Essentials of Computational Chemistry

9.1 PROPERTIES RELATED TO CHARGE DISTRIBUTION 311
(cf. Eq. (9.7)), and f is a damping factor raised to the nth power. New electronic populations
are then computed according to
Q (n)
k
= Q(n−1)
k −
Q (n)
k→k ′ + Q (n)
k ′ →k
k ′ bonded
to k
χ k ′ >χk
k ′ bonded
to k
χk>χ k ′
(9.9)
In the original PEOE method, the damping function f was simply taken to be the constant 0.5,
which led to practical convergence in the atomic partial charges within about five iterations.
No et al. (1990a, 1990b) subsequently proposed a modification in which different damping
factors were used for different bonds (MPEOE) and observed that this, together with some
other minor changes, gave improved charge distributions when compared to known multipole
moments. More recently, Cho et al. (2001) proposed computing the damping factor as
fkk ′ = min
0,
1 −
r vdw
k
rkk ′
+ rvdw
k ′
(9.10)
where rkk ′ is the distance between the two atoms and rvdw is a parametric van der Waals
radius. The use of Eq. (9.10) delivers geometry-dependent atomic charge (GDAC) values,
which were found to improve additionally on computed electrical moments.
Another Class I charge model that is also sensitive to geometry is the QEq charge equilibration
model of Rappé and Goddard (1991). From representing the energy u of an isolated
atom k as a Taylor expansion in its charge truncated at second order, one can derive
uk =ũk + χkqk + 1
2 Jkkq 2 k
(9.11)
where ũ is the energy of the neutral isolated atom, χ is the electronegativity (experimentally
the average of the atomic IP and EA), and J is the idempotential, which is formally equal
to IP − EA. With this formula in hand, we may write the electrostatic energy of a collection
of N atoms as
N
U = (ũk + χkqk) + 1
N N
2
k=1
k=1 k ′ Jkk ′qkqk ′ (9.12)
=1
where J is a matrix of Coulomb integrals for which we have already defined the diagonal
elements as the idempotentials. The off-diagonal elements are computed as (aa|bb)
where a and b are STOs on the centers k and k ′ , respectively (thereby introducing geometry
dependence). QEq charges q are then determined from minimization of U subject to
the constraint that the total molecular charge remain constant. Note the close conceptual
similarities between QEq and SCC-DFTB described in Section 8.4.4.
Eq. (9.12) does not require any specification of bonding – all atoms electrically interact
with all other atoms. Sefcik et al. (2002) have combined QEq electrostatics with Morse
potentials for non-electrostatic non-bonded interactions between all atom pairs to create a