Canonical perturbation expansion of the Hubbard model

We have incorporated the projection operators to the canonical transformation to derive an analytical infinite perturbation-expansion series. This canonical perturbation expansion (CPE) is valid if the unperturbed Hamiltonian H₀ and the perturbation H₁ can be expressed as H₀=Σj^PⱼHPⱼ and H₁=Σj≠k^PⱼHPₖ, where Pⱼ is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies E_jμ with μ=1, 2, , dⱼ, and if |E_jμ-E_jν||E_jω-E_kδ| with j≠k. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian ̃ \~H is obtained when the CPE is applied to the s-band Hubbard model at the atomic limit. An explicit form of ̃ \~H to the eighth order is given, and the magnetic interaction in ̃ \~H is of the form of Heisenberg exchange →Sᵢ·→Sⱼ, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.