Page 154 - Numerical methods for chemical engineering

140 3 Matrix eigenvalue analysis

We first must calculate the integrals for the matrix elements of S and H, and this typically
comprises a significant fraction of the effort of quantum calculations. First, we consider the
elements of the overlap matrix

' 2P
∗
S pq = s mn = χ n (x)χ m (x)dx (3.220)
0
Substituting for the plane wave basis functions,
' 2P ' 2P . $ πx % /
e
S pq = s mn = e −iq n x iq m x dx = exp i (m − n) dx (3.221)
0 0 P
Defining θ = πx/P, this becomes
P iθ(m−n)
' 2π
S pq = s mn = e dθ = 2Pδ mn = 2Pδ pq (3.222)
π 0
Thus, the overlap matrix is merely S = (2P)I, I being the identity matrix.
We next compute the elements of the Hamiltonian matrix, which we break into two contri-
butions, H = T + V , where
h d χ m
¯ 2 ' 2P 2
∗
T pq =− χ (x) 2 dx (3.223)
n
2m e 0 dx
and
' 2P
∗
V pq = χ (x)V (x)χ m (x)dx (3.224)
n
0
In general, we must compute the elements of V numerically. A discussion of numerical
integrationisnotgivenuntilthenextchapter,soherewemerelynotethatweusethetrapezoid
∗
method (trapz in MATLAB) in which we evaluate the integrand f (x) = χ n (x)V (x)χ m (x)
at n G grid points,
2P
x j = ( j − 1)( x) x = j = 1, 2,..., n G (3.225)
n G − 1
We then approximate the integrals (3.224) using the quadrature rule
' 2P n G −1 ' x j+1 n G −1
f (x j+1 ) + f (x j )
f (x)dx = f (x)dx ≈ ( x) (3.226)
0 j=1 x j j=1 2
For the selected basis functions, we can compute the elements of T analytically. Substituting
into (3.223) for the basis functions,

¯ 2 ' 2P 2 ¯ 2 ' 2P
h −iq n x d iq m x h 2 −iq n x iq m x
T pq =− e 2 e dx = q m e e dx (3.227)
2m e 0 dx 2m e 0
- 2P i(q m −q n )x
Using q m = mπ/P and e dx = 2Pδ mn ,wehave
0
¯ 2 2 2
h m π
T pq = δ mn m = p − N − 1 n = q − N − 1 (3.228)
Pm e