The RPA energy

| (12.1) |

consists of the Hartree-Fock exact exchange energy E^{HF} and a correlation energy piece E^{C RPA}.
rirpa computes Eq. (12.1) non-selfconsistently from a given set of converged input orbitals. The
correlation energy

| (12.2) |

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

| (12.3) |

The super-vectors X_{0n} and Y _{0n} are defined on the product space L_{occ} ×L_{virt} and L_{occ} ×L_{virt},
respectively, where L_{occ} and L_{virt} denote the one-particle Hilbert spaces spanned by occupied
and virtual static KS molecular orbitals (MOs). The super-operator

| (12.4) |

contains the so-called orbital rotation Hessians,

(A + B)_{iajb} | = (ϵ_{a} - ϵ_{i})δ_{ij}δ_{ab} + 2(ia|jb), | (12.5) |

(A - B)_{iajb} | = (ϵ_{a} - ϵ_{i})δ_{ij}δ_{ab}. | (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]

| (12.7) |

where the integrand contains N_{aux} × N_{aux} quantities only,

| (12.8) |

N_{aux} is the number of auxiliary basis functions. The integral is approximated using
Clenshaw-Curtiss quadrature.

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 ⟨integer⟩ - 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
- (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.

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.