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# 10.2.2 Population Analysis: Atomic Partial Charges

(December 20, 2021)

## 10.2.2.1 Basic population analysis

The one-electron charge density,

 $\rho({\rm{\bf r}})=\sum\limits_{\mu\nu}{P_{\mu\nu}\,\phi_{\mu}({\rm{\bf r}})\,% \phi_{\nu}({\rm{\bf r}})}\;,$ (10.1)

represents the probability of finding an electron at the point $\mathbf{r}$, 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

 $N=2\sum_{a}^{N/2}\bigl{|}\psi_{a}(\mathbf{r})\bigr{|}^{2}$ (10.2)

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

 $N=\sum_{\mu\nu}P_{\mu\nu}S_{\mu\nu}=\sum_{\mu}\left(\mathbf{PS}\right)_{\mu\mu% }=\mathrm{tr}(\mathbf{PS})$ (10.3)

where we interpret $(\mathbf{PS})_{\mu\mu}$ as the number of electrons associated with $\phi_{\mu}$. 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 $A$:

 $q_{A}=Z_{A}-\sum_{\mu\in A}\left(\mathbf{PS}\right)_{\mu\mu}$ (10.4)

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, 720 Löwdin P.-O.
J. Chem. Phys.
(1950), 18, pp. 365.
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).

POP_MULLIKEN

POP_MULLIKEN
Controls running of Mulliken population analysis.
TYPE:
LOGICAL/INTEGER
DEFAULT:
TRUE (or 1)
OPTIONS:
FALSE (or 0) Do not calculate Mulliken populations. TRUE (or 1) Calculate Mulliken populations. 2 Also calculate shell populations for each occupied orbital. 3 Same output as 2 and also orbital densities at the nuclear centers. $-1$ Calculate Mulliken charges for both the ground state and any CIS, RPA, or TDDFT excited states.
RECOMMENDATION:
Leave as TRUE, unless excited-state charges are desired. Mulliken analysis is a trivial additional calculation, for ground or excited states.

LOWDIN_POPULATION

LOWDIN_POPULATION
Run Löwdin population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate Löwdin populations. TRUE Run Löwdin population analysis.
RECOMMENDATION:
None

