The Natural Orbitals for Chemical Valence (NOCV) approach has been derived from the Nalewajski-Mrozek valence theory [148, 150]. From the mathematical point of view, each NOCV ψi is defined as an eigenvector of the deformation density matrix in the basis of fragment orbitals.
ΔP ψi = νi ψi
Thus, the deformation density Δρ can be expressed in the NOCV representation as a sum of pairs of complimentary eigenfunctions (ψ-k,ψk) corresponding to eigenvalues -νk and νk with the same absolute value but opposite signs:
Δρ(r) = Σ Δρk(r) = Σ νk[-ψ2-k(r)+ψ2k(r)]
here, k goes over the pairs of NOCV's.
In the combined ETS-NOCV scheme the orbital interaction term ΔEorb is expressed in terms of NOCV's as [261, 262]:
ΔEorb = Σ ΔEkorb = Σ νk[-FTS-k+FTSk]
here, FTS-k and FTSk are diagonal transition-state Kohn-Sham matrix elements corresponding to NOCV's with eigenvalues -νk and νk, respectively. The advantage of this expression is that usually only a few complimentary NOCV pairs significantly contribute to the total ΔEorb. Another advantage of this approach is that not only can each Δρk(r) be visualized but there is also a well defined bonding energy contribution ΔEkorb corresponding to it.
Remarks
The ETS-NOCV analysis is only useful when non-atomic fragments are used. No symmetry must be used in the final calculation, thus, use a Symmetry NOSYM keyword if your molecule is symmetric.
Usage
In order to perform the ETS-NOCV analysis, the following two keywords must be specified at the same time:
ETSNOCV RHOKMIN=rhokmin EKMIN=ekmin ENOCV=enocv PRINT {ETSLOWDIN | ETSLOWDIN-Unrestricted}
ETSNOCV
The ETSNOCV keyword specifies thresholds for printing of NOCV-related information. All three arguments are optional and when all three are omitted only the NOCV's corresponding to eigenvalues abs(νk) ≥ 0.05 are included in the analysis.
RHOKMIN
The threshold for population analysis of each deformation density contribution in terms of individual SFO's.
EKMIN
The threshold for orbital interaction energy contributions corresponding to deformation density components originating from each NOCV-pairs.
ENOCV
The threshold for NOCV-eigenvalues.
PRINT {ETSLOWDIN | ETSLOWDIN-Unrestricted}
Only one of the two PRINT options is supposed to be used to activate printing of ETS-NOCV results. The choice depends on the bonding situation.
ETSLOWDIN
If one is interested in a description of bonding between closed-shell molecular fragments, then 'PRINT ETSLOWDIN' keyword must be used. In such a case one set of NOCV's originating from the total deformation density matrix ΔP=(ΔPα+ΔPβ) will be printed out. See the example of carbene bonding between closed shell CH2 and Cr(CO)5.
ETSLOWDIN-Unrestricted
If, however, one is interested in a description of bonding between open-shell molecular fragments then the 'PRINT ETSLOWDIN-Unrestricted' keyword must be used. In this case two sets of NOCV's originating from ΔPα and ΔPβ will be printed out. See the example of CH3-CH3 bonding between two CH3 radicals with opposite spins. This option must also be used when one wants to analyze bonding in a molecule with unpaired electrons.
