Molecule Types <Prev Next>

Symmetric Tops

See Making a Symmetric Top data file for a quick start. For details of the Hamiltonian used, see the symmetric top state section.

PGOPHER will currently calculate rotational structure for symmetric tops for the point groups listed below. Note that electron and nuclear spin effects are not currently implemented.

Quantum Numbers

The following standard quantum numbers are used or displayed for symmetric tops:
J
 Total angular momentum excluding nuclear spin
K
 The projection of J on to the highest order symmetry axis. Note that K will sometimes appear negative, depending on the symmetry of the state.
l
 The vibronic angular momentum; this will have a value of 0 for non-degenerate states, and ±1 for degenerate states.
kl
 The sign of Kl; see Hoy and Mills J. Molec Spectrosc. 46, 333 (1973) for the definition of the value displayed.

Symmetry

The rovibronic symmetry of each level will be given in standard format. The symmetry of non-degenerate levels typically flips between two values for odd and even J, so if JAdjustSym is set True at the Mixture level the symmetry will be read and written as that of the corresponding even J level. The symmetry can also be read (or written) as an integer, which is simply the row index of the character table.

Basis States

The basis states used by  PGOPHER are standard symmetric top functions, and are displayed as:

|Name J K +- l Sym>

where Name is the manifold and state name, J, K and l are the quantum numbers described above and Sym is the rovibronic symmetry. l is omitted for non-degenerate states, where it is 0. If + or - is present, it implies the basis state is the symmetised combination:

1/Sqrt(2){ |JKl> ± |J-K-l> }

For levels with degenerate rovibronic symmetry, only one K, l combination will be shown. The basis states are chosen such that their behaviour under the symmetry operations of the appropriate point group are as in Hegelund, Rasmussen, and Brodersen, J. Raman. Spectrosc. 1, 433 (1973); see also section 12.4 of Bunker and Jensen, Molecular Symmetry and Spectroscopy, 2nd Ed (2006). Note that while the references are specific to vibrational wavefunctions, electronic functions are taken to follow the same phase conventions.

State Labels

The possible contents of state labels are:

Name
 The manifold and state name
J
 The J quantum number
K
 The K quantum number; this is always positive for state labels.
Kl The sign of Kl; this is only present for degenerate vibronic states for K ≠ 0

For example, a E vibronic state in C3v may give the following label:

Excited v=1  3  3 +1 E

where the name is Excited v=1, J = 3, K = 3, l = +1 and the rovibronic symmetry is E.

Further Details

The table below shows the rovibronic symmetries for each point group that have the same statistical weight.

 Dipole wt0
 wt1 wt2 wt3 wt4 wt5 wt6
C3v A2 A1,A2 E
C4v A2 A1,A2 B1,B2 E
C5v A2 A1,A2 E1 E2
C6v A2 A1,A2 B1,B2 E1 E2
C7v A2 A1,A2 E1 E2 E3
C8v A2 A1,A2 B1,B2 E1 E2 E3
D3h A1" A1',A1" A2',A2" E',E"
D4h A1u A1g,A1u A2g,A2u B1g,B1u B2g,B2u Eg,Eu
D5h A1" A1',A1" A2',A2" E1',E1" E2',E2"
D6h A1u A1g,A1u A2g,A2u B1g,B1u B2g,B2u E1g,E1u E2g,E2u
D7h A1" A1',A1" A2',A2" E1',E1" E2',E2" E3',E3"
D8h A1u A1g,A1u A2g,A2u B1g,B1u B2g,B2u E1g,E1u E2g,E2u E3g,E3u
D2d B1 A1,B1 A2,B2 E
D3d A1u A1g,A1u A2g,A2u Eg,Eu
D4d B1 A1,B1 A2,B2 E1,E3 E2
D5d A1u A1g,A1u A2g,A2u E1g,E1u E2g,E2u
D6d B1 A1,B1 A2,B2 E1,E5 E2,E4 E3
D7d A1u A1g,A1u A2g,A2u E1g,E1u E2g,E2u E3g,E3u
D8d B1 A1,B1 A2,B2 E1,E7 E2,E6 E3,E5 E4
C3h A" A',A" E',E"
C4h Au Ag,Au Bg,Bu Eg,Eu
C5h A" A',A" E1',E1" E2',E2"
C6h Au Ag,Au Bg,Bu E1g,E1u E2g,E2u
S4 B A,B E
S6 Au Ag,Au Eg,Eu
S8 B A,B E1,E3 E2
S10 Au Ag,Au E1g,E1u E2g,E2u
S12 B A,B E1,E5 E2,E4 E3
D3 A1 A1 A2 E
D4 A1 A1 A2 B1 B2 E
D5 A1 A1 A2 E1 E2
D6 A1 A1 A2 B1 B2 E1 E2
D7 A1 A1 A2 E1 E2 E3
D8 A1 A1 A2 B1 B2 E1 E2 E3