## 10.2.2.2 Charges derived from the electrostatic potential

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, 132 Breneman C. M., Wiberg K. B.
J. Comput. Chem.
(1990), 11, pp. 361.
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, 132 Breneman C. M., Wiberg K. B.
J. Comput. Chem.
(1990), 11, pp. 361.
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. 456 Herbert J. M. et al.
Phys. Chem. Chem. Phys.
(2012), 14, pp. 7679.
(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. 479 Holden Z. C., Richard R. M., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 244108.
For that reason, an alternative procedure based on atom-centered Lebedev grids is also available, 479 Holden Z. C., Richard R. M., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 244108.
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. CHELPG CHELPG Controls the calculation of CHELPG charges. TYPE: LOGICAL DEFAULT: FALSE OPTIONS: FALSE Do not calculate ChElPG charges. TRUE Compute ChElPG charges. RECOMMENDATION: Set to TRUE if desired. For large molecules, there is some overhead associated with computing ChElPG charges, especially if the number of grid points is large. CHELPG_HEAD CHELPG_HEAD Sets the “head space” 132 Breneman C. M., Wiberg K. B. J. Comput. Chem. (1990), 11, pp. 361. (radial extent) of the ChElPG grid. TYPE: INTEGER DEFAULT: 30 OPTIONS: $N$ Corresponding to a head space of $N/10$, in Å. RECOMMENDATION: Use the default, which is the value recommended by Breneman and Wiberg. 132 Breneman C. M., Wiberg K. B. J. Comput. Chem. (1990), 11, pp. 361. CHELPG_DX CHELPG_DX Sets the rectangular grid spacing for the traditional Cartesian ChElPG grid or the spacing between concentric Lebedev shells (when the variables CHELPG_HA and CHELPG_H are specified as well). TYPE: INTEGER DEFAULT: 6 OPTIONS: $N$ Corresponding to a grid space of $N/20$, in Å. RECOMMENDATION: Use the default, which corresponds to the “dense grid” of Breneman and Wiberg, 132 Breneman C. M., Wiberg K. B. J. Comput. Chem. (1990), 11, pp. 361. , unless the cost is prohibitive, in which case a larger value can be selected. Note that this default value is set with the Cartesian grid in mind and not the Lebedev grid. In the Lebedev case, a larger value can typically be used. CHELPG_H CHELPG_H Sets the Lebedev grid to use for hydrogen atoms. TYPE: INTEGER DEFAULT: NONE OPTIONS: $N$ Corresponding to a number of points in a Lebedev grid. RECOMMENDATION: CHELPG_H must always be less than or equal to CHELPG_HA. If it is greater, it will automatically be set to the value of CHELPG_HA. CHELPG_HA CHELPG_HA Sets the Lebedev grid to use for non-hydrogen atoms. TYPE: INTEGER DEFAULT: NONE OPTIONS: $N$ Corresponding to a number of points in a Lebedev grid (see Section 5.5.2. RECOMMENDATION: None. A closely-related set of ESP-derived charges are the so-called “Merz-Kollman" charges, 1034 Singh U. C., Kollman P. A. J. Comput. Chem. (1984), 5, pp. 129. , 95 Besler B. H., Merz Jr. K. M., Kollman P. A. J. Comput. Chem. (1990), 11, pp. 431. 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. 225 Cornell W. D. et al. J. Am. Chem. Soc. (1993), 115, pp. 9620. 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. ESP_CHARGES ESP_CHARGES Controls the calculations of Merz-Kollman ESP-derived charges. TYPE: INTEGER DEFAULT: NONE OPTIONS: 1 Use Lebedev grid points around each atom. 2 Use spherical harmonics grid points around each atom. RECOMMENDATION: NONE RESP_CHARGES RESP_CHARGES Controls the calculations of RESP charges, where chemically equivalent atoms are restricted to have the same atomic charge value. TYPE: INTEGER DEFAULT: NONE OPTIONS: 1 Use Lebedev grid points around each atom. 2 Use spherical harmonics grid points around each atom. RECOMMENDATION: NONE ESP_SURFACE_DENSITY ESP_SURFACE_DENSITY Controls the spacing between grid points on vdW surfaces. TYPE: INTEGER DEFAULT: 500 OPTIONS: $n$ Spacing of $0.001\times n$ (in Å) RECOMMENDATION: The default corresponds to 0.5 Å spacing. ## 10.2.2.3 Hirshfeld charges Hirshfeld population analysis 473 Hirshfeld F. L. Theor. Chem. Acc. (1977), 44, pp. 129. provides yet another definition of atomic charges in a molecule:  $q_{A}=Z_{A}-\int d{\bf r}\left(\frac{\rho^{0}_{A}({\bf r})}{\sum_{B}\rho^{0}_{% B}({\bf r})}\right)\rho({\bf r}),$ (10.5) where $Z_{A}$ is the nuclear charge of $A$, $\rho^{0}_{B}$ 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), 753 Marenich A. V. et al. J. Chem. Theory Comput. (2012), 8, pp. 527. 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., 139 Bultinck P. et al. J. Chem. Phys. (2007), 126, pp. 144111. , 1153 Vanpoucke D. E. P., Bultinck P., Van Driessche I. J. Comput. Chem. (2013), 34, pp. 405. 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), $\rho_{i}^{0}(\mathbf{r})$ with electronic population $N_{i}^{0}=\int d\mathbf{r}\,\rho_{i}^{0}(\mathbf{r})=Z_{i}$. This affords charges  $q_{i}^{1}=Z_{i}-\int d\mathbf{r}\left(\frac{\rho_{i}^{0}(\mathbf{r})}{\sum_{i}% ^{A}\rho_{i}^{0}(\mathbf{r})}\right)\rho(\mathbf{r})=Z_{i}-N_{i}^{1}$ (10.6) 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 $N_{i}^{1}=\int d\mathbf{r}\,\rho_{i}^{1}(\mathbf{r}_{1})$ 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  $w_{i,\rm{HI}}^{k}(\mathbf{r})=\frac{\rho_{i}^{k-1}(\mathbf{r})}{\sum\limits_{i% \in A}\rho_{i}^{k-1}(\mathbf{r})}\;.$ (10.7) The atomic densities $\rho_{i}^{k}(\mathbf{r})$ with corresponding fractional electron numbers $N_{i}^{k}$ are obtained by linear interpolation between $\rho_{i}^{0,\lfloor N_{i}^{k}\rfloor}(\mathbf{r})$ and $\rho_{i}^{0,\lceil N_{i}^{k}\rceil}(\mathbf{r})$ of the same atom: 139 Bultinck P. et al. J. Chem. Phys. (2007), 126, pp. 144111. , 301 Elking D. M., Perera L., Pedersen L. G. Comput. Phys. Commun. (2012), 183, pp. 390.  $\rho_{i}^{k}(\mathbf{r})=\left(\lceil N_{i}^{k}\rceil-N_{i}^{k}\right)\,\rho_{% i}^{0,\lfloor N_{i}^{k}\rfloor}(\mathbf{r})+\left(N_{i}^{k}-\lfloor N_{i}^{k}% \rfloor\right)\,\rho_{i}^{0,\lceil N_{i}^{k}\rceil}(\mathbf{r})\;,$ (10.8) 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 $\rho_{i}^{0,Z_{A}-2},\rho_{i}^{0,Z_{A}-1},\ldots,\rho_{i}^{0,Z_{A}+2}$ 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 $|N_{i}^{k}-N_{i}^{k-1}|<0.0005e$. 139 Bultinck P. et al. J. Chem. Phys. (2007), 126, pp. 144111. , 1079 Steinmann S. N., Corminbeoeuf C. J. Chem. Theory Comput. (2010), 6, pp. 1990. 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.

