[QM] Notes for spin density

definition of the spin density

A spin density is defined by the difference between an alpha spin one-particle density and a beta spin one-particle density.


physical meaning of the spin density

A spin density describes the probability density of finding twice of the z-component of an N-electron system in real space. The trace of a spin density shows twice of the z-component of an N-electron system or the difference between the number of alpha and beta electrons of an N-electron system.


spin densities in the MO theory

Given a system with equal number of alpha and beta electrons, the spin density of a restricted Slater determinant (RSD) is definitely vanished but the spin density of an unrestricted Slater determinant (USD) is not necessarily zero. The trace (the z-component of total spin) for both densities are definitely zero due to the same number of alpha and beta electrons.

The multiplicity of the former wavefunction (RSD) is singlet and the later one (USD) is possibly not a eigenfunciton of total spin-moment operator (S^2).