Functional-derivative study of the Hubbard model. I. Perturbation method and first-order approximation

In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function $G$ and calculated iteratively in a perturbation scheme which takes the Hubbard I solution $G_0$ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting $G_0$ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\epsilon$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.