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The RPA energy
*E*^{RPA} = *E*^{HF} + *E*^{C RPA} |
(11.1) |

consists of the Hartree-Fock exact exchange energy
*E*^{HF}
and a
correlation energy piece
*E*^{C RPA}
. `rirpa` computes Eq. (11.1)
non-selfconsistently from a given set of converged input orbitals. The correlation
energy
*E*^{C RPA} = *Ω*_{n}^{RPA} - *Ω*_{n}^{TDARPA} |
(11.2) |

is expressed in terms of RPA excitation energies at full coupling
*Ω*_{n}^{RPA}
and
within the Tamm-Dancoff approximation
*Ω*_{n}^{TDARPA}
. The excitation
energies are obtained from time-dependent DFT response theory and are eigenvalues
of the symplectic eigenvalue problem [135,136]
(*Λ* - *Ω*_{0n}*Δ*)| *X*_{0n}, *Y*_{0n}〉 = 0. |
(11.3) |

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
*Λ* = |
(11.4) |

contains the so-called orbital rotation Hessians,
(*A* + *B*)_{iajb} |
= (*ε*_{a} - *ε*_{i})*δ*_{ij}*δ*_{ab} + 2(*ia*| *jb*), |
(11.5) |

(*A* - *B*)_{iajb} |
= (*ε*_{a} - *ε*_{i})*δ*_{ij}*δ*_{ab}. |
(11.6) |

*ε*_{i}
and
*ε*_{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. (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]

*E*^{C RIRPA} = *F*^{C}(*ω*), |
(11.7) |

where the integrand contains
*N*_{aux}×*N*_{aux}
quantities only,
*F*^{C}(*ω*) = trln**I**_{aux} + **Q**(*ω*) - **Q**(*ω*). |
(11.8) |

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

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