Atomic charges via the Electronegativity Equalization Method: EEM

Synopsis

Description

The Electronegativity Equalization Method (EEM) [1Mortier, W. J.; Ghosh, S. K.; Shankar, S.
J. Am. Chem. Soc. 1986, 108, 4315-4320.] enables the determination of atomic charges that are sensitive to the molecule's topology and three-dimensional structure. It is found to be a very powerful way to obtain ab initio quality atomic charges of different kinds in organic molecules, without the computational cost of the ab initio approach (e.g. Mulliken charges,[13Bultinck, P.; Langenaeker, W.; Lahorte, P.; De Proft, F.; Geerlings, P.; Waroquier, M.; Tollenaere, J. P.
J. Phys. Chem. A 2002, 106, 34, 7887-7894.] CHELPG, MK, NPA, Hirshfeld charges,[14Bultinck, P.; Langenaeker, W.; Lahorte, P.; De Proft, F.; Geerlings, P.; Van Alseboy, C.; Tollenaere, J. P.
J. Phys. Chem. A 2002, 106, 34, 7895-7901.] and AIM charges [15Bultinck, P.; Vanholme, R.; Popelier, P. L. A.; De Proft, F.; Geerlings, P.
J. Phys. Chem. A 2004, 108, 46, 10359-10366.]).

Given the 3D structure of a molecule with N atoms and total charge Q, the Electronegativity Equalization Method (EEM) estimates the partial atomic charges q₁ … q_N and the average molecular electronegativity χ via a set of coupled linear equations:

where r_ij is the distance between atoms i and j, and A_i and B_i are EEM parameters for atom i. The factor k, although originally a unit conversion factor [1Mortier, W. J.; Ghosh, S. K.; Shankar, S.
J. Am. Chem. Soc. 1986, 108, 4315-4320.], has been exploited in some EEM models as an adjustable parameter (e.g. [2Ionescu, Crina-Maria; Geidl, Stanislav; Svobodová Vařeková, Radka; Koča, Jaroslav
J. Chem. Inf. Model 2013, 53, 2548-2558., 3Ionescu, Crina-Maria; Sehnal, David; Falginella, Francesco L.; Pant, Purbaj; Pravda, Lukáš; Bouchal, Tomáš; Svobodová Vařeková, Radka; Geidl, Stanislav; Koča, Jaroslav
J. Cheminform 2015, 7, 50-62.]).

The EEM is based on the electronegativity equalization principle, [4Sanderson, R. T.
Science 1951, 114, 670−672., 5Sanderson, R. T.
Science 1955, 121, 207−208.] which has received theoretical grounding within the density functional theory (DFT), [6 Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E.
J. Chem. Phys. 1978, 68, 3801−3807., 7Politzer, P.; Weinstein, H.
J. Chem. Phys. 1979, 71, 4218-4220., 8Parr, R. G.; Bartolotti, L.
J. Am. Chem. Soc. 1982, 104, 3801-3803., 9Parr, R. G.; Pearson, R. G.
J. Am. Chem. Soc. 1983, 105, 7512−7516., 10Nalewajski, R.
J. Phys. Chem. 1985, 89, 2831-2837., 11Parr, R.; Yang, W.
Density-Functional Theory of Atoms and Molecules.
Oxford University Press: New York, 1989, p. 90.] and which states that the electronegativity of all atoms is equalized throughout a molecule:

Within EEM, the electronegativity χ_i of each atom i in a molecule can be approximated as a linear function of several terms:

The first term is the electronegativity of the isolated atom χ0
i, empirically corrected for the presence of the molecular environment (Δχ_i). The second term is the product between the charge of the atom q_i and the hardness of the isolated atom η0
i, empirically corrected for the presence of the molecular environment (Δη_i). The last term, k ∑
i≠j(q_jr_ij), accounts for the electrostatic interaction with every other charged atom j in the molecule. k is an adjusting factor first introduced by Yang and Shen. [12Yang, Z.; Shen, E.
Science in China. Series B, Chemistry 1995, 38, 521−528.] Setting A_i = χ0
i + Δχ_i and B_i = 2 (η0
i + Δη_i), the molecular electronegativity can be formally expressed as

Additionally, the total molecular charge Q is the sum of all partial atomic charges q_i:

Taken all together, eqs (2), (4), and (5) can be expressed as a system of coupled linear equations, eq (1), from which the partial atomic charges q_i and the molecular electronegativity χ can be calculated, provided that the rest of the terms (Q, r_i,j, k, A_i, B_i) are known.

