The concept of “partial ionic character” has proved exceedingly valuable and fruitful as a qualitative description of the electronic structure of molecules; however, the theoretical ideas underlying the empirical relations for the determination of the ionic character of a bond do not bear critical examination. Even Mulliken's population analysis fails in many cases as a means of interpreting quantum-chemical calculations. On the other hand, a graphical representation of the spatial charge density distribution allows an intuitive interpretation even of complicated calculations, and (possibly in combination with an analysis of the forces acting on the atomic nuclei) yields very much farther-reaching information on the polarity of a bond than can be expressed by the values of some parameters on the basis of excessively simplified models.
21Hannay and Smyth based their equation on the dipole moment of HF, which was determined by them for the first time, and which corresponds to an ionic character IHF = 43%, whereas the value estimated by Pauling was IHF = 60%.
29
Only these two states can be obtained as the solution of the Schrödinger equation if a wave function with a covalent and an ionic component is used.
33Ruedenberg [34] refers to the components qμ and 1/2 Σqμv in eq. (14) as the “valence inactive fraction” and the “valence active fraction”, respectively, of the formal charge, since a chemical bond is produced by an accumulation of charge between the atoms.
37
The wave function of a molecule can be expanded in terms of any complete set of basis functions, i.e. also in terms of AOs that are all centered on the same atom A. One-center expansions of this kind have been successfully carried out for molecules with a heavy central atom, such as CH4 or NH3 [38]; results of any desired accuracy can in principle be obtained in this way for physical quantities such as energy and dipole moment, and also for the density function P(1). The population analysis, on the other hand, gives the formal charge pA = N for the atom A and pR = 0 for all other atoms R. According to Mulliken, therefore, a one-center expansion is a physically balanced but formally unbalanced basis.
42
Not only does this emerge from the example of the two Hartree-Fock calculations for the HF molecule (Section 3.2), but it can also be shown in general [41] that the Hartree-Fock density function differs from the true density function at most in terms of the second order of smallness. Bader and Chandra [43] have also shown explicity for the H2 and Li2 molecules how small the changes in P(1) are when one goes beyond the Hartree-Fock approximation.
53
The deviation from zero of the force FA acting on the atomic nuclei A is a measure of how well a given wave function satisfies the Hellmann-Feynman theorem. Since the Hellmann-Feynman theorem is valid not only for the exact wave function but also for the Hartree-Fock function [50], the accuracy with which a given wave function satisfies the Hartree-Fock approximation can be established on the basis of the fA and fB values.
54
It is interesting to compare the bonding in LiH and in HF, which comes closest to the ionic limiting case A−H+. If as in the case of LiH the overlap component of the force acting on the cation is added to the corresponding screening contribution for HF, the resulting value (8.988) is smaller than the nuclear charge ZF = 9, whereas one would expect ZF + 1 = 10 for an ionic bond. While the bond in LiH may be described as ionic with a small covalent component, therefore, the bonds in the series NH, OH, and HF must be regarded as covalent with increasing ionic character.
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