From f3078ed2e217a2c4d73d990d53e6ab552c8b1bbe Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Wed, 16 Dec 2020 11:04:48 +0100 Subject: [PATCH] Put back org file instead of tex --- Notes_CSF/Theory_CFG_CIPSI.org | 181 ++++++++++++++++++++++++++++ Notes_CSF/Theory_CFG_CIPSI.tex | 213 --------------------------------- 2 files changed, 181 insertions(+), 213 deletions(-) create mode 100644 Notes_CSF/Theory_CFG_CIPSI.org delete mode 100644 Notes_CSF/Theory_CFG_CIPSI.tex diff --git a/Notes_CSF/Theory_CFG_CIPSI.org b/Notes_CSF/Theory_CFG_CIPSI.org new file mode 100644 index 00000000..089e296c --- /dev/null +++ b/Notes_CSF/Theory_CFG_CIPSI.org @@ -0,0 +1,181 @@ +#+TITLE: CFG CIPSI +#+AUTHOR: Vijay Gopal Chilkuri (vijay.gopal.c@gmail.com) +#+DATE: 2020-12-08 Tue 08:27 +#+startup: latexpreview + +#+LATEX_HEADER: \usepackage{braket} + +* Biblio +* Theoretical background + + Here we describe the main theoretical background and definitions of the + Configuration (CFG) based CIPSI algorithm. The outline of the document is as follows. + First, we give some definitions of the CFG many-particle basis follwed by the + definitions of the overlap, one-particle, and two-particle matrix-elements. Finally, + an algorithm is presented for the sigma-vector (\( \sigma \)-vector defined later) calculation using + the CFG basis. + + +** Definitinon of CI basis + + In CFG based CIPSI, the wavefunction is represented in CFG basis + as shown in Eq: [[Eq:definebasis1]]. + + #+LATEX: \newcommand{\Ncfg}{N_{\text{CFG}}} + #+LATEX: \newcommand{\Ncsf}{N_{\text{CSF}}} + #+LATEX: \newcommand{\Nsomo}{N_{\text{SOMO}}} + #+NAME: Eq:definebasis1 + \begin{equation} + \ket{\Psi} = \sum_{i=1}^{\Ncfg} \sum_{j=1}^{\Ncsf(i)} c_{ij} {^S\ket{\Phi^j_i}} + \end{equation} + + + where the \(\ket{\Phi^j_i}\) represent Configuration State Functions (CSFs) + which are expanded in terms of Bonded functions (BFs) as shown in + [[Eq:definebasis2]]. + #+NAME: Eq:definebasis2 + \begin{equation} + \ket{\Phi^j_i} = \sum_k O^{\Nsomo(i)}_{kj} \ket{^S\phi_k(i,j)} + \end{equation} + where the functions \(\ket{^S\phi_k(i,j)}\) represent the BFs for the CFG + \(\ket{^S\Phi_i}\). + The coefficients \(O^b_{a,k}\) depend only on the number of SOMOs + in \(\Phi_i\). + + Each CFG contains a list of CSFs related to it which describes the + spin part of the wavefunction (see Eq: [[Eq:definebasis3]]) which is + encoded in the BFs as shown below in Eq: [[Eq:definebasis5]]. + + + #+NAME: Eq:definebasis3 + \begin{equation} + \ket{^S\Phi_i} = \left\{ \ket{^S\Phi^1_i}, \ket{^S\Phi^2_i}, \dots, \ket{^s\Phi^{\Ncsf}_i} \right\} + \end{equation} + + + + #+NAME: Eq:definebasis4 + \begin{equation} + \mathbf{c}_i = \left\{ c^1_i, c^2_i, \dots, c^{\Ncsf}_i \right\} + \end{equation} + + + Each of the CSFs belonging to the CFG \(\ket{^S\Phi_i}\) have coefficients + associated to them as shown in Eq: [[Eq:definebasis4]]. Crucially, the bonded functions + defined in Eq: [[Eq:definebasis5]] are not northogonal to each other. + + + #+NAME: Eq:definebasis5 + \begin{equation} + \ket{^S\phi_k(i,j)} = (a\bar{a})\dots (b\ c) (d (e + \end{equation} + $i$ is the index of the CFG and $j$ determines the coupling. + + + The bonded functions are made up of products of slater determinants. There are + three types of determinants, first, the closed shell pairs \((a\bar{a})\). Second, + the open-shell singlet pairs \((b\ c)\) which are expanded as + \((b\ c) = \frac{\ket{b\bar{c}}-\ket{\bar{b}c}}{\sqrt{2}}\). Third, the + open-shell SOMOs which are coupled parallel and account for the total spin of the + wavefunction \((l (m \dots\). They are shown as open brackets. + +** Overlap of the wavefunction + + Once the wavefunction has been expanded in terms of the CSFs, the most fundamental + operation is to calculate the overlap between two states. The overlap in the + basis of CSFs is defined as shown in Eq: [[Eq:defineovlp1]]. + + + #+NAME: Eq:defineovlp1 + \begin{equation} + \braket{^S\Phi_i|^S\Phi_j} = \sum_{kl} C_i C_j \braket{^S\Psi^k_i|^S\Psi^l_j} + \end{equation} + + + Where the sum is over the CSFs \(k\) and \(l\) corresponding to the \(i\) + and \(j\) CFGs respectively. The overlap between the CSFs can be expanded in terms + of the BFs using the definition given in Eq: [[Eq:definebasis2]] and + Eq: [[Eq:definebasis3]] as given in Eq: [[Eq:defineovlp2]]. + + + #+NAME: Eq:defineovlp2 + \begin{equation} + \braket{^S\Phi^k_i|^S\Phi^l_j} = \sum_m \sum_n \left( O^k_{i,m}\right)^{\dagger} \braket{^S\phi_m(i,k)|^S\phi_n(j,l)} O^l_{j,n} + \end{equation} + + + Therefore, the overlap between two CSFs can be expanded in terms of the overlap + between the constituent BFs. The overlap matrix \(S_{mn}\) is of dimension \(\left( N^k_{N_{BF}} , N^l_{N_{BF}} \right)\). + The equation shown above (Eq: [[Eq:defineovlp2]]) can be written in marix-form as + shown below in Eq: [[Eq:defineovlp3]]. + + #+NAME: Eq:defineovlp3 + \begin{equation} + \braket{^S\Phi_i|^S\Phi_j} = \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{S}_{ij}\cdot\mathbf{O}_j C_{j,1} + \end{equation} + + + Note that the overlap between two CFGs does not depend on the orbital + labels. It only depends on the number of Singly Occupied Molecular Orbitals + (SOMOs) therefore it can be pretabulated. Actually, it is possible to + redefine the CSFs in terms of a linear combination of BFs such that + \(S_{ij}\) becomes the identity matrix. In this case, one needs to store the + orthogonalization matrix \(\mathbf{\tilde{O}}_i\) which is given by + \(\mathbf{O}_i\cdot S^{1/2}_i\) for a given CFG \(i\). Note that the a CFG + \(i\) is by definition of an orthonormal set of MOs automatically orthogonal + to a CFG \(j\) with a different occupation. + +** Definition of matrix-elements + + The matrix-element (ME) evaluation follows a similar logic as the evalulation of + the overlap. However, here the metric is the one-, or two-particle operator \(\hat{E}_{pq}\) + or \(\hat{E}_{pq}\hat{E}_{rs}\) as shown in Eq: [[Eq:defineme1]] and Eq: [[Eq:defineme2]]. + + #+NAME: Eq:defineme1 + \begin{equation} + \braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j} = \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{A}^{pq}_{ij}\cdot\mathbf{O}_j C_{j,1} + \end{equation} + + #+NAME: Eq:defineme2 + \begin{equation} + \braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j} = \sum_K \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{A}^{pq}_{ik}\cdot\mathbf{O}_k \mathbf{O}_k\cdot\mathbf{A}^{rs}_{kj}\cdot\mathbf{O}_j C_{j,1} + \end{equation} + + + Where, \(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\) represent an arbitrary one-, and + two-particle operators respectively. Importantly, the one-, and two-particle + matrix-element evaluation can be recast into an effecient matrix multiplication + form which is crucial for a fast evaluation of the action of the operators + \(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\). The matrix \(\mathbf{A}^{pq}_{ij}\) contains + the result of the operation \(\braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j}\) in terms + of BFs and is therefore of size \(NCSF(i) \textit{x} NBF(i)\). In this formulation, the determinant basis is entirely avoided. + Note that the size and contents of the matrix \(\mathbf{A}^{pq}_{ij}\) depends + only on the