Population Analysis
Population analysis is a mathematical procedure used to partition the total electronic density of a molecule among its constituent atoms or basis functions. This enables the calculation of effective atomic charges, bond orders, and orbital populations, providing physical insights into chemical bonding and polarization.
I. Mulliken Population Analysis
1. Basis Function and Atomic Populations
Mulliken population analysis partitions the total electronic charge by utilizing the non-orthogonal atomic orbital basis set. The net population \(N_\mu\) of basis function \(\chi_\mu\) is computed from the density matrix \(\mathbf{P}\) and overlap matrix \(\mathbf{S}\) as
which corresponds to the diagonal element of the matrix product \(\mathbf{P}\mathbf{S}\). To avoid the expensive computational cost of performing a full matrix multiplication, the codebase implements a trick where it directly evaluates the dot product of the corresponding rows of the density and overlap matrices, extracting the diagonal elements in linear time. The total electron population \(N_A\) associated with atom \(A\) is then obtained by summing the contributions of all basis functions belonging to that atom as
where the shared overlap populations are divided equally between the participating atoms. The codebase maps basis functions to their corresponding atomic centers using a lookup array sys.bf2at representing the basis function to atom association.
2. Net Atomic Charges
The net atomic charge \(q_A\) of atom \(A\) is calculated by subtracting the total electron population \(N_A\) from the nuclear charge \(Z_A\) as
which provides a quantitative measure of charge transfer and molecular dipole moments.