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



Why are the spin components of the total wavefunction orthogonal?


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.


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