Working on a CFG based Configuration Interaction algorithm WIP#143

Theory_CFG_CIPSI.org

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+#+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: