From 621444fd8cf055f1d6b04cecb4112cab5f62cf38 Mon Sep 17 00:00:00 2001 From: vijay Date: Tue, 8 Dec 2020 09:54:14 +0100 Subject: [PATCH] Working on a CFG based Configuration Interaction algorithm WIP#143 --- Theory_CFG_CIPSI.org | 112 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 112 insertions(+) create mode 100644 Theory_CFG_CIPSI.org diff --git a/Theory_CFG_CIPSI.org b/Theory_CFG_CIPSI.org new file mode 100644 index 00000000..3e101b3b --- /dev/null +++ b/Theory_CFG_CIPSI.org @@ -0,0 +1,112 @@ +#+TITLE: CFG CIPSI +#+AUTHOR: Vijay Gopal Chilkuri (vijay.gopal.c@gmail.com) +#+DATE: <2020-12-08 Tue 08:27> + +#+LATEX_HEADER: \usepackage{braket} + +* 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. + + + * Definitino of CI basis + + In CFG based CIPSI, the wavefunction is represented in CFG basis as shown in Eq:\[~\ref{Eq:definebasis1}\]. + + \begin{equation} + \label{Eq:definebasis1} + \ket{\psi} &= \sum_{ij} 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:\[~\ref{Eq:definebasis2}\]. + + \begin{equation} + \label{Eq:definebasis2} + \ket{\Phi^j_i} &= \sum^j_{i,k} O^j_{i,k} \ket{^S\phi_k(i,j)} + \end{equation} + + Where the functions \[\ket{^S\phi_k(i,j)}\] represent the BFs for the CFG + \[i\]. Each CFG contains a list of CSFs related to it which describes the + spin part of the wavefunction (see Eq:~\ref{Eq:definebasis3}) which is + encoded in the BFs as shown below in Eq:~\ref{Eq:definebasis5}. + + \begin{equation}\begin{equation} + \label{Eq:definebasis3} + \ket{^S\Phi_i} = \left\{ \ket{^S\Phi^1_i}, \ket{^S\Phi^2_i}, \dots, \ket{^s\phi^{n_{csf}}_i} \right} + \end{equation} + + \begin{equation}\begin{equation} + \label{eq:definebasis4} + \ket{^s\phi_i} = \left\{ c^1_i, c^1_i, \dots, c^{N_{CSF}}_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:definebasis4}. Crucially, the bonded functions + defined in Eq:~\ref{Eq:definebasis5} are not northogonal to each other. + + \begin{equation} + \label{Eq:definebasis4} + \ket{^S\phi_k(i,j)} = (i\bar{i})\dots (j,k) l m + \end{equation} + + The bonded functions are made up of products of slater determinants. There are + three types of determinants, first, the closed shell pairs \[(i\bar{i})\]. Second, + the open-shell singlet pairs \[(i,j)\] which are expanded as \[(i,j) = \frac{\ket{i\bar{j}}-\ket{\bar{i}j}}{\sqrt{2}}\]. 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:~\ref{Eq:defineovlp1}. + + \begin{equation} + \label{Eq:defineovlp1} + \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:definebasis2} and Eq:~\ref{Eq:definebasis3} + as given in Eq:~\ref{Eq:defineovlp2}. + + \begin{equation} + \label{Eq:defineovlp2} + \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 \[\left( N^k_{N_{BF}} , N^l_{N_{BF}} \rigth)\]. + The equation shown above (Eq:~\ref{Eq:defineovlp2}) can be written in marix-form as + shown below in Eq:~\ref{Eq:defineovlp3}. + + \begin{equation} + \label{Eq:defineovlp3} + \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. + +# Local variables: +# after-save-hook: org-preview-latex-fragment +# end: