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185 lines
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185 lines
8.2 KiB
Org Mode
#+TITLE: CFG CIPSI
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#+AUTHOR: Vijay Gopal Chilkuri
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#+EMAIL: vijay.gopal.c@gmail.com
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#+DATE: 2020-12-08 Tue 08:27
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#+startup: latexpreview
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#+STARTUP: inlineimages
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#+LATEX_HEADER: \usepackage{braket}
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* Biblio
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* Theoretical background
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Here we describe the main theoretical background and definitions of the
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Configuration (CFG) based CIPSI algorithm. The outline of the document is as follows.
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First, we give some definitions of the CFG many-particle basis follwed by the
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definitions of the overlap, one-particle, and two-particle matrix-elements. Finally,
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an algorithm is presented for the sigma-vector (\( \sigma \)-vector defined later) calculation using
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the CFG basis.
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** Definitinon of CI basis
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In CFG based CIPSI, the wavefunction is represented in CFG basis
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as shown in Eq: [[Eq:definebasis1]].
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#+LATEX: \newcommand{\Ncfg}{N_{\text{CFG}}}
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#+LATEX: \newcommand{\Ncsf}{N_{\text{CSF}}}
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#+LATEX: \newcommand{\Nsomo}{N_{\text{SOMO}}}
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#+NAME: Eq:definebasis1
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\begin{equation}
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\ket{\Psi} = \sum_{i=1}^{\Ncfg} \sum_{j=1}^{\Ncsf(i)} c_{ij} {^S\ket{\Phi^j_i}}
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\end{equation}
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where the \(\ket{\Phi^j_i}\) represent Configuration State Functions (CSFs)
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which are expanded in terms of Bonded functions (BFs) as shown in
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[[Eq:definebasis2]].
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#+NAME: Eq:definebasis2
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\begin{equation}
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\ket{\Phi^j_i} = \sum_k O^{\Nsomo(i)}_{kj} \ket{^S\phi_k(i,j)}
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\end{equation}
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where the functions \(\ket{^S\phi_k(i,j)}\) represent the BFs for the CFG
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\(\ket{^S\Phi_i}\).
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The coefficients \(O^b_{a,k}\) depend only on the number of SOMOs
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in \(\Phi_i\).
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Each CFG contains a list of CSFs related to it which describes the
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spin part of the wavefunction (see Eq: [[Eq:definebasis3]]) which is
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encoded in the BFs as shown below in Eq: [[Eq:definebasis5]].
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#+NAME: Eq:definebasis3
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\begin{equation}
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\ket{^S\Phi_i} = \left\{ \ket{^S\Phi^1_i}, \ket{^S\Phi^2_i}, \dots, \ket{^s\Phi^{\Ncsf}_i} \right\}
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\end{equation}
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#+NAME: Eq:definebasis4
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\begin{equation}
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\mathbf{c}_i = \left\{ c^1_i, c^2_i, \dots, c^{\Ncsf}_i \right\}
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\end{equation}
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Each of the CSFs belonging to the CFG \(\ket{^S\Phi_i}\) have coefficients
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associated to them as shown in Eq: [[Eq:definebasis4]]. Crucially, the bonded functions
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defined in Eq: [[Eq:definebasis5]] are not northogonal to each other.
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#+NAME: Eq:definebasis5
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\begin{equation}
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\ket{^S\phi_k(i,j)} = (a\bar{a})\dots (b\ c) (d (e
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\end{equation}
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$i$ is the index of the CFG and $j$ determines the coupling.
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The bonded functions are made up of products of slater determinants. There are
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three types of determinants, first, the closed shell pairs \((a\bar{a})\). Second,
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the open-shell singlet pairs \((b\ c)\) which are expanded as
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\((b\ c) = \frac{\ket{b\bar{c}}-\ket{\bar{b}c}}{\sqrt{2}}\). Third, the
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open-shell SOMOs which are coupled parallel and account for the total spin of the
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wavefunction \((l (m \dots\). They are shown as open brackets.
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** Overlap of the wavefunction
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Once the wavefunction has been expanded in terms of the CSFs, the most fundamental
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operation is to calculate the overlap between two states. The overlap in the
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basis of CSFs is defined as shown in Eq: [[Eq:defineovlp1]].
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#+NAME: Eq:defineovlp1
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\begin{equation}
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\braket{^S\Phi_i|^S\Phi_j} = \sum_{kl} C_i C_j \braket{^S\Psi^k_i|^S\Psi^l_j}
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\end{equation}
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Where the sum is over the CSFs \(k\) and \(l\) corresponding to the \(i\)
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and \(j\) CFGs respectively. The overlap between the CSFs can be expanded in terms
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of the BFs using the definition given in Eq: [[Eq:definebasis2]] and
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Eq: [[Eq:definebasis3]] as given in Eq: [[Eq:defineovlp2]].
