Symmetric Tops

See Making a Symmetric Top data file for a quick start.

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.

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.

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 example, a E vibronic state in C_3v may give the following label:

Excited v=1  3  3 +1 E

where the name is Excited v=0, 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
C_3v	A₂	A₁,A₂	E
C_4v	A₂	A₁,A₂	B₁,B₂	E
C_5v	A₂	A₁,A₂	E₁	E₂
C_6v	A₂	A₁,A₂	B₁,B₂	E₁	E₂
C_7v	A₂	A₁,A₂	E₁	E₂	E₃
C_8v	A₂	A₁,A₂	B₁,B₂	E₁	E₂	E₃
D_3h	A₁"	A₁',A₁"	A₂',A₂"	E',E"
D_4h	A_1u	A_1g,A_1u	A_2g,A_2u	B_1g,B_1u	B_2g,B_2u	Eg,Eu
D_5h	A₁"	A₁',A₁"	A₂',A₂"	E₁',E₁"	E₂',E₂"
D_6h	A_1u	A_1g,A_1u	A_2g,A_2u	B_1g,B_1u	B_2g,B_2u	E_1g,E_1u	E_2g,E_2u
D_7h	A₁"	A₁',A₁"	A₂',A₂"	E₁',E₁"	E₂',E₂"	E₃',E₃"
D_8h	A_1u	A_1g,A_1u	A_2g,A_2u	B_1g,B_1u	B_2g,B_2u	E_1g,E_1u	E_2g,E_2u	E_3g,E_3u
D_2d	B₁	A₁,B₁	A₂,B₂	E
D_3d	A_1u	A_1g,A_1u	A_2g,A_2u	Eg,Eu
D_4d	B₁	A₁,B₁	A₂,B₂	E₁,E₃	E₂
D_5d	A_1u	A_1g,A_1u	A_2g,A_2u	E_1g,E_1u	E_2g,E_2u
D_6d	B₁	A₁,B₁	A₂,B₂	E₁,E₅	E₂,E₄	E₃
D_7d	A_1u	A_1g,A_1u	A_2g,A_2u	E_1g,E_1u	E_2g,E_2u	E_3g,E_3u
D_8d	B₁	A₁,B₁	A₂,B₂	E₁,E₇	E₂,E₆	E₃,E₅	E₄

C_3h is also partially implemented.