13.1 Ground State Energy Theory

The RPA energy \[\begin{equation} E^{\text{RPA}} = E^{\text{HF}} + E^{\text{C RPA}} \end{equation}\](13.1) consists of the Hartree-Fock exact exchange energy $E^{\text{HF}}$ and a correlation energy piece $E^{\text{C RPA}}$. rirpa computes Eq. 13.1 non-selfconsistently from a given set of converged input orbitals. The correlation energy \[\begin{equation} E^{\text{C RPA}} = \frac{1}{2} \sum_n \left( \Omega_{n}^{\text{RPA}} -\Omega_{n}^{\text{TDARPA}} \right) \end{equation}\](13.2) is expressed in terms of RPA excitation energies at full coupling $\Omega_n^{\text{RPA}}$ and within the Tamm-Dancoff approximation $\Omega_n^{\text{TDARPA}}$. The further discussion is restricted to the one-component (nonrelativistic) treatment, for the sake of convenience. For the derivation of the two-component RPA theory see ref. [303]. The excitation energies are obtained from time-dependent DFT response theory and are eigenvalues of the symplectic eigenvalue problem [311, 312] \[\begin{equation} ( \Lambda - \Omega_{0n} \Delta) \ensuremath{|X_{0n},Y_{0n}\rangle} = 0. \end{equation}\](13.3) The super-vectors $X_{0n}$ and $Y_{0n}$ are defined on the product space $L_{\text{occ}} \times L_{\text{virt}}$ and $L_{\text{occ}} \times L_{\text{virt}}$, respectively, where $L_{\text{occ}}$ and $L_{\text{virt}}$ denote the one-particle Hilbert spaces spanned by occupied and virtual static KS molecular orbitals (MOs). The super-operator \[\begin{equation} \Lambda = \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B} & \mathbf{A} \end{pmatrix} \end{equation}\](13.4) contains the so-called orbital rotation Hessians, \[\begin{align} (A+B)_{iajb} & = (\epsilon_a - \epsilon_i) \delta_{ij} \delta_{ab} + 2 (ia|jb) , \\ (A-B)_{iajb} & = (\epsilon_a - \epsilon_i) \delta_{ij} \delta_{ab}. \end{align}\](13.5–13.6) $\epsilon_i$ and $\epsilon_a$ denote the energy eigenvalues of canonical occupied and virtual KS MOs. rirpa computes so-called direct RPA energies only, i.e. no exchange terms are included in Eqs. 13.5 and 13.6.

In RIRPA the two-electron integrals in Eqs 13.5 are approximated by the resolution-of-the-identity approximation. In conjunction with a frequency integration this leads to an efficient scheme for the calculation of RPA correlation energies [298] \[\begin{equation} E^{\text{C RIRPA}} = \int_{-\infty}^{\infty} \frac{d\omega}{2 \pi} F^{\text{C}}(\omega), \end{equation}\](13.7) where the integrand contains $N_{\text{aux}} \times N_{\text{aux}}$ quantities only, \[\begin{equation} F^{\text{C}}(\omega)= \frac{1}{2} \text{tr} \left ( \ln \left ( \mathbf{I}_{\text{aux}} + \mathbf{Q}(\omega) \right) - \mathbf{Q}(\omega) \right ). \end{equation}\](13.8) $N_{\text{aux}}$ is the number of auxiliary basis functions. The integral is approximated using Clenshaw-Curtiss quadrature.

Prerequisites

Calculations with rirpa require

a converged SCF calculation
rirpa-options may be included by adding them in the lines below the keyword $rirpa in the control file. Possible options are:
- npoints $\langle \text{integer} \rangle$ - Number of frequency integration points (default is 60).
- nohxx - HF energy is skipped, (HXX = Hartree + eXact (Fock) eXchange).
- rpaprof - Generates profiling output.
the maximum core memory the program is allowed to allocate should be defined in the data group $maxcor (in MB); the recommended value is ca. 3/4 of the available (physical) core memory at most.
orbitals to be excluded from the correlation treatment have to be specified in data group $freeze
an auxiliary basis defined in the data group $cbas
an auxiliary basis defined in the data group $jbas for the computation of the Coulomb integrals for the Hartree-Fock energy. This is optional if the nohxx option is set.
(optional) an auxiliary basis defined in the data group $jkbas for the computation of the exchange integrals for the Hartree-Fock energy. $rik should be added to the control file for RI-JK to be effective.

To perform a two-component relativistic RIRPA calculation [303] on (Kramers-restricted) closed-shell systems taking into account spin-orbit coupling, the two-component version of ridft has to be run before (see Chapter 6.5) using the keywords $soghf and $kramers. The implementation is currently only available in combination with the nohxx option.