Biomedical Engineering Reference
In-Depth Information
In this chapter I want to show you how we locate stationary points on the molecular
potential energy surface, and howwe characterize them. I will start with the second problem,
because it is straightforward.
5.3 Characterization
For a linear molecule with
N
atoms, there are in total 3
N
degrees of freedom. Three of
these are translational degrees of freedom, three are rotational (two for a linear molecule)
and the remaining
p
=
−
−
5 if linear) are vibrational degrees of freedom. We
therefore need
p
independent variables to describe them, and these are known as the
internal
coordinates
. It is traditional to denote such variables as
q
; if I write them
q
1
,
q
2
, ...,
q
p
then the molecular potential energy will depend on these
q
's and the normal coordinates
will be linear combinations of them. For the minute, do not worry about the nature of these
variables. In Chapter 4, I showed how to collect such variables into the column vector
q
(a matrix of dimension
p
3
N
6(3
N
×
1):
⎛
⎞
q
1
q
2
.
q
p
⎝
⎠
q
=
and I also defined the gradient of
U
,
g
=
grad
U
, as a column vector, and its Hessian
H
as
a symmetric square matrix:
⎛
⎞
∂
U
∂
q
1
∂
U
∂
q
2
.
∂
U
∂
q
p
⎛
⎞
⎝
⎠
∂
2
U
∂
q
1
∂
2
U
∂
q
1
∂
q
p
···
⎝
⎠
.
.
g
=
,
H
=
···
∂
2
U
∂
q
p
∂
q
1
∂
2
U
∂
q
p
···
The molecular potential energy depends on the
q
's, and I will write it as
U
(
q
). At a
stationary point, the gradient is zero: it is as easy as that. In order to characterize the
stationary point, we have to find the eigenvalues of the Hessian calculated at that point. If
the eigenvalues are all positive, the point is a minimum. If the eigenvalues are all negative,
the point is a maximum. Otherwise the point is a saddle point. Saddle points of interest
to chemists are those that are a minimum in all degrees of freedom except one, and it
transpires that the Hessian has just one negative eigenvalues at such a point. They are
sometimes called
first-order saddle points
.
5.4 Finding Minima
MM tends to be concerned with searching the molecular potential energy surfaces of very
large molecules for minima (equilibrium structures). Many mathematicians, scientists and
