Depending on the chosen ghost atoms, the corresponding point group might be a subgroup of the actual point group of the real molecule. However, the used symmetry is in these cases restricted to the symmetry of the supermolecule made up of the real AND ghost atoms. Note that symmetry can be used in calculations involving ghost atoms. This feature is particularly useful for calculations performed to determine the basis set superposition error (BSSE) and has several other potential applications such as describing lone pair electrons of a molecule by additional functions not centered at any of the molecular nuclei. However, unlike dummy atoms ghost atoms serve as a center for basis functions. Ghost atoms, which are specified by the symbol GH have zero nuclear charge. They do not carry a charge nor will be a basis set attached to them. Note that dummy atoms (X) just represent arbitrary points in space which are used to simplify the construction of the Z-matrix. In this way is it possible to place the hydrogen atom on the z-axis by specifying an angle of 90 degrees with respect to the dummy atom together with a dihedral angle for HCXN of 180 degrees. The dummy atom is places with an angle of 90 degrees with respect to the CN bond. The concept of a dihedral angle is best illustrated by the following picture Anti-clockwise rotation with respect to BC as rotation axis is taken as positive the values are restricted to -180 to 180 degrees. It is the angle needed to rotate the projection of the AB vector (in a plane with the BC vector as normal vector) into the projection of the CD vector into the same plane. The dihedral angle for the four atoms ABCD as specified in the example For the assignment of numerical values to the used variables (internal coordinates), see the description given below. Note that (a) variable names are limited to three characters and that (b) one and only one space must separate the different fields on each Z-matrix line. simply a point in space with no charge and no basis functions, is essential. It should be emphasized that this simple example for a Z-matrix works for a tetraatomic molecule such as hydrogen peroxide, but does not suffice for all cases.įor linear molecules (such as acetylene) as well as ring compounds, the use of "dummy" atoms, i.e. The dihedral angle is here defined as the angle between the BCD and ABC planes.įor systems with more than four atoms, the fifth and subsequent lines follow the same pattern as the fourth line of the example given above, i.e., they contain a length, an angle, and a dihedral angle together with the number of three previously specified centers. This line simply states that the fourth atom, say D, has a distance RCD to C, an angle DCB to C and B and a dihedral angle TAU to C,B, and A. Finally for the fourth atom the full set of bond length, bond angle, andĭihedral angle must be given and might in the present case possibly take the form 'D 3 RCD 2 DCB 1 TAU'. No dihedral angle is required, as the third atom is placed in the xz plane. An alternative choice would be the use of the distance between C and B and of the angle CBA. Using the distance between atoms A and C, RAC, and the angle CAB formed by atoms C A B (in this order, see picture), the third line is 'C 1 RAC 2 CAB'. For the specification of the third atom, say C, a distance and an angle are needed. No further specification is necessary for the second atom and it is then placed on the positive z-axis. The second line then contains the atomic symbol B, followed by the number 1 (A is atom number 1) and the distance RAB to this atom. Suppose that the distance between B and A is RAB. The second line specifies the position of the second atom, say B, relative to the first atom (A). There is nothing more to specify for the first atom and it is placed at the origin of the coordinate system. The first line in the Z-matrix just contains the atomic symbol of one of the atoms, say A.
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