Partial atomic charges in systems are calculated by generating all valence bond (resonance) structures for this system and then weighting them on the basis of orbital electronegativities and formal considerations (PEPE = Partial Equalization of electronegativity).
First the system of a molecular structure is determined and those atoms are identified that are starting points of +M or M effects. The various resonance structures are then generated starting at the acceptor or donor atoms. One or more topological weights are assigned to the various resonance structures that depend on the changes in the valence bond structure and in the formal charges of the atoms at both ends of a resonance structure. These topological weight factors have been optimized from sets of data on ^{13}C NMR shifts of twelve monosubstituted benzene derivatives, ^{13}C NMR shifts of twelve carbon atoms in nine substituted pyridines, and C1s ESCA shifts of eleven carbon atoms in seven fluorinated olefines.
shift of charge
topological weight = 1
separation of charge
topological weight = 0.5
If the positive and the negative charge are on adjacent atoms the topological factor has only a value of 0.25
recombination of charge
topological weight = 1.0
Scheme 3. Example for the topological weight factor
The total topological weight factor W_{t} consists of three parts:
W_{t} = f_{Q} f_{B} f_{A}
f_{Q}  factor for separation of charge
f_{B}  factor, if the number of covalent bonds is decreased
f_{A}= 0.3  if an aromatic system is destroyed (e.g. by
formation of a quinoid system)
In the next step, the resonance structures are weighted based on consideration of their electronic nature. The electronic weight W_{e} of each resonance structure is derived from electronegativities _{} and the electrostatic influences of neighbouring atoms:
W_{e} = _{} + f_{e} q_{N}
_{} difference of the electronegativities of those atoms that
change its formal charge
f_{e} q_{N} term for the electron repulsion of neighbouring atom
The electronegativities are dependent on atom type, hybridisation state and charge. The following equation is applied:
_{} = a + b q_{} + c q_{}^{2}
Using the product of the topological and the electronic weight the effect of each resonance structure on the charge equalization process is calculated. The charge is moved along the various system to the atoms of the resonance structures thereby changing their electronegativity. Therefore, the process of weighting the structures and shifting electron density has to be repeated in several cycles with decreasing amounts of charge being shifted.
Values calculated
The following quantities are obtained by the PEPE procedure:
q_{,A} (QPI (A)): charge on atom A
_{}_{,A} (ENPI (A)): electronegativity of atom A
q_{,AB} (DQPI (AB)): Difference on the charges on atoms A and B
_{}_{,AB} (ENPI (A)): Difference in the electronegativities of atoms A and B of a bond
_{LP,A} (ENLP (A)): Lone pair electronegativity of atom A
Example
charge separation 
charge separation of

charge separation of
 
decrease of the number

decrease of the
 
w_{t} 
0.5 
0.1625 
0.1625 
Scheme 4. Determination of topological weights
Changes of charges:
atom number 
1 
2 
3 

_{} 
6.690 
8.568 
3.747 
Cycle 1 
q_{} 
0.01 
0.004 
0.005 

_{} 
6.720 
8.290 
3.977 
Cycle 2 
q_{} 
0.003 
0.026 
0.022 

_{} 
6.895 
7.301 
4.802 
Cycle 8 
q_{} 
0.022 
0.119 
0.097 
Contribution of the various resonance structures:
R_{1}: 213 
R_{1}: 12 
R_{1}: 21 

charge shifted 
0.005 
0.001 
0.001 
cycle 1 
0.017 
0.005 
0.005 
cycle 2  
0.016 
0.004 
0.004 
cycle 3  
0.014 
0.004 
0.004 
cycle 4  
0.010 
0.002 
0.002 
cycle 8 
Scheme 5. Process of charge equalization including electronic weighting
Results
The charges calculated by the PEPE method for various heterocycles containing nitrogen atoms were compared with values from a Mulliken population analysis of ab initio STO3G wave functions.
pyridine derivatives:
Pyridine and 2 and 4hydroxy pyridine derivatives show similar charge patterns. The nitrogen and the carbon atoms at the 3 and 5 position have negative charges in the range of 0 and 0.125 e and the atoms at the 2,4 and 6 position have positive charges in the range of 0 and +0.125. The results of the STO3G and the PEPEcalculations correspond with each other to a reasonable degree.
R 
N_{1} 
C_{2} 
C_{3} 
C_{4} 
C_{5} 
C_{6} 
O 

H 
0.13 
0.07 
0.03 
0.06 
0.03 
0.07 
 
ab initio 
0.08 
0.02 
0.00 
0.04 
0.02 
0.00 
 
PEPE  
2OH 
0.13 
0.06 
0.09 
0.07 
0.08 
0.04 
0.131 
ab initio 
0.11 
0.02 
0.13 
0.05 
0.01 
0.02 
0.051 
PEPE  
3OH 
0.01 
0.09 
0.03 
0.03 
0.02 
0.05 
0.115 
ab initio 
0.08 
0.00 
0.00 
0.02 
0.00 
0.01 
0.044 
PEPE  
4OH 
0.07 
0.04 
0.12 
0.07 
0.08 
0.04 
0.125 
ab initio 
0.10 
0.02 
0.01 
0.03 
0.01 
0.02 
0.057 
PEPE 
Table 8. Comparison of charges in pyridine derivatives calculated by PETRA and
STO3G, respectively (ab initio values from: J. E. Del Bene, J. Comp. Chem. 2, 251260 (1981))
pyrimidine derivates
PEOE charges of pyrimidines, which are substituted at the 4position, a show well agreement with the corresponding values from a Mulliken population analysis of STO3G wave functions (see next page):
R 
N_{1} 
C_{2} 
N_{3} 
C_{4} 
C_{5} 
C_{6} 

H 
0.04 
0.01 
0.04 
0.03 
0.01 
0.03 
ab initio 
0.08 
0.04 
0.08 
0.06 
0.00 
0.06 
PEPE  
Me 
0.06 
0.03 
0.07 
0.07 
0.02 
0.04 
ab initio 
0.08 
0.04 
0.08 
0.06 
0.00 
0.06 
PEPE  
NH_{2} 
0.12 
0.05 
0.15 
0.09 
0.09 
0.06 
ab initio 
0.11 
0.04 
0.12 
0.06 
0.03 
0.06 
PEPE  
OH 
0.09 
0.05 
0.14 
0.06 
0.05 
0.06 
ab initio 
0.10 
0.04 
0.11 
0.05 
0.01 
0.06 
PEPE  
F 
0.07 
0.04 
0.10 
0.03 
0.04 
0.06 
ab initio 
0.09 
0.04 
0.11 
0.05 
0.01 
0.06 
PEPE  
CH=CH_{2} 
0.16 
0.03 
0.07 
0.05 
0.01 
0.04 
ab initio 
0.09 
0.04 
0.10 
0.06 
0.00 
0.06 
PEPE  
CHO 
0.04 
0.02 
0.05 
0.03 
0.02 
0.04 
ab initio 
0.07 
0.04 
0.08 
0.05 
0.02 
0.06 
PEPE  
CN 
0.03 
0.03 
0.04 
0.01 
0.03 
0.05 
ab initio 
0.07 
0.04 
0.08 
0.04 
0.01 
0.07 
PEPE 
Table 9 Comparison of charges in pyrimidine derivatives calculated by PETRA and
STO3G, respectively (ab initio values from: J. E. Del Bene, J. Comp. Chem. 2, 251260 (1981))
Scope and Limitations
The dependence of the electronegativity on charge is parameterized for the following atoms.
(The functions are dependent on the hybridization states)
Applications
A detailed discussion will be given in the next section using the results of the calculations on both the  and charges.
References