total number of SOMOs and the total spin \(S\), therefore, an optimal + prototyping scheme can be deviced for a rapid calculaiton of these matrix contractions. + The resolution of identity (RI) is used to evaluate the two-particle operator since + this alleviates the necessacity to explicity construct two-particle matrix-elements + \(\braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j}\) directly. + +** Sigma-vector evaluation + + Once the \(\mathbf{A}^{pq}_{ij}\) matrices have been constructed for the given + selected list of CFGs, the prototype lists for the \(\mathbf{A}^{pq}_{ij}\) matrices + can be constructed. Following this, one can proceede to the evaluation of the sigma-vector + as defined in the Eq [[Eq:definesigma1]]. + + + #+NAME: Eq:definesigma1 + \begin{equation} + \sigma = \sum_{pq} \tilde{h}_{pq}\hat{E}_{pq}|\ket{^S\Phi^l_j} + \frac{1}{2}\sum_{pq,rs} g_{pq,rs} \hat{E}_{pq}\hat{E}_{rs}|\ket{^S\Phi^l_j} + \end{equation} + + The one-electron part of the sigma-vector can be calculated as shown in Eq: [[Eq:defineme1]] + and the two-electron part can be calculated using the RI as shown in Eq: [[Eq:defineme2]]. + The most expensive part involves the two-particle operator as shown on the RHS of Eq: [[Eq:definesigma1]]. + In this CFG formulation, the cost intensive part of the sigma-vector evaluation has been recast + into an efficient BLAS matrix multiplication operation. Therefore, this formulation is the most efficient + albeit at the cost of storing the prototype matrices \(\mathbf{A}^{pq}_{ij}\). However, where the total spin + is small and the largest number of SOMOs does not exceed 14, the \(\mathbf{A}^{pq}_{ij}\) matrices + can be stored in memory. + + diff --git a/Notes_CSF/Theory_CFG_CIPSI.tex b/Notes_CSF/Theory_CFG_CIPSI.tex deleted file mode 100644 index 7ddbb2bc..00000000 --- a/Notes_CSF/Theory_CFG_CIPSI.tex +++ /dev/null @@ -1,213 +0,0 @@ -% Created 2020-12-14 Mon 14:09 -% Intended LaTeX compiler: pdflatex -\documentclass[11pt]{article} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} -\usepackage{grffile} -\usepackage{longtable} -\usepackage{wrapfig} -\usepackage{rotating} -\usepackage[normalem]{ulem} -\usepackage{amsmath} -\usepackage{textcomp} -\usepackage{amssymb} -\usepackage{capt-of} -\usepackage{hyperref} -\usepackage{minted} -\usepackage{braket} -\author{Vijay Gopal Chilkuri (vijay.gopal.c@gmail.com)} -\date{2020-12-08 Tue 08:27} -\title{CFG CIPSI} -\hypersetup{ - pdfauthor={Vijay Gopal Chilkuri (vijay.gopal.c@gmail.com)}, - pdftitle={CFG CIPSI}, - pdfkeywords={}, - pdfsubject={}, - pdfcreator={Emacs 26.3 (Org mode 9.4)}, - pdflang={English}} -\begin{document} - -\maketitle -\tableofcontents - - -\section{Biblio} -\label{sec:org471cc15} -\section{Theoretical background} -\label{sec:org2af35e7} - -Here we describe the main theoretical background and definitions of the -Configuration (CFG) based CIPSI algorithm. The outline of the document is as follows. -First, we give some definitions of the CFG many-particle basis follwed by the -definitions of the overlap, one-particle, and two-particle matrix-elements. Finally, -an algorithm is presented for the sigma-vector (\(\sigma\)-vector defined later) calculation using -the CFG basis. - - -\subsection{Definitinon of CI basis} -\label{sec:org9ec07e3} - -In CFG based CIPSI, the wavefunction is represented in CFG basis -as shown in Eq: \ref{eq:orgc760af7}. - -\newcommand{\Ncfg}{N_{\text{CFG}}} -\newcommand{\Ncsf}{N_{\text{CSF}}} -\newcommand{\Nsomo}{N_{\text{SOMO}}} -\begin{equation} -\label{eq:orgc760af7} -\ket{\Psi} = \sum_{i=1}^{\Ncfg} \sum_{j=1}^{\Ncsf(i)} c_{ij} {^S\ket{\Phi^j_i}} -\end{equation} - - -where