Levels of Theory

In electronic structure theory, the ``level of theory'' has two degrees of freedom: one is the treatment of ``electron correlation,'' and the other is the basis set. Electron correlation simply refers to the fact that the motion of each electron influences the motion of all other electrons (due to the Coulomb repulsion between each pair of electrons). For more than two electrons, this is an N-body problem which cannot be solved exactly by analytic techniques. It can be solved exactly using numerical methods, but the computational cost is enormous, scaling as the factorial of N. Hence, various approximations must be used; some of these are described below.

The other variable is the basis set; here, we refer to a basis set of one-electron functions (or orbitals), usually centered on the atoms. The electronic Schrödinger equation becomes greatly simplified once we solve it using such a basis. The larger the basis set (the more orbitals it contains), the more accurately we model the Schrödinger equation for the given correlation method. However, even an infinite basis set can give incorrect answers when paired with an approximate treatment of electron correlation. Likewise, even an exact treatment of electron correlation can give terrible answers when paired with a very small basis set. Better and better results can be obtained when one increases the basis set and improves the treatment of correlation. In the limit of an infinite basis set and an exact treatment of electron correlation, the electronic Schrödinger equation would be solved exactly.