The notion of partial charges on the atoms of a molecule is widely used by the chemist. However, it should be kept in mind that this concept is a rather crude reflection of the electron distribution in a molecule and that it has no theoretical foundation. Nevertheless, this model of assigning a nonuniform electron distribution to the individual atoms giving them partial charges has been quite useful.
Partial atomic charges are calculated from orbital electronegativities by consideration of the bond structure (connectivity) of the molecule.
The basis of our approach is the electronegativity concept, electronegativity, , as defined by Mulliken as the mean of the ionization potential and the electron affinity:
= 0.5 (IP + EA)
The various orbitals of an atom are considered separately; each orbital having its own electronegativity value, _{i.} Thus, electronegativity is dependent on the hybridization state of an atom. The initial orbital electronegativity values are those determined by Hinze and Jaffe (references: J. Am. Chem. Soc. 84, 540 (1962); ibid. 85, 148 (1963); J. Phys. Chem. 67, 1501 (1963)).
Electronegativity is not only dependent on the type of orbital being considered but also on its occupation number (n=0,1,2). Electronegativity values can be obtained from the data given by Hinze and Jaffe (see above) for the neutral state (n=1) and for states with a unit positive (n=0) and negative charge (n=2). The occupation of an orbital can be considered as a continous variable, i.e. a continous range of partial charge, q_{A}, in an atom is allowed. With three values fixed for the dependence of orbital electronegativity on charge (n=0,1,2 corresponding to q = +1, 0, 1), a polynomial of degree 2 can be fitted for the dependence of the electronegativity of an orbital _{iA} on the charge, q_{A}, on an atom.
_{iA} = a_{i} + b_{i}q_{A} + c_{i}q_{A}^{2}
Figure 4. Dependence of the electronegativity of an orbital i of atom A on the
charge q_{A}
The three values _{i}^{+}, _{i}^{0}, _{i}^{} allow the determination of the three coefficients, a_{i}, b_{i}, c_{i}, of the dependence of orbital electronegativity on charge.
On bond formation, electron density is transferred from the less electronegative atom, A, to the one, B, with higher electronegativity. This gives atom A a positive charge, thus increasing its electronegativity. Conversely, atom B with higher electronegativity becomes negatively charged, thereby decreasing its electronegativity. In effect, the electronegativities of the atoms bonded together tend to equalize.
But they do so only partially. The electron transfer creates an electrostatic potential that acts against further electron transfer. These ideas form the essence of the method of Partial Equalization of Orbital Electronegativity (PEOE). Partial Equalization of Orbital Electronegativity is realized by an iterative procedure.
This procedure is graphically illustrated by the following figure and scheme.
Figure 5. Iterative procedure for Partial Equalization of Orbital Electronegativity (PEOE)
Step. 1: for each atom A and orbital i
_{iA} = a_{i} + b_{i}q_{A} + c_{i}q_{A}^{2}
Step. 2: for each bond AB
'q_{AB}^{<n>}= (_{iA}^{+})^{1} (_{iA}  _{iB}) _{* }^{n}
q_{AB} = q_{AB }+ q_{AB}^{<n>}
Step 3: for all bonds to an atom
q_{A}^{<n>} = q_{AB}^{<n>}
q_{A} = q_{A} + q_{A}^{<n>}
If n < n_{max} go to Step 1
is a damping factor set to 0.5;
n is the number of the current iteration
Scheme 2. PEOEAlgorithm
The damping mechanism embodied in the factor ^{n} ensures rapid convergence of the procedure; the number of iterations through the loops is set to 10.
The method has been extended to small ring compounds where sizeable changes in hybridization states occur. In this case, the amount of s and pcharacter has to be taken into account in the initial values of the electronegativities (ref. 3).
Values calculated
The following quantities are obtained by the PEOE procedure:
q_{A}_{,} (QSIG(A)):
At the end, for each atom A of a molecule a unique value for its partial charge, q_{A}, is obtained.
_{A,} (ENSIG(A)):
Due to the dependence of electronegativity on charge, this charge q_{A} corresponds to a specific value of the electronegativity of this atom, _{A}.
q_{AB,}_{} (DQSIG (AB)):
Difference in the charges on atoms A and B of a bond.
_{AB,} (DENSIG (AB)):
Difference in the electronegativities of atoms A and B of a bond.
Q_{AB,}_{} (SQIT (AB)):
(Sum of charges (Q) shifted in the ITerations)
The amount of charge shifted across a bond, Q^{}, in the course of all iterations is obtained as an additional parameter. It was found that this value can be taken as a measure of the polarity of a bond, being more characteristic than the difference in the total charges on the two atoms of the bond.
Example
The change in the charges during the iterations is illustrated with fluoromethane as an example.

Figure 6. Fluoromethane as example for the PEOEprocedure
Results
The dissection of the electron distribution in a molecule and its assignment to individual atoms is a drastic oversimplification (where do I have to make the cut in a bond between two atoms?). There is no theoretically sound criterion for the definition of partial atomic charges. Therefore, comparison of the values obtained by our method was made with values calculated by other methods and with physical and chemical data.
