ABSTRACT
As known, the spinless density matrix of a closed shell (singlet) single deter-
minant wave function has the peculiar idempotency property
(DS)2 = 2DS , (7.1)
which has also been utilized several times above. If the wave function is not
a closed shell singlet or if electron correlation has been taken into account,
then this relation is not satisfied, and the number
∆ = Tr[2DS− (DS)2] = ∑ µ
[2DS− (DS)2]µµ , (7.2)
gives a measure of the deviation from the closed shell situation in the
molecule, and it can be called the number of effectively unpaired electrons
[72,81]. An important property of it is that it is always non-negative, which
follows from the fact that matrix D can be expressed in terms of the coeffi-
cient vectors ci of the expansion the natural orbitals ϕi = ∑µ c i µ χµ in terms of
the basis orbitals χµ , and their occupation numbers ni as
By substituting Eq. (7.3) into (7.2) and taking into account the orthonormal-
ization c†i Sc j = δi j of the natural orbitals, and the relationship Tr(cic † i S) =
Tr(c†i Sci) = 1, we get
∆ = ∑ i
ni(2−ni) . (7.4)
As the occupation numbers are in the interval 0≤ ni ≤ 2, neither of the components of this sum can be negative.
