E_{MP2} = t^{ab}_{ij}〈ij|| ab〉, |
(8.1) |

with the t-amplitudes

t^{ab}_{ij} = , |
(8.2) |

MP2 gradients (necessary for optimisation of structure parameters at
the MP2 level) are calculated as analytical derivatives of the MP2
energy with respect to nuclear coordinates; calulation of these
derivatives also yields the first order perturbed wave function,
expressed as "MP2 density matrix", in analogy to the HF density
matrix. MP2 corrections of properties like electric moments or atomic
populations are obtained in the same way as for the HF level, the HF
density matrix is just replaced by the MP2 density
matrix.

The "resolution of the identity (RI) approximation" means expansion of
products of virtual and occupied orbitals by expansions of so-called
"auxiliary functions". Calculation and transformation of
four-center-two-electron integrals (see above) is replaced by that of
three-center integrals, which leads to computational savings of `rimp2`
(compared to `mpgrad`) by a factor of ca. 5 (small basis sets like
SVP) to ca. 10 (large basis sets like TZVPP) or more (for cc-pVQZ
basis sets). The errors (differences to `mpgrad`) of `rimp2` in
connection with optimised auxliliary basis sets are small and well
documented [9,86]. The use of the
`mpgrad` modul is recommended rather for reference calculations or if
suitable auxiliary basis sets are not available.