Fractional "atomic" charges
The concept of polar covalent bond between two different "atoms" of a molecule is of paramount importance for the discussion of molecular properties and chemical reactions. Its electrostatic moments, most noted the "dipole moment", are well defined, measurable and computable for a part or an entire molecule. However, the simplified view of expressing this by partial charges attributed to the "atoms" of a bond has no unique definition. Such charges are not observable and their computation from ground state electron distributions needs arbitrary, non-physical, assumptions.
The reason for this is that the idea of "atoms in molecules" is tenuous. Every unique atom has a large number of properties, well understood and observable, also computable and, for the lighter ones, with good accuracy. When a free atom becomes part of a molecule it looses many of its properties, and we can only uniquely describe the system, at least for the "valence electrons", as an ensemble of nuclei (or, less exactly, as "atomic" cores) in a molecular electron density distribution. This can be probed and measured, e.g. by X-ray diffraction, and computed by quantum mechanics. However, it is not possible to extract a unique value for atomic polarization from experiment or theory, because we don't know how to divide molecular electron density into parcels "belonging" to individual constituting "atoms".
[In Quantum Chemistry every unique property is defined by an operator. When this is applied to the wavefunction, an expectation value for that property can be determined. Many popular properties of chemistry do not have QC operators. Among them are fractional charges on "atoms", Baeyer ring strain, aromaticity, ionic radii, electronegativity, molecular structure and others. For them there is no unique definition nor measurement procedure available. Hence, these concepts are objects of lively and never ending discussions in scientific meetings ...]
In order to help the chemists' assessment of polarity of a bond by traditional partial charges many approximate recipes have been invented. Most electron structure [computing] systems (ESS) offer various schemes for doing this. For a critical review, see Frank Jensen, "Introduction to computational chemistry", 2nd ed, chapter 9, John Wiley, N.J, 2006.
Very popular are the partial charges computed by R.S. Mulliken's and P.-O. Löwdin's [electron] population analyses. Other authors are R.F.W. Bader, F. Weinhold, I. Mayer and many more, see references at end.
Most of the partial charges computed are strongly dependent on the basis functions used. Often the most complete calculation gives the least understandable result, e.g. the "atomic (Mulliken)charge" of the C-atom in CH4 varies from +0.55 to -1.18 for two quite large basis sets (aug-cc-pVDZ vs. aug-cc-pVTZ)! For O in H2O, values of +0.93 to -0.46 are found for Löwdin charges with different bases. What do you want to believe? (Tables 9.1 and 9.2 on pages 312, 313 of Jensen's book, cited above).
Fractional charges estimated by Kimball's model
Although fractional "atomic charges" in molecules have no scientific foundation (i.e. are not falsifiable) they enjoy a traditional value. Practical chemists can draw meaningful conclusions from their values. They play an important role in empirical force field methods used for molecular modeling. Therefore, we also like to present a recipe based on a simple reflection applied to Kimball's model.
We begin with a single bond between C-C, C-N, C-O, and C-F as it is known for CH3CH3, CH3NH2, CH3OH, and CH3F. In order to share the two electrons of the bond between unequal partners we try the following scheme: The bonding spherical Kimball cloud is divided by a plane normal to the bonding direction at a point where a charge element q is equally attracted by either atomic core, C1,C2: q*C1/d12 = q*C2/d22; hence, the distances d1, d2 of the core centers to that probe charge behave like
on the bond axis gives the result: 1.135 of the electron pair are controlled by N and 0.865 by C. Hence, the partial charge on N is -0.135, on C +0.135, compared to equal sharing in a non polar bond. The same treatment of the other cases and Methylborane CH3BH2 is summarized in this table, together with the values obtained with Mulliken's and Löwdin's model (computed with Gamess-US (14 Feb 2018 (R1)), RHF/6-31G(d,p), optimized):
Our numbers are reasonable, i.e. have values and sign a chemist would expect and apply. They are very easily evaluated with Kimball's model. The same holds true for the values of Electronegativity. If you don't need or don't trust quantitative partial charges, differences in electronegativity between the bonded "atoms", often depicted as small charges "δ+" and "δ-", may give you enough guidance to assess the polarity of a chemical bond.
How to estimate a fractional charge for H "atoms"
The recipe described above presupposes an electronic core of the bonded pair of "atoms". This does not exist for the proton. This nucleus is always imbedded in a bonding electron cloud. Hence, a division of an X-H bond into "atomic" fractions becomes even more arbitrary. Linus Pauling also faced a problem, when he tried to determine the electronegativity of hydrogen, see his book "The nature of the chemical bond", 1939. Robert Mulliken's definition: EN equals the average of the sum of ionization energy and (positive taken) electron affinity, gives EN(H)=2.85, EN(C)=2.51, in Paulings scale; i.e. H is more electronegative than C. Pauling found EN(H)=2.2, EN(C)=2.5, i.e. reversed signs! Pauling's values "explain" the usual signs for partial charges in hydrocarbons with H "δ+", C "δ-", as found by Mulliken's population analyses.
In order to use the same recipe as above we move the proton along the bond direction to the rim of the bonding cloud and enhance its charge such that the force on the probe charge is the same as from the correct location. Thus we can avoid to divide the charge within the cloud other than by the plane of equal attraction. This is what we obtain or comparison with Mulliken and Löwdin:
The absolute values are all a bit smaller than Mulliken's but show the same trend. BTW it is known that Mulliken's values are high and not constant for the different C/H "atoms" in normal hydrocarbons, see below. That explains a.o. the differences in chemical reactivity of the end-CH3 groups compared to the -CH2- groups, e.g. in C5H12 (opt. with RHF/6-31G(d,p)):