Chemistry Reference
In-Depth Information
The particle charge
z
has been added to show that these functions depend
strongly on the charge. The equation is discretized at the point
r
i
by employing
Gauss's law for a small cube of volume
V
h
3
, where
h
is the grid spacing:
¼
h
X
6
V
Þ
c
ð
r
i
þ
hj
Þ
c
ð
r
i
Þ
h
1
2
½
y
ð
r
i
þ
hj
Þþ
y
ð
r
i
¼
0
½
57
¼
1
J
where the vectors
j
are directed to the six faces of the cube. The right-hand side
is not zero on coarser levels in an FAS solver due to the defect correction. If Eq.
[55] and [56] are substituted into Eq. [57], and the resulting equation solved
for the density, the following expression is obtained:
h
n
o
i
r
ð
r
i
þ
P
6
z
kT
ð
f
ð
r
i
þ
1
þ
exp
hj
Þ
f
ð
r
i
ÞÞ
hj
Þ
j
¼1
r
ð
r
i
Þ¼
½
58
h
n
o
i
P
6
z
kT
ð
f
ð
r
i
þ
1
þ
exp
hj
Þ
f
ð
r
i
ÞÞ
j
¼
1
This formula involves only the exponential of
differences
of the potential
(between neighboring grid points), and thus it is relatively well behaved. So
long as the potential is relatively smooth, a further approximation can be
made:
h
i
P
6
z
2
kT
ð
f
ð
r
i
1
þ
þ
hj
Þ
f
ð
r
i
ÞÞ
r
ð
r
i
þ
hj
Þ
j
¼
1
r
ð
r
i
Þ¼
½
59
h
i
P
6
z
2
kT
ð
f
ð
r
i
þ
1
hj
Þ
f
ð
r
i
ÞÞ
j
¼
1
Recall that a term for the defect correction must be added to the numerator for
coarse-grid iterations in an FAS V-cycle. On the finer grids, Eq. [59] works
well since the potential does not vary much between grid points. On coarser
grids, however, the linearization may not be feasible, and use of Eq. [58] may
be necessary instead (this, of course, depends on the actual physical problem
being examined). For the steady-state case, Eqs. [58] and [59] are also possible
choices for the interpolation operator. Successive overrelaxation (SOR) was
utilized by us for the relaxation steps, and the old and new charge densities
were mixed with a weighting factor during the self-consistent updates to
ensure stability.
The above-described algorithm was developed and tested successfully on
several simple model systems. The method was further tested on the more
challenging gramicidin A channel; our results correspond well with the pub-
lished results in Refs. 154 and 155. The main conclusion from our work is
that the FAS-MG approach leads to one order-of-magnitude improvement in
