Exchange Interaction and Heisenberg's Spin Hamiltonian

Abstract

By the use of representation matrices $U\left(P\right)\mathrm{}$ of the permutation group, the energy matrix of many-electron systems has been expanded such that $E={\Sigma }_P{J}_P\tilde{U}\mathrm{}\left(P\right)\mathrm{}$, and it has been proved that all the coefficients $J_P$ are bounded and determined uniquely. This means that the expansion is mathematically valid even though nonorthogonal orbitals are used and no matter how large the overlaps between the orbitals are. Furthermore, it has been shown that the nonorthogonality catastrophe which was pointed out by Inglis and Slater does not appear and values of the coefficients can be evaluated correctly even if higher permutations are omitted. Then we find the Heisenberg spin operator as the first-order approximation of the expansion.