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 yields

and

respectively.

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

Substituting eq71a and eq71b in eq72 gives

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,