A Mulliken population analysis (MPA) is the most widely used quantum mechanical method for the derivation of partial atomic charges. In spite of its known weaknesses (improper handling of the overlap population; heavy dependence on the basis set) it is still very much in use.
For an unambiguous comparison of the results of the PEOE method a set of molecules was chosen for which both theoretical and experimental data were available in the literature. As theoretical data, values on atomic charges from a Mulliken population analysis on STO3G wave functions were selected. As experimental data C1s core electron binding energies as obtained from ESCA measurements were chosen for they are known to depend directly on the valence electron distribution.
Table 7 gives a comparison of the PEOE charges with those from the Mulliken population analysis and the C1s core electron binding energy shifts.
PEOE 
ab initio 
C1s  
1 
CH_{4} 
78 
7 
0 
2 
CH_{3}CH_{3} 
68 
26 
0.2 
3 
CH_{2}=CH_{2} 
106 
156 
0.1 
4 
HC=CH 
122 
182 
0.4 
5 
CH_{3}F 
79 
169 
2.8 
6 
CH_{2}F_{2} 
230 
383 
5.6 
7 
CHF_{3} 
380 
532 
8.28 
8 
CF_{4} 
561 
674 
11.0 
9 
*CH_{3}CH_{2}F 
37 
58 
0.2 
10 
CH_{3}*CH_{2}F 
87 
209 
2.4 
11 
*CH_{3}CF_{3} 
39 
99 
1.1 
12 
CH_{3}*CF_{3} 
387 
546 
7.6 
13 
CH_{3}OH 
33 
136 
1.6 
14 
CH_{3}OCH_{3} 
36 
161 
1.4 
15 
H_{2}CO 
115 
167 
3.3 
16 
*CH_{3}CHO 
9 
61 
0.6 
17 
CH_{3}^{*}CHO 
123 
211 
3.2 
18 
*CH_{3}*COCH_{3} 
6 
64 
0.6 
19 
CH_{3}COCH_{3} 
131 
260 
3.1 
20 
HCN 
51 
70 
2.6 
21 
*CH_{3}CN 
3 
62 
2.1 
22 
CH_{3}*CN 
60 
21 
2.1 
Table 7. Partial atomic charges by the PEOE method and a Mulliken population analysis on STO3G wave functions (all in millielectronunits) as well as C1s core electron binding energen (in eV).
The PEOE charges show a fairly strong correlation with the MPA charges with a correlation coefficient of 0.939 (ref. 2). The correlation passes through the zeropoint and has a slope of 1.31 with the MPASTO3G charges having the larger values.
q_{MPA} = 1.31 _{*} q_{PEOE}
To further explore the quality of the two sets of charge values, a comparison was made with the C1s core electron binding energy shifts for the same set of compounds. ESCA shifts are those experimental data probably most closely related to the valence electron distribution.
The PEOE charges give a correlation with the ESCA shifts of 0.987 and a standard deviation of 0.27 eV. The Mulliken population analysis charges (STO3G) show a correlation coefficient of 0.938 and a standard deviation of 0.64 eV. This study showed that the PEOE charges can reproduce the experimental data of C1s ESCA shifts better than the charges from the Mulliken population analysis on STO3G wave functions.
More extensive comparisons of the charge values with theoretical and experimental data are reported for total charges after the calculation of charges has also been presented (next section).
Scope and Limitations
Presently, the parameters for the initial electronegativity values and their charge dependence have been included for the following types of atoms:
C, H, O, N, B, S (II), P (III), F, Cl, Br, I, Li, Be, Na, Mg, Al, Si, Ti.
Applications
Dipole Moments Obtained by Iterative Partial Equilisation of Orbital
Electronegativity
J. Gasteiger, M. D. Guillen
J. Chem. Research (S) 1983, 304305; (M) 1983, 26112624
Prediction of Proton Magnetic Resonance Shifts: The Dependence on Hydrogen Charges obtained by Iterative Partial Equalization of Orbital Electronegativity
J. Gasteiger, M. Marsili
Org. Magn. Resonance 15, 353360 (1981)
Residual Electronegativity  An Empirical Quantification of Polar Influences and its Application to the Proton Affinity of Amines
M. G. Hutchings, J. Gasteiger
Tetrahedron Lett. 24, 25412544 (1983).
These include:
as well as
See reference:
Quantitative Models of GasPhase Proton Transfer Reactions Involving Alcohols, Ethers, and their Thio Analogs. Correlation Analyses Based on Residual Electronegativity and Effective Polarizability
J. Gasteiger, M. G. Hutchings
J. Amer. Chem. Soc. 106, 64896495 (1984)
and
See reference:
A Quantitative Description of Fundamental Polar Reaction Types. Proton and Hydride Transfer Reactions Connecting Alcohols and Carbonyl Compounds in the Gas Phase
M. G. Hutchings, J. Gasteiger J. Chem. Soc. Perkin 2, 1986, 447454
and
See reference:
Correlation Analyses of the Aqueous Phase Acidities of Alcohols and GemDiols, and of Carbonyl Hydration Equlibria, using Electronic and Structural Parameters
M. G. Hutchings, J. Gasteiger
J. Chem. Soc. Perkin 2, 1986, 455462
References: