libuw12 API

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Functionality relating to the UW12 correlation method

The UW12 correlation energy is defined as:

\[E_c^{UW12} = \frac{1}{2} \sum_{ij ab} \left\langle i j \left\vert w^{s_{ij}}_{12} \right\vert \overline{ab} \right\rangle \left\langle a b \left\vert r_{12}^{-1}\right\vert i j \right\rangle \]
Where \(\vert \overline{ab} \rangle = \vert ab \rangle - \vert ba \rangle\), \(ij\) are occupied orbitals, \(ab\) are unoccupied (virtual) orbitals, \( r_{12} = \vert \vec{r}_1 - \vec{r}_2 \vert \), with two-electron geminal operator \( w^s_{12} \) for total spin \(s_{ij} = \delta_{\sigma_i \sigma_j}\).

By removing the sum over the virtual orbitals, the energy can be rewritten as

\[E_c^{UW12} = E_{c, 2el}^{UW12} + E_{c, 3el}^{UW12} + E_{c, 4el}^{UW12}, \]
for two, three and four electron terms defined by:
\[\begin{align*}E_{c, 2el}^{UW12} &= \frac{1}{2} \sum_{ij} \left\langle \overline{ij} \left\vert w^{s_{ij}}_{12} r_{12}^{-1}\right\vert i j \right\rangle \newline E_{c, 3el}^{UW12} &= - \sum_{ijk} \left\langle \overline{ij} k \left\vert w^{s_{ij}}_{12} r_{23}^{-1}\right\vert k j i \right\rangle \newline E_{c, 4 el}^{UW12} &= \frac{1}{2} \sum_{ij kl} \left\langle i j \left\vert w^{s_{ij}}_{12}\right\vert \overline{kl} \right\rangle \left\langle kl \left\vert r_{12}^{-1} \right\vert ij \right\rangle \end{align*}\]

In the frozen core approximation, indices i,j run only over the correlated (active) orbitals; whilst k,l indices summed over all occupied orbitals including the frozen core orbitals.

Fock contributions

Since UW12 is self-consistent, the Fock matrix elements \(F_{\alpha \beta}^{\sigma}\) may also be calculated, where

\[F_{\alpha \beta}^{\sigma} = \frac{1}{2} \left[ \frac{\partial E}{\partial D_{\alpha \beta}^{\sigma}} + \frac{\partial E}{\partial D_{\beta \alpha}^{\sigma}} \right], \]
for real symmetric density matrix \(D_{\alpha \beta}^{\sigma}\). The contributions for which are given by:
\[\begin{align*}\frac{\partial E_{c, 2 el}^{UW 12}}{\partial D_{\alpha \beta}^{\sigma}} &= \sum_{j} \left\langle \overline{\alpha j} \left\vert w^{\delta_{\sigma_j \sigma}}_{12} r_{12}^{-1}\right\vert \beta j \right\rangle \newline \frac{\partial E_{c, 3 el}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= - \sum_{jk} \left\langle \overline{\alpha j} k \left\vert w_{12} r_{23}^{-1} \right\vert k j \beta \right\rangle - \sum_{i k} \left\langle \overline{i \alpha} k \left\vert w_{12} r_{23}^{-1}\right\vert k \beta i \right\rangle - \sum_{i j} \left\langle \overline{i j} \alpha \left\vert w_{12} r_{23}^{-1}\right\vert \beta j i \right\rangle \newline \frac{\partial E_{c, 4el}^{UW12}}{\partial D_{\alpha \beta}^{\sigma}} &= \sum_{jkl} \left\langle \alpha j \left\vert w^{\delta_{\sigma_j \sigma}}_{12} \right\vert \overline{k l} \right\rangle \left\langle k l \left\vert r_{12}^{-1}\right\vert \beta j \right\rangle + \sum_{ijk} \left\langle i j \left\vert w^{\delta_{\sigma_j \sigma}}_{12} \right\vert \overline{\alpha k} \right\rangle \left\langle k \beta \left\vert r_{12}^{-1} \right\vert i j \right\rangle \end{align*}\]

In the full core version, the two terms in the four electron term are the same. This is a result of the symmetry between orbitals i,j and k,l. Similarly for the direct portion of the first and third terms in the three electron Fock contribution. However, for frozen core the terms are no longer the same and must be calculated separately.

Geminal Function

The geminal function \(w_{12}\) is given as a function of the inter-electron distance \(r_{12}\). The function is implemented as a sum of Gaussian terms

\[w^s (r) = \sum_i c_i^s \exp \left[ - \gamma_i r^2 \right] \]
specified by coefficients \(c_i^s\) and exponents \(\gamma_i\). The structure of the theory allows two sets of coefficients for opposite ( \(s = 0\)) and same ( \(s = 1\)) spin contributions. Though the standard implementation has \(w^{s=1} (r) = \kappa w^{s=0} (r)\) for same spin scale factor \(\kappa_1\).

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