HIRSHFELD

HIRSHFELD
Controls running of Hirshfeld population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Calculate Hirshfeld populations. FALSE Do not calculate Hirshfeld populations.
RECOMMENDATION:
None

Switch to force reading in of isolated atomic densities.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Read in isolated atomic densities from previous Hirshfeld calculation from disk. FALSE Generate new isolated atomic densities.
RECOMMENDATION:
Use the default unless system is large. Note, atoms should be in the same order with same basis set used as in the previous Hirshfeld calculation (although coordinates can change). The previous calculation should be run with the -save switch.

HIRSHFELD_SPHAVG

HIRSHFELD_SPHAVG
Controls whether atomic densities should be spherically averaged in pro-molecule.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE Spherically average atomic densities. FALSE Do not spherically average.
RECOMMENDATION:
Use the default.

CM5

CM5
Controls running of CM5 population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Calculate CM5 populations. FALSE Do not calculate CM5 populations.
RECOMMENDATION:
None

HIRSHITER

HIRSHITER
Controls running of iterative Hirshfeld population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Calculate iterative Hirshfeld populations. FALSE Do not calculate iterative Hirshfeld populations.
RECOMMENDATION:
None

HIRSHITER_THRESH

HIRSHITER_THRESH
Controls the convergence criterion of iterative Hirshfeld population analysis.
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
$N$ Corresponding to the convergence criterion of $N/10000$, in $e$.
RECOMMENDATION:
Use the default, which is the value recommended in Ref.  139 Bultinck P. et al.
J. Chem. Phys.
(2007), 126, pp. 144111.

Example 10.1  Iterative Hirshfeld population analysis for $\rm F^{-}(H_{2}O)$

$molecule -1 1 O 1.197566 -0.108087 0.000000 H 1.415397 0.827014 0.000000 H 0.134830 -0.084378 0.000000 F -1.236389 0.012239 0.000000$end

$rem SYM_IGNORE true METHOD B3LYP BASIS 6-31G* HIRSHITER true$end


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