Electron orbital

In quantum chemistry, an electron orbital (or more often just orbital) is a synonym for a quadratically integrable one-electron wave function.

Types of orbitals

Several kinds of orbitals can be distinguished. The first is the atomic orbital (AO). This is a function depending on a single 3-dimensional vector r_A1, which is a vector pointing from point A to electron 1. Generally there is a nucleus at A. The following notation for an AO is frequent,

$\chi _{i}(\mathbf {r} _{A1}),$

but other notations can be found in the literature. Sometimes the center A is added as an index: χ_{A i}. We say that χ_{A i} (or, as the case may be, χ _i) is centered at A. In numerical computations AOs are either taken as Slater type orbitals (STOs) or Gaussian type orbitals (GTOs). Hydrogen-like orbitals are rarely used.

The second kind of orbital is the molecular orbital (MO). Such a one-electron function depends on several vectors: r_A1, r_B1, r_C1, ... where A, B, C, ... are different points in space (usually nuclear positions). The oldest example of an MO (without use of the name MO yet) is in the work of Burrau (1927) on the single-electron ion H₂⁺. Burrau introduced spheroidal coordinates (a bipolar coordinate system) to describe the wavefunction of the electron of H₂⁺. Lennard-Jones (1929) introduced the following linear combination of atomic orbitals (LCAO) way of writing an MO φ:

$\phi (\mathbf {r} _{1})=\sum _{A=1}^{N_{\mathrm {nuc} }}\sum _{i=1}^{n_{A}}c_{iA}\chi _{i}(\mathbf {r} _{A1}),\qquad c_{iA}\in \mathbb {C} ,$

where A runs over N_nuc different points in space (usually A runs over all the nuclei of a molecule, hence the name molecular orbital), and i runs over the n_A different AOs centered at A. The coefficients c _iA are the outcome of calculations based on one of the existing quantum chemical methods.

The AOs and MOs defined so far depend only on the spatial coordinate r_A1 of electron 1. In addition, an electron has a spin coordinate μ, which can have two values: spin-up or spin-down. A complete set of functions of μ consists of two functions only, traditionally these are denoted by α(μ) and β(μ). These functions are eigenfunctions of the z-component s_z of the spin angular momentum operator. The most general spin atomic orbital of electron 1 is of the form

$\chi _{i}^{+}(\mathbf {r} _{A1})\alpha (\mu _{1})+\chi _{i}^{-}(\mathbf {r} _{A1})\beta (\mu _{1})$

which in general is not an eigenfunction of s_z. More common is the use of

$\chi _{i}(\mathbf {r} _{A1})\alpha (\mu _{1})\quad {\hbox{and}}\quad \chi _{i}(\mathbf {r} _{A1})\beta (\mu _{1}),$

which are eigenfunctions of s_z. Since it is rare that different AOs are used for spin-up and spin-down electrons, we dropped the superscripts + and −. Likewise a spin molecular orbital is usually either

$\phi ^{+}(\mathbf {r} _{1})\alpha (\mu _{1})\quad {\hbox{or}}\quad \phi ^{-}(\mathbf {r} _{1})\beta (\mu _{1}).$

Here the superscripts + and − can be necessary, because some quantum chemical methods distinguish the spatial parts of the different spins. These are the so-called different orbitals for different spins (DODS) methods.