The orbitals φi with energies εi are expanded in basis functions χμ, which leads to the definition of density matrices Pi describing orbital densities, from which the total density matrix can be constructed:
φi(r) = ∑μ
χμ(r)
Cμi
ρi (r) =
∫|φi(r)|2 =
∑μν
Pi,μν
χμ(r)
χν(r);
Pi,μν =
Cμi Cνi
ρ (r) =
∑i
ni
ρi (r) =
∑μν
Pμν
χμ(r)
χν(r);
Pμν =
∑iniCμi
Cνi
(3.3.2)
Here μ and μ run over the basis functions, which may be either primitive functions, or combinations of primitive functions, for instance the SCF orbitals of atoms or larger fragments.
The Mulliken population analysis provides a partitioning of either the total charge density or an orbital density. The total density is written as
ρ (r) =
∑μν
Pμν
χμ(r)
χν(r) =
∑A≤B ∑μ∈A ∑ν∈B
Pμν
χμ
χν =
∑A≤B
ρAB
(3.3.3a)
ρAB =
∑μ∈A ∑ν∈B
Pμν
χμ
χν
(3.3.3b)
The total number of electrons, N=∫ ρ(r)d(r), is now partitioned over the atoms by assigning an overlap population PμνSμν + PνμSνμ for one half to the atom A of χμ and one half to atom B of χν,
N = ∫ ρ(r)d(r) =
∑μν
Pμν
Sμν =
∑μ
GPμ
(3.3.4a)
GPμ =
∑ν
Pμν
Sμν
(3.3.4b)
GPμ is the gross population of χμ. It contains the net population Pμμ and half of each total overlap population PμνSμν + PνμSνμ between χμ and χν. Summing the gross populations over the functions μ ∈ A yields the total number of electrons assigned to atom A, or the gross population of atom A, GPA, and hence the gross charge QA of atom A,
GPA =
∑μ∈A
GPμ
(3.3.5a)
QA =
ZA - GPA
(3.3.5b)
The overlap population OPμν between two functions and the overlap population QAB between two atoms are defined in an analogous manner,
OPμν =
PμνSμν +
PνμSνμ
(3.3.6a)
QAB =
∑μ∈A ∑ν∈B
OPμν
(3.3.6b)
These quantities can be evaluated for a single orbital density, N=1=∫|φi(r)|2dr. The gross population GPi,μ of a function in a specific orbital density |φi(r)|2 is then associated with the fraction of the orbital density belonging to that function (or the percentage χμ character of orbital φi, and the overlap population OPi,μν gives an indication of the strength of bonding or antibonding between χμ and χν in orbital φi,
GPiμ =
∑ν
Pi,μνSμν =
∑ν
Cμi Cνi
Sμν
(3.3.7a)
OPi,μν =
Pi,μν Sμν +
Pi,νμ Sνμ =
2 Cμi Cνi
Sμν
(3.3.7b)
