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@ -1,11 +1,14 @@
#+TITLE: CFG CIPSI
#+AUTHOR: Vijay Gopal Chilkuri (vijay.gopal.c@gmail.com)
#+AUTHOR: Vijay Gopal Chilkuri
#+EMAIL: vijay.gopal.c@gmail.com
#+DATE: 2020-12-08 Tue 08:27
#+startup: latexpreview
#+STARTUP: inlineimages
#+LATEX_HEADER: \usepackage{braket}
* Biblio
* Biblio
* Theoretical background
Here we describe the main theoretical background and definitions of the
@ -14,7 +17,7 @@
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
@ -41,7 +44,7 @@
\(\ket{^S\Phi_i}\).
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]].
@ -52,7 +55,7 @@
\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}
@ -130,52 +133,52 @@
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
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
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
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]].
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}
\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]].
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
is small and the largest number of SOMOs does not exceed 14, the \(\mathbf{A}^{pq}_{ij}\) matrices
can be stored in memory.