Spin structure of the first order reduced density matrix and spin‐polarized states

The spin structure of the first order reduced density matrix (RDM‐1) for an arbitrary many‐electron state with zero z ‐projection of the total spin is examined. It is well known that for the state Ψ S 0 ( r 1 σ 1 ,…, r N σ N ), which is an eigenstate of operators ${\widehat {S^2}}$ and ${\widehat S}_z$ with quantum numbers S and M = 0, the matrix elements for spins α and β are equal for any r and r ′: ρ   S 0 α ( r | r ′) = ρ   S 0 β ( r | r ′). In the present article, it is shown that the same is true for any state Φ M = 0 ( r 1 σ 1 ,…, r N ,σ N ) with indefinite total spin if in the expansion Φ M = 0 = ∑ S D S Ψ S 0 only spins S with the same parity are present. To prove the statement, it is shown that the wave function Ψ S 0 acquires the phase factor (−1) N /2− S when all spin functions α(σ i ) are changed for β(σ i ) and vice versa. In the developed proof, the Hamiltonian was not used at all and it was not even assumed that the wave function Ψ S 0 is an eigenfunction of some Hamiltonian. Therefore the obtained result is valid for the stationary and non‐stationary states, ground and excited states, with and without homogeneous magnetic field imposed, exact and approximate wave functions. From the result obtained it follows, in particular, that for the stationary state to be spin‐polarized (ρ   0 α ( r | r ) ≠ ρ   0 β ( r | r )) it is necessary for the Hamiltonian to mix states with different parity spins. The consequences from the proved statement for the antiferromagnetic state are discussed. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008