Quantum Chemistry Laboratory # Z-Matrix

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 provided in Exercise 5.
• 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 counterpoise correction to estimate the Basis Set Superposition Error (BSSE).
• 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.

Atom Distance to atom Angle Dihedral angle
N1 symbol N2 N1-N2 special
option
N3 N1-N2-N3 special
option
N4 N1-N2-N3-N4 special
option
no. no. Å no. degree no. degree
1
2
3
4 Dihedral angle in HOOH