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Thus, the density element D µν is only identical to the matrix element ∆ µν in an orthonormal basis.
Although it could be argued that it would be appropriate to call ∆ the one-electron density matrix in
the AO-basis, we will be consistent with the standard literature and call D the density matrix in the
AO basis, and ∆ the matrix of expectation values of creation-annihilation operators. From the
properties of the one-electron density matrix
D
†
= D
Tr DS = N
DSD = D ,
elec.
(B-6)
one straightforwardly obtains the following relations for ∆
∆
Tr ∆S
−1
†
−1
= ∆
= N
∆S ∆ = ∆.
elec.
(B-7)
Although Eqs. (B-6) and Eqs. (B-7) are formally equivalent, the equations for the standard AO
density matrix D are somewhat simpler to use as they contain the metric S whereas the equations for
∆ involves the inverted metric S -1 . It should be noted that Eqs. (B-7) are necessary and sufficient
conditions, so all three equations are fulfilled if and only if 0 is a normalized single-determinant
wave function.
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