Laplace-transformed SOS-RI-MP2 with scaling costs

The `ricc2` module contains now also a first implementation of SOS-MP2 which exploits
the RI approximation and a Laplace transformation of the orbital energy denominators

to achieve an implementation with scaling costs, opposed to the conventional scaling implementation. In particular for large molecules the Laplace-transformed implementation can reduce a lot the computational costs of SOS-MP2 calculations without loss in accuracy.

The Laplace-transformed implementation for SOS-MP2 calculations is activated with the input

$laplace conv=5where the parameter

`conv`

is a convergence threshold for the numerical integration
in Eq. (9.29). A value of `conv=5`

means that the
numerical integration will be converged to a root mean squared error of
a.u.
Whether the conventional or the Laplace-transformed implementation will be more efficient depends firstly on the system size (the number of occupied orbitals) and secondly on the required accuracy (the number of grid points for the numerical integration in Eq. (9.29)) and can be understood and estimated from the following considerations:

- The computational costs for the most expensive step in (canonical)
RI-MP2 energy calculations for large molecules requires
floating point multiplications, where
and are, respectively, the number occupied and virtual orbitals and
is the number of auxiliary functions for the RI approximation.
For the LT-SOS-RI-MP2 implementation the most expensive step involves
floating point multiplications, where is the number
of grid points for the numerical integration. Thus, the ratio of the computational
costs is approximately

- The threshold
`conv`

for the numerical integration is by default set to the value of`conv`

specified for the ground state energy in the data group`$ricc2`(see Sec. 14.2.13), which is initialized using the threshold`$denconv`, which by default is set conservatively to the tight value of .- For single point energy calculations
`conv`

in`$laplace`can savely be set to 4, which gives SOS-MP2 energies converged within a.u. with computational costs reduced by one third or more compared to calculations with the default settings for these thresholds. - For geometry optimizations with SOS-MP2 we recommend to set
`conv`

in`$laplace`to 5.

- For single point energy calculations
- The spread of the orbital energy denominators depends on the basis sets and the
orbitals included in the correlation treatment.
Most segmented contracted basis sets of triple- or higher accuracy
(as e.g. the TZVPP and QZVPP basis sets) lead to rather high lying
``anti core'' orbitals with orbital energies of
10 a.u. and more.
- For the calculation of SOS-MP2 valence correlation energies
it is recommended to exclude such orbitals from the correlation treatment
(see input for
`$freeze`in Sec. 14). - Alternatively one can use general contracted basis sets, as e.g. the correlation consistent cc-pVXZ basis sets. But note that general contracted basis sets increase the computational costs for the integral evaluation in the Hartree-Fock and, for gradient calculations, also the CPHF equations and related 4-index integral derivatives.
- Also for the calculation of all-electron correlation energies with core-valence basis sets which include uncontracted steep functions it is recommended to check if extremely high-lying anti core orbitals can be excluded.

- For the calculation of SOS-MP2 valence correlation energies
it is recommended to exclude such orbitals from the correlation treatment
(see input for

`nozpreopt`

option in the