# 4: Some Important Tools of Theory

• • Contributed by Jack Simons
• Professor Emeritus and Henry Eyring Scientist (Chemistry) at University of Utah

Learning Objectives

In this Chapter, you should have learned about the following things:

• Rayleigh-Schrödinger perturbation theory with several example applications.
• The variational method for optimizing trial wave functions.
• The use of point group symmetry.
• Time dependent perturbation theory, primarily for sinusoidal perturbations characteristic of electromagnetic radiation.

For all but the most elementary problems, many of which serve as fundamental approximations to the real behavior of molecules (e.g., the Hydrogenic atom, the harmonic oscillator, the rigid rotor, particles in boxes), the Schrödinger equation can not be solved exactly. It is therefore extremely useful to have tools that allow one to approach these insoluble problems by solving other Schrödinger equations that can be trusted to reasonably describe the solutions of the impossible problem. The approaches discussed in this Chapter are the most important tools of this type.

• 4.1: Perturbation Theory
In most practical applications of quantum mechanics to molecular problems, one is faced with the harsh reality that the Schrödinger equation pertinent to the problem at hand cannot be solved exactly. To illustrate how desperate this situation is, I note that neither of the following two Schrödinger equations has ever been solved exactly (meaning analytically):
• 4.2: The Variational Method
The other method that is used to solve Schrödinger equations approximately, the variational method. In this approach, one must again have some reasonable wave function ψ(0) that is used to approximate the true wave function. Within this approximate wave function, one embeds one or more variables that one subsequently varies to achieve a minimum in the energy of ψ(0) computed as an expectation value of the true Hamiltonian H.
• 4.3: Linear Variational Method
The other method that is used to solve Schrödinger equations approximately, the variational method. In this approach, one must again have some reasonable wave function ψ(0) that is used to approximate the true wave function. Within this approximate wave function, one embeds one or more variables that one subsequently varies to achieve a minimum in the energy of ψ(0) computed as an expectation value of the true Hamiltonian H.
• 4.4: Point Group Symmetry
• 4.5: Character Tables
• 4.6: Time Dependent Perturbation Theory