Page 145 - Numerical Methods for Chemical Engineering

134 3 Matrix eigenvalue analysis

The single-shift QR iterations yield upon convergence
 
4 −2.8059 0 0

2.8059 4 0 0  R 11 R 12
[k→∞]

A =   = (3.173)
 0 0 4 0.3564  0 R 22
0 0 −0.3564 4
The eigenvalues of A are then computed easily from the diagonal 2 × 2 submatrices R 11
and R 22 , where for each we have with b ∈ ,

(4 − λ) −b
2 2
R jj − λI = = (4 − λ) + b = 0 (3.174)

b (4 − λ)
The two roots are λ = 4 ± bi, yielding the eigenvalues of (3.171).

Normal mode analysis

We now provide an example in which eigenvalue analysis is of direct interest to a problem
from chemical engineering practice. Let us say that we have some structure (it could be
a molecule or some solid object) whose state is described by the F positional degrees of
F
freedom q ∈ and the corresponding velocities ˙ q. We have some model for the total
potential energy of the system U(q) and some model of the total kinetic energy K (q, ˙ q).
We wish to compute the vibrational frequencies of the structure. Such a normal mode
analysis problem arises when we wish to compute the IR spectra of a molecule (Allen &
Beers, 2005).
First, using the numerical optimization methods outlined in Chapter 5, we identify a state
ˆ q that is a local minimum of the potential energy. That is, it has a lower potential energy
than any neighboring states, and as it is an extremum, ∇U| ˆq = 0 . We wish to describe
the system’s dynamics when it is perturbed slightly from this minimum energy state, and
so define δ = q − ˆq. Expanding U(q) about ˆq as a Taylor series, with ∂U/∂q m | ˆq = 0,
yields

) *
F
F
2
1 ∂ U
U(ˆ q 1 + δ 1 ,..., ˆ q F + δ F ) ≈ U(ˆ q 1 ,..., ˆ q F ) + δ m δ n (3.175)
2 ∂q m ∂q n ˆq

m=1 n=1
Defining the Hessian matrix H, containing the second derivatives of U(q),
2 2
∂ U ∂ U
H mn = = = H nm (3.176)

∂q m ∂q n ˆq ∂q n ∂q m ˆq
the Taylor series for U(q) in the vicinity of ˆq becomes
F F
1
U(ˆ q 1 + δ 1 ,..., ˆ q F + δ F ) ≈ U(ˆ q 1 ,..., ˆ q F ) + δ m H mn δ n (3.177)
2
m=1 n=1
H, which from (3.176) is real symmetric, also must be positive-semidefinite, as for a local
1 T
minimum ˆq, U(ˆq + δ) − U(ˆq) ≈ δ Hδ ≥ 0.
2