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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:
Total angular momentum excluding nuclear spin
Total angular momentum excluding nuclear and electron spin: N = J-S.
The projection of N on to the highest order symmetry axis. Note that K will sometimes appear negative, depending on the symmetry of the state.
Labels the vibronic component; this will have a value of 0 for non-degenerate states, and ±1 for degenerate states. It is related to the vibronic angular momentum, but is defined on the basis of symmetry so that, for example, l=+1 corresponds to the +l levels as defined by Hoy and Mills J. Molec Spectrosc. 46, 333 (1973).
The sign of Kl.


The rovibronic symmetry of each level will be given as one of the irreducible representations of the point group, as listed below. This is required to completely specify some levels, such as the pair of levels making up K = 3 in C3v for non-degenerate vibronic states, which have A1 and A2 symmetry. The energy order of these levels for a given K typically alternates with J so an alternative symmetry notation is available, with 1 and 2 replaced by + and - for even J and - and + for odd J. This applies to all A and B irreducible representations only; E levels are not affected. On output this is enabled if JAdjustSym is set True at the Mixture level; on input either form is accepted. 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 N K +- l Sym>

where Name is the manifold and state name, J, N, K and l are the quantum numbers described above and Sym is the rovibronic symmetry. N is omitted if S = 0. 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:

The manifold and state name
The J quantum number
The N quantum number; omitted if S = 0.
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
The rovibronic symmetry of the state

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. If the CompactKLabels flag is true, then K, l and symmetry are combined to give a single item:

Excited v=1  3  +3E

with the sign omitted if l = 0 and the symmetry omitted as determined by the SymmetryLabels flag.

Further Details

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

Dipole Separably
wt1 wt2 wt3 wt4 wt5 wt6
C3v A2 No
A1,A2 E
C4v A2 No
A1,A2 B1,B2 E
C5v A2 No
A1,A2 E1 E2
C6v A2 No
A1,A2 B1,B2 E1 E2
C7v A2 No
A1,A2 E1 E2 E3
C8v A2 No A1,A2 B1,B2 E1 E2 E3
D3h A1" No A1',A1" A2',A2" E',E"
D4h A1u No A1g,A1u A2g,A2u B1g,B1u B2g,B2u Eg,Eu
D5h A1" No A1',A1" A2',A2" E1',E1" E2',E2"
D6h A1u No A1g,A1u A2g,A2u B1g,B1u B2g,B2u E1g,E1u E2g,E2u
D7h A1" No A1',A1" A2',A2" E1',E1" E2',E2" E3',E3"
D8h A1u No A1g,A1u A2g,A2u B1g,B1u B2g,B2u E1g,E1u E2g,E2u E3g,E3u
D2d B1 No A1,B1 A2,B2 E
D3d A1u No A1g,A1u A2g,A2u Eg,Eu
D4d B1 No A1,B1 A2,B2 E1,E3 E2
D5d A1u No A1g,A1u A2g,A2u E1g,E1u E2g,E2u
D6d B1 No A1,B1 A2,B2 E1,E5 E2,E4 E3
D7d A1u No A1g,A1u A2g,A2u E1g,E1u E2g,E2u E3g,E3u
D8d B1 No A1,B1 A2,B2 E1,E7 E2,E6 E3,E5 E4
C3h A" Yes
A',A" E',E"
C4h Au Yes Ag,Au Bg,Bu Eg,Eu
C5h A" Yes A',A" E1',E1" E2',E2"
C6h Au Yes Ag,Au Bg,Bu E1g,E1u E2g,E2u
S4 B Yes A,B E
S6 Au Yes Ag,Au Eg,Eu
S8 B Yes A,B E1,E3 E2
S10 Au Yes Ag,Au E1g,E1u E2g,E2u
S12 B Yes A,B E1,E5 E2,E4 E3
D3 A1 Yes A1 A2 E
D4 A1 Yes A1 A2 B1 B2 E
D5 A1 Yes A1 A2 E1 E2
D6 A1 Yes A1 A2 B1 B2 E1 E2
D7 A1 Yes A1 A2 E1 E2 E3
D8 A1 Yes A1 A2 B1 B2 E1 E2 E3