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Removed LaTeX blocks since they do not help #143
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@ 18,53 +18,51 @@




In CFG based CIPSI, the wavefunction is represented in CFG basis as shown in Eq:\[~\ref{Eq:definebasis1}\].




#+BEGIN_LaTeX


\begin{equation}


\label{Eq:definebasis1}


\ket{\psi} &= \sum_{ij} c_{ij} ^s\ket{\phi^j_i}


\end{equation}


#+END_LaTeX






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_LaTeX


\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}


#+END_LaTeX






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_LaTeX




\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}


#+END_LaTeX






#+BEGIN_LaTeX




\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}


#+END_LaTeX






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_LaTeX




\begin{equation}


\label{Eq:definebasis4}


\ket{^S\phi_k(i,j)} = (i\bar{i})\dots (j,k) l m


\end{equation}


#+END_LaTeX






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,


@ 78,36 +76,35 @@


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_LaTeX




\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}


#+END_LaTeX






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_LaTeX




\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}


#+END_LaTeX






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 marixform as


shown below in Eq:~\ref{Eq:defineovlp3}.




#+BEGIN_LaTeX


\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}


#+END_LaTeX






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



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