Partial charge

A partial charge is a non-integer charge value when measured in elementary charge units. Partial charge is more commonly called net atomic charge. It is represented by the Greek lowercase letter δ, namely δ− or δ+.

Partial charges are created due to the asymmetric distribution of electrons in chemical bonds. For example, in a polar covalent bond like HCl, the shared electron oscillates between the bonded atoms. The resulting partial charges are a property only of zones within the distribution, and not the assemblage as a whole. For example, chemists often choose to look at a small space surrounding the nucleus of an atom: When an electrically neutral atom bonds chemically to another neutral atom that is more electronegative, its electrons are partially drawn away. This leaves the region about that atom's nucleus with a partial positive charge, and it creates a partial negative charge on the atom to which it is bonded.

${\displaystyle {\ce {H3{\overset {\delta +}{C}}-{\overset {\delta -}{C}}l\qquad H3{\overset {\delta -}{C}}-{\overset {\delta +}{M}}g{\overset {\delta -}{C}}l}}}$

Polarity of chloromethane (left)
and of the related Grignard compound
with indication of the partial charge.

In such a situation, the distributed charges taken as a group always carries a whole number of elementary charge units. Yet one can point to zones within the assemblage where less than a full charge resides, such as the area around an atom's nucleus. This is possible in part because particles are not like mathematical points—which must be either inside a zone or outside it—but are smeared out by the uncertainty principle of quantum mechanics. Because of this smearing effect, if one defines a sufficiently small zone, a fundamental particle may be both partly inside and partly outside it.

Uses

Partial atomic charges are used in molecular mechanics force fields to compute the electrostatic interaction energy using Coulomb's law. They are also often used for a qualitative understanding of the structure and reactivity of molecules.

Determining partial atomic charges

Partial atomic charges can be used to quantify the degree of ionic versus covalent bonding of any compound across the periodic table. The necessity for such quantities arises, for example, in molecular simulations to compute bulk and surface properties in agreement with experiment. Evidence for chemically different compounds shows that available experimental data and chemical understanding lead to justified atomic charges.^[1] Atomic charges for a given compound can be derived in multiple ways, such as:

1. densities
2. measured dipole moments
3. the Extended Born thermodynamic cycle, including an analysis of covalent and ionic bonding contributions
4. the influence of coordination numbers and aggregate state of a given compound on atomic charges
5. the relationship of atomic charges to melting points, solubility, and cleavage energies for a set of similar compounds with similar degree of covalent bonding
6. the relationship of atomic charges to chemical reactivity and reaction mechanisms for similar compounds reported in the literature
7. the relationship between chemical structure and atomic charges for comparable compounds with known atomic charges across the periodic table.

The discussion of individual compounds in prior work has shown convergence in atomic charges, i.e., a high level of consistency between the assigned degree of polarity and the physical-chemical properties mentioned above. The resulting uncertainty in atomic charges is ±0.1e to ±0.2e for highly charged compounds, and <10% for compounds with atomic charges below ±1.0e. Often, the application of one or two of the above concepts already leads to very good values, especially taking into account a growing library of experimental benchmark compounds and compounds with tested force fields.^[2]

The closest equivalent quantum-mechanical (QM) methods to derive atomic charges for molecular simulations are computations of the electron density at a high level of theory in combination with Hirshfeld partitioning of the charge density into approximately spherical atomic basins.^[3] This approach can yield dipole moments as well as Coulomb contributions to cleavage energies and surface tensions in good agreement with measurements. The Hirshfeld method is also a common method to partition electron deformation densities from laboratory measurements in crystallographic studies. Nevertheless, the accuracy of the electron density calculated by QM methods appears to be lower than in experiment. QM methods also require consideration of the correct aggregate state, and particular difficulties arise for electron densities of elements with d and f electrons. Since atomic charges often reflect a minor difference in total electron density, deviations up to multiples compared to the real system may occur. Besides, different partitioning schemes such as by Mulliken or by Bader can lead to additional overestimates in atomic charges by over 30% compared to Hirshfeld charges. Uncertainties up to several 100% have also been reported for Löwdin charges, MP2/CHELPG charges, and M06 derived charges.^[4] Unexpected atomic charges are very difficult to explain and invalidate the entire force field if accepted. Overall, ab-initio studies today suggest atomic charges with uncertainties up to multiples while available laboratory measurements since the 1950s quantify atomic charges consistent with known physical chemical properties within a range of ±5-10%. The possible high uncertainty and limited correlation of QM derived charges with observable properties are the main reason to favor experimental measurements and readily available information about the extent of covalent vs ionic bonding across the periodic table for the assignment of atomic charges, including the extended Born thermodynamic cycle and chemical knowledge.

