Mulliken population analysis

The orbitals φ_i with energies ε_i are expanded in basis functions χ_μ, which leads to the definition of density matrices P_i describing orbital densities, from which the total density matrix can be constructed:

φ_i(r) = ∑_μ χ_μ(r) C_μi
ρ_i (r) = ∫|φ_i(r)|² = ∑_μν P_i,μν χ_μ(r) χ_ν(r);     P_i,μν = C_μi C_νi
ρ (r) = ∑_i n_i ρ_i (r) = ∑_μν P_μν χ_μ(r) χ_ν(r);     P_μν = ∑_in_iC_μi C_νi   (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 P_μν χ_μ(r) χ_ν(r) = ∑_A≤B ∑_μ∈A ∑_ν∈B P_μν χ_μ χ_ν = ∑_A≤B ρ_AB   (3.3.3a)
ρ_AB = ∑_μ∈A ∑_ν∈B P_μν χ_μ χ_ν   (3.3.3b)

The total number of electrons, N=∫ ρ(r)d(r), is now partitioned over the atoms by assigning an overlap population P_μνS_μν + P_νμS_νμ for one half to the atom A of χ_μ and one half to atom B of χ_ν,

N = ∫ ρ(r)d(r) = ∑_μν P_μν S_μν = ∑_μ GP_μ   (3.3.4a)
GP_μ = ∑_ν P_μν S_μν   (3.3.4b)

GP_μ is the gross population of χ_μ. It contains the net population P_μμ and half of each total overlap population P_μνS_μν + P_νμS_νμ between χ_μ and χ_ν. 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, GP_A, and hence the gross charge Q_A of atom A,

GP_A = ∑_μ∈A GP_μ   (3.3.5a)
Q_A = Z_A - GP_A   (3.3.5b)

The overlap population OP_μν between two functions and the overlap population Q_AB between two atoms are defined in an analogous manner,

OP_μν = P_μνS_μν + P_νμS_νμ   (3.3.6a)
Q_AB = ∑_μ∈A ∑_ν∈B OP_μν   (3.3.6b)

These quantities can be evaluated for a single orbital density, N=1=∫|φ_i(r)|²dr. The gross population GP_i,μ of a function in a specific orbital density |φ_i(r)|² is then associated with the fraction of the orbital density belonging to that function (or the percentage χ_μ character of orbital φ_i, and the overlap population OP_i,μν gives an indication of the strength of bonding or antibonding between χ_μ and χ_ν in orbital φ_i,

GP_iμ = ∑_ν P_i,μνS_μν = ∑_ν C_μi C_νi S_μν   (3.3.7a)
OP_i,μν = P_i,μν S_μν + P_i,νμ S_νμ = 2 C_μi C_νi S_μν   (3.3.7b)