Symmetry Number

The (external) symmetry number (denoted as NSYM) is introduced to take into account the over-counting of forbidden states, due to the Pauli exclusion principle, when solving the corresponding Schrödinger equation. According to the Pauli principle, the overall molecular wave function must be symmetric or anti-symmetric with respect to the interchange of two identical atoms of the molecule, depending on whether they are bosons or fermions. A detailed discussion can be found in the work of Philip Bunker [2]. In MSMC, the external symmetry number is automatically determined based on the symmetry point group of the species of interest (cf. Table 1). If the user-provided symmetry number is different from the suggested number, there will be a warning message in the output file.

Table 1: Symmetry Number of Various Point Groups [2]

References:

[1] P.R. Bunker, P. Jensen, Molecular symmetry and spectroscopy, NRC Research Press, Ottawa, 1998.

[2] A. Vincent, Molecular Symmetry and Group Theory, John Wiley & Sons, New York, 1977.