Biomedical Engineering Reference
In-Depth Information
Such objects are usually colour-coded. I have turned the molecule around a little from
the previous plots. Interpretation is obviously a job for experts, and it is necessary to know
the colour-coding.
17.4 Geometry Optimization
The next step might well be a geometry optimization. All the MM considerations of earlier
chapters apply here, but there are twomajor differences.Atypical MMenergyminimization
on a large biological molecule might involve many thousands of variables. Calculation of
the energy at points on the molecular potential energy surface is trivial. A typical HF
calculation will involve fewer variables but the energy calculation for each point on the
molecular potential energy surface is nowmuchmore time consuming because a HF-LCAO
calculation is involved.
I explained in Chapter 16 how the HF energy could be written in terms of the electron
density and various one- and two-electron integrals over the basis functions χ . The HF-
LCAO matrix is
χ
i
(
r
1
)
h
(1)
(
r
1
)
χ
j
(
r
1
)
dτ
lh
ij
=
P
kl
χ
i
(
r
1
)χ
j
(
r
1
)
n
n
+
g
(
r
1
,
r
2
) χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
ˆ
k
=
1
l
=
1
P
kl
χ
i
(
r
1
)χ
k
(
r
1
)
n
n
1
2
−
ˆ
g
(
r
1
,
r
2
) χ
j
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
k
=
1
l
=
1
and the energy gradient will therefore involve terms like
∂χ
i
(
r
1
)
∂
c
∂
P
kl
∂
c
∂χ
i
(
r
1
)
∂
c
h
(
r
1
) χ
j
(
r
1
) dτ
,
and
χ
j
(
r
2
)
g
(
r
1
,
r
2
) χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
where
c
is a parameter to be varied. Advanced methods of geometry optimization usually
require both the gradient and the Hessian, and these can either be calculated numerically
or analytically. It is generally accepted that the analytical methods are to be preferred in
terms of resource consumption, and over the years a great deal of effort has gone into the
elucidation of analytical expressions for the energy gradient and the Hessian. All the major
molecular structure packages now use analytical gradients wherever these are available;
in the case of the HF-LCAO model, the analytical gradients do not involve the LCAO
coefficients directly, just the electron density matrix
P
. This speeds up the calculation by a
large factor.
By default, Gaussian 03 performs geometry optimizations in redundant internal coordin-
ates via the Berny algorithm. The choice of coordinate system for the starting structure
is irrelevant and any efficiency factors concerned with (for example) molecular sym-
metry are of no consequence since all structures are converted to redundant internal
coordinates.
