Biomedical Engineering Reference
In-Depth Information
bond bending force constants, etc. Also, the redundant translational and rotational degrees
of freedom have already been eliminated.
Baker and Hehre (1991) investigated the possibility of performing gradient geometry
optimization directly in terms of Cartesian coordinates. Their key finding concerned the
identification of a suitable initial Hessian, together with a strategy for updates. They reported
on a test set of 20 molecules, and argued that optimization in Cartesian coordinates can be
just as efficient as optimization in internal coordinates, provided due care is taken of the
redundancies.
There are actually twomathematical problems here, both to dowith theNewton-Raphson
formula. Modern optimizations use gradient techniques, and the formula
−
H
(
k
)
−
1
g
(
k
)
X
(
k
+
1)
=
X
(
k
)
demonstrates that we need to know the gradient vector and the Hessian in order to progress
iterations. In the discussion above, I was careful to stress the use of
p
independent variables
q
1
,
q
2
, ...,
q
p
.
As stressed many times, there are
p
6 independent internal coordinates for a
nonlinearmolecule but 3
N
Cartesian coordinates. The familiarWilson
B
matrix relates these
=
3
N
−
q
=
BX
(5.10)
B
has 3
N
rows and
p
columns, and the rows of
B
are linearly dependent. The molecular
potential energy depends on the
q
's, and also on the
X
's but we need to be careful to dis-
tinguish between dependent and independent variables. We can certainly write gradient
vectors in terms of the two sets of variables:
⎛
⎞
⎛
⎞
∂
U
∂
q
1
∂
U
∂
q
2
.
∂
U
∂
q
p
∂
U
∂
X
1
∂
U
∂
X
2
⎝
⎠
⎝
⎠
grad
U
=
,
(grad
U
)
C
=
.
∂
U
∂
X
3
N
where I have added a subscript 'C' to show that the differentiations are with respect to the
3
N
Cartesian coordinates, but it is grad
U
that appears in optimization expressions and not
(grad
U
)
C
. We therefore have to relate the two.
Starting from
q
=
BX
we can show that the gradients are related by
B
T
grad
U
(grad
U
)
C
Remember that we want an expression for grad
U
;if
B
were square and invertible we
would just write
=
=
B
T
−
1
(grad
U
)
C
but unfortunately
B
is rectangular. We therefore appeal to the mathematical concept of a
generalized inverse
. Consider the 3
N
grad
U
×
3
N
matrix
G
=
BuB
T
, where
u
is an arbitrary
nonsingular
p
p
matrix. The generalized inverse of
G
, written
G
−
1
, is a matrix with the
property that
GG
−
1
is a diagonal matrix with a certain number of 1's and a certain number
×
