A spin-orbital expresses the state of an electron as a complete wavefunction, which comprises of a spatial component and a spin component.

It is defined as

where is a composite coordinate consisting of three spatial coordinates and one spin coordinate , while and are spin wavefunctions describing the two possible spin states of an electron.

In the derivation of the canonical Hartree-Fock equations, , where . When we substitute the spin-orbitals in their explicit forms into eq109, we have

Multiplying throughout by and integrating with respect to the spin coordinates, while noting that because of spin orthogonality, we have

###### Question

Why are the spin components of the total wavefunction orthogonal?

###### Answer

From the article on Hermitian operators, we know that two eigenfunctions of a Hermitian operator that correspond to different eigenvalues are orthogonal. Since the spin components of the total wavefunction and correspond to different eigenvalues of the spin Hermitian operator , they are orthogonal.

Hence the spin orbitals reduce to spatial orbitals. This implies that we only need to work with spatial orbitals, e.g. Slater-type orbitals, when we use the Hartree-Fock method to estimate the ground state energies of atoms.