Spin orbitals, spatial orbitals, and the Pauli exclusion principle.

There is a close relation between the properties of a wave function under permutations of particles and the spin of these particles. Electrons have spin s = 1/2 (half-integer spin), and have therefore antisymmetric wave functions. The orbitals in a Slater determinant of the form

specify not only the position of an electron, but also its spin. The common way to talk about spin is to say that an electron has spin up or spin a, or it has spin down or spin b. It's really a bit more complicated, but it will do for the quantum chemistry we are interested in. Because the orbitals above include the spin, they are also called spin-orbitals . If we want to indicate the spin part of a spin-orbital explicitly we write ja and jb, where j is an orbital that only refers to the spatial part of the spin-orbital. These spatial orbitals are the ones that we usually talk about and the ones that we can make drawings of. Most molecules have an even number of electrons. These are almost always paired. This mean that for each spin-orbital ja there is also a corresponding spin-orbital jb in the Slater determinant. The Slater determinant for such a molecule with N electrons is then given by

The Pauli exclusion principle rephrases the Pauli principle when the electronic wave function is a Slater determinant. Such a wave function is already antisymmetric, so we only have to look for cases where it does not give a physically acceptable state, which occurs when it vanishes. As the orbitals forming the Slater determinant are normally taken from an orthonormal set, the Slater determinant is only not acceptable when one or more of these orbitals are used more than once. It is then said that there can be only zero or one electron per orbital. With orbital we then mean a spin-orbital. If the orbitals are of the type ja and jb then it can also be said that there can only be zero, one, or two electrons per orbital. Now orbital, however, refers to the spatial part. If there is one electron in j, it's in ja or in jb. If there are two electrons in j, there is one in ja and one in jb.

The independent-particle model in quantum chemistry.

• Electrons that don't interact.
• The variation principle.
• The independent-particle approximation.
• The Fock equation without exchange.
• The Pauli principle and Slater determinants.
• The Fock equation with exchange.
• Spin orbitals, spatial orbitals, and the Pauli exclusion principle (this page).