Not surprisingly, approaches based on quantum mechanics have therefore questioned the concept of partial atomic charge as somewhat arbitrary. The results depend on the initial guess of the wave function and the method used to delimit between one atom and the next, related to the spatial continuity of the electron density. In spite of experimental data that have shown that boundaries between atoms and shifts in electron density are often well enough defined, many QM procedures have been developed for estimating the partial charges, yet are recommended with caution due to the wide range of assumptions and scatter. According to Cramer (2002), this earlier classification of methods involves the following four classes:^[5]

• Class I charges are those that are not determined from quantum mechanics, but from some intuitive or arbitrary approach. These approaches can be based on experimental data such as dipoles and electronegativities.
• Class II charges are derived from partitioning the molecular wave function using some arbitrary, orbital based scheme.
• Class III charges are based on a partitioning of a physical observable derived from the wave function, such as electron density.
• Class IV charges are derived from a semiempirical mapping of a precursor charge of type II or III to reproduce experimentally determined observables such as dipole moments.

The following is a detailed list of earlier methods, partly based on Meister and Schwarz (1994).^[6]

• Population analysis of wavefunctions
  • Mulliken population analysis
  • Coulson's charges
  • Natural charges
  • CM1, CM2, CM3 charge models
• Partitioning of electron density distributions
  • Bader charges (obtained from an atoms in molecules analysis)
  • Density fitted atomic charges
  • Hirshfeld charges
  • Maslen's corrected Bader charges
  • Politzer's charges
  • Voronoi Deformation Density charges
  • Density Derived Electrostatic and Chemical (DDEC) charges, which simultaneously reproduce the chemical states of atoms in a material and the electrostatic potential surrounding the material's electron density distribution^[7]
• Charges derived from dipole-dependent properties
  • Dipole charges
  • Dipole derivative charges, also called atomic polar tensor (APT) derived charges,^[8] or Born, Callen, or Szigeti effective charges ^[9]
• Charges derived from electrostatic potential
  • Chelp
  • ChelpG (Breneman model)
  • Merz-Singh-Kollman (also known as Merz-Kollman, or MK)
• Charges derived from spectroscopic data
  • Charges from infrared intensities
  • Charges from X-ray photoelectron spectroscopy (ESCA)
  • Charges from X-ray emission spectroscopy
  • Charges from X-ray absorption spectra
  • Charges from ligand-field splittings
  • Charges from UV-vis intensities of transition metal complexes
  • Charges from other spectroscopies, such as NMR, EPR, EQR
• Charges from other experimental data
  • Charges from bandgaps or dielectric constants
  • Apparent charges from the piezoelectric effect
  • Charges derived from adiabatic potential energy curves
  • Electronegativity-based charges
  • Other physicochemical data, such as equilibrium and reaction rate constants, thermochemistry, and liquid densities.
• Formal charges

References

• Frank Jensen. Introduction to Computational Chemistry (2nd ed.). Wiley. ISBN 978-0-470-01187-4.

1. H. Heinz; U. W. Suter (2004). "Atomic Charges for Classical Simulations of Polar Systems". J. Phys. Chem. B. 108: 18341–18352. doi:10.1021/jp048142t.
2. H. Heinz; T. Z. Lin; R. K. Mishra; F. S. Emami (2013). "Thermodynamically Consistent Force Fields for the Assembly of Inorganic, Organic, and Biological Nanostructures: The INTERFACE Force Field". Langmuir. 29: 1754–1765. doi:10.1021/la3038846.
3. F. L. Hirshfeld (1977). "Bonded Atom Fragments for Describing Molecular Charge Densities". Theoret. Chim. Acta (Berl.). 44: 129–138. doi:10.1007/BF00549096.
4. K. C. Gross; P. G. Seybold; C. M. Hadad (2002). "Comparison of Different Atomic Charge Schemes for Predicting pKa Variations in Substituted Anilines and Phenols". Int. J. Quantum Chem. 90: 445–458. doi:10.1002/qua.10108.
5. C. J. Cramer (2002). Essentials of Computational Chemistry: Theories and Methods. Wiley. pp. 278–289.
6. J. Meister; W. H. E. Schwarz (1994). "Principal Components of Ionicity". J. Phys. Chem. 98: 8245–8252. doi:10.1021/j100084a048.
7. T. A. Manz; D. S. Sholl (2012). "Improved Atoms-in-Molecule Charge Partitioning Functional for Simultaneously Reproducing the Electrostatic Potential and Chemical States in Periodic and Nonperiodic Materials". J. Chem. Theory Comput. 8 (8): 2844–2867. doi:10.1021/ct3002199.
8. P. J. Stephens; K. J. Jalkanen; R. W. Kawiecki (1990). "Theory of vibrational rotational strengths: comparison of a priori theory and approximate models". J. Am. Chem. Soc. 112 (18): 6518–6529. doi:10.1021/ja00174a011.
9. Ph. Ghosez; J.-P. Michenaud; X. Gonze (1998). "Dynamical atomic charges: The case of ABO₃ compounds". Phys. Rev. B. 58 (10): 6224–6240. doi:10.1103/PhysRevB.58.6224.