22 | 23 | 24 | 25 | THE S ARRAY 26 | 1 2 27 | 1 0.1000000000E+01 0.4507704116E+00 28 | 2 0.4507704116E+00 0.1000000000E+01 29 | 30 | 31 | 32 | THE X ARRAY 33 | 1 2 34 | 1 0.5870642812E+00 0.9541310722E+00 35 | 2 0.5870642812E+00 -0.9541310722E+00 36 | 37 | 38 | 39 | THE H ARRAY 40 | 1 2 41 | 1 -0.2652744703E+01 -0.1347205024E+01 42 | 2 -0.1347205024E+01 -0.1731828436E+01 43 | 44 | 45 | 46 | ( 1 1 1 1 ) 1.307152 47 | ( 1 1 1 2 ) 0.437279 48 | ( 1 1 2 1 ) 0.437279 49 | ( 1 1 2 2 ) 0.605703 50 | ( 1 2 1 1 ) 0.437279 51 | ( 1 2 1 2 ) 0.177267 52 | ( 1 2 2 1 ) 0.177267 53 | ( 1 2 2 2 ) 0.311795 54 | ( 2 1 1 1 ) 0.437279 55 | ( 2 1 1 2 ) 0.177267 56 | ( 2 1 2 1 ) 0.177267 57 | ( 2 1 2 2 ) 0.311795 58 | ( 2 2 1 1 ) 0.605703 59 | ( 2 2 1 2 ) 0.311795 60 | ( 2 2 2 1 ) 0.311795 61 | ( 2 2 2 2 ) 0.774608 62 | 63 | 64 | 65 | THE P ARRAY 66 | 1 2 67 | 1 0.0000000000E+00 0.0000000000E+00 68 | 2 0.0000000000E+00 0.0000000000E+00 69 | 70 | START OF ITERATION NUMBER = 1 71 | 72 | 73 | 74 | THE G ARRAY 75 | 1 2 76 | 1 0.0000000000E+00 0.0000000000E+00 77 | 2 0.0000000000E+00 0.0000000000E+00 78 | 79 | 80 | 81 | THE F ARRAY 82 | 1 2 83 | 1 -0.2652744703E+01 -0.1347205024E+01 84 | 2 -0.1347205024E+01 -0.1731828436E+01 85 | 86 | 87 | 88 | ELECTRONIC ENERGY = 0.000000000000E+00 89 | 90 | 91 | 92 | THE F' ARRAY 93 | 1 2 94 | 1 -0.2439732411E+01 -0.5158386047E+00 95 | 2 -0.5158386047E+00 -0.1538667186E+01 96 | 97 | 98 | 99 | THE C' ARRAY 100 | 1 2 101 | 1 0.9104452570E+00 0.4136295856E+00 102 | 2 0.4136295856E+00 -0.9104452570E+00 103 | 104 | 105 | 106 | THE E ARRAY 107 | 1 2 108 | 1 -0.2674085994E+01 0.0000000000E+00 109 | 2 0.0000000000E+00 -0.1304313603E+01 110 | 111 | 112 | 113 | THE C ARRAY 114 | 1 2 115 | 1 0.9291467304E+00 -0.6258569539E+00 116 | 2 0.1398330503E+00 0.1111511265E+01 117 | 118 | 119 | 120 | THE P ARRAY 121 | 1 2 122 | 1 0.1726627293E+01 0.2598508430E+00 123 | 2 0.2598508430E+00 0.3910656393E-01 124 | 125 | DELTA(CONVERGENCE OF DENSITY MATRIX) = 0.882867 126 | 127 | 128 | START OF ITERATION NUMBER = 2 129 | 130 | 131 | 132 | THE G ARRAY 133 | 1 2 134 | 1 0.1262330044E+01 0.3740040563E+00 135 | 2 0.3740040563E+00 0.9889530699E+00 136 | 137 | 138 | 139 | THE F ARRAY 140 | 1 2 141 | 1 -0.1390414659E+01 -0.9732009679E+00 142 | 2 -0.9732009679E+00 -0.7428753661E+00 143 | 144 | 145 | 146 | ELECTRONIC ENERGY = -0.414186268681E+01 147 | 148 | 149 | 150 | THE F' ARRAY 151 | 1 2 152 | 1 -0.1406043275E+01 -0.3627102456E+00 153 | 2 -0.3627102456E+00 -0.1701365815E+00 154 | 155 | 156 | 157 | THE C' ARRAY 158 | 1 2 159 | 1 0.9649913726E+00 0.2622816249E+00 160 | 2 0.2622816249E+00 -0.9649913726E+00 161 | 162 | 163 | 164 | THE E ARRAY 165 | 1 2 166 | 1 -0.1504626781E+01 0.0000000000E+00 167 | 2 0.0000000000E+00 -0.7155307568E-01 168 | 169 | 170 | 171 | THE C ARRAY 172 | 1 2 173 | 1 0.8167630145E+00 -0.7667520795E+00 174 | 2 0.3162609186E+00 0.1074704427E+01 175 | 176 | 177 | 178 | THE P ARRAY 179 | 1 2 180 | 1 0.1334203644E+01 0.5166204425E+00 181 | 2 0.5166204425E+00 0.2000419373E+00 182 | 183 | DELTA(CONVERGENCE OF DENSITY MATRIX) = 0.279176 184 | 185 | 186 | START OF ITERATION NUMBER = 3 187 | 188 | 189 | 190 | THE G ARRAY 191 | 1 2 192 | 1 0.1201346300E+01 0.3038061741E+00 193 | 2 0.3038061741E+00 0.9284329600E+00 194 | 195 | 196 | 197 | THE F ARRAY 198 | 1 2 199 | 1 -0.1451398403E+01 -0.1043398850E+01 200 | 2 -0.1043398850E+01 -0.8033954759E+00 201 | 202 | 203 | 204 | ELECTRONIC ENERGY = -0.422649172562E+01 205 | 206 | 207 | 208 | THE F' ARRAY 209 | 1 2 210 | 1 -0.1496305530E+01 -0.3629699437E+00 211 | 2 -0.3629699437E+00 -0.1529380263E+00 212 | 213 | 214 | 215 | THE C' ARRAY 216 | 1 2 217 | 1 0.9694747516E+00 0.2451911622E+00 218 | 2 0.2451911622E+00 -0.9694747516E+00 219 | 220 | 221 | 222 | THE E ARRAY 223 | 1 2 224 | 1 -0.1588104746E+01 0.0000000000E+00 225 | 2 0.0000000000E+00 -0.6113881008E-01 226 | 227 | 228 | 229 | THE C ARRAY 230 | 1 2 231 | 1 0.8030885047E+00 -0.7810630108E+00 232 | 2 0.3351994916E+00 0.1068948958E+01 233 | 234 | 235 | 236 | THE P ARRAY 237 | 1 2 238 | 1 0.1289902293E+01 0.5383897171E+00 239 | 2 0.5383897171E+00 0.2247173984E+00 240 | 241 | DELTA(CONVERGENCE OF DENSITY MATRIX) = 0.029662 242 | 243 | 244 | START OF ITERATION NUMBER = 4 245 | 246 | 247 | 248 | THE G ARRAY 249 | 1 2 250 | 1 0.1194670199E+01 0.2971625826E+00 251 | 2 0.2971625826E+00 0.9218705199E+00 252 | 253 | 254 | 255 | THE F ARRAY 256 | 1 2 257 | 1 -0.1458074504E+01 -0.1050042442E+01 258 | 2 -0.1050042442E+01 -0.8099579160E+00 259 | 260 | 261 | 262 | ELECTRONIC ENERGY = -0.422752275334E+01 263 | 264 | 265 | 266 | THE F' ARRAY 267 | 1 2 268 | 1 -0.1505447474E+01 -0.3630336096E+00 269 | 2 -0.3630336096E+00 -0.1528937446E+00 270 | 271 | 272 | 273 | THE C' ARRAY 274 | 1 2 275 | 1 0.9698136474E+00 0.2438472663E+00 276 | 2 0.2438472663E+00 -0.9698136474E+00 277 | 278 | 279 | 280 | THE E ARRAY 281 | 1 2 282 | 1 -0.1596727643E+01 0.0000000000E+00 283 | 2 0.0000000000E+00 -0.6161357601E-01 284 | 285 | 286 | 287 | THE C ARRAY 288 | 1 2 289 | 1 0.8020052055E+00 -0.7821753152E+00 290 | 2 0.3366806982E+00 0.1068483355E+01 291 | 292 | 293 | 294 | THE P ARRAY 295 | 1 2 296 | 1 0.1286424699E+01 0.5400393450E+00 297 | 2 0.5400393450E+00 0.2267077850E+00 298 | 299 | DELTA(CONVERGENCE OF DENSITY MATRIX) = 0.002318 300 | 301 | 302 | START OF ITERATION NUMBER = 5 303 | 304 | 305 | 306 | THE G ARRAY 307 | 1 2 308 | 1 0.1194147845E+01 0.2966515832E+00 309 | 2 0.2966515832E+00 0.9213575914E+00 310 | 311 | 312 | 313 | THE F ARRAY 314 | 1 2 315 | 1 -0.1458596858E+01 -0.1050553441E+01 316 | 2 -0.1050553441E+01 -0.8104708445E+00 317 | 318 | 319 | 320 | ELECTRONIC ENERGY = -0.422752909612E+01 321 | 322 | 323 | 324 | THE F' ARRAY 325 | 1 2 326 | 1 -0.1506156505E+01 -0.3630388891E+00 327 | 2 -0.3630388891E+00 -0.1529058377E+00 328 | 329 | 330 | 331 | THE C' ARRAY 332 | 1 2 333 | 1 0.9698390734E+00 0.2437461212E+00 334 | 2 0.2437461212E+00 -0.9698390734E+00 335 | 336 | 337 | 338 | THE E ARRAY 339 | 1 2 340 | 1 -0.1597397746E+01 0.0000000000E+00 341 | 2 0.0000000000E+00 -0.6166459619E-01 342 | 343 | 344 | 345 | THE C ARRAY 346 | 1 2 347 | 1 0.8019236265E+00 -0.7822589536E+00 348 | 2 0.3367921305E+00 0.1068448236E+01 349 | 350 | 351 | 352 | THE P ARRAY 353 | 1 2 354 | 1 0.1286163006E+01 0.5401631334E+00 355 | 2 0.5401631334E+00 0.2268578784E+00 356 | 357 | DELTA(CONVERGENCE OF DENSITY MATRIX) = 0.000174 358 | 359 | 360 | START OF ITERATION NUMBER = 6 361 | 362 | 363 | 364 | THE G ARRAY 365 | 1 2 366 | 1 0.1194108547E+01 0.2966131916E+00 367 | 2 0.2966131916E+00 0.9213190058E+00 368 | 369 | 370 | 371 | THE F ARRAY 372 | 1 2 373 | 1 -0.1458636156E+01 -0.1050591833E+01 374 | 2 -0.1050591833E+01 -0.8105094301E+00 375 | 376 | 377 | 378 | ELECTRONIC ENERGY = -0.422752913203E+01 379 | 380 | 381 | 382 | THE F' ARRAY 383 | 1 2 384 | 1 -0.1506209810E+01 -0.3630392881E+00 385 | 2 -0.3630392881E+00 -0.1529068392E+00 386 | 387 | 388 | 389 | THE C' ARRAY 390 | 1 2 391 | 1 0.9698409800E+00 0.2437385353E+00 392 | 2 0.2437385353E+00 -0.9698409800E+00 393 | 394 | 395 | 396 | THE E ARRAY 397 | 1 2 398 | 1 -0.1597448132E+01 0.0000000000E+00 399 | 2 0.0000000000E+00 -0.6166851652E-01 400 | 401 | 402 | 403 | THE C ARRAY 404 | 1 2 405 | 1 0.8019175078E+00 -0.7822652261E+00 406 | 2 0.3368004878E+00 0.1068445602E+01 407 | 408 | 409 | 410 | THE P ARRAY 411 | 1 2 412 | 1 0.1286143379E+01 0.5401724156E+00 413 | 2 0.5401724156E+00 0.2268691372E+00 414 | 415 | DELTA(CONVERGENCE OF DENSITY MATRIX) = 0.000013 416 | 417 | 418 | 419 | CALCULATION CONVERGED 420 | 421 | ELECTRONIC ENERGY = -0.422752913203E+01 422 | 423 | TOTAL ENERGY = -0.286066199152E+01 424 | 425 | 426 | 427 | THE PS ARRAY 428 | 1 2 429 | 1 0.1529637121E+01 0.1119927796E+01 430 | 2 0.6424383099E+00 0.4703628793E+00 -------------------------------------------------------------------------------- /Pictures/heh-tot-ener.txt: -------------------------------------------------------------------------------- 1 | 0.5000000000 -1.6894184057 2 | 0.5100000000 -1.7447916214 3 | 0.5200000000 -1.7974358803 4 | 0.5300000000 -1.8475134873 5 | 0.5400000000 -1.8951743299 6 | 0.5500000000 -1.9405570506 7 | 0.5600000000 -1.9837900907 8 | 0.5700000000 -2.0249926215 9 | 0.5800000000 -2.0642753760 10 | 0.5900000000 -2.1017413937 11 | 0.6000000000 -2.1374866881 12 | 0.6100000000 -2.1716008467 13 | 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-2.6690880186 283 | 3.3100000000 -2.6686232154 284 | 3.3200000000 -2.6681648581 285 | 3.3300000000 -2.6677128861 286 | 3.3400000000 -2.6672672386 287 | 3.3500000000 -2.6668278548 288 | 3.3600000000 -2.6663946736 289 | 3.3700000000 -2.6659676343 290 | 3.3800000000 -2.6655466759 291 | 3.3900000000 -2.6651317376 292 | 3.4000000000 -2.6647227584 293 | 3.4100000000 -2.6643196776 294 | 3.4200000000 -2.6639224344 295 | 3.4300000000 -2.6635309682 296 | 3.4400000000 -2.6631452183 297 | 3.4500000000 -2.6627651244 298 | 3.4600000000 -2.6623906261 299 | 3.4700000000 -2.6620216633 300 | 3.4800000000 -2.6616581759 301 | 3.4900000000 -2.6613001042 302 | 3.5000000000 -2.6609473884 303 | 3.5100000000 -2.6605999692 304 | -------------------------------------------------------------------------------- /Chaps/AppendixA.tex: -------------------------------------------------------------------------------- 1 | \chapter{$1s$原初高斯函数的积分计算方法} 2 | \label{appendix:a} 3 | 4 | 5 | 大多数分子计算使用固定的分子坐标系，使得基函数以该坐标系中的位置矢量 6 | $\bo{R}_A$为中心，如\autoref{figA.1}所示。