Remarks

The main limitation of the EEM approach is inherent to its empirical nature. EEM relies on empirical parameters fitted to reference QM data. As such, when employing a particular set of EEM parameters, it is important to consider the nature of the reference QM data, as well as the particular fitting strategy used in the development of the set of EEM parameters. In general, one cannot expect that EEM charges will outperform QM charges.

References

1. "Electronegativity Equalization Method for the Calculation of Atomic Charges in Molecules"
   Mortier, W. J.; Ghosh, S. K.; Shankar, S. J. Am. Chem. Soc. 1986, 108, 4315-4320.
2. "Rapid Calculation of Accurate Atomic Charges for Proteins via the Electronegativity Equalization Method"
   Ionescu, Crina-Maria; Geidl, Stanislav; Svobodová Vařeková, Radka; Koča, Jaroslav J. Chem. Inf. Model 2013, 53, 2548-2558.
3. "AtomicChargeCalculator: interactive web‐based calculation of atomic charges in large biomolecular complexes and drug-like molecules"
   Ionescu, Crina-Maria; Sehnal, David; Falginella, Francesco L.; Pant, Purbaj; Pravda, Lukáš Bouchal, Tomáš Svobodová Vařeková, Radka; Geidl, Stanislav; Koča, Jaroslav J. Cheminform 2015, 7, 50-62.
4. "An Interpretation of Bond Lengths and a Classification of Bonds"
   Sanderson, R. T. Science 1951, 114, 670−672.
5. "Partial Charges on Atoms in Organic Compounds"
   Sanderson, R. T. Science 1955, 121, 207−208.
6. "Electronegativity: The density functional viewpoint"
   Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801−3807.
7. "Molecular Electronegativity in Density Functional Theory"
   Politzer, P.; Weinstein, H. J. Chem. Phys. 1979, 71, 4218-4220.
8. "On the Geometric Mean Principle for Electronegativity Equalization"
   Parr, R. G.; Bartolotti, L. J. Am. Chem. Soc. 1982, 104, 3801-3803.
9. "Absolute hardness: companion parameter to absolute electronegativity"
   Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512−7516.
10. "A Study of Electronegativity Equalization"
    Nalewajski, R. J. Phys. Chem. 1985, 89, 2831-2837.
11. Parr, R.; Yang, W. Density-Functional Theory of Atoms and Molecules. Oxford University Press: New York, 1989, p. 90.
12. "Molecular electronegativity in density functional theory (I)"
    Yang, Z.; Shen, E. Science in China. Series B, Chemistry 1995, 38, 521−528.
13. "The Electronegativity Equalization Method I: Parametrization and Validation for Atomic Charge Calculations"
    Bultinck, P.; Langenaeker, W.; Lahorte, P.; De Proft, F.; Geerlings, P.; Waroquier, M.; Tollenaere, J. P. J. Phys. Chem. A 2002, 106, 34, 7887–7894. DOI: 10.1021/jp0205463
14. "The Electronegativity Equalization Method II: Applicability of Different Atomic Charge Schemes"
    Bultinck, P.; Langenaeker, W.; Lahorte, P.; De Proft, F.; Geerlings, P.; Van Alseboy, C.; Tollenaere, J. P. J. Phys. Chem. A 2002, 106, 34, 7895–7901. DOI: 10.1021/jp020547v
15. "High-Speed Calculation of AIM Charges through the Electronegativity Equalization Method"
    Bultinck, P.; Vanholme, R.; Popelier, P. L. A.; De Proft, F.; Geerlings, P. J. Phys. Chem. A 2004, 108, 46, 10359-10366. DOI: 10.1021/jp046928l

Links

1. AtomicChargeCalculator: Conformationally Dependent Atomic Charges of Quantum Mechanics Quality
2. EEM SOLVER and ABEEM SOLVER, Version 1.0 (accessed November 12, 2015)
3. Electronegativity (Wikipedia) (accessed November 12, 2015)