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.
The EEM approach 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. [Mulliken: J. Phys. Chem. A 2002, 106, 7887-7894. CHELPG, MK, NPA, Hirshfeld: J. Phys. Chem. A 2002, 106, 7895-7901. AIM: J. Phys. Chem. A 2004, 108, 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 … qN and the average molecular electronegativity χ via a set of coupled linear equations:
where ri,j is the distance between atoms i and
j, and Ai and Bi 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.]).
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:
|χ1 = χ2 = … = χi = … = χ||(2)|
Within EEM, the electronegativity χi of each atom i in a molecule can be approximated as a linear function of several terms:
|χi = (χi0 + Δχi) + 2 (ηi0 + Δηi) qi + k N ∑ i≠j qj / ri,j||(3)|
The first term is the electronegativity of the isolated atom
χi0, empirically corrected for the presence
of the molecular environment (Δχi).
The second term is the product between the charge of the atom
qi and the hardness of the isolated atom
ηi0, empirically corrected for the presence
of the molecular environment (Δηi).
The last term k
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 Ai = χi0 + Δχi and Bi = 2 (ηi0 + Δηi), the molecular electronegativity can be formally expressed as
|χ = Ai + Biqi + k N ∑ i≠j qj / ri,j||(4)|
Additionally, the total molecular charge Q is the sum of all partial atomic charges qi:
|Q = N ∑ i=1 qi||(5)|
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 qi and the molecular electronegativity χ can be calculated, provided that the rest of the terms (Q, ri,j, k, Ai, Bi) are known.
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.