Molecule Types <Prev Next>

Linear Molecules

See Making a linear molecule data file and the worked example: The Schumann-Runge Bands of O2 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.

Quantum Numbers

The following standard quantum numbers are used for linear molecules:
Total angular momentum excluding nuclear spin
Total angular momentum
Total electron spin angular momentum. This must be set for each State
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.
The notation F1, F2, F3 ... is an alternative notation for the components of a multiplet, ordered by energy with with F1 being the lowest. This numbering scheme can also be defined in terms of the N quantum number, with the F1 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 F1, whereas the definition in terms of energy omits F1 for inverted states and the highest numbered component for regular states. For example, in a regular (A>0) 2Π state Ω=1/2 will be the lower energy component, and thus be labelled F1 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 F2 if N is used for labelling. By default PGOPHER uses energy ordering to assign Fn 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 X3Σg− state of O2. The lowest level J = 0, N = 1 is F1 by default, but often F3 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:
|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.

State Labels

The possible contents of state labels are:
The manifold and state name
The J quantum number; not shown if ShowJ is false at the Molecule level
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
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.
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 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.

Branch Labels

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

For example, a 2Π - 2Π 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 ), F1 - F1 (1), e-e, J" = 6.5.

Further Details