12.1 Ground State Energy Theory

The RPA energy

 RPA     HF    C RPA
E    = E   + E
(12.1)

consists of the Hartree-Fock exact exchange energy EHF and a correlation energy piece EC RPA. rirpa computes Eq. (12.1) non-selfconsistently from a given set of converged input orbitals. The correlation energy

  C RPA   1∑  ( RPA    TDARPA )
E      =  2    Ωn   - Ωn
            n
(12.2)

is expressed in terms of RPA excitation energies at full coupling ΩnRPA and within the Tamm-Dancoff approximation ΩnTDARPA. 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. [154]. The excitation energies are obtained from time-dependent DFT response theory and are eigenvalues of the symplectic eigenvalue problem  [155,156]

(Λ - Ω0nΔ )|X0n,Y0n⟩ = 0.
(12.3)

The super-vectors X0n and Y 0n 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

     (     )
      A   B
Λ =   B   A
(12.4)

contains the so-called orbital rotation Hessians,

(A + B)iajb = (ϵa - ϵi)δijδab + 2(ia|jb), (12.5)
(A - B)iajb = (ϵa - ϵi)δijδab. (12.6)
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. (12.5) and (12.6).

In RIRPA the two-electron integrals in Eqs (12.5) 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  [151]

          ∫ ∞
EC RIRPA =     dωF C(ω),
           - ∞ 2π
(12.7)

where the integrand contains Naux × Naux quantities only,

FC (ω ) = 1tr(ln (Iaux + Q (ω )) - Q(ω)).
        2
(12.8)

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

Prerequisites

Calculations with rirpa require

Effective core potentials (ECPs) are not presently compatible with the HF energy at the KS reference as computed in rirpa. The nohxx option must therefore be included for systems where ECPs were used to obtain the reference KS orbitals in order to skip the HF energy calculation and compute solely the correlation energy.

To perform a two-component relativistic RIRPA calculation [154] 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 6.4) using the keywords $soghf and $kramers. The implementation is currently only available in combination with the nohxx option.