the \(\ket{\Phi^j_i}\) represent Configuration State Functions (CSFs) -which are expanded in terms of Bonded functions (BFs) as shown in -\ref{eq:org9412d75}. -\begin{equation} -\label{eq:org9412d75} -\ket{\Phi^j_i} = \sum_k O^{\Nsomo(i)}_{kj} \ket{^S\phi_k(i,j)} -\end{equation} -where the functions \(\ket{^S\phi_k(i,j)}\) represent the BFs for the CFG -\(\ket{^S\Phi_i}\). -The coefficients \(O^b_{a,k}\) depend only on the number of SOMOs -in \(\Phi_i\). - -Each CFG contains a list of CSFs related to it which describes the -spin part of the wavefunction (see Eq: \ref{eq:org9a932e6}) which is -encoded in the BFs as shown below in Eq: \ref{eq:orgd7ae34a}. - - -\begin{equation} -\label{eq:org9a932e6} -\ket{^S\Phi_i} = \left\{ \ket{^S\Phi^1_i}, \ket{^S\Phi^2_i}, \dots, \ket{^s\Phi^{\Ncsf}_i} \right\} -\end{equation} - - - -\begin{equation} -\label{eq:org1d5619f} -\mathbf{c}_i = \left\{ c^1_i, c^2_i, \dots, c^{\Ncsf}_i \right\} -\end{equation} - - -Each of the CSFs belonging to the CFG \(\ket{^S\Phi_i}\) have coefficients -associated to them as shown in Eq: \ref{eq:org1d5619f}. Crucially, the bonded functions -defined in Eq: \ref{eq:orgd7ae34a} are not northogonal to each other. - - -\begin{equation} -\label{eq:orgd7ae34a} -\ket{^S\phi_k(i,j)} = (a\bar{a})\dots (b\ c) (d (e -\end{equation} -\(i\) is the index of the CFG and \(j\) determines the coupling. - - -The bonded functions are made up of products of slater determinants. There are -three types of determinants, first, the closed shell pairs \((a\bar{a})\). Second, -the open-shell singlet pairs \((b\ c)\) which are expanded as -\((b\ c) = \frac{\ket{b\bar{c}}-\ket{\bar{b}c}}{\sqrt{2}}\). Third, the -open-shell SOMOs which are coupled parallel and account for the total spin of the -wavefunction \((l (m \dots\). They are shown as open brackets. - -\subsection{Overlap of the wavefunction} -\label{sec:orgf2a5d5d} - -Once the wavefunction has been expanded in terms of the CSFs, the most fundamental -operation is to calculate the overlap between two states. The overlap in the -basis of CSFs is defined as shown in Eq: \ref{eq:org48d76cc}. - - -\begin{equation} -\label{eq:org48d76cc} -\braket{^S\Phi_i|^S\Phi_j} = \sum_{kl} C_i C_j \braket{^S\Psi^k_i|^S\Psi^l_j} -\end{equation} - - -Where the sum is over the CSFs \(k\) and \(l\) corresponding to the \(i\) -and \(j\) CFGs respectively. The overlap between the CSFs can be expanded in terms -of the BFs using the definition given in Eq: \ref{eq:org9412d75} and -Eq: \ref{eq:org9a932e6} as given in Eq: \ref{eq:orge9a0815}. - - -\begin{equation} -\label{eq:orge9a0815} -\braket{^S\Phi^k_i|^S\Phi^l_j} = \sum_m \sum_n \left( O^k_{i,m}\right)^{\dagger} \braket{^S\phi_m(i,k)|^S\phi_n(j,l)} O^l_{j,n} -\end{equation} - - -Therefore, the overlap between two CSFs can be expanded in terms of the overlap -between the constituent BFs. The overlap matrix \(S_{mn}\) is of dimension \(\left( N^k_{N_{BF}} , N^l_{N_{BF}} \right)\). -The equation shown above (Eq: \ref{eq:orge9a0815}) can be written in marix-form as -shown below in Eq: \ref{eq:orgd608801}. - -\begin{equation} -\label{eq:orgd608801} -\braket{^S\Phi_i|^S\Phi_j} = \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{S}_{ij}\cdot\mathbf{O}_j C_{j,1} -\end{equation} - - -Note that the overlap between two CFGs does not depend on the orbital -labels. It only depends on the number of Singly Occupied Molecular Orbitals -(SOMOs) therefore it can be pretabulated. Actually, it is possible to -redefine the CSFs in terms of a linear combination of BFs such that -\(S_{ij}\) becomes the identity matrix. In this case, one needs to store