The Molecule as a Many-Electron System: Electron Densities and Molecular Properties

Abstract

A molecule is, in essence, a collection of electrons and (much heavier) nuclei which may exist in certain stationary states, possessing certain invariant geometrical and behavioural features which bestow upon the system a ‘chemical indentity’. The location in space of the various particles is determined only in a statistical sense; every state is characterized by means of a wavefunction Ψ(q₁, q₂, q₃, ...), where q _i represents collectively all the variables (or ‘generalized coordinates’) associated with particle i, and |Ψ(q₁, q₂, q₃,...)|²dq₁ dq₂ dq₃ ... gives the probability of finding particle 1 in volume element dq₁ enclosing ‘point q₁’, particle 2 simultaneously in dq2 at point 2, and so on. In other words | Ψ |2, evaluated for any values of the variables, determines the probability of a corresponding instantaneous ‘configuration’ of the particles. For a system in a stationary state, Ψ contains only a trivial time factor exp(-iEt/ℏ) [where E is the energy of the system and ℏ is the rationalized Planck constant (h/2π)] which is usually discarded since it disappears in forming |Ψ|² (= ΨΨ*) and is of no observational interest. The wavefunction then satisfies the time-independent Schrödinger equation

$$ \hat H{\rm{\psi }} = E{\rm{\psi }} $$

(1.1)

where Ĥ is the Hamiltonian operator, constructed in the usual way [1]. It is only because of the large mass ratio between nuclei and electrons that a molecule has a shape, the nuclei executing comparatively slow ‘internal’ motions around hypothetical equilibrium positions, in the presence of the much lighter and faster moving electrons.