一个以$\bo{R}_A$为中心的 7 | 位置矢量$\bo{r}$的值将会依赖于$\bo{r}-\bo{R}_A$，因此我们可以写一个一般化的基函数 8 | $\phi_\mu(\bo{r}-\bo{R}_A)$来代表它是以$\bo{R}_A$为中心的。在一个分子计算的过程中 9 | 我们需要计算数量庞大的包含不同中心$\phi_\mu(\bo{r}-\bo{R}_A)$的单电子和双电子积分。 10 | 如果我们使用的基函数包括四个或者更多的中心，那么我们的双电子积分将会包含$1-$、$2-$、$3-$和 11 | $4-$中心积分。核吸引势积分最多只能处理到三中心。这些多中心（超过$2$）的积分对于Slater型函数 12 | 来说非常难以处理，但是对于Gaussian型函数来说处理起来相对简单。因此许多多原子计算使用Gaussian函数。 13 | \begin{figure}[h] 14 | \begin{tikzpicture}[scale=2,inner sep=0,arrows=-latex] 15 | \draw (0,0)--(0,2); 16 | \draw (0,0)--(-1.2,-1.2); 17 | \draw (0,0)--(2,0); 18 | \draw (2,1) circle[radius=0.3]; 19 | \node (A) at (2,1) {\bf A}; 20 | \node (i) at (.7,1.8) {} ; 21 | \fill (i) circle(1pt) ; 22 | \draw (0,0)--node[right=13pt]{$\mathbf{R_A}$}(A); 23 | \draw (0,0)--node[right=5pt]{$\mathbf{r}$}(i); 24 | \draw (A)--node[right=.2cm]{$\mathbf{r-R_A}$}(i); 25 | \end{tikzpicture} 26 | \caption{分子坐标系} 27 | \label{figA.1} 28 | \end{figure} 29 | 30 | 在高斯函数的计算中，收缩型的高斯基函数$\phi_\mu^{CGF}$一般由一系列的原初高斯函数 31 | $g\equiv\phi_p^{GF}$（见\autoref{3.283}或\autoref{3.212}） 展开，因此我们只需要考虑 32 | 计算这些原初函数之间的积分即可。收缩函数$\phi_\mu^{CGF}$的积分只需要通过对原初函数的积分 33 | 用恰当的收缩系数进行求和就能得到。一般常用$1s$、$2p$、$3d$等原初高斯函数，因为任何$s$、$p$、 34 | $d$的函数能够类似的用这些高斯函数展开。这里我们仅考虑$1s$原初高斯函数积分的计算。 35 | Shavitt 36 | \endnote{I.Shavitt,The Gaussian function in calculations of statistical mechanics and quantummechanics,in \textit{Methods in Computational Physics} 37 | ,B.Aledr,S.Fernbach,and M.Rotenberg (Eds.),Academic Press,New York,1963.} 38 | 已经给出了一个相似的但是可以互通有无的关于高斯函数积分计算的讨论。包含$p$、$d$等类似函数的积分 39 | 可以通过把我们结果对中心$\bo{R}_A$、$\bo{R}_B$等在笛卡尔坐标系中的坐标进行微分得到。但是， 40 | 高角量子数的积分计算通过基于Rys多项式的新方法更高效的实现。\endnote{ 41 | M. Dupuis,J. Rys,and H. King,Evaluation of molecular integrals over Gaussian basis functions,\textit{J. Chem. Phys.} \textbf{65}:111 (1976).} 42 | 43 | 以$\bo{R}_A$未归一化的$1s$原初高斯函数为 44 | \begin{align} 45 | \label{A.1} 46 | \tilde{g}_{1s}(\bo{r}-\bo{R}_A)=e^{-\alpha\lvert \bo{r}-\bo{R}_A \rvert ^2} 47 | \end{align} 48 | 我们将用$\alpha$、$\beta$、$\gamma$和$\delta$分别表示$\bo{R}_A$、 49 | $\bo{R}_B$、$\bo{R}_C$、$\bo{R}_D$的轨道指数。高斯函数能够简化多中心电子积分的原因是 50 | 两个不同中心的$1s$高斯函数的乘积可以成比例的化成1个在第三个中心的$1s$高斯函数。因此 51 | \begin{align} 52 | \label{A.2} 53 | \tilde{g}_{1s}(\bo{r}-\bo{R}_A)\tilde{g}_{1s}(\bo{r}-\bo{R}_B) 54 | =\tilde{K}\tilde{g}_{1s}(\bo{r}-\bo{R}_P) 55 | \end{align} 56 | 其中，比例系数$\tilde{K}$为 57 | \begin{align} 58 | \label{A.3} 59 | \tilde{K}=\exp[-\alpha\beta/(\alpha+\beta)\lvert \bo{R}_A-\bo{R}_B \rvert ^2] 60 | \end{align} 61 | 第三个中心$P$落在中心$A$和$B$的连线上， 62 | \begin{align} 63 | \label{A.4} 64 | \bo{R}_P=(\alpha\bo{R}_A+\beta \bo{R}_B)/(\alpha+\beta) 65 | \end{align} 66 | 中心在$\bo{R}_P$的新高斯函数的指数为 67 | \begin{align} 68 | \label{A.5} 69 | p=\alpha+\beta 70 | \end{align} 71 | 因此对于$1s$高斯函数的任何2-中心分布，例如，两个不同中心的函数的乘积，能够马上转化成一个单中心 72 | 分布的高斯函数。我们现在计算任何从头算计算过程都需要的仅收缩$1s$原初高斯函数的基本积分。我们将 73 | 计算未归一化的函数的积分，并且如果我们需要归一化的函数的积分，我们可以简单的乘一个合适的归一化 74 | 常数来得到它。 75 | 76 | 首先考虑2-中心的重叠积分 77 | \begin{align} 78 | \label{A.6} 79 | (A|B)=\int \emph{d}\bo{r}_1\,\tilde{g}_{1s}(\bo{r}_1-\bo{R}_A)\tilde{g}_{1s}(\bo{r}_1-\bo{R}_B) 80 | \end{align} 81 | 通过\autoref{A.2},我们马上得到 82 | \begin{align} 83 | \label{A.7} 84 | (A|B)=\tilde{K}\int \emph{d}\bo{r}_1\,\tilde{g}_{1s}(\bo{r}_1-\bo{R}_P) 85 | =\tilde{K}\int \emph{d}\bo{r}_1\, \exp[-p\lvert \bo{r}_1-\bo{R}_P \rvert ^2] 86 | \end{align} 87 | 如果我们使$\bo{r}=\bo{r}_1-\bo{R}_p$并且$\emph{d}\bo{r}\,=\emph{d}\bo{r}_1\,$,则 88 | \begin{align} 89 | \label{A.8} 90 | (A|B)=\tilde{K}\int \emph{d}\bo{r}\,\,e^{-p\bo{r}^2}=4\pi\tilde{K}\int_0^{\infty} \emph{d}r\,r^2e^{-pr^2} 91 | \end{align} 92 | 最后一个积分的值恰好是$(\pi/p)^{\frac{3}{2}}/{4\pi}$，所以 93 | \begin{align} 94 | \label{A.9} 95 | (A|B)=[\pi/(\alpha+\beta)]^{\frac{3}{2}}\exp[-\alpha\beta/(\alpha+\beta)\lvert \bo{R}_A-\bo{R}_B \rvert ^2] 96 | \end{align} 97 | 98 | 动能积分为 99 | \begin{align} 100 | \label{A.10} 101 | (A|-\frac{1}{2}\nabla^2|B)=\int \emph{d}\bo{r}_1\,\tilde{g}_{1s}(\bo{r}_1-\bo{R}_A)(-\frac{1}{2}\nabla_1^2)\tilde{g}_{1s}(\bo{r}_1-\bo{R}_B) 102 | \end{align} 103 | 在算符$\nabla_1^2$作用完成后，上式能够通过非常类似的方式进行计算，得到 104 | \begin{align} 105 | \label{A.11} 106 | (A|-\frac{1}{2}\nabla^2|B)=&\alpha\beta/(\alpha+\beta)[3-2\alpha\beta/(\alpha+\beta)\lvert \bo{R}_A-\bo{R}_B \rvert ^2][\pi/(\alpha+\beta)]^{\frac{3}{2}}\nonumber\\ 107 | &\times\exp[-\alpha\beta/(\alpha+\beta)\lvert \bo{R}_A-\bo{R}_B \rvert ^2] 108 | \end{align} 109 | 110 | 为了计算核吸引势能积分和双电子排斥积分，现在我们在这里介绍一种强力的并常用的，对于计算积分 111 | 尤其是我们将要讨论的类型的积分往往有效的方法。这种方法是将每一个我们需要积分的变量替换为它的 112 | 傅里叶变换的形式。如果我们给定了一个矢量$\bo{r}$的函数$f(\bo{r})$,那么它的三维坐标傅里叶变换 113 | $F(\bo{k})$定义为 114 | \begin{align} 115 | \label{A.12} 116 | F(\bo{k})=\int \emph{d}\bo{r}\,f(\bo{r})e^{-i\bo{k}\bo{r}} 117 | \end{align} 118 | 其中矢量$\bo{k}$是变换变量。傅里叶积分定理说明 119 | \begin{align} 120 | \label{A.13} 121 | f(\bo{r})=(2\pi)^{-3}\int \emph{d}\bo{k}\,F(\bo{k})e^{i\bo{k}\bo{r}} 122 | \end{align} 123 | \begin{table}[h!] 124 | 125 | \caption{Fourier transform pairs} 126 | \centering 127 | \begin{tabular}{lll} 128 | \hline 129 | $f(\bm{r})$ & ~~~~ ~~~~& $F(\bm{k})$ \\ \hline \\ 130 | $\frac{1}{r}$& ~~~~ ~~~~& $\frac{4\pi}{k^2}$ \\ 131 | \\ 132 | $e^{-\alpha^2}$ & ~~~~ ~~~~ & $(\frac{\pi}{\alpha})^{\frac{3}{2}}e^{-k^2/{4\alpha}}$\\ 133 | \\ 134 | $\delta(\bo{r})$& ~~~~ ~~~~ &1 135 | \\ \hline 136 | \end{tabular} 137 | \label{tA.1} 138 | \end{table} 139 | $F(\bo{k})$和$f(\bo{r})$被称为一个傅里叶变换对。所有的我们将要用到的傅里叶变换对都在 140 | \autoref{tA.1}给出。特别的，傅里叶表示下的三维坐标狄拉克$delta$函数是 141 | \begin{align} 142 | \label{A.14} 143 | \delta(\bo{r}_1-\bo{r}_2)=(2\pi)^{-3}\int \emph{d}\bo{k}\,F(\bo{k})e^{i\bo{k}(\bo{r}_1-\bo{r_2})} 144 | \end{align} 145 | 146 | 正如之前提到的，狄拉克$\delta$函数有以下性质 147 | \begin{align} 148 | \label{A.15} 149 | \int \emph{d}\bo{r}_1\,\delta(\bo{r}_1-\bo{r}_2)h(\bo{r}_1)=h(\bo{r}_2) 150 | \end{align} 151 | 上式对于任意一个函数$h(\bo{r})$都成立。 152 | 153 | 让我们现在考虑$1s$原初高斯函数的核吸引势积分， 154 | \begin{align} 155 | \label{A.16} 156 | (A|-Z_C/r_{1C}|B)=&\int\emph{d}\bo{r}_1\,\tilde{g}_{1s}(\bo{r}_1-\bo{R}_A) 157 | [-Z_C/|\bo{r}_1-\bo{R}_C|^{-1}]\tilde{g}_{1s}(\bo{r}_1-\bo{R}_B) \nonumber\\ 158 | =&-Z_C\int\emph{d}\bo{r}_1\,e^{-\alpha\lvert \bo{r}-\bo{R}_A \rvert ^2}|\bo{r}_1-\bo{R}_C|^{-1} 159 | e^{-\beta\lvert \bo{r}-\bo{R}_B \rvert ^2} 160 | \end{align} 161 | 我们首先把两个高斯函数结合，得到一个中心在$\bo{R}_P$的高斯函数， 162 | \begin{align} 163 | \label{A.17} 164 | (A|-Z_C/r_{1C}|B)=-Z_C\tilde{K}\int\emph{d}\bo{r}_1\,e^{-p\lvert \bo{r}-\bo{R}_P \rvert ^2}|\bo{r}_1-\bo{R}_C|^{-1} 165 | \end{align} 166 | 然后，我们用\autoref{A.13}和\autoref{tA.1}中的变换形式替换上面的两项， 167 | \begin{align} 168 | \label{A.18} 169 | (A|-Z_C/r_{1C}|B)=&-Z_C\tilde{K}(2\pi)^{-6}(\pi/p)^{3/2} 170 | \int\emph{d}\bo{r}_1\emph{d}\bo{k}_1\emph{d}\bo{k}_2\,e^{-\bo{k}_1^2/4p}\nonumber\\ 171 | &\times e^{i\bo{k}_1\cdot(\bo{r}_1-\bo{R}_P)}4\pi k_2^{-2}e^{i\bo{k}_2\cdot(\bo{r}_1-\bo{R}_C)} 172 | \end{align} 173 | 如果我们合并所有含有原来积分变量$\bo{r}_1$的指数项，我们得到 174 | \begin{align} 175 | \label{A.19} 176 | (A|-Z_C/r_{1C}|B)=&-4\pi Z_C\tilde{K}(2\pi)^{-6}(\pi/p)^{3/2} 177 | \int\emph{d}\bo{r}_1\emph{d}\bo{k}_1\emph{d}\bo{k}_2\,k_2^{-2}e^{-\bo{k}_1^2/4p}\nonumber\\ 178 | &\times e^{-i\bo{k}_1\cdot\bo{R}_P}e^{-i\bo{k}_2\cdot\bo{R}_C}e^{i\bo{r}_1\cdot(\bo{k}_1+\bo{k}_2)} 179 | \end{align} 180 | 我们现在可以用\autoref{A.14}定义的$\delta$函数处理$\bo{r}_1$的积分，得到 181 | \begin{align} 182 | \label{A.20} 183 | (A|-Z_C/r_{1C}|B)=&-4\pi Z_C\tilde{K}(2\pi)^{-3}(\pi/p)^{3/2} 184 | \int\emph{d}\bo{k}_1\emph{d}\bo{k}_2\,e^{-\bo{k}_1^2/4p}k_2^{-2}\nonumber\\ 185 | &\times e^{i\bo{k}_1\cdot\bo{R}_P}e^{-i\bo{k}_2\cdot\bo{R}_C}\delta(\bo{k}_1+\bo{k}_2)\nonumber\\ 186 | &=-Z_C\tilde{K}(2\pi^{2})(\pi/p)^{3/2}\int\emph{d}\bo{k}\,e^{-\bo{k}^2/4p}k^{-2} e^{-i\bo{k}\cdot(\bo{R}_P-\bo{R}_C)} 187 | \end{align} 188 | 由于$\delta$函数的存在，这里我们令$\bo{k}_2=-\bo{k}_1$，并且重新把变量记为$\bo{k}$。