Partial Charges

TeraChem can compute several common partial-charge analyses for any SCF wavefunction. Each scheme answers a slightly different question — "how much electron density sits in the basis functions centered on atom $A$" (Mulliken, Löwdin), "how much density crosses into atom $A$'s spatial neighborhood" (VDD), "what set of point charges best reproduces the molecular electrostatic potential outside the molecule" (RESP), or "how much population sits in a minimal set of intrinsic atomic orbitals on atom $A$" (IAO) — and they are complementary rather than competing.

In the expressions below, $\rho(\mathbf{r})$ is the SCF electron density, $\mathbf{P}$ is the one-particle density matrix in the AO basis $\{\chi_\mu\}$, $\mathbf{S}$ is the AO overlap matrix ($S_{\mu\nu}=\langle\chi_\mu|\chi_\nu\rangle$), $Z_A$ is the nuclear charge of atom $A$, and the notation $\mu\in A$ means "AO $\chi_\mu$ is centered on atom $A$".

Mulliken charges

Mulliken population analysis¹ partitions each off-diagonal density-matrix element evenly between the two atoms whose basis functions are involved. The gross atomic population on $A$ is

\[ N_A^{\text{Mull}} \;=\; \sum_{\mu \in A} \big(\mathbf{P}\mathbf{S}\big)_{\mu\mu} \;=\; \sum_{\mu \in A} \sum_\nu P_{\mu\nu}\,S_{\mu\nu}, \]

and the Mulliken partial charge follows from

\[ q_A^{\text{Mull}} \;=\; Z_A \;-\; N_A^{\text{Mull}}. \]

Mulliken charges are inexpensive (they reuse $\mathbf{P}$ and $\mathbf{S}$ already in memory) but are notoriously sensitive to the choice of basis set — diffuse functions in particular cause the values to drift with no well-defined limit. They are most useful as a quick sanity check or for comparing across calculations that use identical basis sets.

Löwdin charges

Löwdin's scheme² removes the basis-overlap bias of Mulliken by first transforming to a symmetrically-orthogonalized AO basis, $\tilde\chi_\mu = \sum_\nu (\mathbf{S}^{-1/2})_{\nu\mu}\,\chi_\nu$. In that basis the density matrix becomes $\tilde{\mathbf{P}} = \mathbf{S}^{1/2}\mathbf{P}\mathbf{S}^{1/2}$ and atomic populations are read directly from its diagonal:

\[ N_A^{\text{Löw}} \;=\; \sum_{\mu \in A} \big(\mathbf{S}^{1/2}\mathbf{P}\mathbf{S}^{1/2}\big)_{\mu\mu}, \qquad q_A^{\text{Löw}} \;=\; Z_A \;-\; N_A^{\text{Löw}}. \]

Löwdin charges remove the worst Mulliken pathologies and are noticeably less basis-dependent for non-diffuse basis sets, though they still inherit the basic limitation that "atomic population" is defined in terms of basis functions rather than space.

Voronoi Deformation Density (VDD) charges

VDD charges³ are real-space and basis-set-independent. Space is partitioned by Voronoi cells $V_A$ — the set of points closer to nucleus $A$ than to any other nucleus — and the VDD charge is the integral over $V_A$ of the deformation density, i.e. the SCF density minus the promolecular density built from a superposition of spherically-averaged free-atom densities:

\[ q_A^{\text{VDD}} \;=\; -\int_{V_A} \big[\rho(\mathbf{r})\;-\;\rho^{\text{prom}}(\mathbf{r})\big]\,d\mathbf{r}, \qquad \rho^{\text{prom}}(\mathbf{r}) \;=\; \sum_B \rho_B^{\text{free}}(\mathbf{r}). \]

The interpretation is direct: $q_A^{\text{VDD}}$ measures how much electron density flows into ($q_A<0$) or out of ($q_A>0$) the Voronoi cell of $A$ when the isolated atoms are assembled into the molecule. Because the promolecular reference is subtracted, the values are typically small and chemically intuitive, and they converge smoothly with basis-set size.

Restrained electrostatic potential (RESP) charges

RESP charges⁴ are obtained by fitting a single point charge per atom to reproduce the molecular electrostatic potential $V(\mathbf{r}) = \sum_B Z_B/|\mathbf{r}-\mathbf{R}_B| - \int \rho(\mathbf{r}')/|\mathbf{r}-\mathbf{r}'|\,d\mathbf{r}'$ evaluated on a grid of points $\{\mathbf{r}_g\}$ outside the van-der-Waals surface. To prevent buried atoms (whose charges are nearly redundant with those of their neighbors) from taking on arbitrary values, a hyperbolic restraint is added that drives small charges toward zero. The fit minimizes

\[ L \;=\; \sum_g \Big(V(\mathbf{r}_g) \;-\; \hat V(\mathbf{r}_g)\Big)^2 \;+\; a\sum_A\!\left[\big(q_A^2+b^2\big)^{1/2}-b\right], \qquad \hat V(\mathbf{r}_g) \;=\; \sum_A \frac{q_A}{|\mathbf{r}_g-\mathbf{R}_A|}, \]

subject to the hard constraint $\sum_A q_A = Q_{\text{total}}$. The restraint parameters $a$ and $b$ are typically $a=5\times10^{-4}$ a.u. and $b=0.1\,e^-$; $a=0$ recovers the unrestrained ESP fit. Because they are defined by the electrostatic potential the molecule actually generates, RESP charges are the natural choice for fitting empirical (Amber, CHARMM, GAFF, …) force fields, and they are far less basis-set-dependent than Mulliken or Löwdin charges.

See the dedicated RESP page for the grid choices, restraint parameters, ESP-only mode, and complete keyword reference.

Intrinsic atomic orbital (IAO) charges

Intrinsic atomic orbitals (IAOs)⁵ are a small, exact set of atom-centered orbitals that span the occupied space; the partial charges they yield are robust and largely basis-set insensitive. The IAOs $A$ are built from the converged occupied orbitals and a minimal "MinAO" basis, and the partial charge of atom $X$ is its nuclear charge minus the IAO population summed over occupied orbitals $i$ and the MinAO functions $\mu$ centered on $X$:

\[ q_X = Z_X - 2\sum_{i}\sum_{\mu\in X}\big(\mathbf{C}^{T}\mathbf{S}\,A\big)^2_{i\mu}. \]

The charges sum to the total molecular charge. For water (cc-pvdz) this gives $q_\mathrm{O} = -0.694$, $q_\mathrm{H} = +0.347$.

IAO charges are requested by turning on localization with loctype iao:

localization  yes
loctype       iao

Projection (MinAO) basis

The IAO construction projects the occupied orbitals onto a minimal cc-pvdz-minao basis, shipped with TeraChem at $TeraChem/basis/cc-pvdz-minao (independent of the main basis you choose). It is a normal TeraChem basis file and may also be selected as a basis set in its own right.

The same IAO machinery underlies the intrinsic bond orbitals (IBOs) — see the Localization page for IBO localization and the other loctype options.

R. S. Mulliken, J. Chem. Phys. 23, 1833 (1955). ↩
P.-O. Löwdin, J. Chem. Phys. 18, 365 (1950). ↩
C. F. Guerra, J.-W. Handgraaf, E. J. Baerends, and F. M. Bickelhaupt, J. Comp. Chem. 25, 189 (2004). ↩
C. I. Bayly, P. Cieplak, W. D. Cornell, and P. A. Kollman, J. Phys. Chem. 97, 10269 (1993). ↩
G. Knizia, Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts, J. Chem. Theory Comput. 9, 4834 (2013). ↩