Overlap integral Partition functions

The DV-Xa program outputs the overlap integrals, the Fock matrix and MO coefficients into the f08 file. The author developed the postprocessor program which reads data listed above from the f08 file and orthonormalizes basis functions and calculates bond indices, valencies and partitioned energies. DV-Xa calculation should be done with actual atomic orbitals instead of the symmetry adapted basis funtiones to get all overlap inegrals. 

The partitioning functions have been chosen variously as Gaussians, atomic-like densities (proatoms, [30]), and inverse powers of r the latter is implemented in the DV program. One advantage of this approach, as emphasized by Becke [31], is that classical product rules can be applied to the overlapping multicenter density, to obtain integrals of high precision. 

Figure 11. Numerical evaluation of geometric integrals for the two-site case [79]. represent the probability that there is no core overlap in a triatomic chain and ring integrals P > are proportional to the partition function of an isolated ring of size m.

Although based on integration over small volumes, the spin-pair composition yields a continuous function (i.e., the integration volumes overlap). In contrast, ELI is a discrete distribution of values, which formally can be made continuous in the so-called limit after rescaling (cf. Sect. 4). Formally, the spin-pair composition of Silvi could be seen as being based on the (/-restricted space partitioning, i.e., as ratio of same-spin and opposite-spin pairs in micro-cells enclosing fixed electron population. 

---