Chemistry Reference
In-Depth Information
Next, we replace the denominator on the left side
and call this the
time-step size. Eq. [23] can then be written as an update equation:
ð
d
t
Þ
by
t
x
i
Þþ
h
2
½
f
n
þ
1
n
n
n
n
f
ð
x
i
Þ¼
f
ð
ð
x
i
1
Þ
2
f
ð
x
i
Þþ
f
ð
x
i
þ1
Þþ
4
ptr
ð
x
i
Þ ½
24
where
n
labels the iteration number. This equation can be rearranged to give
x
i
Þþ
2
½
f
n
þ
1
n
n
n
h
2
f
ð
x
i
Þ¼ð
1
o
Þ
f
ð
ð
x
i
1
Þþ
f
ð
x
i
þ1
Þþ
4
pr
ð
x
i
Þ
½
25
h
2
. The update equation above is called the weighted Jacobi
where
o
¼
2
t
=
1, it is termed the Jacobi method.
99,104,105
As will be shown
below, there is a limit on the value of
method. If
o
¼
; if it becomes too large, the values
of the function diverge during repeated iterations.
One simple modification leads to two other update schemes. As the sol-
ver scans through the grid, the values at grid points are typically updated in
sequence. Thus, when we update the function at the (
i
) point, the
o
ð
i
1
Þ
value
has already been changed. Eq. [25] can then be altered to
x
i
Þþ
2
½
f
n
þ
1
n
n
þ
1
n
h
2
f
ð
x
i
Þ¼ð
1
o
Þ
f
ð
ð
x
i
1
Þþ
f
ð
x
i
þ1
Þþ
4
pr
ð
x
i
Þ
½
26
This relaxation scheme is named successive overrelaxation (SOR), and if
o
¼
1, it is the Gauss-Seidel method. This slight change alters the underlying
spectral properties of the update matrix, most often favorably. It turns out that
Gauss-Seidel iterations are particularly well suited for multigrid calcula-
tions—they tend to efficiently smooth the errors on wavelengths characteristic
for a given grid level.
105
The same methods can be applied to the Schr ¨ dinger equation, but the
eigenvalue
E
needs to be updated after each iteration, and the wave function
must be normalized.
106
If the lowest
n
states are desired, the wave functions
must be orthogonalized (Gram-Schmidt procedure) and then normalized.
Given normalized approximate eigenfunctions, the approximate eigenvalues
are given by Eq. [27]:
ð
1
2
d
2
dx
2
þ
E
i
¼
c
i
V
c
i
dx
¼h
c
i
j
H
j
c
i
i
½
27
This formula can be approximated on the grid by utilizing the FD methods
discussed above.
What Are the Limitations of Real-Space Methods on a
Single Fine Grid?
If we iterate toward the solution as discussed above, the residual
decreases quickly at first, but then the convergence slows considerably. This