如果我们 189 | 令$\bo{R}_P-\bo{R}_C$落在$z$轴上，那么$\bo{k}\cdot(\bo{R}_P-\bo{R}_C)=k|\bo{R}_P-\bo{R}_C|\cos(\theta)$, 190 | 之后我们可以简单的处理角度部分在$\bo{k}$上的积分， 191 | \begin{align} 192 | \label{A.21} 193 | (A|-Z_C/r_{1C}|B)=&N\int^{\infty}_0\emph{d}k\,e^{-\bo{k}^2/4p}k^{-1}\sin{(k|\bo{R}_P-\bo{R}_C|)} 194 | \end{align} 195 | \begin{align} 196 | \label{A.22} 197 | N=-2Z_C\tilde{K}(\pi |\bo{R}_P-\bo{R}_C|)^{-1}(\pi/p)^{3/2} 198 | \end{align} 199 | 为了计算\autoref{A.21}中的积分，我们考虑下列一般化的积分： 200 | \begin{align} 201 | \label{A.23} 202 | I(x)=\int^{\infty}_0\emph{d}k\,e^{-a k^2}k^{-1}\sin{kx} 203 | \end{align} 204 | 上式对$x$的导数为 205 | \begin{align} 206 | \label{A.24} 207 | I`(x)=\int^{\infty}_0\emph{d}k\,e^{-a k^2}\cos{kx} 208 | \end{align} 209 | 注意\autoref{A.24}是$k$的偶函数。由于$I(0)=0$，我们得到 210 | \begin{align} 211 | \label{A.25} 212 | \int^{x}_0 \emph{d}yI`(y)=I(x)-I(0)=I(x) 213 | \end{align} 214 | 注意到$\cos(\theta)$是$e^{i\theta}$的实数部分（例如，$\cos(\theta)=\Re e[e^{i\theta}]$）， 215 | \autoref{A.24}中的积分可以这样来计算 216 | \begin{align} 217 | \label{A.26} 218 | I`(x)=\frac{1}{2}\int^{\infty}_{-\infty}\emph{d}k\,e^{-a k^2}\cos{kx}=\frac{1}{2}\Re e\biggl[\int^{\infty}_{-\infty}\emph{d}k\,e^{-a k^2}e^{ikx}\biggr] 219 | \end{align} 220 | 通过凑平方，我们得到 221 | \begin{align} 222 | \label{A.27} 223 | I`(x)=\frac{1}{2}e^{-x^2/4a}\Re e\biggl[\int^{\infty}_{-\infty}\emph{d}k\,e^{-(a^{1/2}k-1/2i a^{-1/2}x)^2}\biggr] 224 | \end{align} 225 | 令$u=a^{1/2}k-1/2i a^{-1/2}x$,我们得到 226 | \begin{align} 227 | \label{A.28} 228 | I`(x)=\frac{1}{2}e^{-x^2/4a}a^{-1/2}\int^{\infty}_{-\infty}\emph{d}u\,e^{-u^2}=\frac{1}{2}(\pi /a)^{1/2}e^{-x^2/4a} 229 | \end{align} 230 | 故 231 | \begin{align} 232 | \label{A.29} 233 | I(x)=\frac{1}{2}(\pi /a)^{1/2}\int^{x}_{0}\emph{d}ye^{-y^2/4a} 234 | \end{align} 235 | 因此 236 | \begin{align} 237 | \label{A.30} 238 | (A|-Z_C/r_{1C}|B)=&-2\pi Z_C\tilde{K}(p|\bo{R}_P-\bo{R}_C|)^{-1}\int^{|\bo{R}_P-\bo{R}_C|}_{0}\emph{d}y\,e^{-py^2} 239 | \nonumber\\ 240 | =&-2\pi Z_C\tilde{K}p^{-1}(p^{1/2}|\bo{R}_P-\bo{R}_C|)^{-1}\int^{p^{1/2}|\bo{R}_P-\bo{R}_C|}_{0}\emph{d}y\,e^{-y^2} 241 | \end{align} 242 | 我们现在引入$F_0$函数，定义为 243 | \begin{align} 244 | \label{A.31} 245 | F_0(t)=t^{-1/2}\int_0^{t^{1/2}}\emph{d}y\,e^{-y^2} 246 | \end{align} 247 | 它可以通过下式与误差函数关联起来 248 | \begin{align} 249 | \label{A.32} 250 | F_0(t)=\frac{1}{2}(\pi/t)^{1/2}erf(t^{1/2}) 251 | \end{align} 252 | 因此，我们的核吸引势积分可以写成包含$F_0$的形式 253 | \begin{align} 254 | \label{A.33} 255 | (A|-Z_C/r_{1C}|B)=&-2\pi/(\alpha+\beta) Z_C\exp[-\alpha\beta/(\alpha+\beta)|\bo{R}_A-\bo{R}_B|^2] 256 | \nonumber\\ 257 | \times&F_0[(\alpha+\beta)|\bo{R}_P-\bo{R}_C|^2] 258 | \end{align} 259 | 误差函数（在这里是$F_0$）是一个IBM FORTRAN编译器里的内置函数。 260 | 261 | 我们现在开始考虑双电子积分 262 | \begin{align} 263 | \label{A.34} 264 | (AB|CD)=\int\emph{d}\bo{r}_1\emph{d}\bo{r}_2\,\tilde{g}_{1s}(\bo{r}_1-\bo{R}_A) 265 | \tilde{g}_{1s}(\bo{r}_1-\bo{R}_B)r_{12}^{-1}\tilde{g}_{1s}(\bo{r}_2-\bo{R}_C)\tilde{g}_{1s}(\bo{r}_2-\bo{R}_D) 266 | \end{align} 267 | 第一步是将中心在$\bo{R}_A$和$\bo{R}_B$的高斯函数合并成一个新的、中心在$\bo{R}_P$高斯函数， 268 | 以及将中心在$\bo{R}_C$和$\bo{R}_D$的高斯函数合并成一个新的、中心在$\bo{R}_Q$高斯函数，如 269 | \autoref{figA.2}所示，则\autoref{A.34}化为 270 | \begin{align} 271 | \label{A.35} 272 | (AB|CD)=&\exp[-\alpha\beta/(\alpha+\beta)|\bo{R}_A-\bo{R}_B|^2-\gamma\delta/(\gamma+\delta)|\bo{R}_C-\bo{R}_D|^2] 273 | \nonumber\\ 274 | \times&\int\emph{d}\bo{r}_1\emph{d}\bo{r}_2\,e^{-p\lvert \bo{r}_1-\bo{R}_P \rvert ^2}r_{12}^{-1}e^{-q\lvert \bo{r}_2-\bo{R}_Q \rvert ^2} 275 | \end{align} 276 | 277 | \begin{figure}[h] 278 | \begin{tikzpicture}[scale=2,inner sep=0,arrows=-latex] 279 | \draw (0,0)--(0,2); 280 | \draw (0,0)--(-1.2,-1.2); 281 | \draw (0,0)--(2,0); 282 | \coordinate(a) at(0.9,1.4); 283 | \coordinate(b) at(0.5,0.3); 284 | \coordinate(c) at(1.4,1.4); 285 | \coordinate(d) at(1.9,0.25); 286 | \node (A) at (1,1.5) {\bf A}; 287 | \node (B) at (0.6,0.2) {\bf B}; 288 | \node (C) at (1.5,1.5) {\bf C}; 289 | \node (D) at (2,0.15) {\bf D}; 290 | \fill (a) circle(1pt) ; 291 | \fill (b) circle(1pt) ; 292 | \fill (c) circle(1pt) ; 293 | \fill (d) circle(1pt) ; 294 | \draw (a) -- (b); 295 | \draw (c) -- (d); 296 | \coordinate(p) at(0.6,0.575); 297 | \node (P) at (0.47,0.6) {\bf P}; 298 | \fill (p) circle(1pt) ; 299 | \coordinate(q) at(1.5,1.17); 300 | \node (Q) at (1.6,1.2) {\bf Q}; 301 | \fill (q) circle(1pt) ; 302 | \draw (p)--node[below=0.4cm]{$\mathbf{R_Q-R_P}$}(q); 303 | \end{tikzpicture} 304 | \caption{双电子积分中包含的六个中心} 305 | \label{figA.2} 306 | \end{figure} 307 | 把积分中的三项替换为它们的傅里叶变换形式，得 308 | \begin{align} 309 | \label{A.36} 310 | (AB|CD)=&M(2\pi)^{-9}\int\emph{d}\bo{r}_1\emph{d}\bo{r}_2\emph{d}\bo{k}_1\emph{d}\bo{k}_2\emph{d}\bo{k}_3\, 311 | (\pi/p)^{3/2}e^{-k_1^2/{4p}}e^{i\bo{k}_1\cdot(\bo{r}_1-\bo{R}_P)} 312 | \nonumber\\ 313 | \times&4\pi k_2^{-2}e^{i\bo{k}_2\cdot(\bo{r}_1-\bo{r}_2)} 314 | (\pi/q)^{3/2}e^{-k_3^2/{4q}}e^{i\bo{k}_3\cdot(\bo{r}_2-\bo{R}_Q)} 315 | \end{align} 316 | 其中 317 | \begin{align} 318 | \label{A.37} 319 | M=\exp[-\alpha\beta/(\alpha+\beta)|\bo{R}_A-\bo{R}_B|^2-\gamma\delta/(\gamma+\delta)|\bo{R}_C-\bo{R}_D|^2] 320 | \end{align} 321 | 把初始变量$\bo{r}_1$和$\bo{r}_2$的指数项合并，得到 322 | \begin{align} 323 | \label{A.38} 324 | (AB|CD)=&4\pi M(2\pi)^{-9}(\pi^2/p)q^{3/2}\int\emph{d}\bo{r}_1\emph{d}\bo{r}_2\emph{d}\bo{k}_1\emph{d}\bo{k}_2\emph{d}\bo{k}_3\, 325 | e^{-k_1^2/{4p}}e^{-k_3^2/{4q}} 326 | \nonumber\\ 327 | \times&k_2^{-2}e^{-i\bo{k}_1\cdot\bo{R}_P}e^{-i\bo{k}_3\cdot\bo{R}_Q}e^{i\bo{r}_1\cdot(k_1+k_2)} 328 | e^{i\bo{r}_2\cdot(k_3-k_2)} 329 | \end{align} 330 | 现在对在$\bo{r}_1$和$\bo{r}_2$上的积分引入两个delta函数\autoref{A.14} 331 | \begin{align} 332 | \label{A.39} 333 | (AB|CD)=&4\pi M(2\pi)^{-9}(\pi^2/p)q^{3/2}\int\emph{d}\bo{k}_1\emph{d}\bo{k}_2\emph{d}\bo{k}_3\, 334 | e^{-k_1^2/{4p}}e^{-k_3^2/{4q}} 335 | \nonumber\\ 336 | \times&k_2^{-2}e^{-i\bo{k}_1\cdot\bo{R}_P}e^{-i\bo{k}_3\cdot\bo{R}_Q} 337 | \delta(k_1+k_2)\delta(k_3-k_2) 338 | \end{align} 339 | 现在我们令$k_1=-k_2$和$k_3=k_2$，并重新把$k_2$记为$k$,得 340 | \begin{align} 341 | \label{A.40} 342 | (AB|CD)=&4\pi M(2\pi)^{-9}(\pi^2/p)q^{3/2}\int\emph{d}\bo{k}\, 343 | k^{-2}e^{-(p+q)k^2/4pq}e^{i\bo{k}\cdot(\bo{R}_P-\bo{R}_Q)} 344 | \end{align} 345 | 这个积分等同于我们在计算核吸引势积分时遇到的\autoref{A.20}的形式。进行剩余的代数过程 346 | 我们最终得到 347 | \begin{align} 348 | \label{A.41} 349 | (AB|CD)=&2\pi ^{5/2}/[(\alpha+\beta)(\gamma+\delta)(\alpha+\beta+\gamma+\delta)^{1/2}] 350 | \nonumber\\ 351 | \times&\exp[-\alpha\beta/(\alpha+\beta)|\bo{R}_A-\bo{R}_B|^2-\gamma\delta/(\gamma+\delta)|\bo{R}_C-\bo{R}_D|^2] 352 | \nonumber\\ 353 | \times&F_0[(\alpha+\beta)(\gamma+\delta)(\alpha+\beta+\gamma+\delta)|\bo{R}_P-\bo{R}_Q|^2] 354 | \end{align} 355 | 以上这些就是我们仅用$1s$类型的原初高斯函数进行Hartree-Fock计算所需要的所有积分的精确方程。 356 | 它们被用在附录B中计算机程序中。 357 | 358 | \newpage 359 | \theendnotes 360 | \addcontentsline{toc}{section}{注释} 361 | -------------------------------------------------------------------------------- /Chaps/AppendixC.tex: -------------------------------------------------------------------------------- 1 | \chapter{几何结构优化与解析导数法} 2 | 3 | Michael C.Zerner 4 | 5 | 量子理论项目 6 | 7 | 佛罗里达大学 8 | 9 | \section{引言} 10 | 11 | 分子构象的再现和预测是分子量子力学的最成功的应用之一。 12 | 许多情况下，对于很多简单的分子轨道模型或者最小基组的从头算方法，键长可以重现到$\pm 0.02$\r A， 13 | 键角可以重现到$5^{\circ}$。 14 | 对于大一点的基组，尤其是那些双$\zeta$加极化的基组；和包含电子相关能的方法，现在生成的几何构型已经 15 | 可以达到晶体学精度了。 16 | 随着构象计算的日益成功，人们甚至可以选择孤立分子的计算结果，而不是在凝聚介质中获得的实验结果，因为前者可能更适合气相。 17 | 18 | 除了产生关于势能面全局最小点的信息外，量子力学计算还可以产生局域最小点的信息，这些局域最小点可能不会被直接观测到，但是 19 | 很可能会被包含在反应路径中。类似的，关于过渡态和能垒的信息也可以得到，这些信息一般很难甚至不可能通过其他方式获得。 20 | 21 | 收集一个势能面上这些所有的信息是很困难的。对于N个原子的体系，它的能量是一个具有$3N-6$(或者$3N-5$)个自由度的函数。 22 | 为了进行详细的统计计算，人们可能不得不面对这个“3N”问题，并访问势能面上所有热力学可得的区域。 23 | 然而，本附录只涉及势能面的一小部分：那些对应于极小值的点，代表稳态或亚稳态的构象，以及对应于过渡态的点。 24 | %%%%%%% 25 | \section{概论} 26 | \label{secC.2} 27 | 在Born-Oppenheimer近似下得到的分子体系的能量$E$是一个以核坐标为参数的函数，记核坐标为 28 | $\mathbf{X}^{\dagger}=(X_1,X_2,\dots,X_{3N})$。 29 | 我们希望从$E(\mathbf{X})$进而得到$E(\mathbf{X_1})$，$\mathbf{q=(X_1-X)}$。 30 | 我们将能量对$\mathbf{X}$进行泰勒展开： 31 | \begin{align} 32 | \label{C.1} 33 | E(\mathbf{X_1})=E(\mathbf{X})+\mathbf{q}^{\dagger}\mathbf{f(X)} 34 | +\frac{1}{2}\mathbf{q}^{\dagger}\mathbf{H(X)}\mathbf{q}+\dots 35 | \end{align} 36 | 式中梯度为 37 | \begin{align} 38 | f_i=\frac{\partial E(\mathbf{X})}{\partial X_i} 39 | \nonumber 40 | \end{align} 41 | Hessian矩阵元为 42 | \begin{align} 43 | H_{ij}=\frac{\partial E(\mathbf{X})}{\partial X_i\partial X_j} 44 | \nonumber 45 | \end{align} 46 | 注意，列矩阵的下标表示不同的矩阵，如$\mathbf{X_1}$、$\mathbf{X_2}$等等；而$X_i$表示矩阵$\mathbf{X}$第i个元素。 47 | 虽然泰勒展开是无穷项的，但是接近极值时，我们希望二阶展开是足够的；例如，对于$\mathbf{X}=\mathbf{X_e}$，其中$\mathbf{X_e}$ 48 | 表示一个驻点，根据定义$\mathbf{f(X_e)=0}$，则 49 | \begin{align} 50 | \nonumber 51 | E(\mathbf{X_1})=E(\mathbf{X_e})+\frac{1}{2}\mathbf{q}^{\dagger}\mathbf{H(X_e)}\mathbf{q} 52 | \end{align} 53 | 类似的， 54 | \begin{align} 55 | \label{C.2} 56 | \mathbf{f(X_1)}=\mathbf{f(X)}+\mathbf{H(X)}\mathbf{q} 57 | \end{align} 58 | 对点$\mathbf{X_1}=\mathbf{X_e}$ 59 | \begin{align} 60 | \label{C.3} 61 | \mathbf{f(X)}=-\mathbf{H(X)}\mathbf{q} 62 | \end{align} 63 | 64 | \autoref{C.3}的解是不显含$\mathbf{X}$的$E(\mathbf{X})$泛函寻找多变量函数极值的最高效方法的初始点。 