Molecule Types | <Prev Next> |

See Making a linear molecule data file
and the worked example: The Schumann-Runge
Bands of O_{2} for an introduction to working with
linear molecules.

PGOPHER will calculate Hund's cases (a) and (b) exactly, and will work with the other possible cases, though these typically require more work to set up.

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 |
Total angular
momentum excluding nuclear and electron spin: 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; Ω = Λ + Σ where Σ is the
projection of S
onto the axis of the molecule. |

F_{n} |
The notation F_{1}, F_{2,
}F_{3} ... is an alternative notation for the
components of a multiplet, ordered by energy with with F_{1}
being the lowest. This numbering scheme can also be defined
in terms of the N quantum number, with the F_{1}
level having the lowest N for a given J and
the higher numbered components having higher values of N.
The two schemes are equivalent, except where J <
S, where there can be differences in the omitted
label. A definition in terms of N would always omit
F_{1}, whereas the definition in terms of energy
omits F_{1} for inverted states and the highest
numbered component for regular states. For example, in a
regular (A>0) ^{2}Π state Ω=1/2
will be the lower energy component, and thus be labelled F_{1}
by PGOPHER and will also correlate with the lower
value of N=J−1/2. However the J=1/2
state must have N=1 and would thus be F_{2}
if N is used for labelling. By default PGOPHER uses
energy ordering to assign F_{n} as it is the
more common usage in the literature, but this choice is not
universal. The OmegaOrder
setting can be used to override the default for an
individual state. This is most likely to be an issue for
Σ states with spin 1 or higher.The one common case where the default is in disagreement with a significant amount of the literature is for the X ^{3}Σg−
state of O2. The lowest level J = 0, N = 1
is F_{1} by default, but often F_{3} in the
literature. To force the latter, set OmegaOrder to Inverted. |

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. Note
that it is not possible to have gerade
and ungerade states in
the same manifold.

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 True at the Mixture level. Either form can be used on input, and in addition 0 for + and 1 for − parity. Note that JAdjustSym should be set to False if simulating hyperfine structure as otherwise confusing results can be obtained.## 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: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 True at the Mixture level. Either form can be used on input, and in addition 0 for + and 1 for − parity. Note that JAdjustSym should be set to False if simulating hyperfine structure as otherwise confusing results can be obtained.

|Name J +- Omega>where Name is the manifold and state name. If hyperfine structure is included in the calculation then F (and intermediate quantum numbers if there is more than one nucleus) is added to the end.

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. |

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).

## Branch Labels

Note that the only guaranteed quantum numbers are the total
angular momentum and symmetry; while PGOPHER 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 most commonly arises in the case of
perturbations, or where S
> J. The algorithm
used can be adjusted by the EigenSearch and LimitSearch settings at the Manifold level and the `OmegaOrder`
setting at the State level;
the default values (`True`, `True` and `Auto`)
are recommended for the most consistent quantum number
assignment.. Variations in the quantum number assignment does
not affect other parts of the calculation, so the simulated
positions and intensities are not affected by these
considerations.

The general format is ΔNΔJ_{Fn'Fn"p"}(J) though, as for the state
labels above some elements may be omitted:

ΔN | The change in the N quantum number expressed as a P, Q or R; not shown if ShowN is false at the Molecule level or all states are singlet states |

ΔJ | The change J quantum number, expressed as P, Q or R. |

Fn'Fn" | The upper and lower
(spin-orbit) component number. If the two numbers are the
same, only one number is shown. |

p" | The lower state parity,
expressed as e
or f. |

F',F |
If nuclear spin is included, the upper and lower state hyperfine (F) quantum numbers are added. |

For example, a ^{2}Π - ^{2}Π band may
give the following transition:

rR1e(6.5) A v=0 7.5 7 F1e - X v=0 6.5 6 F1e

implying ΔN = +1
(^{ }r^{ }),
ΔJ = +1 (^{ }R^{ }),
F_{1} - F_{1} (1), e-e, J" = 6.5.