13.4 \(\sigma\)-Functionals

\(\sigma\)-functionals offer a significant improvement over pure direct (i.e., not self-consistent) RPA at similar computational cost.

Formally, \(\sigma\)-functionals are rooted in perturbation theory along the adiabatic connection which yields an expression for the total energy which closely resembles the RPA energy expression (eq. 13.1, 13.7).[307, 308, 309] \[\begin{equation} E^{\sigma f} = E^{\text{HF}} + E^{\text{C}, \sigma f} \end{equation}\](13.24) with \[\begin{equation} E^{\text{C}, \sigma f} = \int_{-\infty}^{\infty} \frac{d\omega}{2 \pi} F^{\text{C}}(\omega), \end{equation}\](13.25) and \[\begin{equation} F^{\text{C}}(\omega)= \frac{1}{2} \text{tr} \left ( \ln \left ( \mathbf{I}_{\text{aux}} + \mathbf{Q}(\omega) \right) - \mathbf{Q}(\omega) - H(\boldsymbol{\sigma}(\omega)) \right ). \end{equation}\](13.26) where \(\mathbf{Q}(\omega)\) is the negative response matrix at the imaginary frequency \(\mathrm{i}\omega\) represented in an auxiliary basis, and \(\boldsymbol{\sigma}(\omega)\) is a diagonal matrix containing the eigenvalues of \(\mathbf{Q}(\omega)\). The function \(H\) is represented by cubic splines and has been optimized for different DFT functionals and test sets.[308, 309] Currently, only so-called unscaled \(\sigma\)-functionals are supported.

The present implementation features both ground state energies and analytical gradients facilitating geometry optimizations and calculation of numerical vibrational frequencies.[310]

Prerequisites

A \(\sigma\)-functional calculation can be envoked by adding

sigmafunc parametrization

to the rirpa input section where parametrization can be one of pbe_s1, pbe_s2, pbe_w1, pbe0_s1, pbe0_s2, pbe0_w1, b3-lyp_w1, and tpss_w1, see Ref. [308, 309] for more details about the different parametrizations. If parametrization is omitted, the default is pbe_s1. It is recommended to use orbitals and eigenvalues generated with the DFT functional indicated in the parametrization. For example, a \(\sigma\)-functional calculation with parametrization pbe_s1 should be carried out using PBE orbitals and eigenvalues from a preceeding ridft run.

Since the function \(H\) introduces additional structure to \(F^{\text{C}}(\omega)\), it is recommended to switch on Gauss–Legendre quadrature for the imaginary frequeny integration where the default number of 60 frequency points typically yields very well converged results; experience shows that usually 35 – 50 points suffice in case of Gauss–Legendre quadrature. For small-gap and open-shell systems, somewhat more points may be required. See the keyword section 25.2.20 for more details.

Otherwise, the general notes given for the RPA energy / gradient apply.