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
&\phi_2(x_1)&\cdots &\phi_n(x_1)\\ \phi_1(x_2)&\phi_2(x_2)& \cdots &\phi_n(x_2)\\ \vdots&\vdots &\ddots &\vdots \\ \phi_1(x_n)&\phi_2(x_n)&\cdots &\phi_n(x_n) \end{pmatrix})
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
=\sum_{i=1}^n\phi_i(x)u_{iq})
=\sum_{i=1}^n\phi_i^*(x)u_{ir}^*)
=\sum_{j=1}^n\phi_j(x)u_{js})
=\sum_{j=1}^n\phi_j^*(x)u_{jt}^*)
where we have replaced the dummy variable within the parentheses with 
.
The reverse transformations are
=\sum_{q=1}^n\phi_q^{'}(x)u_{iq}^*\;\;\;\;\;\;\;\;(74))
=\sum_{r=1}^n\phi_r^{'*}(x)u_{ir}\;\;\;\;\;\;\;\;(75))
=\sum_{s=1}^n\phi_s^{'}(x)u_{js}^*\;\;\;\;\;\;\;\;(76))
=\sum_{t=1}^n\phi_t^{'*}(x)u_{jt}\;\;\;\;\;\;\;\;(77))
With reference to eq71,
_{qi}(U)_{ir}=(U^{\dagger}U)_{qr}=\delta_{qr}\;\;\;\;\;\;\;\;(82))
_{sj}(U)_{jt}=(U^{\dagger}U)_{st}=\delta_{st}\;\;\;\;\;\;\;\;(83))
_{sj}\Lambda_{ji}^*(U)_{ir}=(U^{\dagger}\Lambda U)_{sr}=\lambda_{sr}^{'}\;\;\;\;\;\;\;\;(84))
