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Added the part on sigma-vector (#143)
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** Definition of matrix-elements
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The matrix-element (ME) evaluation follows a similar logic.
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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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