Localized Molecular Orbitals¶

Boys-Foster and Pipek-Mezey localization¶

With the LOCORB key ADF provides the Boys-Foster method and Pipek-Mezey method for localization of Molecular Orbitals [1] [2] [3]. 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.

These localization methods can not be used in case of spin-orbit coupling.

The goal of orbital-localization lies in analysis: the localized orbitals provide an easier-to-interpret picture. The localized molecular orbitals can be visualized with the ADF-GUI.

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. The Pipek-Mezey localization maximizes the sum of orbital-dependent partial charges on the nuclei, see literature for details. Both Boys-Foster localization method and Pipek-Mezey localization method have been implemented in ADF, see Ref. [1].

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}
  {Criterion <BOYS|PM>}
  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 adf.rkf (TAPE21).

Criterion <BOYS|PM>

By default the Boys-Foster localization method is used if Criterion is not specified and in case ‘Criterion BOYS’ is specified. If ‘Criterion PM’ is specified, the Pipek-Mezey localization method is used.

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.

LOCORB ..
  ..
End
AORESPONSE
  ..
End
PERTLOC
  efield
  bfield
  alfa
  gprime
  beta
  <static|dynamic>
  <diag|fulltens>
End

Reduction of Orbital Space Extent (ROSE)¶

ROSE (Reduction of Orbital Space Extent) aims at constructing intrinsic fragment orbitals (IFOs), see Ref. [5], and recanonicalized intrinsic localized molecular orbitals (RILMOs), see Ref. [6]. With the external ROSE package (see https://gitlab.com/quantum_rose/rose, this package is not distributed by SCM), one can calculate such orbitals for different types of basis functions. In AMS2025 the ROSE functionality relating to Slater-Type Orbitals has been made available inside ADF.

The basic idea used in ROSE is to define an intrinsic minimal basis set of polarized orbitals which span the occupied space exactly and also covers the valence part of the unoccupied, or virtual, orbital space. These intrinsic fragment orbitals are localized and not very basis set dependent and therefore form a good starting point for analysis. They are furthermore used as a reference in a localization procedure in which the occupied orbitals are unitarily transformed into RILMOs that typically resemble the occupied MOs of the reference fragment calculations. The same localization procedure is also applied to localize the valence virtuals that together with the occupied RILMOs span the IFO space. As the thus generated virtual space is too small to reach quantitative accuracy in TDDFT calculations, it is necessary to also add the so-called ‘hard’ virtuals, see Ref. [6], to the virtual space. This is done, slightly different from the procedure reported in Ref. [6], by defining a set of reference virtuals on the fragments. The generation of localized hard virtuals follows then in mathematically exactly the same way as the construction of valence intrinsic fragment orbitals. The difference is that an energy-truncated space, complementary to the already localized valence space, is now the target. This procedure combines energy localization and spatial localization to allow for subsystem TDDFT treatments, see section on exciton transfer integrals with ROSE and Ref. [6].

Next we only discuss the functionality of ROSE that is available inside ADF. In this implementation spin-orbit coupling can not be used, spin-restricted orbitals should be used, and all electron basis sets should be used.

Starting point for ROSE calculations is always to divide the target molecule or complex into 2 or more chemically relevant fragments. One needs to calculate these fragments using SYMMETRY NOSYM, and the fragment nuclear coordinates should match the corresponding coordinates in the full molecule or complex. The subsequent ADF calculation on the full molecule or complex should also use symmetry NOSYM. It is convenient to use the same basis set as used in the fragments to make sure that the reference virtual space is large enough to cover the (energy-truncated) supermolecular virtual space.

An input that calculates localized orbitals with the ROSE implementation in ADF could look like

AMS_JOBNAME=Frag1 $AMSBIN/ams <<eor
System
  atoms
    coordinates Frag1
  end
end
Task SinglePoint
Engine ADF
  basis
    ...
    core None
  end
  symmetry nosym
  ...
EndEngine
eor

AMS_JOBNAME=Frag2 $AMSBIN/ams <<eor
System
  atoms
    coordinates Frag2
  end
end
Task SinglePoint
Engine ADF
  basis
    ...
    core None
  end
  symmetry nosym
  ...
EndEngine
eor

$AMSBIN/ams <<eor
System
  atoms
    coordinates Frag1
    coordinates Frag2
  end
end
Task SinglePoint
Engine ADF
  basis
    ...
    core None
  end
  symmetry nosym
  ROSE
    fragmentfile Frag1.results/adf.rkf
    fragmentfile Frag2.results/adf.rkf
  End
  ...
EndEngine
eor

The calculated intrinsic orbitals can be visualized with the AMS-GUI.

The ROSE block key should contain the FragmentFile key. Other keys are optional.

ROSE
   Additional_virtuals_cutoff float
   Exponent integer
   Frag_bias float_list
   Frag_threshold float
   Frag_valence integer_list
   FragmentFile string
   Version [Stndrd_2013 | Simple_2013 | Simple_2014]
   nfragments integer
End

ROSE

Type:

Block

Description:

Options for orbital localization

Additional_virtuals_cutoff

Type:

Float

Default value:

2.0

Unit:

Hartree

GUI name:

Additional virtuals cutoff

Description:

Specifies the energy threshold to determine the number of additional hard virtuals to localize for the supermolecule. If not specified the default value is used.

Exponent

Type:

Integer

Description:

The value of the exponent (for the default Pipek-Mezey type of localization, the exponent is equal to 2. A higher value of 3 or 4 is sometimes useful.).

Frag_bias

Type:

Float List

Description:

Expert key. Specifies the bias applied to each fragment when assigning the localized orbitals to a particular fragment. Use a negative value for default behavior for a given fragment. If not specified, or if the number of items is unequal the number of fragments, a default bias is used.

Frag_threshold

Type:

Float

Default value:

10.0

Unit:

Hartree

GUI name:

Frag threshold

Description:

Specifies energy threshold to determine number of reference hard virtuals to use for the fragments (needed to localize the additional hard virtuals). The total number of reference virtuals should be larger than the number of virtual MOs to localize (similar to having more reference valence orbitals than occupied valence orbitals to localize), so it is advisable to use a relatively high value for this parameter. If not specified the default value is used.

Frag_valence

Type:

Integer List

Description:

Expert key. Specifies the number of valence orbitals for each fragment. Use a negative value for default behavior for a given fragment. If not specified, or if the number of items is unequal the number of fragments, by default the code generates the valence space for each fragment as the sum of valence orbitals of individual atoms within each fragment.

FragmentFile

Type:

String

Recurring:

True

Description:

Path (either absolute or relative) of each fragment file. The number of fragments used to project on will be number of occurrences of this key.

Version

Type:

Multiple Choice

Options:

[Stndrd_2013, Simple_2013, Simple_2014]

Description:

Expert key, specifies version of the IAO generation. If not specified a default value is used.

nfragments

Type:

Integer

Default value:

2

Description:

Expert key, preferably use the FragmentFile key for each fragment instead, and do not use this key. If the FragmentFile key is not used, then with this key one can specify the number of fragments used to project on. If this key is used the fragment file names should be called frag1.rkf, frag2.rkf, etcetera, and should be present at the directory where ams is started.

Natural Bond Orbitals (NBO)¶

See the section on NBO analysis for localized orbitals that are calculated in case of NBO.

References