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Ground State Energy Theory

The RPA energy

ERPA = EHF + EC RPA (11.1)

consists of the Hartree-Fock exact exchange energy EHF and a correlation energy piece EC RPA . rirpa computes Eq. (11.1) non-selfconsistently from a given set of converged input orbitals. The correlation energy

EC RPA = $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \sum_{n}^{}$$\displaystyle \left(\vphantom{ \Omega_{n}^{\text{RPA}} -\Omega_{n}^{\text{TDARPA}} }\right.$ΩnRPA - ΩnTDARPA$\displaystyle \left.\vphantom{ \Omega_{n}^{\text{RPA}} -\Omega_{n}^{\text{TDARPA}} }\right)$ (11.2)

is expressed in terms of RPA excitation energies at full coupling ΩnRPA and within the Tamm-Dancoff approximation ΩnTDARPA . The excitation energies are obtained from time-dependent DFT response theory and are eigenvalues of the symplectic eigenvalue problem [135,136]

(Λ - Ω0nΔ)| X0n, Y0n〉 = 0. (11.3)

The super-vectors X0n and Y0n are defined on the product space Locc×Lvirt and Locc×Lvirt , respectively, where Locc and Lvirt denote the one-particle Hilbert spaces spanned by occupied and virtual static KS molecular orbitals (MOs). The super-operator

Λ = $\displaystyle \begin{pmatrix}\mathbf{A} & \mathbf{B} \  \mathbf{B} & \mathbf{A} \end{pmatrix}$ (11.4)

contains the so-called orbital rotation Hessians,

(A + B)iajb = (εa - εi)δijδab + 2(ia| jb), (11.5)
(A - B)iajb = (εa - εi)δijδab. (11.6)

εi and εa 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. (11.5) and (11.6).

In RIRPA the two-electron integrals in Eqs (11.5) 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 [133]

EC RIRPA = $\displaystyle \int_{{-\infty}}^{{\infty}}$$\displaystyle {\frac{{d\omega}}{{2 \pi}}}$FC(ω), (11.7)

where the integrand contains Naux×Naux quantities only,

FC(ω) = $\displaystyle {\frac{{1}}{{2}}}$tr$\displaystyle \left(\vphantom{ \ln \left ( \mathbf{I}_{\text{aux}} + \mathbf{Q}(\omega) \right) - \mathbf{Q}(\omega) }\right.$ln$\displaystyle \left(\vphantom{ \mathbf{I}_{\text{aux}} + \mathbf{Q}(\omega) }\right.$Iaux + Q(ω)$\displaystyle \left.\vphantom{ \mathbf{I}_{\text{aux}} + \mathbf{Q}(\omega) }\right)$ - Q(ω)$\displaystyle \left.\vphantom{ \ln \left ( \mathbf{I}_{\text{aux}} + \mathbf{Q}(\omega) \right) - \mathbf{Q}(\omega) }\right)$. (11.8)

Naux is the number of auxiliary basis functions. The integral is approximated using Clenshaw-Curtiss quadrature.


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