Publisher Summary The chapter presents a study on the recursive solution of the Schrodinger equation. The local environment approach to the electronic structure of solids requires an alternative to band theory for solving the Schrodinger equation. In the limit of weak interaction between electrons and atoms, for example in simple metals, the electronic structure is determined by the long-range periodicity of the atomic potentials. Band theory exploits this aspect of the physics to express electronic properties of the solid as a coherent superposition of the electronic properties of all the atoms. When the electrons interact strongly with the atoms, this picture breaks down and properties no longer depend on long-range periodicity, but rather on only the first few shells of neighbors of each atom. The d electrons in transition metals are prime examples of this regime. While band theory is still a valid, formal solution to the Schrodinger equation, the physics is better understood by means of a solution that explicitly accounts for the role of local environment. This chapter presents a discussion on such a method of solution of the Schrodinger equation. The recursive solution of the Schrodinger equation is well adapted to the digital computer. The chapter is organized in order of sophistication of the applications. The elementary part finishes with Green functions. The topics apply to particular kinds of calculations, such as local density of states, energy differences, and photocurrents.

The Recursive Solution of the Schrodinger Equation

The recursive solution of the Schrödinger equation

Nuclear Electric Dipole Moment of 3He

The recursion method is applied to the Anderson model of disorder on a Cayley tree, i.e. a Bethe lattice. For small disorder an infinite order perturbation theory is applied which determines the distributions of parameters of a tight-binding linear chain which is exactly equivalent to the original system. The linear chain possesses localized states and extended states separated in energy by mobility edges. Analysis of the linear chain gives the position of the mobility edges, the nature of the singularity at the mobility edge, and the spatial extent of the localized states. A universal property of the Gaussian distribution is demonstrated and comments are made about a Cauchy distribution.

Study of a mobility edge by a new perturbation theory

Stochastic estimation of nuclear level density in the nuclear shell model: An application to parity-dependent level density in 58Ni

The result of the inverse of a linear operator on a vector is given by a recursion method which involves only three vectors at a time. The method is independent of any inner product.

https://iopscience.iop.org/article/10.1088/0305-4470/7/17/006/pdf

R Haydock

Papers

The inverse of a linear operator