To specify the structure of a molecule it is convenient to use
internal coordinates, such as bond distances, bond angles, and
dihedral angles. A widely used system of internal co-ordinates
is the Z-matrix where the six redundant co-ordinates are
eliminated by providing no structure information for the first
atom (3 redundant parameters), the bond distance only for
the second atom (2 redundant parameters), and only bond distance
as well as bond angle for the third atom (1 redundant parameters).
For all other atoms, one specifies bond distance, bond angle,
and dihedral ange. To set up a Z-matrix, the following input
parameters have to be specified:
- Atom symbols are designated by the usual element symbols
H, He, Li, Be,
B, etc. by and the following special symbols
- Dummy atoms (symbol X) facilitate
the definition of a complicated molecular structure.
They neither carry a positive charged nucleus nor a
basis set (AO). In the electronic structure
calculation proper, they are totally ignored.
Examples for the advantageous use of dummy atoms will be
- Ghost atoms (symbol Bq) have no
positively charged nucleus, but the do carry atomic orbitals.
For example, they are used to define bond orbitals, i.e.
orbitals centered at the middle of a bond. Another important
usage is when carrying out a
to estimate the Basis Set Superposition
- The bond distance N1-N2 is that between atoms number N1 and atom
number N2. Any previously defined atom N2 can be specified.
- The angle N1-N2-N3 defines the bond angle
a located at atom
number N2. The atoms N1 and N3 lie at the two sides of
the angle. The allowed range for bond angles is
0° < a < 180°.
- The dihedral angle N1-N2-N3-N4 specifies the angle between
the two planes [N1,N2,N3] and [N2,N3,N4]. The value of a dihedral
angle has to fall in the range from -180° to +180°.
Have a look at the example hydrogen peroxide H2O2.
Dihedral angle in HOOH
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Technische Universität München