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

Calculations of the ground state of inhomogeneous many-electron systems involve a solving of the Poisson equation for Coulomb potential and the Schrödinger 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 renewed by means of the Schrödinger equation. Rigorously speaking, it means the iteration process cannot be continued. The resolution of this difficulty is proposed for both infinite conducting systems in jellium approximation and periodic structures. 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. It is proposed to insert a limiting transition into the definition of scalar product specifying the Hilbert space. 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 the orthogonality of the 'right' and 'left' eigenfunctions belonging to twofold degenerate eigenvalues. 1. Problem definition Ground-state calculations of inhomogeneous many-electron systems involve generally a solving of the Poisson equation for averaged Coulomb potential u(r) at given spatial electron density n(r) and the Schrödinger equation for single-particle orbitals ψ E (r) in the potential u ef f accounting by some approximation for the difference between u(r) and the microscopic local field. In the density functional theory the corresponding set of Kohn-Sham equations for the spin-unpolarized electron gas has the form (in atomic units |e| = m = = 1)