10.2.2 Population Analysis: Atomic Partial Charges

The one-electron charge density,

represents the probability of finding an electron at the point $𝐫$, but implies little regarding the number of electrons associated with a given nucleus in a molecule. However, since the number of electrons $N$ is related to the occupied orbitals ${\psi }_i$ by

We can substitute the atomic orbital (AO) basis expansion of ${\psi }_a$ into Eq. (10.2) to obtain

where we interpret ${(\mathrm{𝐏𝐒})}_{\mu \mu }$ as the number of electrons associated with ${\varphi }_{\mu }$. If the basis functions are atom-centered, the number of electrons associated with a given atom can be obtained by summing over all the basis functions. This leads to the Mulliken formula for the net charge of the atom $A$:

where $Z_A$ is the atom’s nuclear charge. This is called a Mulliken population analysis, and it is performed by default.

Although conceptually simple, Mulliken population analyses suffer from a strong dependence on the basis set used, as well as the possibility of producing unphysical negative numbers of electrons. An alternative is that of Löwdin population analysis, ⁷⁵⁸ Löwdin P.-O.
J. Chem. Phys.
(1950), 18, pp. 365. Link which uses the Löwdin symmetrically orthogonalized basis set (which is still atom-tagged) to assign the electron density. This shows a reduced basis set dependence, but maintains the same essential features.

While Mulliken and Löwdin population analyses are commonly employed, and can be used to produce information about changes in electron density and also localized spin polarizations, they should not be interpreted as oxidation states of the atoms in the system. For such information we would recommend a bonding analysis technique (LOBA or NBO).

A more stable alternative to Mulliken or Löwdin charges are charges derived from the electrostatic potential (ESP), of which there are several different types. So-called “ChElPG” charges, ¹³⁷ Breneman C. M., Wiberg K. B.
J. Comput. Chem.
(1990), 11, pp. 361. Link whose name is an acronym for “Charges from the Electrostatic Potential on a Grid”, are perhaps the most conceptually straightforward of the various ESP-derived charge schemes. By definition, the ChElPG atomic charges are the ones that provide the best fit to the molecular electrostatic potential, evaluated on a real-space grid outside of the van der Waals region and subject to the constraint that the sum of the ChElPG charges must equal the molecular charge. Q-Chem’s implementation of the ChElPG algorithm differs slightly from the one originally algorithm described by Breneman and Wiberg, ¹³⁷ Breneman C. M., Wiberg K. B.
J. Comput. Chem.
(1990), 11, pp. 361. Link in that Q-Chem weights the grid points with a smoothing function to ensure that the ChElPG charges vary continuously as the nuclei are displaced. ⁴⁸³ Herbert J. M. et al.
Phys. Chem. Chem. Phys.
(2012), 14, pp. 7679. Link (For any particular geometry, however, numerical values of the charges are quite similar to those obtained using the original algorithm.) Note also that the Breneman-Wiberg approach uses a Cartesian grid and becomes expensive for large systems, especially when ChElPG charges are used in QM/MM-Ewald calculations. ⁵⁰⁸ Holden Z. C., Richard R. M., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 244108. Link For that reason, an alternative procedure based on atom-centered Lebedev grids is also available, ⁵⁰⁸ Holden Z. C., Richard R. M., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 244108. Link which provides very similar charges using far fewer grid points. In order to use the Lebedev grid implementation the $rem variables CHELPG_H and CHELPG_HA must be set, which specify the number of Lebedev grid points for the hydrogen atoms and the heavy atoms, respectively.

A closely-related set of ESP-derived charges are the so-called “Merz-Kollman" charges, ¹⁰⁹⁹ Singh U. C., Kollman P. A.
J. Comput. Chem.
(1984), 5, pp. 129. Link ^{, 99} Besler B. H., Merz Jr. K. M., Kollman P. A.
J. Comput. Chem.
(1990), 11, pp. 431. Link in which the atom-centered charges are fit to reproduce the ESP on a small number of concentric atomic spheres (or van der Waals surfaces of the molecule), and in this respect the Merz-Kollman algorithm is similar to Q-Chem’s Lebedev-based implementation of the ChElPG charges. Q-Chem’s algorithm for computing Merz-Kollman charges uses surfaces constructed from atomic spheres whose radii are 1.4$\times$, 1.6$\times$, 1.8$\times$, and 2.0$\times$ the atomic van der Waals radii. Lebedev or spherical-harmonics grid points are placed on each surface with a 0.5 Å default spacing between these grid points. These charges can be restricted to satisfy “chemical symmetry”, where chemically equivalent atoms have the same atomic charge value, leading to the so-called “RESP” charges. ²³⁵ Cornell W. D. et al.
J. Am. Chem. Soc.
(1993), 115, pp. 9620. Link

Note:  Both ESP_CHARGES and RESP_CHARGES can be used to compute the atomic charges of any singlet excited state from a CIS or TDDFT calculation (RPA or TDA). For excited-state popular analysis, it is recommended to turn on CIS_RELAXED_DENSITY. Physically, the external electrostatic environment should feel the relaxed excited state density not the unrelaxed density.

Hirshfeld population analysis ⁵⁰² Hirshfeld F. L.
Theor. Chem. Acc.
(1977), 44, pp. 129. Link provides yet another definition of atomic charges in a molecule:

where $Z_A$ is the nuclear charge of $A$, ${\rho }_B^0$ is the isolated ground-state atomic density of atom $B$, and $\rho$ is the molecular density. The sum goes over all atoms in the molecule. Thus computing Hirshfeld charges requires a self-consistent calculation of the isolated atomic densities (the promolecule) as well as the total molecule. Prior to the SCF calculation, the Hirshfeld atomic density matrix is constructed. After SCF convergence, numerical quadrature is used to evaluate the integral in Eq. (10.5). Neutral ground-state atoms are used, as the choice of appropriate reference for a charged molecule is ambiguous (such jobs will crash). As numerical integration (with default quadrature grid) is used, charges may not sum precisely to zero. A larger XC_GRID may be used to improve the accuracy of the integration, but the magnitude of the Hirshfeld charges should be largely independent of grid choice.

The charges (and corresponding molecular dipole moments) obtained using Hirshfeld charges are typically underestimated as compared to other charge schemes or experimental data. To correct this, Marenich et al. introduced “Charge Model 5” (CM5), ⁷⁹⁵ Marenich A. V. et al.
J. Chem. Theory Comput.
(2012), 8, pp. 527. Link which employs a single set of parameters to map the Hirshfeld charges onto a more reasonable representation of the electrostatic potential. CM5 charges generally lead to more accurate dipole moments as compared to the original Hirshfeld charges, at negligible additional cost. CM5 is available for molecules composed of elements H–Ca, Zn, Ge–Br, and I.

The use of neutral ground-state atoms to define the promolecular density in Hirshfeld scheme has no strict theoretical basis and there is no unique way to construct the promolecular densities. For example, Li${}^0$F${}^0$, Li${}^{+}$F${}^{-}$, or Li${}^{-}$F${}^{+}$ could each be used to construct the promolecular densities for LiF. Furthermore, the choice of appropriate reference for a charged molecule is ambiguous, and for this reason Hirshfeld analysis is disabled in Q-Chem for any molecule with a net charge. A solution for charged molecules is to use the iterative “Hirshfeld-I” partitioning scheme proposed by Bultinck et al., ¹⁴⁵ Bultinck P. et al.
J. Chem. Phys.
(2007), 126, pp. 144111. Link ^{, 1220} Vanpoucke D. E. P., Bultinck P., Van Driessche I.
J. Comput. Chem.
(2013), 34, pp. 405. Link in which the reference state is not predefined but rather determined self-consistently, thus eliminating the arbitrariness. The final self-consistent reference state for Hirshfeld-I partitioning usually consists of non-integer atomic populations.

In the first iteration, the Hirshfeld-I method uses neutral atomic densities (as in the original Hirshfeld scheme), ${\rho }_i^0(𝐫)$ with electronic population $N_i^0=\int 𝑑𝐫{\rho }_i^0(𝐫)=Z_i$. This affords charges

on the first iteration. The new electronic population (number of electrons) for atom $i$ is $N_i^1$, and is derived from the promolecular populations $N_i^0$. One then computes new isolated atomic densities with $N_i^1=\int 𝑑𝐫{\rho }_i^1({𝐫}_1)$ and uses them to construct the promolecular densities in the next iteration. In general, the new weighting function for atom $i$ in the $k$th iteration is

The atomic densities ${\rho }_i^k(𝐫)$ with corresponding fractional electron numbers $N_i^k$ are obtained by linear interpolation between ${\rho }_i^{0,\lfloor N_i^k\rfloor }(𝐫)$ and ${\rho }_i^{0,\lceil N_i^k\rceil }(𝐫)$ of the same atom: ¹⁴⁵ Bultinck P. et al.
J. Chem. Phys.
(2007), 126, pp. 144111. Link ^{, 314} Elking D. M., Perera L., Pedersen L. G.
Comput. Phys. Commun.
(2012), 183, pp. 390. Link

where $\lfloor N_i^k\rfloor$ and $\lceil N_i^k\rceil$ denote the integers that bracket $N_i^k.$ The two atomic densities on the right side of Eq. (10.8) are obtained from densities ${\rho }_i^{0,Z_A-2},{\rho }_i^{0,Z_A-1},\mathrm{\dots },{\rho }_i^{0,Z_A+2}$ that are computed in advance. (That is, the method uses the neutral atomic density along with the densities for the singly- and doubly-charged cations and anions of the element in equation.) The Hirshfeld-I iterations are converged once the atomic populations change insignificantly between iterations, say . ¹⁴⁵ Bultinck P. et al.
J. Chem. Phys.
(2007), 126, pp. 144111. Link ^{, 1145} Steinmann S. N., Corminbeoeuf C.
J. Chem. Theory Comput.
(2010), 6, pp. 1990. Link

The iterative Hirshfeld scheme generally affords more reasonable charges as compared to the original Hirshfeld scheme. In LiF, for example, the original Hirshfeld scheme predicts atomic charges of $\pm$0.57 while the iterative scheme increases these charges to $\pm$0.93. The integral in Eq. (10.6) is evaluated by numerical quadrature, and the cost of each iteration of Hirshfeld-I is equal to the cost of computing the original Hirshfeld charges. The $rem variable SYM_IGNORE must be set to TRUE for Hirshfeld-I analysis.