65 | 如果$\mathbf{H}$是非奇异的，则有 66 | \begin{align} 67 | \label{C.4} 68 | \mathbf{q}=-\mathbf{H^{-1}(X)}\mathbf{f(X)} 69 | \end{align} 70 | 这能够从任意一点$\mathbf{X}$解得$\mathbf{X_e}$，从而使能量函数接近二阶展开。同样的，一个预测的能量$\mathbf{E(H_e)}$ 71 | \footnote{译者注：原文如此，此处可能是笔误，结合上下文来看，应为$\mathbf{E(X_e)}$} 72 | 可以从下式得到 73 | \begin{align} 74 | \label{C.5} 75 | \mathbf{E(X_e)}&=\mathbf{E(X)}-\frac{1}{2}\mathbf{f(X)}^{\dagger}\mathbf{H^{-1}(X)}\mathbf{f(X)} 76 | \nonumber\\ 77 | &=\mathbf{E(X)}-\frac{1}{2}\mathbf{q}^{\dagger}\mathbf{H(X_e)}\mathbf{q} 78 | \end{align} 79 | 对于特殊的势能面寻找极值的问题，我们必须要指出：除非将代表$\mathbf{H}$的零特征值的旋转和平移移除， 80 | 否则$\mathbf{H^{-1}(X)}$将不存在。这个工作可以通过Wilson和Eliashevich提出的$\mathbf{B}$矩阵实现: 81 | \endnote{See,for example,E.B.Wilson,J.C.Decius,and P.C.Cross,$Molecular Vibrations$, 82 | McGraw-Hill,New York,1955.} 83 | \begin{align} 84 | \label{C.6} 85 | \mathbf{Y}^{\dagger}=\mathbf{X}^{\dagger}\mathbf{B} 86 | \end{align} 87 | 式中$\mathbf{X}$是$3N$维，$\mathbf{B}$是$3N\times 3(N-6)$维，关联内坐标和原本的笛卡尔坐标$\mathbf{X}$。 88 | 揭示这些新坐标下的Taylor级数结构的最简单的方法是考虑功$w$，因为它与坐标系的选择是独立的。 89 | 功是力和距离的乘积，能够在任何坐标系下表示为 90 | \begin{align} 91 | \nonumber 92 | w=\mathbf{f}^{\dagger}\mathbf{q}=\mathbf{f_y}^{\dagger}\mathbf{q_y}=\mathbf{f_y}^{\dagger}\mathbf{B}^{\dagger}\mathbf{q} 93 | \end{align} 94 | \begin{align} 95 | \nonumber 96 | \mathbf{f}^{\dagger}=\mathbf{f_y}^{\dagger}\mathbf{B}^{\dagger} 97 | \end{align} 98 | 或者 99 | \begin{align} 100 | \label{C.7} 101 | \mathbf{f_y}=\mathbf{f}^{\dagger}(\mathbf{B}^{\dagger})^{-1} 102 | \end{align} 103 | 式中$(\mathbf{B}^{\dagger})^{-1}$满足 104 | \begin{align} 105 | \nonumber 106 | \mathbf{B}^{\dagger}(\mathbf{B}^{\dagger})^{-1}=\mathbf{1} 107 | \end{align} 108 | 下面给出上述方程的一般解为 109 | \begin{align} 110 | \nonumber 111 | (\mathbf{B}^{\dagger})^{-1}=\mathbf{mB}(\mathbf{B^{\dagger}mB})^{-1} 112 | \end{align} 113 | 式中$\mathbf{m}$是一个任意的$3N\times 3N$矩阵，经常选择一个对角矩阵, 114 | 矩阵元素是每一个对应位置上原子的原子质量的倒数的三倍。 115 | 也可以选择一个单位矩阵，其中6（或5）个元素选择为0去阻止平动和转动。 116 | 这种类型一个简单的选择是将原子1放在原点，原子2在z轴上，原子3在xz平面上。 117 | 于是这被移除的6（或5）个坐标为$x_1=y_1=z_1=0$，$x_2=y_2=0$，$y_3=0$。 118 | 如果$y_3=0$对于任意选择的第三个原子来说都意味着$x_3=0$，那么这个分子是线性的并且只有5个自由度被选择。 119 | 实际操作中，求$\mathbf{H}$的逆矩阵被证明有一点困难。 120 | 接下来要讨论的更新的方法直接构建$\mathbf{H}^{-1}$，并且从不更新平动和转动。 121 | 在解析求解$\mathbf{H}$的时候，平动和转动可以像之前讨论的那样被移除，或者，如果通过对角化求逆的话， 122 | $\mathbf{H}$的6（或5）个为0的特征值会被一个任意的大数替换掉，本质上在求逆中解耦这些状态。 123 | %%%%% 124 | \section{解析导数} 125 | 我们现在来考虑量子化学中$\mathbf{f}$和$\mathbf{H}$的计算。一个直接的方式是简单的数值微分能量。 126 | 取而代之的是，我们可以通过解析求导的方式得到这些导数。让我们来举例说明Hartree-Fock计算框架中的 127 | 一些基本思想。 128 | 129 | Hartree-Fock能量显示的依赖占据轨道系数$\mathbf{C}$和$\mathbf{X}$。它的导数由下式给出 130 | \begin{align} 131 | \label{C.8} 132 | \frac{\partial E}{\partial X_A}=\frac{\partial {\tilde{E} }}{\partial X_A} 133 | +\sum_{\mu a}\frac{\partial E}{\partial C_{\mu a}}\frac{\partial C_{\mu a}}{\partial X_A} 134 | \end{align} 135 | 式中，$\frac{\partial {\tilde{E} }}{\partial X_A}$表示所有显式依赖核坐标$X_A$的项的导数， 136 | 并且其中链式规则项源于分子轨道系数对几何结构的隐含依赖性。因为$\frac{\partial E}{\partial C_{\mu a}}=0$ 137 | 是Hartree-Fock解的条件(\autoref{sec3.2.1})， 138 | \begin{align} 139 | \label{C.9} 140 | \frac{\partial E}{\partial X_A}=\frac{\partial {\tilde{E} }}{\partial X_A} 141 | \end{align} 142 | 这种实现允许我们忽略分子轨道系数相对于几何结构变化的一阶变化。 143 | 一个闭壳层体系在Hartree-Fock近似下的总能量可以表示为 144 | \begin{align} 145 | \label{C.10} 146 | E=\sum_{\mu \nu}P_{\nu \mu}H_{\mu \nu}^{core} 147 | +\frac{1}{2}\sum_{\mu \nu \lambda \sigma }P_{\nu \mu}P_{\lambda \sigma}(\mu \nu||\sigma \lambda) 148 | +V_{NN} 149 | \end{align} 150 | (见\autoref{3.184}，\autoref{3.185}和\autoref{3.154})式中我们已经定义 151 | \begin{align} 152 | \nonumber 153 | V_{NN}=\sum_{A}\sum_{A \ge B}\frac{Z_A Z_B}{R_{AB}} 154 | \end{align} 155 | 以及介绍了速记符号 156 | \begin{align} 157 | \nonumber 158 | (\mu \nu||\sigma \lambda)=(\mu \nu|\sigma \lambda)-\frac{1}{2}(\mu \lambda ||\sigma \nu) 159 | \end{align} 160 | 对于实轨道，密度矩阵表示为 161 | \begin{align} 162 | \nonumber 163 | P_{\nu \mu}=2\sum_{a}^{N/2}C_{\mu a}C_{\nu a} 164 | \end{align} 165 | (见\autoref{3.145})。微分\autoref{C.10}得 166 | \begin{align} 167 | \label{C.11} 168 | \frac{\partial E}{\partial X_A}= 169 | &\sum_{\mu \nu}P_{\nu \mu}\frac{\partial H_{\mu \nu}^{core}}{\partial X_A} 170 | +\frac{1}{2}\sum_{\mu \nu\lambda \sigma }P_{\nu \mu}P_{\lambda \sigma} 171 | \frac{\partial (\mu \nu|| \sigma\lambda)}{\partial X_A} 172 | +\frac{\partial V_{NN}}{\partial X_A} 173 | \\ \nonumber 174 | &+\sum_{\mu \nu}\frac{\partial P_{\nu \mu}}{\partial X_A} H_{\mu \nu}^{core} 175 | +\sum_{\mu \nu\lambda \sigma }\frac{\partial P_{\nu \mu} }{\partial X_A}P_{\lambda \sigma}(\mu \nu|| \sigma\lambda) 176 | \end{align} 177 | \autoref{C.11}表明需要求取组合系数的导数，然而\autoref{C.9}不需要！ 178 | 展开\autoref{C.11}的后两项得到 179 | \begin{align} 180 | \nonumber 181 | =&4\sum_{\mu \nu}\sum_{a}^{N/2}\frac{\partial C_{\mu a}}{\partial X_A}H_{\mu \nu}^{core}C_{\nu a} 182 | +4\sum_{\mu \nu\lambda \sigma }\sum_{a}^{N/2}\frac{\partial C_{\mu a}}{\partial X_A} P_{\lambda \sigma}(\mu \nu|| \sigma\lambda) C_{\nu a} 183 | \\ \nonumber 184 | =&4\sum_{\mu \nu}\sum_{a}^{N/2}\frac{\partial C_{\mu a}}{\partial X_A} 185 | [H_{\mu \nu}^{core}-\sum_{\lambda \sigma } P_{\lambda \sigma}(\mu \nu|| \sigma\lambda)]C_{\nu a} 186 | \\ \nonumber 187 | =&4\sum_{\mu \nu}\sum_{a}^{N/2}\frac{\partial C_{\mu a}}{\partial X_A}F_{\mu \nu}C_{\nu a} 188 | \\ \nonumber 189 | =&4\sum_{a}^{N/2} \varepsilon_a \sum_{\mu \nu}\frac{\partial C_{\mu a}}{\partial X_A}S_{\mu \nu}C_{\nu a} 190 | \end{align} 191 | 为了计算系数的导数，我们回顾一下分子轨道的正交归一化条件，即 192 | \begin{align} 193 | \nonumber 194 | \sum_{\mu \nu}C_{\mu a}S_{\mu \nu}C_{\nu a}=\delta_{ab} 195 | \end{align} 196 | （见练习3.10）。微分上式得到 197 | \begin{align} 198 | \nonumber 199 | 2\sum_{\mu \nu}\frac{\partial C_{\mu a}}{\partial X_A}S_{\mu \nu}C_{\nu a} 200 | =-\sum_{\mu \nu}C_{\mu a}C_{\nu a}\frac{\partial S_{\mu \nu}}{\partial X_A} 201 | \end{align} 202 | 组合这些表达式的结果得 203 | \begin{align} 204 | \label{C.12} 205 | \frac{\partial E}{\partial X_A}= 206 | \sum_{\mu \nu}P_{\nu \mu}\frac{\partial H_{\mu \nu}^{core}}{\partial X_A} 207 | +\frac{1}{2}\sum_{\mu \nu\lambda \sigma }P_{\nu \mu}P_{\lambda \sigma} 208 | \frac{\partial (\mu \nu|| \sigma\lambda)}{\partial X_A} 209 | -\sum_{\mu \nu}Q_{\nu \mu} \frac{\partial S_{\mu \nu}}{\partial X_A} 210 | +\frac{\partial V_{NN}}{\partial X_A} 211 | \end{align} 212 | 式中我们定义 213 | \begin{align} 214 | \nonumber 215 | Q_{\nu \mu}=2\sum_{a}^{N/2}\varepsilon_a C_{\mu a}C_{\nu a} 216 | \end{align} 217 | 因此能量的导数可以通过分子轨道系数和重叠积分以及单电子、双电子积分的导数计算得到。 218 | 219 | 以高效的方式获取电子结构理论中出现的积分的导数是一个有点专业化的领域，但我们很容易理解其一般思路。 220 | 许多从头计算都是使用笛卡尔高斯函数进行的 221 | \begin{align} 222 | \nonumber 223 | \phi_{lmn}^{GF}=N_{lmn}x_{a}^{l}y_{a}^{m}z_{a}^{n} e^{-\alpha|\mathbf{r}-\mathbf{R_{A}}|^2} 224 | \end{align} 225 | 见\autoref{sec:3.6}，式中$N_{lmn}$是归一化参数，表达式为 226 | \begin{align} 227 | \nonumber 228 | N_{lmn}=\bigg[\frac{(8\alpha)^{l+m+n}l!m!n!}{(2l)!(2m)!(2n)!} \bigg]^{1/2}(\frac{2\alpha}{\pi})^{3/4} 229 | \end{align} 230 | 其中小写字母a表示从原子核A测量的电子坐标。$\phi_{lmn}^{GF}$对核坐标的导数是可以显式直接得到的，例如 231 | \begin{align} 232 | \nonumber 233 | \frac{\partial \phi_{lmn}^{GF}}{\partial X_A}=[(2l+1)a]^{1/2}\phi_{l+1.mn}^{GF}-2l(\frac{a}{2l-1})^{1/2}\phi_{l-1.mn}^{GF} 234 | \end{align} 235 | 式中，现在的$X_A$特指核A的x坐标。需要指出的是当$l=0$时，不考虑上式第二项。 236 | 237 | 所有单中心积分的导数为零，因为通常假设中心A上的所有轨道都遵循中心A的位移。动能算符和电子排斥势算符$r_{12}^{-1}$不是核坐标的函数。 238 | 核排斥势能$V_{NN}$的导数也可以直接得到 239 | \begin{align} 240 | \nonumber 241 | \frac{\partial V_{NN}}{\partial X_A}=Z_A\sum_{B}\frac{Z_B(X_B-X_A)}{R_{AB}^{3}} 242 | \end{align} 243 | 对于核-电子吸引项，可以得到 244 | \begin{align} 245 | \nonumber 246 | \frac{\partial V_{Ne}}{\partial X_A}=-Z_A\sum_{i}\frac{X_i-X_A}{r_{iA}^{3}} 247 | \end{align} 248 | 上面的公式已经足够去求解电子结构计算中遇到的电子积分的导数了。但在实际计算中，需要使用许多技巧去减少计算量。 249 | 使用Slater函数去计算梯度会更困难(见\autoref{sec:3.5}和\autoref{sec:3.6})，但是至少对于经常在半经验模型中出现的 250 | 双中心积分而言，可以沿着与高斯函数相同的方式进行推导。 251 | 252 | \autoref{C.12}相对简单，因为$\mathbf{P}$的导数没有出现，这不应与Hellmann-Feynman定理 253 | \endnote{H.Hellmann,$Einf\ddot{u}hrung in du Quantenchmie$,Franz Deuticke,Leipzig,1937; 254 | R.P.Feynman $ Phys.Rev.\bf{41}:$721(1939); 255 | A.C.Hurley,$Proc.Roy.Soc.\bf{A226:}$170,179(1954)} 256 | 混淆。现在有 257 | \begin{align} 258 | \nonumber 259 | E= \braket{\Phi |\mathcal{H} |\Phi } 260 | \end{align} 261 | 且$\braket{\Phi |\Phi }=1$，则 262 | \begin{align} 263 | \label{C.13} 264 | \frac{\partial E}{\partial X_A}=\braket{\frac{\partial \Phi}{\partial X_A} |\mathcal{H} |\Phi }+ 265 | \braket{\Phi|\mathcal{H} | \frac{\partial\Phi}{\partial X_A} }+ 266 | \braket{\Phi|\frac{\partial \mathcal{H}}{\partial X_A} | \Phi } 267 | \end{align} 268 | Hellmann-Feynman条件如下 269 | \begin{align} 270 | \label{C.14} 271 | \braket{\frac{\partial \Phi}{\partial X_A} |\mathcal{H} |\Phi }+ \braket{\Phi|\mathcal{H} | \frac{\partial\Phi}{\partial X_A} }=0 272 | \end{align} 273 | 上式只在精确解或者特定类型的试探函数的时候才成立。