Chemistry Reference
In-Depth Information
During the solution process, the progress can be monitored by comput-
ing the residual
r
¼
f
Lu
½
15
and plotting its magnitude averaged over the domain. As the residual is
reduced in magnitude toward 0, we approach the exact solution on the grid
(but not the exact solution of the PDE, since there are numerical errors in
our FD approximation.) The lowercase
u
is used to signify the
current approx-
imation
to the exact solution
U.
This chapter is directed at describing a new development in computa-
tional chemistry, but it should be noted that, already in Pauling and Wilson's
classic quantum chemistry text from 1935,
101
there is a section on variational
methods that includes a discussion of FD approximations for the Schr ¨ dinger
equation!
Finite-Element Representations
Here again we consider a simple Poisson problem, but the same general
procedure can be applied to the eigenvalue or total energy minimization pro-
blems. As we discussed above, one approach is to minimize the action func-
tional of Eq. [3], using an FD representation of the Laplacian inside the
integral (or summation on a grid when the integral is discretized) and specified
boundary conditions (e.g., fixed or periodic). Alternatively, we can integrate
by parts the action term involving the Laplacian in Eq. [3] to obtain
ð
2
dx
ð
1
2
d
dx
S
½
f
¼
4
p
rf
dx
½
16
assuming that the values of the potential or its derivative vanish on the bound-
aries.
65
We can then minimize this alternate form of the action with respect to
variations in the potential
.
98
Once the minimum is reached, the potential
f
f
is the solution. If we assume that we have that minimizing solution
f
,an
approximate form of the potential can be represented as
v
, where
v
is
any function that vanishes on the boundaries. If we substitute this general
form into Eq. [16], take the derivative with respect to
f
þ
e
e
, and set
e
equal to
0, we obtain the following variational expression:
dv
dx
dx
ð
ð
d
dx
¼
4
p
r
vdx
½
17
If Eq. [17] is true for all arbitrary
v
, then
satisfies Poisson's equation. This
equation is called the variational boundary value problem for the Poisson
f