the -orthogonalization matrix \(\mathbf{\tilde{O}}_i\) which is given by -\(\mathbf{O}_i\cdot S^{1/2}_i\) for a given CFG \(i\). Note that the a CFG -\(i\) is by definition of an orthonormal set of MOs automatically orthogonal -to a CFG \(j\) with a different occupation. - -\subsection{Definition of matrix-elements} -\label{sec:org4e0679d} - -The matrix-element (ME) evaluation follows a similar logic as the evalulation of -the overlap. However, here the metric is the one-, or two-particle operator \(\hat{E}_{pq}\) -or \(\hat{E}_{pq}\hat{E}_{rs}\) as shown in Eq: \ref{eq:org68e7cb8} and Eq: \ref{eq:orge4248f3}. - -\begin{equation} -\label{eq:org68e7cb8} -\braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j} = \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{A}^{pq}_{ij}\cdot\mathbf{O}_j C_{j,1} -\end{equation} - -\begin{equation} -\label{eq:orge4248f3} -\braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j} = \sum_K \left( C_{i,1} \right)^{\dagger} \mathbf{O}_i\cdot\mathbf{A}^{pq}_{ik}\cdot\mathbf{O}_k \mathbf{O}_k\cdot\mathbf{A}^{rs}_{kj}\cdot\mathbf{O}_j C_{j,1} -\end{equation} - - -Where, \(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\) represent an arbitrary one-, and -two-particle operators respectively. Importantly, the one-, and two-particle -matrix-element evaluation can be recast into an effecient matrix multiplication -form which is crucial for a fast evaluation of the action of the operators -\(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\). The matrix \(\mathbf{A}^{pq}_{ij}\) contains -the result of the operation \(\braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j}\) in terms -of BFs and is therefore of size \(NCSF(i) \textit{x} NBF(i)\). In this formulation, the determinant basis is entirely avoided. -Note that the size and contents of the matrix \(\mathbf{A}^{pq}_{ij}\) depends -only on the total number of SOMOs and the total spin \(S\), therefore, an optimal -prototyping scheme can be deviced for a rapid calculaiton of these matrix contractions. -The resolution of identity (RI) is used to evaluate the two-particle operator since -this alleviates the necessacity to explicity construct two-particle matrix-elements -\(\braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j}\) directly. - -\subsection{Sigma-vector evaluation} -\label{sec:orge1be4a8} - -Once the \(\mathbf{A}^{pq}_{ij}\) matrices have been constructed for the given -selected list of CFGs, the prototype lists for the \(\mathbf{A}^{pq}_{ij}\) matrices -can be constructed. Following this, one can proceede to the evaluation of the sigma-vector -as defined in the Eq \ref{eq:org3f76f3d}. - - -\begin{equation} -\label{eq:org3f76f3d} -\sigma = \sum_{pq} \tilde{h}_{pq}\hat{E}_{pq}|\ket{^S\Phi^l_j} + \frac{1}{2}\sum_{pq,rs} g_{pq,rs} \hat{E}_{pq}\hat{E}_{rs}|\ket{^S\Phi^l_j} -\end{equation} - -The one-electron part of the sigma-vector can be calculated as shown in Eq: \ref{eq:org68e7cb8} -and the two-electron part can be calculated using the RI as shown in Eq: \ref{eq:orge4248f3}. -The most expensive part involves the two-particle operator as shown on the RHS of Eq: \ref{eq:org3f76f3d}. -In this CFG formulation, the cost intensive part of the sigma-vector evaluation has been recast -into an efficient BLAS matrix multiplication operation. Therefore, this formulation is the most efficient -albeit at the cost of storing the prototype matrices \(\mathbf{A}^{pq}_{ij}\). However, where the total spin -is small and the largest number of SOMOs does not exceed 14, the \(\mathbf{A}^{pq}_{ij}\) matrices -can be stored in memory. -\end{document} \ No newline at end of file