Localized Molecular Orbitals¶

ADF provides the Boys-Foster method for localization of Molecular Orbitals [122-124]. This implies a unitary transformation of the occupied molecular orbitals as computed in the SCF procedure, with the objective to obtain a (transformed) set of orbitals that represent exactly the same charge density but with molecular orbitals that are more localized in space than the original MOs.

The goal of orbital-localization lies in analysis: the localized orbitals provide an easier-to-interpret picture.

Orbital localization procedures require a measure of the localization of the orbitals which can then be optimized in the space of the allowed unitary transformations. Methods advocated in the literature differ in the definition of this measure. The Boys-Foster method minimizes the mean extension of the occupied orbitals around their center of gravity; see the literature for details.

Occasionally it is useful to apply the localization only to a subset of the MOs, with the objective to expose certain features better. This is accomplished by performing the localization in a number of distinct steps, where at each step the localization is restricted by keeping a subset of the MOs frozen. A case is worked out in the Examples document.

The computation of localized orbitals is controlled with the block-type key. By default (if the key is not supplied in input) no orbital localization is carried out.

LOCORB {nopop store}
  Spintype FrozenMOs
  Spintype FrozenMOs
  ...
end

nopop

Specifies that no SFO population analysis is to be carried out on the localized MOs. By default this population analysis will be printed in the output file.

store

Specifies that the transformation from MOs to localized MOs is stored on TAPE21.

Spintype

Must be either alfa or beta (not case sensitive) and refers to spin-A and spin-B orbitals respectively. In a spin-restricted run beta records are meaningless and must not be used.

FrozenMOs

A list (possibly empty) of integers, referring to a list of MOs from the SCF, and/or labels of irreducible representations. The integers and/or labels may be given in any order.

Each record Spintype FrozenMOs in the data block defines a localization cycle in which the localization procedure is carried out on all orbitals (of the indicated spin), except those indicated by the FrozenMOs.

For either spin at least one localization cycle is carried out. If no data record for that spin is found in the data block, a full localization is performed, without any MOs excluded.

The data block may be completely empty (but the record end must be supplied since the key is block-type) and would be equivalent with specifying two records, one for either spin, without any FrozenMOs:

LOCORB {nopop}
end

is equivalent with

LOCORB {nopop}
  alfa
  beta
end

The integers in FrozenMOs refer to an overall list of SCF MOs consisting of all valence MOs in each symmetry representation up to and including the highest non-empty one. So, when for instance in the first irrep MO #4 is the highest non-empty one and in the second irrep mo #2 is the highest non-empty one, then in the overall list the first 4 are the orbitals of the first irrep, the no.s 5 and 6 are from the second irrep, et cetera.

Each symmetry label in FrozenMOs collectively denotes in one stroke all molecular orbitals of that representation up to and including the highest occupied one (in that symmetry). The label may be the name of an irreducible representation or of a subspecies. In the former case all partner representations are denoted collectively. In an atom symmetry for instance, specifying P would be equivalent to P:x P:y P:z.

Note that if the final SCF has in any symmetry representation empty orbitals below the highest non-empty orbital in that symmetry - violating the Aufbau principle - then these empty orbitals are included in the above-defined overall list and hence a FrozenMOs specification is necessary, namely to avoid mixing MOs with different occupation numbers in the localization.

In the output file the localized MOs are printed as expansions in SFOs and (optionally) a population analysis is given, again in terms of the SFOs. Furthermore, each localized MO has associated with it an energy value and an occupation number. The energy is the expectation value of the Fock operator for the orbital. The occupation number is obtained as a weighted sum from the SCF MOs that were combined into the localized orbital. As mentioned before one should combine only SCF MOs with identical occupations into a localized orbital, in which case its occupation number will be the same. The printout of the occupation number of the localized orbital allows therefore a verification that a correct localization procedure has been carried out.