Population Analysis (Mulliken & Löwdin)

Population analysis extracts per-atom partial charges from the molecular orbital coefficients produced by the EHT solver. sci-form implements both Mulliken and Löwdin partitioning schemes.

Overview

Given the density matrix $P$ and overlap matrix $S$ from the EHT solution, population analysis answers: how many electrons does each atom "own"?

Mulliken Population Analysis

Density Matrix

The density matrix is constructed from occupied molecular orbital coefficients:

$$P_{\mu \nu }=\sum _i^{\text{occ}}{n}_i{C}_{\mu i}{C}_{\nu i}$$

where $n_i$ is the occupation number (2 for closed-shell doubly occupied orbitals) and $C_{\mu i}$ is the coefficient of atomic orbital $\mu$ in molecular orbital $i$.

Mulliken Charge Formula

The Mulliken charge on atom $A$ is:

$$q_A=Z_A-\sum _{\mu \in A}(PS{)}_{\mu \mu }$$

where:

• $Z_A$ is the nuclear charge (number of valence electrons)
• $(PS{)}_{\mu \mu }=\sum _{\nu }{P}_{\mu \nu }{S}_{\mu \nu }$ is the gross orbital population for orbital $\mu$
• The sum runs over all atomic orbitals $\mu$ centered on atom $A$

Charge Conservation

The total Mulliken charges always sum to the net molecular charge:

$$\sum _A{q}_A=Q_{\text{total}}$$

This follows from $\text{Tr}(PS)=N_{\text{electrons}}$.

Löwdin Population Analysis

Symmetric Orthogonalization

Löwdin analysis first orthogonalizes the basis by diagonalizing the overlap matrix:

$$S=U\mathrm{\Lambda }{U}^T\Rightarrow S^{-1/2}=U{\mathrm{\Lambda }}^{-1/2}{U}^T$$

Löwdin Charge Formula

The Löwdin density matrix in the orthogonalized basis is:

$$\tilde{P}=S^{-1/2}P{S}^{-1/2}$$

The Löwdin charge on atom $A$:

$$q_A=Z_A-\sum _{\mu \in A}{\tilde{P}}_{\mu \mu }$$

Advantages Over Mulliken

Property	Mulliken	Löwdin
Basis-set dependence	Strong — charges shift with basis size	Weaker — more stable
Orbital populations	Can be negative	Always $0\le {\tilde{P}}_{\mu \mu }\le 2$
Overlap handling	Splits 50/50 between atoms	Orthogonalizes first
Rotational invariance	No	Yes

API

Rust

use sci_form::compute_population;

let result = compute_population("O", None);
// result.mulliken_charges: Vec<f64>
// result.lowdin_charges: Vec<f64>
// result.gross_populations: Vec<f64>
// result.bond_orders: Vec<Vec<f64>>

CLI

sci-form population "O"
# Output: JSON with mulliken_charges, lowdin_charges, etc.

Python

import sci_form
result = sci_form.population("O")
print(result.mulliken_charges)  # [-0.33, 0.17, 0.17]

WASM

import { compute_population } from 'sci-form';
const result = compute_population("O");
// result.mulliken_charges, result.lowdin_charges

Validation

• Charge conservation: $\sum _i{q}_i=Q_{\text{total}}$ tested for all molecules
• Electronegativity ordering: O charges < C charges < H charges (expected from EN)
• Equivalence: Symmetry-equivalent atoms get equal charges (e.g., H atoms in CH₄)
• Boundedness: Löwdin populations always in $[0,2]$