在\autoref{C.14}的约束下，\autoref{C.13}简化为 274 | \begin{align} 275 | \label{C.15} 276 | \frac{\partial E}{\partial X_A}=\braket{\Phi|\frac{\partial \mathcal{H}}{\partial X_A} | \Phi } 277 | \end{align} 278 | \autoref{C.15}是一个简单的单电子算符的期望值加上核排斥项的导数项。但是\autoref{C.12}并不依赖\autoref{C.14}。 279 | $\frac{\partial H^{core}}{\partial X_A}$和$ \frac{\partial (\mu \nu|| \sigma\lambda)}{\partial X_A}$中的积分 280 | 通过“原子轨道跟随”包含了波函数——例如，$\frac{\partial \phi_{\mu}^{A} }{\partial X_A}$，其中$\phi_{\mu}^{A}$ 281 | 是一个以$A$为中心的原子轨道——并且比\autoref{C.15}复杂的多。事实上，即便在\autoref{C.13}下受力应该为0， 282 | 但通过\autoref{C.15}算出来的受力也会很大，因此代表了能量函数的极值。然而，\autoref{C.15}的简洁很有吸引力， 283 | 人们想知道，当目标是几何优化时，满足\autoref{C.14}的条件所增加的不便是否不像在使用\autoref{C.15}那样快。 284 | 285 | 对于一个组态相互作用(CI)波函数，含有行列式$\ket{\Psi _I}$， 286 | \begin{align} 287 | \nonumber 288 | \ket{\Phi _I}=\sum_{I}c_I\ket{\Psi _I} 289 | \end{align} 290 | 我们可以得到它能量的导数为 291 | \begin{align} 292 | \label{C.16} 293 | \frac{\partial E}{\partial X_A}=\frac{\partial {\tilde{E} }}{\partial X_A} 294 | +\sum_{\mu i}\frac{\partial E}{\partial C_{\mu i}}\frac{\partial C_{\mu i}}{\partial X_A} 295 | +\sum_{I}\frac{\partial E}{\partial c_{I}}\frac{\partial c_{I}}{\partial X_A} 296 | \end{align} 297 | 式中第一个求和遍历所有的分子轨道系数。在这个情况下，只有1个多组态自洽场(MCSCF)函数时$\frac{\partial E}{\partial X_A}=\frac{\partial {\tilde{E} }}{\partial X_A}$。 298 | 对于一般的Hartree-Fock和CI波函数，$\frac{\partial E}{\partial c_{I}}$，则 299 | \begin{align} 300 | \label{C.17} 301 | \frac{\partial E}{\partial X_A}=\frac{\partial {\tilde{E} }}{\partial X_A} 302 | +\sum_{\mu i}\frac{\partial E}{\partial C_{\mu i}}\frac{\partial C_{\mu i}}{\partial X_A} 303 | \end{align} 304 | 计算$\frac{\partial C_{\mu i}}{\partial X_A}$很复杂，但是可以通过微扰理论 305 | \endnote{See,for example,J.A.Pople,H.Krishnan,H.B.Schlegel,and J.S.Binkley, 306 | $Int.J.Quantum Chem.\\ \bf{S13:}$225(1979),and refences therein.}解决。 307 | 对于一个大的CI系统能量对$\mathbf{C}$的依赖性降低时；或者对于没有大量极性键的系统时；或者对于分子轨道由对称性决定的系统时，第二项对受力的贡献可能很小。 308 | 在这种情况下，对于势能面的初始搜索可以使用\autoref{C.9}，但是对于精确的结果来说，依赖于这种近似是不令人满意的。 309 | 310 | Hartree-Fock能量的二阶导数可以从\autoref{C.12}直接得到 311 | \begin{align} 312 | \nonumber 313 | \frac{\partial^2 E}{\partial X_A\partial X_B}=&\sum_{ \mu \nu }P_{\nu \mu}\frac{\partial^2 H_{\mu \nu}^{core}}{\partial X_A\partial X_B} 314 | +\frac{1}{2}\sum_{\mu \nu \sigma \lambda }P_{ \nu \mu }P_{\lambda \sigma}\frac{\partial^2 (\mu \nu|| \sigma\lambda)}{\partial X_A\partial X_B} 315 | \\ \nonumber & 316 | -\sum_{\mu \nu }Q_{ \nu \mu}\frac{\partial^2 S}{\partial X_A\partial X_B} 317 | +\frac{\partial^2 V_{NN}}{\partial X_A\partial X_B} 318 | +\sum_{\mu \nu }\frac{\partial P_{ \nu\mu}}{\partial X_B}\frac{\partial H_{\mu \nu}^{core}}{\partial X_A} 319 | \\ \nonumber & 320 | +\sum_{\mu \nu \sigma \lambda }\frac{\partial P_{ \nu \mu }}{\partial X_B}P_{\lambda \sigma}\frac{\partial (\mu \nu|| \sigma\lambda)}{\partial X_A} 321 | -\sum_{\mu \nu }\frac{\partial Q_{ \nu\mu}}{\partial X_B}\frac{\partial S_{\mu \nu}}{\partial X_A} 322 | \end{align} 323 | 表达式的最后三项包含了分子轨道系数的导数并且不能用简单的方式避开。它们可以通过耦合微扰Hartree-Fock理论(CPHF)$^3$得到。 324 | %%%%% 325 | \section{优化技术} 326 | 对一个多变量函数求驻点的数值方法是一个很宏大的课题。 327 | 这些方法可以分成下列4类： 328 | \begin{enumerate} 329 | \item[(a) ] 不用梯度的方法 330 | \item[(b) ] 使用数值梯度和数值二阶导数的方法 331 | \item[(c) ] 使用解析梯度和数值二阶导数的方法 332 | \item[(d) ] 使用解析梯度和解析二阶导数的方法 333 | \end{enumerate} 334 | 等等。 335 | 336 | 上述方法除了第一种之外都是基于能量$E$的泰勒展开，它们的导数$\mathbf{f}$已经在\autoref{secC.2}给出。 337 | 在实际应用上，这些方法可以用在“估算”技术或者“迭代”技术上。 338 | 339 | d类型的方法可能是首选，因为它使用了最丰富的信息，但只有当解析的一阶导数和二阶导数能够同时获得，并且与能量$E$一样容易时，才会有这种情况。 340 | 然而，很明显，只要我们对极值处的几何结构的初始估计在价键力场的二次区域内（\autoref{C.6}的$\mathbf{Y}$坐标）， 341 | 单一的应用\autoref{C.4}和\autoref{C.5}就给出$\mathbf{X_e}$和$\mathbf{E(X_e)}$。 342 | 这一对于\autoref{C.4}的单一应用，我们把它称为\textit{估算}。 343 | 如果我们不在势的二次区域内，估算可能不是非常准确，可能需要迭代；也就是说，从初猜$\mathbf{X_0}$已经确定了一个新的集合$\mathbf{X_1}$， 344 | 我们可以通过求解\autoref{secC.2}的方程得到$\mathbf{X_2}$。这要求$\mathbf{f(X_1)}$和$\mathbf{H^{-1}(X_1)}$。 345 | 这个流程可能需要一直重复，直到$E_n-E_{n-1}$小于给定的阈值，$\sigma =\mathbf{f(X_n)}^{\dagger}\mathbf{f(X_n)}$小于给定的阈值， 346 | $\mathbf{q_n}^{\dagger}\mathbf{q_n}$小于给定的阈值，或者全部三个都满足。 347 | 348 | 实际应用中，由于在获得解析二阶导数的困难，d类型算法没有被普遍的使用在几何搜索上。另一方面，一阶导数的计算时间一般与能量计算相当。 349 | 因为这个原因，c类型算法是最流行的，而且被应用在大多数的现代量子化学程序中。不幸的是，多数的过渡态搜索算法需要解析的二阶导数，因此 350 | 需要非常昂贵的计算机资源。解析的二阶导数也被用在确定一个极值点是不是势能面的最小值点（Hessian矩阵所有的本征值为正）或者过渡态 351 | （Hessian矩阵的本征值有且只有一个为负）上；如果需要的话，也会用在生成一个最小值点的振动光谱上。 352 | 353 | %%%%% 354 | \section{一些优化算法} 355 | %%%%% 356 | \section{过渡态} 357 | %%%%% 358 | \section{约束变分} 359 | %%%%% 360 | \newpage 361 | \theendnotes 362 | \addcontentsline{toc}{section}{注释} 363 | -------------------------------------------------------------------------------- /code/AppendixBcode.f90: -------------------------------------------------------------------------------- 1 | C******************************************************************** 2 | C 3 | C MINIMAL BASIS STO-3G CALCULATION ON HEH+ 4 | C 5 | C THIS IS A LITTLE DUMMY MAIN PROGRAM WHICH CALLS HFCALC 6 | C 7 | C********************************************************************* 8 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 9 | IOP=2 10 | N=3 11 | R=1.4632D0 12 | ZETA1=2.0925D0 13 | ZETA2=1.24D0 14 | ZA=2.0D0 15 | ZB=1.0D0 16 | CALL HFCALC(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 17 | END 18 | C********************************************************************* 19 | SUBROUTINE HFCALC(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 20 | C 21 | C DOES A HARTREE-FOCK CALCULATION FOR A TWO-ELECTRON DIATOMIC 22 | C USING THE 1S MINIMAL STO-NG BASIS SET 23 | C MINIMAL BASIS SET HAS BASIS FUNCTIONS 1 AND 2 ON NUCLEI A AND B 24 | C 25 | C IOP=0 NO PRINTING WHATSOEVER (TO OPTIMIZE EXPONENTS, SAY) 26 | C IOP=1 PRINT ONLY CONVERGED RESULTS 27 | C IOP=2 PRINT EVERY ITERATION 28 | C N STO-NG CALCULATION (N=1,2 OR 3) 29 | C R BONDLENGTH (AU) 30 | C ZETA1 SLATER ORBITAL EXPONENT (FUNCTION 1) 31 | C ZETA2 SLATER ORBITAL EXPONENT (FUNCTION 2) 32 | C ZA ATOMIC NUMBER (ATOM A) 33 | C ZB ATOMIC NUMBER (ATOM B) 34 | C 35 | C********************************************************************* 36 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 37 | IF (IOP.EQ.0) GO TO 20 38 | PRINT 10,N,ZA,ZB 39 | 10 FORMAT(' ',2X,'STO-',I1,'G FOR ATOMIC NUMBERS ',F5.2,' AND ', 40 | $ F5.2//) 41 | 20 CONTINUE 42 | C CALCULATE ALL THE ONE AND TWO ELECTRON INTEGRALS 43 | CALL INTGRL(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 44 | C BE INEFFICIENT AND PUT ALL INTEGRALS IN PRETTY ARRAYS 45 | CALL COLECT(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 46 | C PERFORM THE SCF CALCULATION 47 | CALL SCF(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 48 | RETURN 49 | END 50 | C********************************************************************* 51 | SUBROUTINE INTGRL(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 52 | C 53 | C CALCULATES ALL THE BASIC INTEGRALS NEEDED FOR SCF CALCULATION 54 | C 55 | C********************************************************************* 56 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 57 | COMMON/INT/S12,T11,T12,T22,V11A,V12A,V22A,V11B,V12B,V22B, 58 | $ V1111,V2111,V2121,V2211,V2221,V2222 59 | DIMENSION COEF(3,3),EXPON(3,3),D1(3),A1(3),D2(3),A2(3) 60 | DATA PI/3.1415926535898D0/ 61 | C THESE ARE THE CONTRACTION COEFFICIENTS AND EXPONENTS FOR 62 | C A NORMALIZED SLATER ORBITAL WITH EXPONENT 1.0 IN TERMS OF 63 | C NORMALIZED 1S PRIMITIVE GAUSSIANS 64 | DATA COEF,EXPON/1.0D0,2*0.0D0,0.678914D0,0.430129D0,0.0D0, 65 | $ 0.444635D0,0.535328D0,0.154329D0,0.270950D0,2*0.0D0,0.151623D0, 66 | $ 0.851819D0,0.0D0,0.109818D0,0.405771D0,2.22766D0/ 67 | R2=R*R 68 | C SCALE THE EXPONENTS (A) OF PRIMITIVE GAUSSIANS 69 | C INCLUDE NORMALIZATION IN CONTRACTION COEFFICIENTS (D) 70 | DO 10 I=1,N 71 | A1(I)=EXPON(I,N)*(ZETA1**2) 72 | D1(I)=COEF(I,N)*((2.0D0*A1(I)/PI)**0.75D0) 73 | A2(I)=EXPON(I,N)*(ZETA2**2) 74 | D2(I)=COEF(I,N)*((2.0D0*A2(I)/PI)**0.75D0) 75 | 10 CONTINUE 76 | C D AND A ARE NOW THE CONTRACTION COEFFICIENTS AND EXPONENTS 77 | C IN TERMS OF UNNORMALIZED PRIMITIVE GAUSSIANS 78 | S12=0.0D0 79 | T11=0.0D0 80 | T12=0.0D0 81 | T22=0.0D0 82 | V11A=0.0D0 83 | V12A=0.0D0 84 | V22A=0.0D0 85 | V11B=0.0D0 86 | V12B=0.0D0 87 | V22B=0.0D0 88 | V1111=0.0D0 89 | V2111=0.0D0 90 | V2121=0.0D0 91 | V2211=0.0D0 92 | V2221=0.0D0 93 | V2222=0.0D0 94 | C CALCULATE ONE-ELECTRON INTEGRALS 95 | C CENTER A IS FIRST ATOM, CETER B IS SECOND ATOM 96 | C ORIGIN IS ON CENTER A 97 | C V12A = OFF-DIAGONAL NUCLEAR ATTRACTION TO CENTER A, ETC. 98 | DO 20 I=1,N 99 | DO 20 J=1,N 100 | C RAP2 = SQUARED DISTANCE BETWEEN CENTER A AND CENTER P, ETC. 101 | RAP=A2(J)*R/(A1(I)+A2(J)) 102 | RAP2=RAP**2 103 | RBP2=(R-RAP)**2 104 | S12=S12+S(A1(I),A2(J),R2)*D1(I)*D2(J) 105 | T11=T11+T(A1(I),A1(J),0.0D0)*D1(I)*D1(J) 106 | T12=T12+T(A1(I),A2(J),R2)*D1(I)*D2(J) 107 | T22=T22+T(A2(I),A2(J),0.0D0)*D2(I)*D2(J) 108 | V11A=V11A+V(A1(I),A1(J),0.0D0,0.0D0,ZA)*D1(I)*D1(J) 109 | V12A=V12A+V(A1(I),A2(J),R2,RAP2,ZA)*D1(I)*D2(J) 110 | V22A=V22A+V(A2(I),A2(J),0.0D0,R2,ZA)*D2(I)*D2(J) 111 | V11B=V11B+V(A1(I),A1(J),0.0D0,R2,ZB)*D1(I)*D1(J) 112 | V12B=V12B+V(A1(I),A2(J),R2,RBP2,ZB)*D1(I)*D2(J) 113 | V22B=V22B+V(A2(I),A2(J),0.0D0,0.0D0,ZB)*D2(I)*D2(J) 114 | 20 CONTINUE 115 | C CALCULATE TWO-ELECTRON INTEGRALS 116 | DO 30 I=1,N 117 | DO 30 J=1,N 118 | DO 30 K=1,N 119 | DO 30 L=1,N 120 | RAP=A2(I)*R/(A2(I)+A1(J)) 121 | RBP=R-RAP 122 | RAQ=A2(K)*R/(A2(K)+A1(L)) 123 | RBQ=R-RAQ 124 | RPQ=RAP-RAQ 125 | RAP2=RAP*RAP 126 | RBP2=RBP*RBP 127 | RAQ2=RAQ*RAQ 128 | RBQ2=RBQ*RBQ 129 | RPQ2=RPQ*RPQ 130 | V1111=V1111+TWOE(A1(I),A1(J),A1(K),A1(L),0.0D0,0.0D0,0.0D0) 131 | $ *D1(I)*D1(J)*D1(K)*D1(L) 132 | V2111=V2111+TWOE(A2(I),A1(J),A1(K),A1(L),R2,0.0D0,RAP2) 133 | $ *D2(I)*D1(J)*D1(K)*D1(L) 134 | V2121=V2121+TWOE(A2(I),A1(J),A2(K),A1(L),R2,R2,RPQ2) 135 | $ *D2(I)*D1(J)*D2(K)*D1(L) 136 | V2211=V2211+TWOE(A2(I),A2(J),A1(K),A1(L),0.0D0,0.0D0,R2) 137 | $ *D2(I)*D2(J)*D1(K)*D1(L) 138 | V2221=V2221+TWOE(A2(I),A2(J),A2(K),A1(L),0.0D0,R2,RBQ2) 139 | $ *D2(I)*D2(J)*D2(K)*D1(L) 140 | V2222=V2222+TWOE(A2(I),A2(J),A2(K),A2(L),0.0D0,0.0D0,0.0D0) 141 | $ *D2(I)*D2(J)*D2(K)*D2(L) 142 | 30 CONTINUE 143 | IF (IOP.EQ.0) GO TO 90 144 | PRINT 40 145 | 40 FORMAT(3X,'R',10X,'ZETA1',6X,'ZETA2',6X,'S12',8X,'T11'/) 146 | PRINT 50, R,ZETA1,ZETA2,S12,T11 147 | 50 FORMAT(5F11.6//) 148 | PRINT 60 149 | 60 FORMAT(3X,'T12',8X,'T22',8X,'V11A',7X,'V12A',7X,'V22A'/) 150 | PRINT 50, T12,T22,V11A,V12A,V22A 151 | PRINT 70 152 | 70 FORMAT(3X,4HV11B,7X,4HV12B,7X,4HV22B,7X,'V1111',6X,'V2111'/) 153 | PRINT 50, V11B,V12B,V22B,V1111,V2111 154 | PRINT 80 155 | 80 FORMAT(3X,5HV2121,6X,5HV2211,6X,5HV2221,6X,5HV2222/) 156 | PRINT 50, V2121,V2211,V2221,V2222 157 | 90 RETURN 158 | END 159 | C********************************************************************* 160 | FUNCTION F0(ARG) 161 | C 162 | C CALCULATES THE F FUNCTION 163 | C FO ONLY (S-TYPE ORBITALS) 164 | C 165 | C********************************************************************* 166 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 167 | DATA PI/3.1415926535898D0/ 168 | IF (ARG.LT.1.0D-6) GO TO 10 169 | C F0 IN TERMS OF THE ERROR FUNCTION 170 | F0=DSQRT(PI/ARG)*DERFOTHER(DSQRT(ARG))/2.0D0 171 | GO TO 20 172 | C ASYMPTOTIC VALUE FOR SMALL ARGUMENTS 173 | 10 F0=1.0D0-ARG/3.0D0 174 | 20 CONTINUE 175 | RETURN 176 | END 177 | C********************************************************************* 178 | FUNCTION DERFOTHER(ARG) 179 | C 180 | C CALCULATES THE ERROR FUNCTION ACCORDING TO A RATIONAL 181 | C APPROXIMATION FROM M. ARBRAMOWITZ AND I.A. STEGUN, 182 | C HANDBOOK OF MATHEMATICAL FUNCTIONS, DOVER. 183 | C ABSOLUTE ERROR IS LESS THAN 1.5*10**(-7) 184 | C CAN BE REPLACED BY A BUILT-IN FUNCTION ON SOME MACHINES 185 | C 186 | C********************************************************************* 187 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 188 | DIMENSION A(5) 189 | DATA P/0.3275911D0/ 190 | DATA A/0.254829592D0,-0.284496736D0,1.421413741D0, 191 | $ -1.453152027D0,1.061405429D0/ 192 | T=1.0D0/(1.0D0+P*ARG) 193 | TN=T 194 | POLY=A(1)*TN 195 | DO 10 I=2,5 196 | TN=TN*T 197 | POLY=POLY+A(I)*TN 198 | 10 CONTINUE 199 | DERFOTHER=1.0D0-POLY*DEXP(-ARG*ARG) 200 | RETURN 201 | END 202 | C********************************************************************* 203 | FUNCTION S(A,B,RAB2) 204 | C 205 | C CALCULATES OVERLAPS FOR UN-NORMALIZED PRIMITIVES 206 | C 207 | C********************************************************************* 208 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 209 | DATA PI/3.1415926535898D0/ 210 | S=(PI/(A+B))**1.5D0*DEXP(-A*B*RAB2/(A+B)) 211 | RETURN 212 | END 213 | C********************************************************************* 214 | FUNCTION T(A,B,RAB2) 215 | C 216 | C CALCULATES KINETIC ENERGY INTEGRALS FOR UN-NORMALIZED PRIMITIVES 217 | C 218 | C********************************************************************* 219 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 220 | DATA PI/3.1415926535898D0/ 221 | T=A*B/(A+B)*(3.0D0-2.0D0*A*B*RAB2/(A+B))*(PI/(A+B))**1.5D0 222 | $ *DEXP(-A*B*RAB2/(A+B)) 223 | RETURN 224 | END 225 | C********************************************************************* 226 | FUNCTION V(A,B,RAB2,RCP2,ZC) 227 | C 228 | C CALCULATES UN-NORMALIZED NUCLEAR ATTRACTION INTEGRALS 229 | C 230 | C********************************************************************* 231 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 232 | DATA PI/3.1415926535898D0/ 233 | V=2.0D0*PI/(A+B)*F0((A+B)*RCP2)*DEXP(-A*B*RAB2/(A+B)) 234 | V=-V*ZC 235 | RETURN 236 | END 237 | C********************************************************************* 238 | FUNCTION TWOE(A,B,C,D,RAB2,RCD2,RPQ2) 239 | C 240 | C CALCULATES TWO-ELECTRON INTEGRALS FOR UN-NORMALIZED PRIMITIVES 241 | C A,B,C,D ARE THE EXPONENTS ALPHA, BETA, ETC. 242 | C RAB2 EQUALS SQUARED DISTANCE BETWEEN CENTER A AND CENTER B, ETC. 243 | C********************************************************************* 244 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 245 | DATA PI/3.1415926535898D0/ 246 | TWOE=2.0D0*(PI**2.5D0)/((A+B)*(C+D)*DSQRT(A+B+C+D)) 247 | $ *F0((A+B)*(C+D)*RPQ2/(A+B+C+D)) 248 | $ *DEXP(-A*B*RAB2/(A+B)-C*D*RCD2/(C+D)) 249 | RETURN 250 | END 251 | C********************************************************************* 252 | SUBROUTINE COLECT(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 253 | C 254 | C THIS TAKES THE BASIC INTEGRALS FROM COMMON AND ASSEMBLES THE 255 | C RELEVENT MATRICES, THAT IS S,H,X,XT, AND TWO-ELECTRON INTEGRALS 256 | C 257 | C********************************************************************* 258 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 259 | COMMON/MATRIX/S(2,2),X(2,2),XT(2,2),H(2,2),F(2,2),G(2,2),C(2,2), 260 | $ FPRIME(2,2),CPRIME(2,2),P(2,2),OLDP(2,2),TT(2,2,2,2),E(2,2) 261 | COMMON/INT/S12,T11,T12,T22,V11A,V12A,V22A,V11B,V12B,V22B, 262 | $ V1111,V2111,V2121,V2211,V2221,V2222 263 | C FORM CORE HAMILTONIAN 264 | H(1,1)=T11+V11A+V11B 265 | H(1,2)=T12+V12A+V12B 266 | H(2,1)=H(1,2) 267 | H(2,2)=T22+V22A+V22B 268 | C FORM OVERLAP MATRIX 269 | S(1,1)=1.0D0 270 | S(1,2)=S12 271 | S(2,1)=S(1,2) 272 | S(2,2)=1.0D0 273 | C USE CANONICAL ORTHOGONALIZATION 274 | X(1,1)=1.0D0/DSQRT(2.0D0*(1.0D0+S12)) 275 | X(2,1)=X(1,1) 276 | X(1,2)=1.0D0/DSQRT(2.0D0*(1.0D0-S12)) 277 | X(2,2)=-X(1,2) 278 | C TRANSPOSE OF TRANSFORMATION MATRIX 279 | XT(1,1)=X(1,1) 280 | XT(1,2)=X(2,1) 281 | XT(2,1)=X(1,2) 282 | XT(2,2)=X(2,2) 283 | C MATRIX OF TWO-ELE�CTRON INTEGRALS 284 | TT(1,1,1,1)=V1111 285 | TT(2,1,1,1)=V2111 286 | TT(1,2,1,1)=V2111 287 | TT(1,1,2,1)=V2111 288 | TT(1,1,1,2)=V2111 289 | TT(2,1,2,1)=V2121 290 | TT(1,2,2,1)=V2121 291 | TT(2,1,1,2)=V2121 292 | TT(1,2,1,2)=V2121 293 | TT(2,2,1,1)=V2211 294 | TT(1,1,2,2)=V2211 295 | TT(2,2,2,1)=V2221 296 | TT(2,2,1,2)=V2221 297 | TT(2,1,2,2)=V2221 298 | TT(1,2,2,2)=V2221 299 | TT(2,2,2,2)=V2222 300 | IF (IOP.EQ.0) GO TO 40 301 | CALL MATOUT(S,2,2,2,2,4HS ) 302 | CALL MATOUT(X,2,2,2,2,4HX ) 303 | CALL MATOUT(H,2,2,2,2,4HH ) 304 | PRINT 10 305 | 10 FORMAT(//) 306 | DO 30 I=1,2 307 | DO 30 J=1,2 308 | DO 30 K=1,2 309 | DO 30 L=1,2 310 | PRINT 20, I,J,K,L,TT(I,J,K,L) 311 | 20 FORMAT(3X,1H(,4I2,2H ),F10.6) 312 | 30 CONTINUE 313 | 40 RETURN 314 | END 315 | C********************************************************************* 316 | SUBROUTINE SCF(IOP,N,R,ZETA1,ZETA2,ZA,ZB) 317 | C 318 | C PERFORMS THE SCF ITERATIONS 319 | C 320 | C********************************************************************* 321 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 322 | COMMON/MATRIX/S(2,2),X(2,2),XT(2,2),H(2,2),F(2,2),G(2,2),C(2,2), 323 | $ FPRIME(2,2),CPRIME(2,2),P(2,2),OLDP(2,2),TT(2,2,2,2),E(2,2) 324 | DATA PI/3.1415926535898D0/ 325 | C CONVERGENCE CRITERION FOR DENSITY MATRIX 326 | DATA CRIT/1.0D-4/ 327 | C MAXIMUM NUMBER OF ITERATIONS 328 | DATA MAXIT/25/ 329 | C ITERATION NUMBER 330 | ITER=0 331 | C USE CORE-HAMILTONIAN FOR INITIAL GUESS AT F, I.E. (P=0) 332 | DO 10 I=1,2 333 | DO 10 J=1,2 334 | 10 P(I,J)=0.0D0 335 | IF (IOP.LT.2) GO TO 20 336 | CALL MATOUT(P,2,2,2,2,4HP ) 337 | C START OF ITERATION LOOP 338 | 20 ITER=ITER+1 339 | IF (IOP.LT.2) GO TO 40 340 | PRINT 30, ITER 341 | 30 FORMAT(/,4X,28HSTART OF ITERATION NUMBER = ,I2) 342 | 40 CONTINUE 343 | C FORM TWO-ELECTRON PART OF FOCK MATRIX FROM P 344 | CALL FORMG 345 | IF (IOP.LT.2) GO TO 50 346 | CALL MATOUT(G,2,2,2,2,4HG ) 347 | 50 CONTINUE 