Quantum mechanical methods all are based on the Schroedinger equation:

The expression in the square brackets, which is called the "Hamiltonian", represents the kinetic and potential energy of an electron of mass m at a distance r from a nuclear charge Z.

• The L is the Laplacian, or gradient, operator, the derivative with respect to Cartesian coordinates.

• The electronic charge is e, and h is Planck's constant.

• Y is the wavefunction, which describes the motion of the electron,

• E is the energy.

The wavefunctions of the hydrogen atom are the familiar s, p, d, etc., orbitals. The square of the wavefunction multiplied by a volume element yields the probability of finding the electron inside the volume element.

• This quantity also is referred to as the electron density in the volume element.

• It is the same quantity measured by X-ray diffraction.

The Schroedinger equation, written above for a hydrogen atom, can be generalized to a multinuclear, multielectron molecule, resulting in the familiar T-shirt equation:

Y is now a many-electron wavefunction, and H represents the Hamiltonian operator, more complex now than in the case of the hydrogen atom:

• The first term (blue brace) in this expression represents the kinetic energy of the nuclei, while the second gives the kinetic energy of the electrons.

• The third describes the electrostatic (Coulombic) interactions between the nuclei and the electrons, and the final two terms represent the internuclear and interelectronic repulsions, respectively.

• Ms are nuclear masses, r and R are interparticle distances, e is the electronic charge, and Z is the nuclear charge.

To date, the mathematical tools do not exist to solve the many-electron Schroedinger equation. Approximations to provide practical sollutions are described on the following pages.