Biomedical Engineering Reference
In-Depth Information
5.10 The Z-Matrix
=
−
For a nonlinear molecule of
N
atoms, there are
p
6 vibrational degrees of freedom
that should be described by
p
independent coordinates
q
1
,
q
2
, ...,
q
p
and at first sight
all should be varied in a geometry optimization. These coordinates are often defined to
modelling packages using the so-called Z-matrix, which gives a way of building up a
molecular geometry in terms of valence descriptors such as bond lengths, bond angles and
dihedral angles. Two examples will help to make things clear. First of all water, for which I
will take a bond length of 95.6 pm (0.956 Å) and an H-O-H bond angle of 104
◦
(Figure 5.9).
3
N
O
H
1
H
2
Z-Matrix
Oxy
H1 Oxy ROH
H2 Oxy ROH H1 Ang
ROH 0.956
Ang 104.
Figure 5.9
Z-matrix for water
I can start my Z-matrix at any atom, and that atom occupies a line by itself. I choose
oxygen, which I have called Oxy. H1 is joined to oxygen with a bond length of
ROH (0.956 Å). I have assumed that the bond lengths are equal. H2 is also joined to
oxygen with a bond length of 0.956 Å, and the H2-oxygen-H1 bond angle is Ang (104
◦
).
Notice that I have subconsciously added a
symmetry constraint
by requiring that the two
bond lengths are equal. This means that I am effectively reducing the number of degrees
of freedom, for the purposes of geometry optimization. We will see later that this is not
necessary.
I had no particular software in mind when I wrote this example (although it would be
fine for Gaussian 03, as we will see in later chapters). There are many Z-matrix pits for the
unwary to fall into, some simple and some subtle. Two simple ones: I did not call the first
atom 'Oxygen'because many packages limit you to a small number of characters for atomic
descriptors; also, I was careful to include a decimal point for the starting values of ROH
and Ang, since many packages still expect floating-point numbers rather than integers.
For molecules having more than three atoms, we also have to define the dihedral angles
in addition to bond lengths and bond angles. Figure 5.10 shows a Z-matrix for ethene, where
I have made a lot of assumptions about the molecular geometry. All the C-H bond lengths
are equal, and all the HCC bond angles are also taken to be equal. I have also assumed that
the molecule is planar (as indeed it is, in its electronic ground state).
I should admit that Z-matrices are not easy things to write for large molecules, and in
any case downloads of molecular geometries (for example, from the Protein Data Bank)
are usually in Cartesian coordinates. There is also a problem to be addressed, namely that
