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See Making a Symmetric Top data file for a quick start. For details of the Hamiltonian used, see the symmetric top state section.
J |
Total angular momentum
excluding nuclear spin |
N |
Total angular momentum excluding nuclear and electron spin: N = J-S. |
K |
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. |
l |
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). |
kl |
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 C_{3v} for non-degenerate vibronic states, which
have A_{1} and A_{2} 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.
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:
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.
The possible contents of state labels are:
Name |
The manifold and state name |
J |
The J quantum number |
N |
The N quantum number; omitted if S
= 0. |
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 |
symmetry |
The rovibronic symmetry of the state |
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=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.
Dipole | Separably Degenerate |
wt0 |
wt1 | wt2 | wt3 | wt4 | wt5 | wt6 | |
C_{3v} | A_{2} | No |
A_{1},A_{2} | E | |||||
C_{4v} | A_{2} | No |
A_{1},A_{2} | B_{1},B_{2} | E | ||||
C_{5v} | A_{2} | No |
A_{1},A_{2} | E_{1} | E_{2} | ||||
C_{6v} | A_{2} | No |
A_{1},A_{2} | B_{1},B_{2} | E_{1} | E_{2} | |||
C_{7v} | A_{2} | No |
A_{1},A_{2} | E_{1} | E_{2} | E_{3} | |||
C_{8v} | A_{2} | No | A_{1},A_{2} | B_{1},B_{2} | E_{1} | E_{2} | E_{3} | ||
D_{3h} | A_{1}" | No | A_{1}',A_{1}" | A_{2}',A_{2}" | E',E" | ||||
D_{4h} | A_{1u} | No | A_{1g},A_{1u} | A_{2g},A_{2u} | B_{1g},B_{1u} | B_{2g},B_{2u} | Eg,Eu | ||
D_{5h} | A_{1}" | No | A_{1}',A_{1}" | A_{2}',A_{2}" | E_{1}',E_{1}" | E_{2}',E_{2}" | |||
D_{6h} | A_{1u} | No | 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} | A_{1}" | No | A_{1}',A_{1}" | A_{2}',A_{2}" | E_{1}',E_{1}" | E_{2}',E_{2}" | E_{3}',E_{3}" | ||
D_{8h} | A_{1u} | No | 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} | B_{1} | No | A_{1},B_{1} | A_{2},B_{2} | E | ||||
D_{3d} | A_{1u} | No | A_{1g},A_{1u} | A_{2g},A_{2u} | Eg,Eu | ||||
D_{4d} | B_{1} | No | A_{1},B_{1} | A_{2},B_{2} | E_{1},E_{3} | E_{2} | |||
D_{5d} | A_{1u} | No | A_{1g},A_{1u} | A_{2g},A_{2u} | E_{1g},E_{1u} | E_{2g},E_{2u} | |||
D_{6d} | B_{1} | No | A_{1},B_{1} | A_{2},B_{2} | E_{1},E_{5} | E_{2},E_{4} | E_{3} | ||
D_{7d} | A_{1u} | No | A_{1g},A_{1u} | A_{2g},A_{2u} | E_{1g},E_{1u} | E_{2g},E_{2u} | E_{3g},E_{3u} | ||
D_{8d} | B_{1} | No | A_{1},B_{1} | A_{2},B_{2} | E_{1},E_{7} | E_{2},E_{6} | E_{3},E_{5} | E_{4} | |
C_{3h} | A" | Yes |
A',A" | E',E" | |||||
C_{4h} | A_{u} | Yes | A_{g},A_{u} | Bg,Bu | Eg,Eu | ||||
C_{5h} | A" | Yes | A',A" | E_{1}',E_{1}" | E_{2}',E_{2}" | ||||
C_{6h} | A_{u} | Yes | A_{g},A_{u} | B_{g},B_{u} | E_{1}_{g},E_{1}_{u} | E_{2}_{g},E_{2}_{u} | |||
S_{4} | B | Yes | A,B | E | |||||
S_{6} | A_{u} | Yes | A_{g},A_{u} | E_{g},E_{u} | |||||
S_{8} | B | Yes | A,B | E_{1},E_{3} | E_{2} | ||||
S_{10} | A_{u} | Yes | A_{g},A_{u} | E_{1g},E_{1u} | E_{2g},E_{2u} | ||||
S_{12} | B | Yes | A,B | E_{1},E_{5} | E_{2},E_{4} | E_{3} | |||
D_{3} | A_{1} | Yes | A_{1} | A_{2} | E | ||||
D_{4} | A_{1} | Yes | A_{1} | A_{2} | B_{1} | B_{2} | E | ||
D_{5} | A_{1} | Yes | A_{1} | A_{2} | E_{1} | E_{2} | |||
D_{6} | A_{1} | Yes | A_{1} | A_{2} | B_{1} | B_{2} | E_{1} | E_{2} | |
D_{7} | A_{1} | Yes | A_{1} | A_{2} | E_{1} | E_{2} | E_{3} | ||
D_{8} | A_{1} | Yes | A1 | A_{2} | B_{1} | B_{2} | E_{1} | E_{2} | E_{3} |