348 | C ADD CORE HAMILTONIAN TO GET FOCK MATRIX 349 | DO 60 I=1,2 350 | DO 60 J=1,2 351 | F(I,J) = H(I,J)+G(I,J) 352 | 60 CONTINUE 353 | C CALCULATE ELECTRONIC ENERGY 354 | EN=0.0D0 355 | DO 70 I=1,2 356 | DO 70 J=1,2 357 | EN=EN+0.5D0*P(I,J)*(H(I,J)+F(I,J)) 358 | 70 CONTINUE 359 | IF (IOP.LT.2) GO TO 90 360 | CALL MATOUT(F,2,2,2,2,4HF ) 361 | PRINT 80, EN 362 | 80 FORMAT(///,4X,20HELECTRONIC ENERGY = ,D20.12) 363 | 90 CONTINUE 364 | C TRANSFORM FOCK MATRIX USING G FOR TEMPORARY STORAGE 365 | CALL MULT(F,X,G,2,2) 366 | CALL MULT(XT,G,FPRIME,2,2) 367 | C DIAGONALIZE TRANSFORMED FOCK MATRIX 368 | CALL DIAG(FPRIME,CPRIME,E) 369 | C TRANSFORM EIGENVECTORS TO GET MATRIX C 370 | CALL MULT(X,CPRIME,C,2,2) 371 | C FORM NEW DENSITY MATRIX 372 | DO 100 I=1,2 373 | DO 100 J=1,2 374 | C SAVE PRESENT DENSITY MATRIX 375 | C BEFORE CREATING NEW ONE 376 | OLDP(I,J)=P(I,J) 377 | P(I,J)=0.0D0 378 | DO 100 K=1,1 379 | P(I,J)=P(I,J)+2.0D0*C(I,K)*C(J,K) 380 | 100 CONTINUE 381 | IF (IOP.LT.2) GO TO 110 382 | CALL MATOUT(FPRIME,2,2,2,2,"F' ") 383 | CALL MATOUT(CPRIME,2,2,2,2,"C' ") 384 | CALL MATOUT(E,2,2,2,2,'E ') 385 | CALL MATOUT(C,2,2,2,2,'C ') 386 | CALL MATOUT(P,2,2,2,2,'P ') 387 | 110 CONTINUE 388 | C CALCULATE DELTA 389 | DELTA=0.0D0 390 | DO 120 I=1,2 391 | DO 120 J=1,2 392 | DELTA=DELTA+(P(I,J)-OLDP(I,J))**2 393 | 120 CONTINUE 394 | DELTA=DSQRT(DELTA/4.0D0) 395 | IF (IOP.EQ.0) GO TO 140 396 | PRINT 130, DELTA 397 | 130 FORMAT(/,4X,39HDELTA(CONVERGENCE OF DENSITY MATRIX) = 398 | $F10.6,/) 399 | 140 CONTINUE 400 | C CHECK FOR CONVERGENCE 401 | IF (DELTA.LT.CRIT) GO TO 160 402 | C NOT YET CONVERGED 403 | C TEST FOR MAXIMUM NUMBER OF ITERATIONS 404 | C IF MAXIMUM NUMBER NOT YET REACHED 405 | C GO BACK FOR ANOTHER ITERATION 406 | IF(ITER.LT.MAXIT) GO TO 20 407 | C SOMETHING WRONG HERE 408 | PRINT 150 409 | 150 FORMAT(4X,21HNO CONVERGENCE IN SCF) 410 | STOP 411 | 160 CONTINUE 412 | C CALCULATION CONVERGED IF IT GOT HERE 413 | C ADD NUCLEAR REPULSION TO GET TOTAL ENERGY 414 | ENT=EN+ZA*ZB/R 415 | IF (IOP.EQ.0) GO TO 180 416 | PRINT 170, EN, ENT 417 | 170 FORMAT(//,4X,21HCALCULATION CONVERGED,//, 418 | $4X,20HELECTRONIC ENERGY = ,D20.12,//, 419 | $4X,20HTOTAL ENERGY = ,D20.12 ) 420 | 180 CONTINUE 421 | IF (IOP.NE.1) GO TO 190 422 | C PRINT OUT THE FINAL RESULTS IF 423 | C HAVE NOT DONE SO ALREADY 424 | CALL MATOUT(G,2,2,2,2,4HG ) 425 | CALL MATOUT(F,2,2,2,2,4HF ) 426 | CALL MATOUT(E,2,2,2,2,4HE ) 427 | CALL MATOUT(C,2,2,2,2,4HC ) 428 | CALL MATOUT(P,2,2,2,2,4HP ) 429 | 190 CONTINUE 430 | C PS MATRIX HAS MULLIKEN POPULATIONS 431 | CALL MULT(P,S,OLDP,2,2) 432 | IF(IOP.EQ.0) GO TO 200 433 | CALL MATOUT(OLDP,2,2,2,2,4HPS ) 434 | 200 CONTINUE 435 | RETURN 436 | END 437 | C********************************************************************* 438 | SUBROUTINE FORMG 439 | C 440 | C CALCULATES THE G MATRIX FROM THE DENSITY MATRIX 441 | C AND TWO-ELECTRON INTEGRALS 442 | C 443 | C********************************************************************* 444 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 445 | COMMON/MATRIX/S(2,2),X(2,2),XT(2,2),H(2,2),F(2,2),G(2,2),C(2,2), 446 | $FPRIME(2,2),CPRIME(2,2),P(2,2),OLDP(2,2),TT(2,2,2,2),E(2,2) 447 | DO 10 I=1,2 448 | DO 10 J=1,2 449 | G(I,J)=0.0D0 450 | DO 10 K=1,2 451 | DO 10 L=1,2 452 | G(I,J)=G(I,J)+P(K,L)*(TT(I,J,K,L)-0.5D0*TT(I,L,K,J)) 453 | 10 CONTINUE 454 | RETURN 455 | END 456 | C********************************************************************* 457 | SUBROUTINE DIAG(F,C,E) 458 | C 459 | C DIAGONALIZES F TO GIVE EIGENVECTORS IN C AND EIGENVALUES IN E 460 | C THETA IS THE ANGLE DESCRIBING SOLUTION 461 | C 462 | C********************************************************************* 463 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 464 | DIMENSION F(2,2),C(2,2),E(2,2) 465 | DATA PI/3.1415926535898D0/ 466 | IF (DABS(F(1,1)-F(2,2)).GT.1.0D-20) GO TO 10 467 | C HERE IS SYMMETRY DETERMINED SOLUTION (HOMONUCLEAR DIATOMIC) 468 | THETA=PI/4.0D0 469 | GO TO 20 470 | 10 CONTINUE 471 | C SOLUTION FOR HETERONUCLEAR DIATOMIC 472 | THETA=0.5D0*DATAN(2.0D0*F(1,2)/(F(1,1)-F(2,2))) 473 | 20 CONTINUE 474 | C(1,1)=DCOS(THETA) 475 | C(2,1)=DSIN(THETA) 476 | C(1,2)=DSIN(THETA) 477 | C(2,2)=-DCOS(THETA) 478 | E(1,1)=F(1,1)*DCOS(THETA)**2+F(2,2)*DSIN(THETA)**2 479 | $ +F(1,2)*DSIN(2.0D0*THETA) 480 | E(2,2)=F(2,2)*DCOS(THETA)**2+F(1,1)*DSIN(THETA)**2 481 | $ -F(1,2)*DSIN(2.0D0*THETA) 482 | E(2,1)=0.0D0 483 | E(1,2)=0.0D0 484 | C ORDER EIGENVALUES AND EIGENVECTORS 485 | IF (E(2,2).GT.E(1,1)) GO TO 30 486 | TEMP=E(2,2) 487 | E(2,2)=E(1,1) 488 | E(1,1)=TEMP 489 | TEMP=C(1,2) 490 | C(1,2)=C(1,1) 491 | C(1,1)=TEMP 492 | TEMP=C(2,2) 493 | C(2,2)=C(2,1) 494 | C(2,1)=TEMP 495 | 30 RETURN 496 | END 497 | C********************************************************************* 498 | SUBROUTINE MULT(A,B,C,IM,M) 499 | C 500 | C MULTIPLIES TWO SQUARE MATRICES A AND B TO GET C 501 | C 502 | C********************************************************************* 503 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 504 | DIMENSION A(IM,IM),B(IM,IM),C(IM,IM) 505 | DO 10 I=1,M 506 | DO 10 J=1,M 507 | C(I,J)=0.0D0 508 | DO 10 K=1,M 509 | 10 C(I,J)=C(I,J)+A(I,K)*B(K,J) 510 | RETURN 511 | END 512 | C********************************************************************* 513 | SUBROUTINE MATOUT(A,IM,IN,M,N,LABEL) 514 | C 515 | C PRINT MATRICES OF SIZE M BY N 516 | C 517 | C********************************************************************* 518 | IMPLICIT DOUBLE PRECISION(A-H,O-Z) 519 | DIMENSION A(IM,IN) 520 | IHIGH=0 521 | 10 LOW=IHIGH+1 522 | IHIGH=IHIGH+5 523 | IHIGH=MIN(IHIGH,N) 524 | PRINT 20, LABEL,(I,I=LOW,IHIGH) 525 | 20 FORMAT(///,3X,5H THE ,A4,6H ARRAY,/,15X,5(10X,I3,6X)//) 526 | DO 30 I=1,M 527 | 30 PRINT 40, I,(A(I,J),J=LOW,IHIGH) 528 | 40 FORMAT(I10,5X,5(1X,D18.10)) 529 | IF (N-IHIGH) 50,50,10 530 | 50 RETURN 531 | END 532 | -------------------------------------------------------------------------------- /Chaps/Chap5.tex: -------------------------------------------------------------------------------- 1 | \chapter{对理论与耦合对理论} 2 | 第四章中我们已经看到， 3 | 若仅使用双激发组态相互作用(DCI)计算, 4 | 则$N$个无相互作用$\hd$分子的关联能正比于$N^{-1/2}$(当$N$大时). 5 | 由于宏观系统的能量是热力学广度性质, 6 | 它必须正比于粒子数目， 7 | 所以DCI无法良好地处理大体系. 8 | 举个例子, 9 | 若用DCI计算晶体中每个原子所占的相关能, 10 | 结果会是零！很明显, 11 | 要描述无限大体系的相关效应, 12 | 必须使用能使能量正比于粒子数的方法. 13 | 即使对有限体系, 14 | 使用近似方法时也我们希望能给出在不同大小分子下可比较的结果. 15 | 举个例子, 16 | 研究分子解离时, 17 | 要对整个分子以及解离碎片使用(某种意义上的)同等质量的方法. 18 | 一种近似方法, 19 | 若用其计算所得的能量在体系增大时按粒子数目线性变化, 20 | 那么就说该近似是\emph{大小一致的(size consistent)}. 21 | 之前的特例——$N$个闭壳层无相互作用的``单体''组成的超分子中, 22 | 若用大小一致的办法, 23 | 则超分子的能量就等于$N$乘以单体的能量. 24 | 25 | 26 | 虽然大小一致性看似是个普通的要求, 27 | 但除了full CI外的所有CI方法——前者当然是精确的——都不满足这个性质. 28 | 本征我们要考虑对理论及对耦合理论, 29 | 它们有大小一致性, 30 | 下一章要讨论的一种微扰论形式也具有这个性质. 31 | 对理论和微扰论具有大小一致性, 32 | 但代价就是, 33 | 与DCI不同, 34 | 前两个方法不是变分的, 35 | 因此由它们所得的总能可以比真实能量低. 36 | 比如, 37 | 对理论在特定情形下会给出$120\%$的相关能. 38 | 39 | 40 | 5.1节讲独立电子对近似(the independent electron pair approximation, IEPA). 我们用一种快速的方式先构建计算框架, 但是这种方式可能会让读者错误地认为IEPA是DCI的近似. 展示了\phrase{对计算}中的细节后, 我们回到该方法的物理基础上并证明实际上IEPA和DCI是\emph{full CI}的不同近似. 5.1.1节指出了IEPA的一个缺点, 该缺点在DCI和微扰论中都不存在：IEPA在简并分子轨道的酉变换下并非不变. 5.1.2节列出了一些数值结果, 以说明IEPA用在小分子上比较精确, 而在大分子上有严重缺陷. 41 | 42 | 5.2节中考虑如何超越IEPA：加入不同电子对间的耦合. 我们讨论耦合对多电子理论(CPMET), 它有时也叫作耦合簇近似(coupled cluster approximation, CCA). 接下来介绍一系列的对这个复杂方法的简化方案, 特别地，我们要考察耦合电子对近似(coupled electron pair approximation, CEPA). 最后在 5.2.4节介绍一阶\phrase{耦合对理论}的一些数值例子. 43 | 44 | 由于\phrase{耦合对理论}非常重要但又较复杂, 45 | 我们在5. 46 | 3节(为教学计)用这些方法来计算一个特殊的$N$-电子体系的能量, 47 | 该体系的哈密顿仅包含单粒子作用. 48 | 这个问题很容易用基础的方法精确求解. 49 | 但是, 50 | 通过观察``大马力''的手段在这种简单问题中工作的方式, 51 | 我们可以洞察这些近似方法的本质. 52 | 特别地, 53 | 由此可以清晰地知道各个多电子理论间的关系. 54 | 作为多电子方法应用到单电子哈密顿体系中的一个具体例子, 55 | 我们在5. 56 | 3. 57 | 2节考虑H\"ucekl框架下环多烯的共振能. 58 | 这里的主要目的不是要主张用H\"uckel理论或是多电子方法得到共振能. 59 | 而是， 60 | 要利用共振能和相关能的类似性来提供一个可以解析研究的模型, 61 | 通过它我们能够说明各个多电子方法在在计算层面上的一些事情。 62 | 63 | 64 | \section{独立电子对近似(The Independent Electron Pair Approximation, IEPA)} 65 | 在前面一章我们已经知道, 66 | 用中间归一化full CI波函数(由\hft 行列式中的自旋轨道激发出的所有可能组态构成)可以得到相关能： 67 | \begin{align} 68 | E_\mathrm{corr} = \sum_{a2$也不是精确的。 398 | 因此，将四激发系数近似为双激发系数的平方，并不只是对DQCI的近似，实际上还隐含了将六激发系数近似为双激发系数的立方，等等。 399 | CCA在这个理想模型中给出精确解的原因是，\emph{所有}更高（六、八，等等）激发的系数都是双激发系数的乘积。 400 | 在下一节中将会以一个更基本的视角（也是历史上CCA被提出的方式）清楚地展示CCA的这个特点。 401 | 这一节会用一些二次量子化的记号，可以不失连续性地跳过。 402 | 403 | \subsection{波函数的簇展开} 404 | 双激发行列式$\ket{\Psi^{rs}_{ab}}$可以写成二次量子化的形式： 405 | \begin{align*} 406 | \ket{\Psi^{rs}_{ab}}=a^{\dagger}_ra^{\dagger}_sa_ba_a\ket{\Psi_0} 407 | \end{align*} 408 | 其中$a_a,a_b$从HF行列式中移除了一个占据轨道，取而代之，$a^{\dagger}_r,a^{\dagger}_s$添加了一个非占据的自旋轨道。 409 | 因此双激发CI波函数可以写成： 410 | \begin{align*} 411 | \ket{\Psi_{DCI}}=\left(1+\frac{1}{4}\sum_{abrs}c^{rs}_{ab}a^{\dagger}_ra^{\dagger}_sa_ba_a\ket{\Phi_0}\right) 412 | \end{align*} 413 | 现引入一个波函数，它包含双激发、四激发、六激发等等，其中2n激发的系数是n个双激发系数的乘积。