Biomedical Engineering Reference
In-Depth Information
H
1
H
3
C
1
C
2
H
2
H
4
Z-Matrix
C1
C2 C1 RCC
H1 C1 RCH C2 AngleHCC
H2 C2 RCH C1 AngleHCC H1 180.
H3 C2 RCH C1 AngleHCC H1 0.
H4 C1 RCH C2 AngleHCC H3 180.
RCC 1.34
RCH 1.08
AngleHCC 120.
Figure 5.10
Ethene Z-matrix
there are at most 3
N
6 vibrational degrees of freedom for a nonlinear molecule that is
described by 3
N
Cartesian (or other) coordinates. The Z-matrix has to be correctly written
in order to describe these 3
N
−
6 internal coordinates, and a frequent cause of problems in
optimizations specified by a Z-matrix was (and still is) the specification of too many or too
few internal coordinates. Too few coordinates correspond to a constraint on the geometry
and means that the full surface is not searched. Too many coordinates result in a redundancy
leading to zero eigenvalues in the Hessian (which cannot then be inverted).
In the early days of geometry optimizations it was usual to take all kinds of shortcuts in
order to save on computer resources. For example, we would routinely take all the C-H
bond lengths equal in substituted benzenes, yet there are subtle, systematic differences and
these should come out of a respectable calculation. Computer resources are much cheaper
than was once the case, and the assumptions of constant bond length, etc. are no longer
needed.
The Z-matrix method is often claimed to be intuitive to chemists because it uses the
everyday concepts of bond lengths, bond angles and so on. It was always an acquired
taste and in any case many databases give molecular geometries in terms of Cartesian
coordinates, not internal ones.
−
5.11 The End of the Z-Matrix
Geometry optimization is of major importance in modern molecular modelling. Most of the
early packages used internal coordinates as input by the Z-matrix. Practically all modern
(gradient) optimization procedures require calculation of the Hessian
H
and/or its inverse.
In practice, it is usual to make an estimate and update these estimates at every iteration.
Sometimes, the initial Hessian is taken to be the unit matrix, sometimes not.Agreat strength
of the internal coordinate method is that construction of the initial Hessian can be based on
chemical ideas; the individual diagonal elements of
H
are identified as bond stretching and
