Molecule Types Linear Molecules | <Prev Next> |

The linear molecule Hamiltonian
used by PGOPHER is
designed to be absolutely standard. IUPAC has produced a set of
recommendations for fine and hyperfine parameters (Hirota *et
al*, 1994) which are followed by PGOPHER as far as possible. The basic form
ignoring centrifugal distortion is the sum of the terms in the
following terms in the table:

For all states | Rotation | |

S > 0 |
Spin-Rotation | |

Λ > 0, S > 0 |
Spin-Orbit | |

S > 1/2 |
Spin-Spin | |

S > 1 |
Spin-Rotation | |

Λ > 0, S > 1 |
Spin-Orbit | |

S > 3/2 |
Spin-Spin | |

Π states | Λ doubling | |

Π states, S > 0 |
Λ doubling | |

Π states, S > 1/2 |
Λ doubling | |

Δ states |
See Λ type doubling for
Δ and higher states. |

See the individual sections below for a more detailed discussion. Note that any operators not listed above (such as alternate forms of operators or higher powers of centrifugal distortion) can be generated using a perturbation diagonal in electronic state.

though formally, it should only involve the rotational angular momentum of the nuclear framework, and thus be written:

PGOPHER can use either form, as controlled by the "RSquaredH" flag at the molecule level. This flag affects many of the terms in the Hamiltonian; for all the linear molecule terms read for if RSquaredH=True; to follow the IUPAC recommendations the form must be used.

The reason both forms are in use is that each has some problems
in evaluation, as discussed by Brown *et al*, 1987. Strictly
the second form is correct, but full evaluation is not possible
because of the terms involving .
Consider evaluating the term:

The diagonal terms are:

which becomes:

The term must be discarded, leaving:

which is the form used. The equivalent expression in is:

i.e. simply an additional Λ^{2}
term compared to . The main
practical difference is a shift in the effective band origin of *B*Λ^{2}
(so they are identical for Σ states), but it is not clear
that one is any more correct than the other as both ignore the
term .
This latter term is only likely to be worth considering if isotope
shifts of vibrational band origins are required, and the former
leads to different definitions of the band origin, so it is worth
checking all published constants to see which is used. The matrix
elements of higher powers of
or are evaluated by
evaluating the matrix of or
as above and taking the
appropriate power of the matrix. This means that the RSquaredH flag will cause
small changes which will affect most of the constants so, for
example, *B* will change by 2Λ^{2}*D*.

Of the off-diagonal terms, is the same in both forms, and their inclusion allows Hund's case (b) to be accurately modelled. The terms off diagonal in Λ:

are normally discarded in both forms of the
Hamiltonian as they only connect completely different electronic
states so they are not normally important. They can be included as
a perturbation if required (see * Luncouple*)

where:

Note
that, for ^{2}Π states *A** _{D}*
and the spin rotation constant γ (see below) have the same
effect on the energy levels so their effects can only be
distinguished by special methods (Veseth, 1971 and Brown and
Watson, 1977). For this reason only one of

For states with *S* > 1 an additional term is required:

as described in Brown *et al*, 1981.

For *S* > 1/2 the spin-spin interaction can contribute:

is also in use. The term
in θ only contributes for states with *S* > 3/2;
again see references in Brown *et al*, 1987. Read for if RSquaredH=True.

If RSquaredH=True, read for but note the term is unchanged. For *S*
> 1 an additional term is also required:

This form is from Cheung et al, 1984. It is
perhaps clearer in terms of its (only) Hund’s case (a)
matrix elements as given by Brown and Milton, 1976:

The IUPAC form is:

Of these terms *q *will contribute for any Π state; for
^{1}Π states it is effectively the difference between
the rotational constants for *e* and *f* parity
states. *p* requires *S* > 0 also and *o*
will only contribute for *S *> 1/2. The *e*^{±2iφ}
terms are shorthand to ensure that the Λ-doubling operators
only connect the two halves of a Π state:

If RSquaredH=True, read for in the centrifugal distortion terms, and leave the other operators unchanged. There are alternative Λ doubling parameters, in use including:

This arises naturally from expressing the Hamiltonian in terms of
*J*, rather than *N* as can be seen by making the
replacement:

See Brown and Merer, 1979 for a discussion of this. This also
explains why under some circumstances only *p* + 2*q*
is determined rather than *p* and *q* individually. (A
term equivalent to this can be generated using perturbations;
see the CrO sample file for an example of
this.)

Λ doubling in Δ states is discussed by Brown et al, 1987. They propose two different forms, a case (b) form:

and a case (a) form:

The case (a) form is programmed into `PGOPHER`
with the non-standard names for the parameters: *m*̃_{Δ}
= `o`, *n*̃_{Δ} = `oD`, *o*̃_{Δ}
= `oH`, *p*̃_{Δ} = `p`
and *q*̃_{Δ} = `q`, but does not
include any centrifugal distortion terms. The case (b) form, and
centrifugal distortion of either can be generated using perturbation objects,
and Λ doubling operators for higher values of Λ can
also be generated in this way.

- "Spin-dependent interactions for linear molecules in Σ
states of quartet and higher multiplicity", J. M. Brown and D.
J. Milton, Mol. Phys,
**31**, 409 (1976).

- "Lambda-type doubling parameters for molecules in Π
electronic states of triplet and higher multiplicity", J. M.
Brown and A. J. Merer, J. Mol. Spec.,
**74**, 488 (1979). - "Higher-order fine structure of the a
^{4}Π_{u}state of O_{2}^{+}. J. M. Brown, D. J. Milton, J. K. G. Watson, R. N. Zare, D. L. Albritton, M. Horani and J Rostas, J. Mol. Spec.,**90**, 139 (1981).

- "Λ-type doubling parameters for molecules in Δ
electronic states", J. M. Brown, A. S-C. Cheung and A. J. Merer,
J. Mol. Spec.,
**124**, 464 (1987). - "Spin-orbit and spin-rotation coupling in doublet states of
diatomic molecules", J. M. Brown and J. K. G. Watson, J.
Mol. Spec.,
**65**, 65 (1977). - "Fourier transform spectroscopy of CrO: Rotational analysis of
the A
^{5}Σ – X^{5}Π (0,0) band near 8000 cm^{–1}.", A. S-C. Cheung, W. Zyrnicki and A. J. Merer, J. Mol. Spec.,**104**, 315 (1984).

- "Symbols for fine and hyperfine parameters", E. Hirota, J. M.
Brown, J. T. Hougen, T. Shida and N. Hirota, Pure & Appl.
Chem.,
**66**, 571 (1994). - "Corrections to the spin-orbit splitting in
^{2}Π states of diatomic molecules", L. Veseth, J. Mol. Spec.,**38**, 228 (1971).