Genuine converging solution of self-consistent field equations for extended many-electron systems

TL;DR

This paper presents a rigorous mathematical framework for solving self-consistent field equations in extended many-electron systems, addressing boundary condition issues and the continuous spectrum of Hamiltonians.

Contribution

It introduces a new approach to ensure the existence of self-consistent solutions and defines a Hilbert space basis suitable for unbounded systems with continuous spectra.

Findings

Provides a resolution to boundary condition incompatibilities in infinite systems.

Defines a scalar product with a limiting transition for continuous spectra.

Proves orthogonality and normalization properties of eigenfunctions in the new framework.

Abstract

Calculations of the ground state of inhomogeneous many-electron systems involve a solving of the Poisson equation for Coulomb potential and the Schroedinger equation for single-particle orbitals. Due to nonlinearity and complexity this set of equations, one believes in the iterative method for the solution that should consist in consecutive improvement of the potential and the electron density until the self-consistency is attained. Though this approach exists for a long time there are two grave problems accompanying its implementation to infinitely extended systems. The first of them is related with the Poisson equation and lies in possible incompatibility of the boundary conditions for the potential with the electron density distribution. The analysis of this difficulty and suggested resolution are presented for both infinite conducting systems in jellium approximation and periodic solids. It provides the existence of self-consistent solution for the potential at every iteration step due to realization of a screening effect. The second problem results from the existence of continuous spectrum of Hamiltonian eigenvalues for unbounded systems. It needs to have a definition of Hilbert space basis with eigenfunctions of continuous spectrum as elements, which would be convenient in numerical applications. The definition of scalar product specifying the Hilbert space is proposed that incorporates a limiting transition. It provides self-adjointness of Hamiltonian and, respectively, the orthogonality of eigenfunctions corresponding to the different eigenvalues. In addition, it allows to normalize them effectively to delta-function and to prove in the general case the orthogonality of the 'right' and 'left' eigenfunctions belonging to twofold degenerate eigenvalues.