The Mayer bond order between two atoms is calculated from the density and the overlap matrices (key EXTENDEDPOPAN), see Ref. [140].
The bond order analysis with the key BONDORDER produces the output in which the bond order values are printed for each pair of atoms for which the Nalewajski-Mrozek bond order value is larger than the threshold that can be specified with the keyword BONDORDER. For convenience the printed bond orders are accompanied by the corresponding inter-atomic distance. In the Nalewajski-Mrozek approach [148-153] the bond order indices bAB are calculated based on the one- and two-center valence indices
bAB = VAB + wABA VA + wABB VB
with the weighting factors for one-center indices given by
wXYX = VcovXY/∑ZVcovXZ
Unlike other definitions of covalent bond orders, the Nalewajski-Mrozek valence indices comprise both, covalent and ionic contributions. There exist three alternative sets of the Nalewajski-Mrozek valence indices, [148-153, 140]. The bond order indices calculated from each set of the valence indices differ slightly due to arbitrariness in the way of splitting the one-center terms between bonds. More detailed description of alternative valence indices and their physical meaning is summarized in [148]; see also original papers [149-153]
By default the bond order indices based on the valence indices obtained from partitioning of Tr(PΔP) are printed in the ADF output. Note that in this version the covalent two-center part (also printed in the output) is equal to the Gopinathan-Jug [153] bond order. The default values are:
VA = VionA + VcovA
VionA = ∑a∈A {Pαaa ΔPαaa + Pβaa ΔPβaa}
VcovA = 2 ∑a∈A ∑a'∈A,a<a' {Pαaa' ΔPαa'a + Pβaa' ΔPβa'a}
VcovAB = 2 ∑a∈A ∑b∈B {Pαab ΔPαba + Pβab ΔPβba}
To produce the values from all alternative versions of Nalewajski-Mrozek valence indices, accompanied by the Gopinathan-Jug [153] and Mayer [140] bond orders, see the keyword BONDORDER.
The Mayer [140] bond orders can also be calculated using the keyword EXTENDEDPOPAN. The two implementations of calculating the Mayer bond orders differ slightly if one uses frozen cores. They should agree exactly in all electron calculations.
