Chemistry Reference
In-Depth Information
The expression on the right side of Eq. [4] is called a functional derivative. The
easiest way to think about the functional derivative is to develop an FD repre-
sentation for the right-hand side of Eq. [4] (see below). To do this we take the
usual derivative with respect to the value of the potential at one point on the
grid, divide by the grid spacing, and take the limit of decreasing grid spacing.
When those steps are taken, we get Eq. [5]:
2
qf
qt
¼
q
f
x
2
þ
4
pr
½
5
q
This equation, when iterated, will repeatedly move downhill on the action
surface until the minimum is reached. Notice that this equation looks like a
diffusion equation with a source term. It can be proved mathematically that,
for this case, there is only one extremum, and it is a minimum. At the mini-
mum (which is the point where the potential stops changing),
2
qf
qt
¼
q
f
x
2
þ
4
pr
¼
0
½
6
q
and we obtain the Poisson equation. PDEs like the Poisson equation are called
elliptic equations. Other PDEs of importance are parabolic (diffusion) and
hyperbolic (wave) equations.
99
Solution of the Schr ¨ dinger equation can be viewed similarly.
65
The
quantum energy functional (analog of the Poisson action functional above) is
ð
ð
d
2
1
2
c
dx
2
dx
c
c
V
E
½
c
¼
c
dx
½
7
We seek to minimize this energy, but for this case we need a constraint
during the minimization, namely for the normalization of the wave function.
(If we are seeking many wave functions, we then need to have constraints
for orthonormality—each eigenfunction must be normalized and orthogonal
to all the other wave functions.) The augmented functional with the Lagrange
multiplier constraint is
ð
ð
E
ð
1
2
d
2
c
dx
2
dx
c
c
V
c
c
E
c
½
c
¼
þ
c
dx
dx
½
8
Now our steepest-descent update equation becomes
2
qc
qt
¼
d
E
c
½
c
dc
¼
1
2
q
c
x
2
½
V
E
c
½
9
q
which when iterated to the minimum, yields the Schr ¨ dinger equation.
