16 Density fitting approximations

Density fitting can be used to approximate the integrals in spin restricted Hartree-Fock (`HF`), density functional theory (`KS`),
second-order Møller-Plesset perturbation theory (`MP2` and `RMP2`),
explicitly correlated MP2 (MP2-F12), all levels of closed-shell local
correlation methods (`LCC2`, `LMP2`-`LMP4`, `LQCISD(T)`, `LCCSD(T)`), as well as for `CASSCF` and `CASPT2` methods.
In the literature the corresponding methods are often also denoted as RI-HF, RI-DFT, or RI-MP2 etc. We prefer
the acronym DF since RI is used for another purpose in explicitly correlated methods (cf. section 18).

These methods are either direct or semi-direct, i.e. only relatively small sets of transformed four-index integrals are stored. They are very much faster than standard direct calculations and highly recommended, in particular for calculations on larger molecules. The errors caused by density fitting approximations are generally negligible.

Density fitting is
invoked by adding the prefix `DF-` to the command name, e.g. `DF-HF`, `DF-KS`, `DF-MP2`
and so on. Gradients are available for `DF-HF`, `DF-KS`, and `DF-LMP2`. Symmetry is not implemented
for density fitting programs. Therefore, symmetry is turned off automatically if DF- is found in the input.

By default, a fitting basis set will be chosen automatically that corresponds to the current orbital basis set and is
appropriate for the method. For instance, if the orbital basis set is `VTZ`, the default fitting
basis is `VTZ/JKFIT` for `DF-HF` or `DF-KS`, and `VTZ/MP2FIT` for `DF-MP2`.
Other fitting basis sets from the library can be chosen using the `DF_BASIS` option, e.g.

BASIS=VTZ !use VTZ orbital basis DF-HF,DF_BASIS=VQZ !use VQZ/JKFIT fitting basis DF-MP2,DF_BASIS=VQZ !use VQZ/MP2FIT fitting basis

The program then chooses automatically the set which is appropriate for the method. Optionally, the basis type can appended to the basis name and then this supercedes the default, e.g.

BASIS=VTZ !use VTZ orbital basis DF-HF,DF_BASIS=VQZ/JKFIT !use VQZ/JKFIT fitting basis DF-MP2,DF_BASIS=VQZ/MP2FIT !use VQZ/MP2FIT fitting basisis equivalent to the previous example.

If default fitting basis sets are not available for a given orbital basis and atom, it is recommended to use the TZVPP or QZVPP basis sets (they are identical). These are the def2 sets from Turbomole, which are available for most atoms.

In density fitted local coupled cluster methods [DF-LCCSD(T)] (see section 17) there
is an additional option `df_basis_ccsd` that may optionally be used to specify a basis
for the 4-external integrals, e.g.

BASIS=VTZ !use VTZ orbital basis DF-HF !use VTZ/JKFIT default fitting basis DF-LCCSD(T),DF_BASIS_CCSD=VQZ !use VTZ/MP2FIT basis for 0-3 external integrals !and VQZ/MP2FIT basis for 4-external integrals

If accurate absolute values of the correlation energies are needed, the cardinal number of
`df_basis_ccsd` should be one higher than that of the orbital basis. For energy differences
this has a neglible effect, and therefore the default is `df_basis_ccsd`=`df_basis`.

Special basis set definitions may also be needed in explicitly correlated calculations, see section 18.

There are many other options affecting, e.g. screening and other details, but the default values should normally be appropriate. A full description can be found in the MOLPRO users manual.

molpro@molpro.net 2018-09-24