Function uw12::three_el::form_fock_three_el_term_df_ri

Function Documentation

utils::FockMatrixAndEnergy uw12::three_el::form_fock_three_el_term_df_ri(const integrals::Integrals &W, const integrals::Integrals &V, const ri::ABSProjectors &abs_projectors, bool indirect_term, bool calculate_fock, double scale_opp_spin, double scale_same_spin)

Calculates the UW12 three electron energy and Fock matrix contribution using a density-fitted RI approximation

This algorithm calculates the UW12 three-electron energy given by:

\[E_{c, 3el}^{UW12} = - \left\langle \overline{ij} k \left\vert w^{s_{ij}}_{12} r_{23}^{-1}\right\vert k j i \right\rangle \]
(repeated index summation)

Using the RI approximation, these integrals may be approximated as

\[E_{c, 3el}^{UW12, RI} = - \left\langle \overline{ij} \left\vert w^{s_{ij}}_{12} \right\vert k q \right\rangle \left\langle k q \left\vert r_{12}^{-1} \right\vert ij \right\rangle, \]
using the resolution of the identity \(| q ><q | \approx 1\).

Resolution of the identity

To construct the resolution of the identity, an ABS+ approach is used: https://doi.org/10.1016/j.cplett.2004.07.061 In this approach, the orthonormal basis q is constructed from the union of the ao basis and an auxiliary basis set (ABS) ri. This basis is not constructed explicitly but calculated using the inverse overlap for the combined ao and ri space. The projector may be written as \(|q \rangle \langle q | = |\tilde{\mu} \rangle [S^{-1}]_{\tilde{\mu}\tilde{\nu}} \langle \tilde{\nu} |\) for indices \(\tilde{\mu},\tilde{\nu}\) in the union of the two basis sets. After computing the inverse of \(S\) for the full union using singular value decomposition to remove linearly dependent functions, the inverse may be split into four sub-matrices for each possible component basis combination:

\[ | q \rangle \langle q | = | \mu \rangle [S^{-1}]_{\mu\nu} \langle \nu | + | \rho \rangle [S^{-1}]_{\rho\nu} \langle \nu | + | \mu \rangle [S^{-1}]_{\mu \sigma} \langle \sigma | + | \rho \rangle [S^{-1}]_{\rho\sigma} \langle \sigma | \]
for \(\mu, \nu\) in ri and \(\rho, \sigma\) in ao. The projectors \(S^{-1}\) are taken from abs_projectors.

In order to evaluate the integrals efficiently, a density-fitting approach is used to approximate the two-electron integrals. Therefore only three-index objects must be stored.

Direct Energy

The direct energy is then given by:

\[E_{c, 3el, +}^{UW12} = - X_{AB} \tilde{t}_{AB} \]
for density fitting basis indices \(A,B\), where:
\[\begin{align*}X_{AB} &= (A|w_{12}|j \mu) [S^{-1}]_{\mu\nu} (\nu j| r_{12}^{-1} |B) + (A|w_{12}|j \rho) [S^{-1}]_{\rho\nu} (\nu j| r_{12}^{-1} |B) \newline &+ (A|w_{12}|j \mu) [S^{-1}]_{\mu\sigma} (\sigma j| r_{12}^{-1} |B) + (A|w_{12}|j \rho) [S^{-1}]_{\rho\sigma} (\sigma j| r_{12}^{-1} |B) \end{align*}\]
for active (occupied) orbitals j. Similarly:
\[\tilde{t}_{AB} = (\tilde{A} | w_{12} | ik ) ( ki | r_{12}^{-1} | \tilde{B}) \]
for active occupied orbitals i and all occupied orbitals k with \((\tilde{A} | w_{12} | ik ) = (A|w_{12}|B)^{-1} (B|w_{12}|ik)\). Three-index integrals \((A|x_{12}|j\rho)\) and \((A|x_{12}|ik)\) are the one and two mo-transformed integrals respectively, while the three-index mo-transformed ri integrals are \((A|w_{12}|j \mu)\).

