qp2/Theory_CFG_CIPSI.org

#+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_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,
    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_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 marix-form 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
    (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.
    
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