RI-MP2-R12 Calculations

To obtain the R12 correction to the MP2 energy, the keyword $rir12 must be added to the control file. A typical run will include the keywords:

$ricc2
  mp2
$rir12

The MP2-R12 ground-state energy is

$$E_{\hbox{\tiny MP2-R12}} = E_{\rm MP2} + E_{\rm R12} ,$$ (9.24)

where $E_{\rm MP2}$ mathend000# is the conventional MP2 energy and $E_{\rm R12}$ mathend000# the correction from explicitly-correlated theory. The R12 correction is obtained by minimizing the functional

$$F_{\rm R12} = \sum_{i<j} \left\{ {\mathbf c}_{ij}^T {\mathbf B}_{ij} {\mathbf c}_{ij} + 2{\mathbf c}_{ij}^T {\mathbf v}_{ij}\right\}$$ (9.25)

with respect to the amplitudes collected in the vector ${\mathbf c}_{ij}$ mathend000#. The vectors ${\mathbf v}_{ij}$ mathend000# and the matrices ${\mathbf B}_{ij}$ mathend000# are defined as

$${\mathbf v}_{ij}(kl) = \langle kl\vert r_{12}\hat Q_{12} r_{12}^{-1}\vert ij\rangle ,$$ (9.26)

$${\mathbf B}_{ij}(kl,mn) = \langle kl\vert r_{12}\hat Q_{12} (\hat f_1 + \hat f_2 - \varepsilon_i - \varepsilon_j ) \hat Q_{12} r_{12}\vert mn\rangle ,$$ (9.27)

in the spin-orbital formalism ($m,n$ mathend000# denote spin orbitals and $\vert mn\rangle$ mathend000# is a two-electron determinant). $\hat f_\mu$ mathend000# is the Fock operator for electron $\mu$ mathend000# and $\varepsilon_k$ mathend000# is a canonical Hartree-Fock orbital energy. $\hat Q_{12} = (1-\hat O_1)(1-\hat O_2)$ mathend000# is the strong-orthogonality projection operator, with $\hat O_\mu = \sum_{k} \vert\varphi_k(\mu)\rangle\langle\varphi_k(\mu)\vert$ mathend000# the projection operator onto the space spanned by the occupied spin orbitals $\varphi_k$ mathend000#.

The present implementation of the MP2-R12 method computes the vectors ${\mathbf v}_{ij}$ mathend000# and matrices ${\mathbf B}_{ij}$ mathend000# in the following manner:

• It uses either approximation A (default) or approximation A$^\prime$ mathend000# to compute the vectors ${\mathbf v}_{ij}$ mathend000# and the matrices ${\mathbf B}_{ij}$ mathend000#. These approximations are described in detail in Ref. [92]. It is recommended to use approximation A. The keyword r12model must be used for calculations in the framework of approximation A$^\prime$ mathend000#.
• The calculation is based on the orbital-invariant "ijkl" Ansatz of Ref. [93]. The keyword noinv must be used if only the original orbital-dependent diagonal "ijij" Ansatz of Ref. [94] shall be applied (not recommended).
• It uses either canonical or localized Hartree-Fock orbitals. Both the Boys [95] and Pipek-Mezey [96] methods can be used to localize the orbitals (keyword: local ). The diagonal "ijij" Ansatz can be used in conjunction with localized orbitals, but be aware of the dependence of the results on the orbitals. For example, spin-adapted singlet and triplet pairs "ij" are taken for RHF cases while $\alpha\alpha$ mathend000#, $\alpha\beta$ mathend000#, $\beta\alpha$ mathend000#, and $\beta\beta$ mathend000# pairs "ij" are taken for UHF cases, yielding different results even for identical RHF and UHF determinants.
• It uses the robust fitting techniques of Ref. [97].
• The (approximate) completeness relations of R12 theory that avoid four- and three-electron integrals are inserted in terms of the same orbital basis that is used to expand the wave function. This implies that the basis set must be chosen with special care.