33 EXPLICITLY CORRELATED METHODS

Explicitly correlated MP2-R12 and MP2-F12 calculations can be performed using density fitting for the necessary integrals. Currently the available Ansätze are restricted to the 2A type. Methods are available in local (DF-LMP2-R12,DF-LMP2-F12) and canonical (DF-MP2-R12, DF-MP2-F12) versions, detailed below. Symmetry is not implemented for any of these methods, and therefore the NOSYM option must be given in the geometry block.

For DF-MP2-F12 the correlation factor is a frozen expansion $f_{12}$ of Gaussian type geminals. By default the geminal is built from six Gaussian functions, and the exponents and coefficients are optimized to obtain the best least squares fit to $f_{12}=\exp({-\beta\,r_{12}})$ using a suitable weight function. If correlation consistent basis sets are used, a suitable density fitting (DF) basis is automatically chosen. In the case of R12 methods, the default for the RI basis is the AO basis set, while for F12 methods Hartree-Fock JK-fitting bases are used by default (e.g., VTZ/JKFIT is used for orbital basis VTZ).

In general, only the F12 methods are recommended, since these lead to much more accurate results and converge better with respect to the AO, DF, and RI basis sets than the R12 methods.

Options for canonical and local versions:

DF_BASIS=basis

Select the basis for density fitting (see section 11 for details). basis can either refer to a set name defined in the basis block, or to a default MP2 fitting basis (e.g., DF_BASIS=VTZ generates the VTZ/MP2FIT basis). See section 11 for more details.

RI_BASIS=basis

Select the basis for the resolution of the identity (RI). In case of R12 methods, this should be chosen to be a large uncontracted AO basis (at least AVQZ). For F12 methods we have found that the Hartree-Fock JKFIT basis sets perform well for the RI, despite having been optimized for other purposes.

ANSATZ=ansatz

Select the explicitly correlated ansatz ansatz for the canonical methods. The ansatz takes the form 2A, 2*A, or 2A`. The optional * invokes additional approximations (based on the extended Brillouin approximation) that result in increased efficiency. The optional backward quote ` (standing in for `prime') results in the inclusion of some small terms required for full orbital invariance. Since the terms are cheap to compute, the flexibility not to include them is provided for historical reasons. Whatever ansatz is chosen, all levels of theory are computed that do not entail the evaluation of additional integrals. Currently only ansatz 2*A is implemented in the local version, with the additional approximation that only ''diagonal'' $(ijij)$ pairs are included in the correlation factor.

GEM_BASIS

Basis set name for geminal expansion; atom labels are ignored. This can either be OPTFULL (full nonlinear fit of the geminal expansion), EVEN (even tempered fit), or refer to a set name defined in a previous BASIS block. Default is OPTFULL.

GEM_TYPE

Frozen geminal type: LINEAR or SLATER, default is SLATER.

GEM_NUMBER

Number of geminal functions (default 6).

GEM_CENTRE

Centre of even tempered geminal exponents, if GEM_BASIS=EVEN (default 1.0).

GEM_RATIO

Ratio of even tempered geminal exponents, if GEM_BASIS=EVEN (default 3.0).

GEM_BETA

Exponent for Slater-type frozen geminal, or parameter for weight function in other frozen geminal models (default 1.4).

GEM_OMEGA

Exponent for weighting function (default -1, which means a value derived from GEM_BETA.

GEM_MOM

Exponent for r in omega fitting (default 0).

GEM_M

Exponent for r in weighting function (default 0).

GEM_MAXIT

Max iteration in geminal optimization (default 200).

GEM_PRINT

Print parameter for geminal optimization (default 0).

GEM_DEBUG

Debug option for geminal optimization (default 0).

GEM_ACC

Convergence threshold for geminal line search (default 0.001).

GEM_FAC

Scaling factor for exponents in geminal optimization (default 1.0).

GEM_METHOD

Geminal optimization method (augmented Hessian (AH) or Newton-Raphson (NR), default AH).

GEM_TRUST

Trust ratio in AH geminal optimization (default 0.4).

GEM_SHIFT

Hessian shift in AH geminal optimization (default 0).

GEM_NUMERICAL

Flags numerical integration in geminal optimization (default 0).

GEM_PLOT

Geminal plot file (default blank).

Options only available for the canonical version:

PRINT=ipri

Select output level for canonical methods:

ipri=0

Standard output

ipri=1

Standard output plus pair energies plus basis information

ipri=2

Debugging output

THRBINV

Threshold below which non-physical eigenvalues are projected from approximate B matrices

THRINT

Threshold for integral screening

Local variants of the DF-MP2-F12 methods are available invoked by the commands DF-LMP2-F12 or DF-LMP2-R12.

Special options for these local variants are:

PAIRS

Specifies which pairs to be treated by R12 or F12
(STRONG|CLOSE|WEAK|ALL; pairs up to the given level are included). The default is STRONG.

DEBUG

Parameter for debug print

LOCFIT_F12

If set to one, use local fitting. Default is no local fitting (LOCFIT_F12=0)

LOCFIT_R12

Alias for LOCFIT_F12. Local fitting is not recommended in R12 calculations.

FITDOM

Determine how the base fitting domains are determined:
0: Fitdomains based on united operator domains;
1: Fitdomains based in orbital domains;
2: Fitdomains based on united pair domains using strong pairs;
3: Fitdomains based on united pair domains using strong, close and weak pairs (default);

RDOMAUX

Distance criterion for density fitting domain extensions in case of local fitting. The default depends on FITDOM.

IDOMAUX

Connectivity criterion for density fitting domain extensions in case of local fitting.

RAODOM

Distance criterion for RI domain extensions. Zero means full RI basis (default). If this parameter is chosen to be nonzero, it must be rather large to achieve sufficient accuracy. Values of at least 10 bohr have been found to work reasonably well (only for F12!).

IAODOM

Connectivity criterion for RI domain extensions. Zero means full RI basis (default). Values greater or equal to 6 should lead to sufficiently accurate results.

THRAO

Screening threshold for integrals in the AO or RI basis.

THRMO

Screening threshold for half transformed integrals.

THRPROD

Product screening threshold in the first half transformation.

Further options for density fitting are described in section 11. The use of local DF and RI domains is still experimental and is not recommended yet for general use.

Published work arising from these methods should cite the following:
F. R. Manby, J. Chem. Phys. 119 4607 (2003) (for canonical DF-MP2-R12)
A. J. May and F R Manby, J. Chem. Phys. 121 4479 (2004) (for canonical DF-MP2-F12)
H.-J. Werner and F R Manby, J. Chem. Phys. 124 054114 (2006) (for local DF-LMP2-R12)
F. R. Manby, H.-J. Werner, T. B. Adler and A. J. May, J. Chem. Phys. 124 094103 (2006) (for local DF-LMP2-F12).