The Slater determinant of unitarily transformed spin orbitals is related to the Slater determinant of the original spin orbitals by the product of and the determinant of the unitary matrix .

A unitary matrix is defined by eq70, where . Let , where

Substituting and in eq66, we have

and

respectively.

Using the identity of for square matrices (see this article for proof),

Substitute eq71a and eq71b in eq72,

From eq70, . Since (see this article for proof), we have . Therefore, (or ) and

The average value of a physical property of an electron, which is described by , is , which is the same when the electron is described by . In other words, any expectation value of a physical property of an electron is invariant to a unitary transformation of the electron’s wavefunction that is expressed as a Slater determinant. Such a consequence is used to derive the Hartree-Fock equations.

Lastly, if , and are the matrix elements of , and , we have

where we have replaced the dummy variable within the parentheses with .

The reverse transformations are

With reference to eq71,