这样一个波函数$\ket{\Phi_{CCA}}$可以写成： 414 | \begin{align} 415 | \ket{\Phi_{CCA}}=\exp(\mathscr{T}_2)\ket{\Psi_0}\tag{5.61a} 416 | \end{align} 417 | \addtocounter{equation}{1} 418 | 其中 419 | \begin{align} 420 | \exp(\mathscr{T}_2)=\frac{1}{4}\sum_{abrs}c^{rs}_{ab}a^{\dagger}_ra^{\dagger}_sa_ba_a\tag{5.61b} 421 | \end{align} 422 | 这叫做波函数的簇形式。为了找到一些感觉，我们将指数展开为$\exp(x)=1+x+\frac{1}{2}x^2+\cdots$，得到 423 | \begin{align*} 424 | \ket{\Phi_{CCA}}&=\left(1+\frac{1}{4}\sum_{abrs}c^{rs}_{ab}a^{\dagger}_ra^{\dagger}_sa_ba_a +\frac{1}{32}\sum_{\substack{abcd\\rstu}}c^{rs}_{ab}c^{tu}_{cd}a^{\dagger}_ra^{\dagger}_sa_ba_aa^{\dagger}_ta^{\dagger}_ua_da_c+\cdots\right)\ket{\Psi_0}\\ 425 | &=\ket{\Psi_0}+\frac{1}{4}\sum_{abrs}c^{rs}_{ab}\ket{\Psi^{rs}_{ab}}+\frac{1}{32}\sum_{\substack{abcd\\rstu}}c^{rs}_{ab}c^{tu}_{cd}\ket{\Psi^{rstu}_{abcd}}+\cdots 426 | \end{align*} 427 | 经过一些冗长的运算，它可以写成： 428 | \begin{align} 429 | \label{5.62} 430 | \ket{\Phi_{CCA}}=\ket{\Psi_0}+\sum_{\substack{a\m@ne 148 | \if@mainmatter 149 | \@chapapp\ \thechapter. \ % 150 | \fi 151 | \fi 152 | ##1}}{}}% 153 | \def\sectionmark##1{% 154 | \markright {\MakeUppercase{% 155 | \ifnum \c@secnumdepth >\z@ 156 | \thesection. \ % 157 | \fi 158 | ##1}}}} 159 | \else 160 | \def\ps@headings{% 161 | \let\@oddfoot\@empty 162 | \def\@oddhead{{\slshape\rightmark}\hfil\thepage}% 163 | \let\@mkboth\markboth 164 | \def\chaptermark##1{% 165 | \markright {\MakeUppercase{% 166 | \ifnum \c@secnumdepth >\m@ne 167 | \if@mainmatter 168 | \@chapapp\ \thechapter. \ % 169 | \fi 170 | \fi 171 | ##1}}}} 172 | \fi 173 | \def\ps@myheadings{% 174 | \let\@oddfoot\@empty\let\@evenfoot\@empty 175 | \def\@evenhead{\thepage\hfil\slshape\leftmark}% 176 | \def\@oddhead{{\slshape\rightmark}\hfil\thepage}% 177 | \let\@mkboth\@gobbletwo 178 | \let\chaptermark\@gobble 179 | \let\sectionmark\@gobble 180 | } 181 | \if@titlepage 182 | \newcommand\maketitle{\begin{titlepage}% 183 | \let\footnotesize\small 184 | \let\footnoterule\relax 185 | \let \footnote \thanks 186 | \null\vfil 187 | \vskip 60\p@ 188 | \begin{center}% 189 | {\LARGE \@title \par}% 190 | \vskip 3em% 191 | {\large 192 | \lineskip .75em% 193 | \begin{tabular}[t]{c}% 194 | \@author 195 | \end{tabular}\par}% 196 | \vskip 1.5em% 197 | {\large \@date \par}% % Set date in \large size. 198 | \end{center}\par 199 | \@thanks 200 | \vfil\null 201 | \end{titlepage}% 202 | \setcounter{footnote}{0}% 203 | \global\let\thanks\relax 204 | \global\let\maketitle\relax 205 | \global\let\@thanks\@empty 206 | \global\let\@author\@empty 207 | \global\let\@date\@empty 208 | \global\let\@title\@empty 209 | \global\let\title\relax 210 | 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388 | \vskip 40\p@ 389 | }} 390 | \def\@schapter#1{\if@twocolumn 391 | \@topnewpage[\@makeschapterhead{#1}]% 392 | \else 393 | \@makeschapterhead{#1}% 394 | \@afterheading 395 | \fi} 396 | \def\@makeschapterhead#1{% 397 | \vspace*{50\p@}% 398 | {\parindent \z@ \raggedright 399 | \normalfont 400 | \interlinepenalty\@M 401 | \fontsize{12pt}{\baselineskip}\selectfont \bfseries #1\par\nobreak 402 | \vskip 40\p@ 403 | }} 404 | \newcommand\section{\@startsection {section}{1}{\z@}% 405 | {-3.5ex \@plus -1ex \@minus -.2ex}% 406 | {2.3ex \@plus.2ex}% 407 | {\normalfont\fontsize{8.5pt}{\baselineskip}\selectfont\bfseries}} 408 | \newcommand\subsection{\@startsection{subsection}{2}{\z@}% 409 | {-3.25ex\@plus -1ex \@minus -.2ex}% 410 | {1.5ex \@plus .2ex}% 411 | {\normalfont\fontsize{8pt}{\baselineskip}\selectfont\bfseries}} 412 | \newcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% 413 | {-3.25ex\@plus -1ex \@minus -.2ex}% 414 | {1.5ex \@plus .2ex}% 415 | 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\@mkboth{% 596 | \MakeUppercase\contentsname}{\MakeUppercase\contentsname}}% 597 | \@starttoc{toc}% 598 | \if@restonecol\twocolumn\fi 599 | } 600 | \newcommand*\l@part[2]{% 601 | \ifnum \c@tocdepth >-2\relax 602 | \addpenalty{-\@highpenalty}% 603 | \addvspace{2.25em \@plus\p@}% 604 | \setlength\@tempdima{3em}% 605 | \begingroup 606 | \parindent \z@ \rightskip \@pnumwidth 607 | \parfillskip -\@pnumwidth 608 | {\leavevmode 609 | \large \bfseries #1\hfil \hb@xt@\@pnumwidth{\hss #2}}\par 610 | \nobreak 611 | \global\@nobreaktrue 612 | \everypar{\global\@nobreakfalse\everypar{}}% 613 | \endgroup 614 | \fi} 615 | \newcommand*\l@chapter[2]{% 616 | \ifnum \c@tocdepth >\m@ne 617 | \addpenalty{-\@highpenalty}% 618 | \vskip 1.0em \@plus\p@ 619 | \setlength\@tempdima{1.5em}% 620 | \begingroup 621 | \parindent \z@ \rightskip \@pnumwidth 622 | \parfillskip -\@pnumwidth 623 | \leavevmode \bfseries 624 | \advance\leftskip\@tempdima 625 | \hskip -\leftskip 626 | #1\nobreak\hfil \nobreak\hb@xt@\@pnumwidth{\hss #2}\par 627 | \penalty\@highpenalty 628 | \endgroup 629 | \fi} 630 | \newcommand*\l@section{\@dottedtocline{1}{1.5em}{2.3em}} 631 | \newcommand*\l@subsection{\@dottedtocline{2}{3.8em}{3.2em}} 632 | \newcommand*\l@subsubsection{\@dottedtocline{3}{7.0em}{4.1em}} 633 | \newcommand*\l@paragraph{\@dottedtocline{4}{10em}{5em}} 634 | \newcommand*\l@subparagraph{\@dottedtocline{5}{12em}{6em}} 635 | \newcommand\listoffigures{% 636 | \if@twocolumn 637 | \@restonecoltrue\onecolumn 638 | \else 639 | \@restonecolfalse 640 | \fi 641 | \chapter*{\listfigurename}% 642 | \@mkboth{\MakeUppercase\listfigurename}% 643 | {\MakeUppercase\listfigurename}% 644 | \@starttoc{lof}% 645 | \if@restonecol\twocolumn\fi 646 | } 647 | \newcommand*\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} 648 | \newcommand\listoftables{% 649 | \if@twocolumn 650 | \@restonecoltrue\onecolumn 651 | \else 652 | \@restonecolfalse 653 | \fi 654 | \chapter*{\listtablename}% 655 | \@mkboth{% 656 | \MakeUppercase\listtablename}% 657 | {\MakeUppercase\listtablename}% 658 | \@starttoc{lot}% 659 | \if@restonecol\twocolumn\fi 660 | } 661 | \let\l@table\l@figure 662 | \newdimen\bibindent 663 | \setlength\bibindent{1.5em} 664 | \newenvironment{thebibliography}[1] 665 | {\chapter*{\bibname}% 666 | \@mkboth{\MakeUppercase\bibname}{\MakeUppercase\bibname}% 667 | \list{\@biblabel{\@arabic\c@enumiv}}% 668 | {\settowidth\labelwidth{\@biblabel{#1}}% 669 | \leftmargin\labelwidth 670 | \advance\leftmargin\labelsep 671 | \@openbib@code 672 | \usecounter{enumiv}% 673 | \let\p@enumiv\@empty 674 | \renewcommand\theenumiv{\@arabic\c@enumiv}}% 675 | \sloppy 676 | \clubpenalty4000 677 | \@clubpenalty \clubpenalty 678 | \widowpenalty4000% 679 | \sfcode`\.\@m} 680 | {\def\@noitemerr 681 | {\@latex@warning{Empty `thebibliography' environment}}% 682 | \endlist} 683 | \newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em} 684 | \let\@openbib@code\@empty 685 | \newenvironment{theindex} 686 | {\if@twocolumn 687 | \@restonecolfalse 688 | \else 689 | \@restonecoltrue 690 | \fi 691 | \twocolumn[\@makeschapterhead{\indexname}]% 692 | \@mkboth{\MakeUppercase\indexname}% 693 | {\MakeUppercase\indexname}% 694 | \thispagestyle{plain}\parindent\z@ 695 | \parskip\z@ \@plus .3\p@\relax 696 | \columnseprule \z@ 697 | \columnsep 35\p@ 698 | \let\item\@idxitem} 699 | {\if@restonecol\onecolumn\else\clearpage\fi} 700 | \newcommand\@idxitem{\par\hangindent 40\p@} 701 | \newcommand\subitem{\@idxitem \hspace*{20\p@}} 702 | \newcommand\subsubitem{\@idxitem \hspace*{30\p@}} 703 | \newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} 704 | \renewcommand\footnoterule{% 705 | \kern-3\p@ 706 | \hrule\@width.4\columnwidth 707 | \kern2.6\p@} 708 | \@addtoreset{footnote}{chapter} 709 | \newcommand\@makefntext[1]{% 710 | \parindent 1em% 711 | \noindent 712 | \hb@xt@1.8em{\hss\@makefnmark}#1} 713 | \newcommand\contentsname{Contents} 714 | \newcommand\listfigurename{List of Figures} 715 | \newcommand\listtablename{List of Tables} 716 | \newcommand\bibname{Bibliography} 717 | \newcommand\indexname{Index} 718 | \newcommand\figurename{Figure} 719 | \newcommand\tablename{Table} 720 | \newcommand\partname{Part} 721 | \newcommand\chaptername{Chapter} 722 | \newcommand\appendixname{Appendix} 723 | \def\today{\ifcase\month\or 724 | January\or February\or March\or April\or May\or June\or 725 | July\or August\or September\or October\or November\or December\fi 726 | \space\number\day, \number\year} 727 | \setlength\columnsep{10\p@} 728 | \setlength\columnseprule{0\p@} 729 | \pagestyle{headings} 730 | \pagenumbering{arabic} 731 | \if@twoside 732 | \else 733 | \raggedbottom 734 | \fi 735 | \if@twocolumn 736 | \twocolumn 737 | \sloppy 738 | \flushbottom 739 | \else 740 | \onecolumn 741 | \fi 742 | \endinput 743 | %% 744 | %% End of file `book.cls'. 745 | --------------------------------------------------------------------------------