Atomic Clusters

Abstract

Atoms may bind together according to the quantal structure of their single-electron energy shells. Electrons in the valence upper shells participate in the chemical bond, leaving behind a background of positive ions. The hole density of the ionic background is given, in its most general pair-wise form, by

$$ \rho (r) = \sum\limits_{ia;jb} {(\sum\limits_s {\beta _s^2 c_{ia}^{s*} c_{jb}^s } )\chi _{ia}^* (r)\chi _{jb} (r),} $$

((1))

where χ _ia (r) = χ _a (r - R_i is the a-th atomic-like orbital of the i-th ion placed at R _i , c ^s _ia are the coefficients of the s-th linear combination of atomic-like orbitals, and β _s is the content of the chemical-bond orbital Φ _s in the molecular-like orbital ψ _s ; ψ _s = α _s ϕ _s + β _s Φ _s , ϕ _s = ∑ _ia c ^s _ia χ _ia , α ² _s + β ² _s = 1. Averaging out the c ^s _ia -coefficient in (1), and in virtue of the orthogonality of the eTa-matrix, one gets the ionic density

$$ \rho (r) = \beta ^2 \sum\limits_{ia} {\left| {\chi _{ia} \left( r \right)} \right|^2 ;} $$

((2))

hence, the total number of electronic holes of the ionic background is given by

$$ \int {dr \cdot \rho (r)} = \beta ^2 \sum\limits_i {z_i ,} $$

((3))

where z _i is the nominal valence of the i-th atom. One can see that each ion participate in the chemical bond with an effective valence

$$ z_i^* = \beta ^2 z_i , $$

((4))

smaller than the nominal valence z _i ; and the same is true for each valence atomic-like orbital ia.