 Objects Mixture Species Molecule Manifold

# Custom Transition Moment Functions

To allow for a variation in the vibronic transition dipole moment with rotational state, commonly described in terms of Herman-Wallis factors, an arbitrary expression can be associated with any Transition Moment object. To do this, create a Custom Transition Function under the transition moment of interest (right click on the transition moment, select "Add New" and then "Custom Transition Function") and enter the required expression in the "Expression" field of the object. To enter a typical form for a parallel band in a linear molecule (see Watson 1987) use:
Value*(1+A1*m+A2*m^2)
Here m is N+1 for an R branch and -N for a P branch. Important Note: the upper state is taken to be the one on the left (Bra) side of the object. To use the form suggested by Watson for a perpendicular band a more complicated form is required:
Value*(1+A1*m+APR2*m^2+(J'=J")*AQ2*J"*(J"+1))
For linear molecules with S > 0 be aware that the selection rules on Σ are not enforced, so a starting point for a parallel band might be:
Value*(Sigma'=Sigma")*(1+A1*m+A2*m^2)
This illustrates the use of a conditional expression Sigma'=Sigma", which evaluates to 1 if true and 0 otherwise, thus enforcing the normal ΔΣ = 0 selection rule. Note that Signed must be set to True to use Sigma.

Notes:
• The expression is used to correct the value of the transition moment matrix element in the basis used, not the transition moment matrix elements after diagonalization.
• The function should be written to correct the transition moment matrix element in terms of Value, the value before correction. (As this is a matrix element, the intensity will normally vary as the square of the value.)
• Most standard mathematical forms and functions are available; see expressions for more information.
• The comparison operators =, <>, >, <, >= and <= evaluate to 1 for true, 0 for false. Note the use in the perpendicular band formula above.
• A built in polynomial function, poly, is also available, which may be more convenient if many terms are used; for a quadratic function use poly(x,a0,a1,a2) = a0+a1.x+a2.x2. Add more coefficients for a higher order polynomial. A Table object may also be useful.
• The function is held in the comment field for the object; in the constants window an enlarged text window replaces the normal comments window. (If you want to add comments to the function use # or // to start the comment, or enclose it in {...} or /* ... */.)
• Any unknown variables in the function are automatically created as parameters on simulating, or on right clicking on the node and selecting "Check Variables"
• Unwanted variables that are no longer in the expression can be removed by right clicking on the parameter and selecting remove.
• ' and " are used to indicate quantum numbers for the bra and ket as displayed, respectively. The available quantum numbers are listed below. Note that these are for the basis state, not the final state after diagonalisation.

## Built In Variables

 Value The value PGOPHER would use if Active were set to false, though see the note about a special case for linear molecules below. m N+1 for an R branch and -N for a P branch. 0 for a Q branch. J', J" J values for the states involved N', N" N values for the states involved F', F" F values for the states involved Phase', Phase" Where basis functions are expressed as, for example |J,K> ± |J,-K> for symmetrization, phase gives the relative sign of two parts of wavefunction. It takes values of +1, -1 or 0 where 0 indicates only one part.

Additional quantum numbers are also available, depending on the molecule type. As for the variables listed above, use ' and " to indicate bra and ket (left and right) quantum numbers respectively.

For asymmetric tops the standard quantum numbers Ka and Kc are available.

For linear molecules:
 Omega The Ω quantum number; note this is evaluated here taking Λ ≥ 0. Fn The spin component, 1 for F1, 2 for F2, etc. Note that for versions before 8.0.171 the values were divided by 2.
If Signed is true, then additional variables are available:
 Lambda', Lambda" Λ for the bra (') and ket (") states. Sigma', Sigma" Σ for the bra (') and ket (") states.
As a special case for linear molecules, the selection rule normally applied on Σ is not enforced, so for S > 0 it will be necessary set Signed to True and use Value*(Sigma'=Sigma") instead of simply Value to reproduce the calculation PGOPHER would otherwise perform.

For symmetric tops:
 K The absolute value of K Kl The sign of Kl

For vibrating molecules:
 v1', v2', ... Bra vibrational quantum numbers v1", v2", ... Ket vibrational quantum numbers l1', l2', ... Bra vibrational angular momentum quantum number, for degenerate modes only. l1", l2", ... Ket vibrational angular momentum quantum number, for degenerate modes only. Lambda' Bra electronic orbital angular momentum, Λ'. Lambda" Ket electronic orbital angular momentum, Λ". Omega' Bra Electronic angular momentum, Ω'. Omega" Ket Electronic angular momentum, Ω".

## Settings

 Active Set true to use the function provided to calculate the transition moment. nDebug If non zero, write the values of all the variables and the value of the function on each evaluation to the log window. See debugging for options on the output. To see any output the expression must not be blank - simply set to "Value" to see the defaults in action. Signed True to apply to unsymmetrized basis functions, as in |J,K> and |J,-K> separately above rather then |J,K> ± |J,-K>.

## References

• "Quadratic Herman-Wallis factors in the fundamental bands of linear molecules", J K G Watson, J Mol Spec, 125 428 (1987).