On fractional charge in molecules and materials

Abstract

Determining the charge on an atom in molecules and materials is a long-standing open issue. Quantitative analysis with clear physical picture of charge transfer, electrostatic and covalent contributions to bond energy is highly desirable to advance the understanding of both charge and ionicity. However, the consensus among computations from quantum mechanics to semiempirical methods and state-of-the-art experimental measurement on fractional charge remains elusive. In this work, we generalize Born–Haber thermochemical cycle to compute fractional charge on an atom by delicately balancing the interplay of all interacting components, including charge transfer, electrostatic–covalent interactions and bond energy in molecules. The fractional charge of an atom in solid materials can be also computed straightforwardly with this scheme.

Appendix

1.1 A graphic solution of Eq. (13)

The Eq. (13) can be solved graphically by plotting algebraic functions at both sides as a function of fractional charge. The point of intersection yields the fractional charge which optimized charge transfer, ionic, polarization and covalent interactions in bond energy. The same scheme can be applied to solve Eq. (11) to compute fractional charge of an atom in solid materials (Fig. 3).

Fig. 3

A graphic method to determine fractional charge $q$ by searching the intersection point of two curves. Both curves represent fractional charge-dependent binding energy of diatomic molecules

1.2 An approximate solution of Eq. (17)

Assuming the total polarization of the anion and cation can be approximated as:

$$\alpha^{*} = \alpha_{m, + Q} + \alpha_{n, - Q}$$

(22)

Equation (13) is approximated into the following quadratic one:

$$\left( {E_{\text{i}} \left( Q \right) - \alpha^{*} E^{2} - E_{\text{c}} } \right)q^{2} + \left( {{\text{IP}} + {\text{EA}}} \right)q + E_{\text{c}} + E_{\text{bond}} = 0$$

(23)

The solution of this equation is

$$q = - \frac{1}{2}\left( {\frac{{{\text{IP}} + {\text{EA}}}}{{E_{\text{i}} \left( Q \right) - \alpha^{*} E^{2} - E_{\text{c}} }}} \right) + \frac{1}{2}\sqrt {\left( {\frac{{{\text{IP}} + {\text{EA}}}}{{E_{\text{i}} \left( Q \right) - \alpha^{*} E^{2} - E_{\text{c}} }} } \right)^{2} - 4\left( {\frac{{E_{\text{c}} + E_{\text{bond}} }}{{E_{\text{i}} \left( Q \right) - \alpha^{*} E^{2} - E_{\text{c}} }}} \right) }$$

(24)

The approximate solution in Eq. (24) explicitly highlights that the fractional charge results from the delicate balance among charge transfer, electrostatic–covalent interactions and bond energy in molecules.

1.3 Calculations by density functional theory-based methods

All geometric optimization and energy calculations were performed by Gaussian 09 [89] (RevA.02) using B3LYP functional [90,91,92] and DGDZVP basis set [93, 94]. Optimizations were performed with cut-off values for maximum force of 0.00015, root-mean-square (RMS) force of 0.0001, maximum displacement of 0.0018 and RMS displacement of 0.0012 (all in atomic unit). The bond energy was calculated by E_bond = (E_M + E_X) − E_MX; electron affinity (EA) was calculated by $\text {EA}=E_{{X}^{-}}-E_X$; and ionization potential (IP) was calculated as IP = $\text {IP}=E_{{M}^{+}}-E_M$.