9.6 Laplace-transformed SOS-RI-MP2 with (4) scaling costs
The ricc2 module contains an implementation of SOS-MP2 which exploits the RI approximation
and a Laplace transformation of the orbital energy denominators
to achieve an implementation with (4) scaling costs, opposed to the conventional
(5) 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
The Laplace-transformed implementation for SOS-MP2 calculations is activated with the
where the parameter conv is a convergence threshold for the numerical integration in Eq. (9.8). A
value of conv=5 means that the numerical integration will be converged to a root mean squared
error of ≈ 10-5 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. (9.8)) 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 O2V 2N
x floating point multiplications,
where O and V are, respectively, the number occupied and virtual orbitals and
Nx is the number of auxiliary functions for the RI approximation. For the
LT-SOS-RI-MP2 implementation the most expensive step involves nLOV Nx2 floating
point multiplications, where nL is the number of grid points for the numerical
integration. Thus, the ratio of the computational costs is approximately
where for the last step Nx ≈ 3V has been assumed. Thus, the Laplace-transformed
implementation will be faster than the conventional implementation if O > 6nL.
The number of grid points nL depends on the requested accuracy and the spread of the orbital
energy denominators in Eq. (9.8). The efficiency of Laplace-transformed SOS-RI-MP2 calculations
can therefore (in difference to conventional RI-MP2 calculations) be enhanced significantly by a
carefull choice of the thresholds, the basis set, and the orbitals included in the correlation
- 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. 21.2.17), which is
initialized using the threshold $denconv, which by default is set conservatively to the tight
value of 10-7.
- For single point energy calculations conv in $laplace can savely be set
to 4, which gives SOS-MP2 energies converged within ≈ 10-4 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.
- 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-ζ 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
- 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. 21).
- 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
- 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.
Note that for large molecules it is recommended to disable for geometry optimizations (or for
gradient or property calculations in general) the preoptimization for the Z vector equations with
the nozpreopt option in the $response data group (see Sec. 21.2.17).
- The Laplace-transformed SOS-MP2 implementation in ricc2 is presently only
parallelized with MPI. The OpenMP parallelization is not (yet) recognized by the
LT-SOS-RI-MP2 related program parts.
- It is presently not compatible with the calculation of the D1 and D2 diagnostics.
The respective options will be ignored by program if the Laplace-transformed
implementation is used.