Meta-GGA and hybrid energy functionals

In the ADF2004.01 version, several GGA [26-28,35-39], meta-GGA [40-46], hybrid GGA (for example B3LYP), and hybrid meta-GGA energy xc functionals have been implemented that have been shown to give good results for energies. The present implementation enables only energies to be calculated with these functionals. The xc potentials have not been implemented, so that for example geometry optimizations or property calculations cannot be performed with these functionals.

The implementation in ADF of the calculation of exact exchange (Hartree Fock exchange), which is needed for the hybrid functionals, is based on work by Watson et al., Ref. [138]. The difference with their method is the way in which ADF the orbital densities are fitted.

The calculation of a large, prespecified list of LDA, GGA, and meta-GGA energy functionals is invoked by specifying

METAGGA

as a separate keyword. The hybrid GGA and hybrid meta-GGA energy functionals are calculated if in addition to the METAGGA key, the key

HFEXCHANGE

is included.

The keys METAGGA and HFEXCHANGE can be used in combination with any xc potential. Note that iat the moment hybrid functionals can not be used in combination with frozen cores or spin-orbit coupling. Some METAGGA functionals will also give wrong results if used in combination with frozen cores. One should include the HFEXCHANGE keyword also in the create runs of the atoms. In ADF the hybrid energies only make sense if the calculation is performed with completely filled orbitals (ROHF is not implemented in ADF, only UHF).

The Examples document describes an application to the OH molecule for the METAGGA option. More output, on the total xc energy of the system, can be obtained by specifying

PRINT METAGGA

This latter option is intended for debugging purposes mainly and is not recommended for general use.

The implementation calculates the total xc energy for a system and writes it to a file. This is always done in Create runs. If the basic fragments are atoms, the keyword

ENERGYFRAG
ATOM [filename]
ATOM [filename]
 ... ...
END 

specifies that different atomic fragment files are to be used in the meta-GGA energy analysis than the regular atomic fragment files from the create runs. This keyword cannot be used for molecular fragment files. In order to compare meta-GGA energy differences between molecular fragments and the total molecule, results from the various calculations need to be combined by hand.

In such situations, it is advisable to use a somewhat higher integration accuracy than one would normally do, at least for the smaller fragments, as there is no error cancellation as in a regular ADF bond energy analysis.

A general comment is that some functionals show a more stable behavior than others (at least in our current implementation). In general, the functionals which are dependent on the Laplacian of the density may display a large variation with respect to basis set changes or different numerical integration accuracy. For this reason we currently recommend FT97 in favor of FT98. Similarly, the results with the BmTau1 functional should still be carefully checked. In our test calculations on the G2 set of molecules, the VS98 showed best performance, both for the average error and for the maximum error. The G2 set consists only of small molecules with elements up to Cl. The relative performance for transition metals and heavy elements is unknown and may well be very different from the ordering for the G2 set.