Function uw12::two_el::form_fock_two_el_df

Function Documentation

utils::FockMatrixAndEnergy uw12::two_el::form_fock_two_el_df(const integrals::BaseIntegrals &WV, const utils::Orbitals &active_Co, bool indirect_term, bool calculate_fock, double scale_opp_spin, double scale_same_spin)

Find the contribution to the energy and Fock matrix of the two electron term of UW12.

The two electron UW12 Fock matrix and energy are calculated using density fitting. In this method, the direct (+) and indirect (-) contributions are calculated separately with the energies given by

\[\begin{align*}E_{c, 2el,+}^{UW12} &= \frac{1}{2} \sum_{C} \sum_{ij} \left( ii \left| w_{12}^{s_{ij}} r_{12}^{-1} \right| C \right) \left( \tilde{C} \left| w_{12}^{s_{i j}} r_{12}^{-1} \right| jj \right) \newline E_{c, 2el,-}^{UW12} &= - \frac{1}{2} \sum_{C} \sum_{ij} \left( ij \left| w_{12}^{1} r_{12}^{-1} \right| C \right) \left( \tilde{C} \left| w_{12}^{1} r_{12}^{-1} \right| ij \right), \end{align*}\]
where:
\[\left( \tilde{A} \left| w_{12}^{s_{i j}} r_{12}^{-1} \right| ij \right) = \sum_A M_{AB}^{-1} \left( B \left| w_{12}^{s_{i j}} r_{12}^{-1} \right| ij \right) \]
The corresponding Fock matrix contributions are given by:
\[\begin{align*}\frac{\partial E_{c, 2el,+}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= \sum_{C} \sum_{j} \left( \alpha \beta \left| w_{12}^{\delta_{\sigma_{j} \sigma}} r_{12}^{-1} \right| C \right) \left( \tilde{C} \left| w_{12}^{\delta_{\sigma_{j} \sigma}} r_{12}^{-1} \right| jj \right) \newline \frac{\partial E_{c, 2el, -}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= - \sum_{j} \delta_{\sigma_{j} \sigma} \sum_{C} \left( \alpha j \left| w_{12}^{1} r_{12}^{-1} \right| C \right) \left( \tilde{C} \left| w_{12}^{1} r_{12}^{-1} \right| \beta j \right) \end{align*}\]

The indirect term contains only same-spin contributions. Therefore, for \(w^{s=1}(r) = 0\) (opposite spin only), there is no indirect contribution.

Parameters

  • WV – Integrals \((ab|r^{-1}w(r)|A)\) and \((A|r^{-1}w(r)|B)\)

  • active_Co – Frozen core occupation weighted orbitals

  • indirect_term – Whether to compute the indirect term

  • calculate_fock – Whether to calculate the Fock matrix contribution

  • scale_opp_spin – Scale factor for \( w^0 (r) \)

  • scale_same_spin – Scale factor for \( w^1 (r) \)

Returns

Two electron UW12 Fock matrix and energy contribution