spinorbitals

A spinorbital is a one-electron function that incorporates both spatial and spin degrees of freedom. In nonrelativistic quantum chemistry, a spinorbital can be written as the product of a spatial orbital φ(r) and a two-component spin function χ(σ), typically α for spin up or β for spin down. Each spatial orbital thus corresponds to two spinorbitals: φ(r)α and φ(r)β. Spinorbitals form an orthonormal basis for the one-electron Hilbert space, which is the product space L^2(R^3) ⊗ C^2.

In practice, many-electron wavefunctions are built as antisymmetric Slater determinants of spinorbitals. Because spinorbitals fix the spin projection of each electron, determinants constructed from them have well-defined determinants with respect to the z-component of spin, but they do not necessarily have a definite total spin quantum number S^2 in all contexts, especially in unrestricted formalisms. The use of spinorbitals doubles the size of the one-electron basis compared to purely spatial orbitals, reflecting the two possible spin states for each spatial orbital.

Spinorbitals are central to many computational methods, including Hartree-Fock, configuration interaction, and coupled-cluster theories, as well as Kohn-Sham procedures in density functional theory. Restricted schemes pair electrons in opposite spins within the same spatial orbital, while unrestricted schemes treat alpha and beta spinorbitals separately, which can introduce spin contamination but offer flexibility for open-shell species.

In relativistic quantum chemistry, the concept extends to spinors that couple spin and orbital motion more intimately, often described by four-component spinors in the Dirac formalism. Spinorbitals, in either nonrelativistic or relativistic contexts, provide the single-electron basis from which many-electron wavefunctions and properties are computed.