Essentials of Computational Chemistry

9.1 PROPERTIES RELATED TO CHARGE DISTRIBUTION 321
The Mayer (1983) bond order is defined as
Bkk ′ =
µ∈k ν∈k ′
(PS)µν(PS)νµ
(9.26)
where P and S are the usual density and overlap matrices, respectively. This definition of bond
order proves to be quite robust across a wide variety of bonding situations (for an example of
its use to settle a controversy over alternative modes of bonding in a silylpalladium complex,
see Sherer et al. 2002).
CM2 and CM3 charges are then defined as
q CM2/CM3
k
= q (0)
k
+
k=k ′
Bkk ′(CZkZ k ′ + DZkZk ′ Bkk ′) (9.27)
where C and D are model parameters specific to pairs of atoms (as opposed to individual
atoms, as in CM1), and charge normalization is assured simply by taking
CZkZ k ′ =−CZ k ′ Zk and DZkZ k ′ =−DZ k ′ Zk (9.28)
CM2 and CM3 mappings have to date been defined for many different levels of theory,
including AM1, PM3, SCC-DFTB, HF, and DFT.
Table 9.1 provides several molecular dipole moments as computed by a variety of different
charge models and electronic structure methods, and compares them to experiment. The
expectation value of the dipole moment operator evaluated for MP2/6-31G(d) wave functions
has an RMS error compared to experiment of 0.21 D. The same expectation value at the HF
level shows the expected increase in error from the tendency of the HF level to overestimate
dipole moments. Dipole moments computed using Eq. (9.22) and ESP charges have about
the same accuracy as the operator expectation value (indeed, it is possible to constrain the
ESP fit so that the expectation value of the dipole moment is exactly reproduced; of course,
this is not necessarily desirable if one knows the expectation value to suffer from a systematic
error because of the level of theory). Eq. (9.22) used with either Mulliken or NPA charges
shows rather high errors. Indeed, the error associated with the NPA charges is larger than
the dispersion in the data (the dispersion is the RMS error for the simple model that assumes
every dipole moment to be 2.11 D, which is the mean of the experimental data). At the PM3
level, not only are the Mulliken charges rather bad, but the expectation value of the dipole
moment operator is not particularly good either. However, the CM1 mapping corrects for
the errors in the PM3 electronic structures sufficiently well that the RMS error for the CM1P
model is lower than that for the MP2 expectation value. The CM1 model with the AM1
Hamiltonian, the CM2 model for the BPW91/MIDI! level of theory, and the CM3 model
for a tight-binding DFT level also do well. Note also that the CMx models, the last four
columns of the table, represent the four fastest methodologies listed.
Jakalian, Jack, and Bayly (2002) have described a scheme similar in spirit to the CMn
models insofar as AM1 charges are corrected in a bond-dependent fashion. In their AM1
bond charge corrections (AM1-BCC) model, however, each bond is assigned by the chemist