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#+NAME: Eq:defineovlp2
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\begin{equation}
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\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}
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\end{equation}
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Therefore, the overlap between two CSFs can be expanded in terms of the overlap
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between the constituent BFs. The overlap matrix \(S_{mn}\) is of dimension \(\left( N^k_{N_{BF}} , N^l_{N_{BF}} \right)\).
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The equation shown above (Eq: [[Eq:defineovlp2]]) can be written in marix-form as
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shown below in Eq: [[Eq:defineovlp3]].
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#+NAME: Eq:defineovlp3
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\begin{equation}
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\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}
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\end{equation}
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Note that the overlap between two CFGs does not depend on the orbital
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labels. It only depends on the number of Singly Occupied Molecular Orbitals
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(SOMOs) therefore it can be pretabulated. Actually, it is possible to
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redefine the CSFs in terms of a linear combination of BFs such that
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\(S_{ij}\) becomes the identity matrix. In this case, one needs to store the
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orthogonalization matrix \(\mathbf{\tilde{O}}_i\) which is given by
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\(\mathbf{O}_i\cdot S^{1/2}_i\) for a given CFG \(i\). Note that the a CFG
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\(i\) is by definition of an orthonormal set of MOs automatically orthogonal
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to a CFG \(j\) with a different occupation.
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** Definition of matrix-elements
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The matrix-element (ME) evaluation follows a similar logic as the evalulation of
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the overlap. However, here the metric is the one-, or two-particle operator \(\hat{E}_{pq}\)
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or \(\hat{E}_{pq}\hat{E}_{rs}\) as shown in Eq: [[Eq:defineme1]] and Eq: [[Eq:defineme2]].
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#+NAME: Eq:defineme1
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\begin{equation}
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\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}
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\end{equation}
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#+NAME: Eq:defineme2
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\begin{equation}
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\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}
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\end{equation}
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Where, \(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\) represent an arbitrary one-, and
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two-particle operators respectively. Importantly, the one-, and two-particle
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matrix-element evaluation can be recast into an effecient matrix multiplication
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form which is crucial for a fast evaluation of the action of the operators
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\(\hat{O}_{pq}\) and \(hat{O}_{pq,rs}\). The matrix \(\mathbf{A}^{pq}_{ij}\) contains
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the result of the operation \(\braket{^S\Phi^k_i|\hat{O}_{pq}|^S\Phi^l_j}\) in terms
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of BFs and is therefore of size \(NCSF(i) \textit{x} NBF(i)\). In this formulation, the determinant basis is entirely avoided.
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Note that the size and contents of the matrix \(\mathbf{A}^{pq}_{ij}\) depends
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only on the total number of SOMOs and the total spin \(S\), therefore, an optimal
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prototyping scheme can be deviced for a rapid calculaiton of these matrix contractions.
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The resolution of identity (RI) is used to evaluate the two-particle operator since
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this alleviates the necessacity to explicity construct two-particle matrix-elements
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\(\braket{^S\Phi^k_i|\hat{O}_{pq,rs}|^S\Phi^l_j}\) directly.
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** Sigma-vector evaluation
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Once the \(\mathbf{A}^{pq}_{ij}\) matrices have been constructed for the given
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selected list of CFGs, the prototype lists for the \(\mathbf{A}^{pq}_{ij}\) matrices
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can be constructed. Following this, one can proceede to the evaluation of the sigma-vector
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as defined in the Eq [[Eq:definesigma1]].
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#+NAME: Eq:definesigma1
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\begin{equation}
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\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}
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\end{equation}
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The one-electron part of the sigma-vector can be calculated as shown in Eq: [[Eq:defineme1]]
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and the two-electron part can be calculated using the RI as shown in Eq: [[Eq:defineme2]].
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The most expensive part involves the two-particle operator as shown on the RHS of Eq: [[Eq:definesigma1]].
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In this CFG formulation, the cost intensive part of the sigma-vector evaluation has been recast
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into an efficient BLAS matrix multiplication operation. Therefore, this formulation is the most efficient
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albeit at the cost of storing the prototype matrices \(\mathbf{A}^{pq}_{ij}\). However, where the total spin
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is small and the largest number of SOMOs does not exceed 14, the \(\mathbf{A}^{pq}_{ij}\) matrices
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can be stored in memory.
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