Exact solutions to the Schrödinger equation for realistic nanoscale systems are beyond reach, hence, different strategies for approximating the solutions are necessary. Perturbation theory relies on a “zero-order Hamiltonian” to express the desired eigenvalues and eigenvectors of “the full Hamiltonian.” We derive working equations for the Rayleigh–Schrödinger perturbation theory, and the validity of the approach is analyzed for a generic two-level system. Applications are given to atoms perturbed by point charges or static fields, and for electrons in quantum wells. An alternative strategy is the variation method, which replaces exact solutions by their projection on a reduced space of “trial functions,” varied to minimize the associated error. Particularly important is the method of linear variation, which can potentially converge to the exact Hamiltonian eigenstates. The mean-field approximation, commonly used for many-particle systems, is derived by optimizing a trial function in the form of product of single-particle functions.

Optical nonlinearity emerges from nonlinear interaction of light with matter. In this chapter, the basic concept and formulation of light‒matter interaction are discussed through a semiclassical approach with the behavior of the optical field classically described by Maxwell’s equations and the state of the material quantum mechanically described by a wave function governed by the Hamiltonian of the material. An optical field interacts with a material through its interaction with the electrons in the material. A Schrödinger electron is nonrelativistic with a nonzero mass, and a Dirac electron is relativistic with a zero mass. The interaction Hamiltonian can be expressed in terms of the vector and scalar potentials by using the Coulomb gauge. It can be expressed in terms of the electric and magnetic fields through multipole expansion as a series of electric and magnetic multipole interactions, with the first term being the electric dipole interaction. The electric polarization of a material induced by an optical field is obtained through density matrix analysis. The optical susceptibility of the material is then obtained from the electric polarization.