Linear Molecules

Quantum Numbers

The following quantum numbers are used or displayed:
J
Total angular momentum excluding nuclear spin
F
Total angular momentum
S
Total electron spin angular momentum. This must be  - set for each State
N
J-S
Λ The projection of the electronic orbital angular momentum onto the z axis of the molecule. This must be set for each State
Ω The projection of J onto the axis of the molecule; also = Λ + Σ where Σ is the projection of S onto the axis of the molecule.
Fn
The notation F1, F2, F3 ... is an alternative notation for the components of a multiplet, ordered by energy with with F1 being he lowest.

Symmetry

For molecules with a centre of symmetry, Symmetric must be set at the Molecule level, and gerade set true or false for each State. If Symmetric is false, then gerade is ignored.

The overall parity of a particular state is displayed or read as + or -. In addition the J adjusted parity, e or f is also displayed in most circumstances if JAdjustSym is set at the Mixture level. Either form can be used on input, and in addition 0 or + and 1 for - parity.

Basis States

The basis states used by  PGOPHER are Hund's case (a) though, as discussed under State, it will correctly calculate any Hund's case. The basis states are displayed as:
|Name J +- Omega>
where Name is the manifold and state name, +- is the parity. If hyperfine is included in the calculation then F (and intermediate quantum numbers if there is more than one nucleus) is added to the end.

State Labels

The possible contents of state labels are:
Name
The manifold and state name
J
The J quantum number; not shown if ShowJ is false at the Molecule level
N
The N quantum number; not shown if ShowN is false at the Molecule level or all states are singlet states
Ω The Ω quantum number; not shown if ShowOmega is false at the Molecule level (the default) or all states are singlet states
Fn
The component of the multiplet numbered from 1 in order of increasing energy; not shown if ShowFNumber is false at the Molecule level or all states are singlet states. This contains the same information as the Ω quantum number, so it does not usually make sense to show both.
e/f
The parity; not shown if Showef is false at the Molecule level.

Hyperfine quantum numbers are added at the end as required.
For example, a regular 2Π state may give the following label:
X v=0 7.5  7 F1e
where the name is X v=0, J = 7.5, N = 7 the parity is e and it is the F1 component (Ω = 1/2). Note that the only guaranteed quantum numbers are the total angular momentum and symmetry; while the program tries to work out sensible assignments of the other quantum numbers there are cases where this is not possible, or the choice the program makes is not the same as other programs. This typically arises in the case of perturbations. The algorithm used is controlled by the EigenSearch and LimitSearch settings at the Manifold level. Note that even if the quantum numbers are ill defined it is still possible to use the labels generated as the program only rearranges the set generated in the absence of mixing between the basis states. Small changes in parameters can cause such labels to swap between related states.

Details