Comparison of Loge and Virial Methods of Partitioning Molecular Charge Distributions

Abstract

The properties of fragments of molecular systems defined by the loge and virial partitioning methods are compared. In the cases LiH⁺(X ²Σ⁺) and LiH(X ¹Σ⁺), where the localization of the charge density is well-defined by deep minima between the fragments, the virial fragments are also the ‘best’ loges, i.e., the missing information function and the fluctuations in the average populations of the fragments are minimized by the surfaces of zero-flux in ▽ϱ(r). In the ground state distributions of BeH, BH and BeH₂, where the minima in ϱ(r) in the zero-flux surfaces are less pronounced, the virial fragments appear as close approximations to the ‘best’ loges. In BeH⁺ (X ¹Σ⁺) and BeH (A ²Π_r) where there is no pronounced minimum in ϱ(r) in the outer regions of the zero-flux surface, the virial fragments are poor approximations to the ‘best’ loges. The valence density in BeH(A ²Π_r) is, in fact, not partitionable by the loge criterion, the missing information being minimized only for a loge which contains all three valence electrons.

The fluctuation in the average population of a loge, $\Lambda (\bar{N},\Omega )$, appears to attain a minimum value for the same boundary which minimizes the missing information function. Thus, the ‘best’ loges could be found and defined by varying their boundaries to attain a minimum in their fluctuations, $\Lambda (\bar{N},\Omega )$. It is demonstrated that the fluctuation Λ ($\Lambda (\bar{N},\Omega )$ for a region Ω is a measure of the extent to which the motion of the electrons within Ω are internally correlated, and free of any correlative interactions with electrons outside of Ω. Specifically, defining the correlation factor f(r₁, r₂) in terms of the pair density P ₂(r ₁, r ₂) and number density P ₁(r ₁) as

$$ {{P}_{2}}({{r}_{1}},{{r}_{2}}) = {{P}_{1}}({{r}_{1}}){{P}_{1}}({{r}_{2}})[1 + f({{r}_{1}},{{r}_{2}})], $$

then, it is shown that

$$ \Lambda (\bar{N},\Omega ) = \bar{N},(\Omega ) + \int\limits_{\Omega } {d{{r}_{1}}} \int\limits_{\Omega } {d{{r}_{2}}} {{P}_{1}}({{r}_{1}}){{P}_{1}}({{r}_{2}})f({{r}_{1}},{{r}_{2}}), $$

where the double integral is a measure of the total correlation hole for all the electrons in Ω. The limiting value of this correlation hole is $(\bar{N},\Omega )$. Thus, a minimization of $\Lambda (\bar{N},\Omega )$ defines a region of space within which the magnitude of the correlation hole for the electrons in Ω is maximized, or alternatively, it defines a region of space such that the correlation of the motions of the electrons within the loge with those in the remainder of the system has been reduced to a minimum.