Derivation of model Hamiltonians: Exchange Hamiltonian forH2

Abstract

The derivation of model Hamiltonians such as crystal-field and spin Hamiltonians requires the decoupling of electrons, which may be done by defining an appropriate equivalent Hamiltonian $H_{\mathrm{eq}}$. The connection between $H_{\mathrm{eq}}$ and the original Hamiltonian, and between the respective eigenfunctions, is explicitly given. The eigenfunctions of $H_{\mathrm{eq}}$ are linear combinations of simple products of atomic spin orbitals, the latter being chosen in such a way that its antisymmetrization leads to a linearly independent set. The problem is then reformulated in a chosen subspace (effective space) via the effective-Hamiltonian formalism. The model Hamiltonian $H_{\mathrm{mod}}$ is then obtained by projecting out the undesired degrees of freedom. If in the two-atom two-electron case one keeps in $H_{\mathrm{mod}}$ only first-order terms in the interatomic interaction, one gets the usual Heitler-London approximation. For this reason the formulation has been called the extended Heitler-London method. As an illustration the formalism is applied to the ${\mathrm{H}}_2$ molecule, where the usual Heisenberg Hamiltonian is rigorously derived and the values of the energy baricenter and magnetic exchange constant are given, up to the second order of perturbation theory.