Direct Fock

The direct fock matrix contribution is calculated from:

\[\frac{\partial E_{c, 3el,+}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} = \frac{\partial X_{AB}}{\partial D_{\alpha \beta}^{\sigma}} \tilde{t}_{AB} + X_{AB} \frac{\partial \tilde{t}_{AB}}{\partial D_{\alpha \beta}^{\sigma}} \]
where
\[\begin{align*}\frac{\partial X_{AB}}{\partial D_{\alpha \beta}^{\sigma}} &= (A|w_{12}|\alpha \mu) [S^{-1}]_{\mu\nu} (\nu \beta| r_{12}^{-1} |B) + (A|w_{12}|\alpha \rho) [S^{-1}]_{\rho\nu} (\nu \beta| r_{12}^{-1} |B) \newline &+ (A|w_{12}|\alpha \mu) [S^{-1}]_{\mu\sigma} (\sigma \beta| r_{12}^{-1} |B) + (A|w_{12}|\alpha \rho) [S^{-1}]_{\rho\sigma} (\sigma \beta| r_{12}^{-1} |B) \end{align*}\]
and
\[ \frac{\partial \tilde{t}_{AB}}{\partial D_{\alpha \beta}^{\sigma}} = (\tilde{A} | w_{12} | \alpha k ) ( k \beta | r_{12}^{-1} | \tilde{B}) + (\tilde{A} | w_{12} | i \beta ) ( \alpha i | r_{12}^{-1} | \tilde{B}) \]

Indirect Energy

The indirect energy is given by:

\[E_{c, 3el, -}^{UW12} = - \sum_{ij} \sum_{AB} X_{AB}^{ij} \tilde{t}_{AB}^{ij} \]
for density fitting basis indices \(A,B\), where:
\[\begin{align*}X_{AB}^{ij} &= \sum_{\mu \nu} (A|w_{12}|i \mu) [S^{-1}]_{\mu\nu} (\nu j| r_{12}^{-1} |B) + (A|w_{12}|i \rho) [S^{-1}]_{\rho\nu} (\nu j| r_{12}^{-1} |B) \newline &+ (A|w_{12}|i \mu) [S^{-1}]_{\mu\sigma} (\sigma j| r_{12}^{-1} |B) + (A|w_{12}|i \rho) [S^{-1}]_{\rho\sigma} (\sigma j| r_{12}^{-1} |B) \end{align*}\]
\[\tilde{t}_{AB}^{ij} = \sum_{k} (\tilde{A} | w_{12} | jk ) ( ki | r_{12}^{-1} | \tilde{B}) \]
This operation is performed using parallelisation over indices i,j.

Indirect Fock

Writing \(|q \rangle\langle q| = |\mu' \rangle[S^{-1}]_{\mu'\nu'}\langle \nu'|\), where \(\mu',\nu'\) represent the complete set of ri basis function (ao and aux indices), the indirect fock contribution is given by (in four-index notation):

\[\begin{align*}\frac{\partial E_{c, 3el,-}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= \sum_{jk} \sum_{\mu' \nu'} (jk | w_{12} | \alpha \mu' ) [S^{-1}]_{\mu'\nu'} (\nu' j | r_{12}^{-1} | k \beta) \newline &+ \sum_{ik} \sum_{\mu'\nu'} (\alpha k | w_{12} | i \mu') [S^{-1}]_{\mu'\nu'} (\nu' \beta | r_{12}^{-1} | k i) \newline &+ \sum_{ij} \sum_{\mu'\nu'}( j \beta | w_{12} | i\mu')[S^{-1}]_{\mu' \nu'} (\nu' j | r_{12}^{-1} | \alpha i ) \end{align*}\]
Where we refer to these separate terms as the i, j, and k contributions corresponding to the index which has been removed from the summation by differentiation. Each of these terms is made up of four separate terms corresponding to the four combinations of ri basis components:
\[|\mu' \rangle[S^{-1}]_{\mu'\nu'}\langle \nu'| = | \mu \rangle [S^{-1}]_{\mu\nu} \langle \nu | + | \rho \rangle [S^{-1}]_{\rho\nu} \langle \nu | + | \mu \rangle [S^{-1}]_{\mu \sigma} \langle \sigma | + | \rho \rangle [S^{-1}]_{\rho\sigma} \langle \sigma | \]
for \(\mu, \nu\) in aux, \(\rho, \sigma\) in ao, and \(\mu', \nu'\) in the union of the two. Four-index integrals are evaluated from the density fitting integrals for a single pair ij, ik, or jk with parallelisation over these.

Parameters

  • W – Integrals for \(w_{12}\)

  • V – Integrals for \(r_{12}^{-1}\)

  • abs_projectors – projectors \(S^{-1}\) for each subspace

  • indirect_term – calculate the indirect term

  • calculate_fock – calculate the fock matrix contribution

  • scale_opp_spin – scale factor for opposite spin contribution

  • scale_same_spin – scale factor for same spin contribution

Returns

Fock matrix and energy contributions for the three electron term