6.2 Electronic Structure Theory of CRK
127
Fig. 6.2 Schematic picture
of the electrostatic potential
at R G (n)
R R G (n)
Q a
Z c
R R (c)
R
N
(r)
(a)
{R G (n)} to evaluate the electrostatic potential in outer region of the molecule. The
locations of {R G (n)} are arbitrary, though they are usually distributed evenly on the
van der Waals envelope (green line in Fig. 6.1) or in the outer region. The electrons
and nuclei of the molecule generate the electrostatic potential φ(R G (n)) at the point
R G (n) by
φ(R G (n)) = −e
ρ(r)
|r − R G (n)|
dr +
nuc
c
Z c e
|R N (c) − R G (n)|
= −e
AO
p,q
D pq
p|
1
|r − R G (n)|
|q
+
nuc
c
Z c e
|R N (c) − R G (n)|
,
(6.10)
where r is the electron coordinate, and ρ(r) is the electron density at r. The matrix
element
p|
1
|r − R G (n)|
|q
is equivalent to the conventional one-electron integral
for the nuclear attraction in the quantum chemistry, by replacing R G (n) with the
nuclear coordinate.
Instead of the quantum chemical formula of Eq. (6.10), the electrostatic potential
at R G (n) is alternatively represented using a set of partial charges {Q a } located
at R(a). The electrostatic potential is presented with the model of partial charges
{Q a } by
φ
model (R G (n)) =
site
a
Q a
|R(a) − R G (n)|
.
(6.11)
(We again note that R(a) is arbitrary in the above formula, though it is often located
at the nuclear position, R N (c).) The set of ESP charges {Q a } are determined so
that φ model (R G (n)) in Eq. (6.11) coincides with φ(R G (n)) in Eq. (6.10) as much as
possible. This procedure of optimization of {Q a } is carried out with the least square
fitting that minimizes the following square displacement L,
L({Q a }) =
grid
n
φ
model (R G (n)) − φ(R G (n))
2
.
(6.12)