External Fields - The Zeeman and Stark Effects

The effect of a static external electric and/or magnetic field can be included in PGOPHER simulations. Setting up a data file in principle only requires making sure the required electric or magnetic dipole moments for the state(s) of interest are present. As the underlying physical interaction is the same as absorption or emission of radiation, these are set as transition moments. The only difference in the treatment of static fields is that only transition moments acting within a manifold are considered. For sample data files see:

The presence of a static field has significant consequences for the way calculations are performed, and the way various windows work.

Spectra

To simulate a spectra in the presence of a field, just set EField and/or BField as required in the Simulation object using the Constants window but note:

Calculations in the presence of a field are more time consuming than at zero field, so care may be necessary in the selection of the basis state.
In the Linelist window, the "J" columns (the 3rd and 7th) will show M rather than the total angular momentum, and the symmetry columns will be blank.
The Transitions window will allow M and the change in M to be selected, rather than the total angular momentum, and the symmetry selection will be ignored.
By default, the simulated spectra will assume unpolarized light, giving ΔM = 0, ±1 selection rules. To control this, add one or more Polarization objects under the simulation object.
As currently implemented, static electric and magnetic fields can be applied simultaneously, but must be parallel.

Basis State

The only good quantum number in the presence of a field is the projection of the total angular momentum along the field, M, so the basis used for any one diagonalization includes all states with the required M. The M quantum number will be added onto the end of any other quantum numbers in the state and basis labels. As this is a notionally infinite number of states, the range for the total angular momentum included in the basis is taken from Jmin to Jmax. (All rovibronic symmetries are included in the same matrix.) A full matrix diagonalization is used, not a perturbation based approach which is also in common use. The results will be exact, provided a large enough range of total angular momentum is used, though possibly slow.

The slow speed arises because the matrices to be diagonalized are much larger matrix than for zero field calculations. This can be mitigated by choosing Jmax (and possibly Jmin) carefully. The range must not be set too small as it controls the accuracy of the calculations. As a rule of thumb a range including total angular momentum one higher and one lower than the states of interest will typically give the correct trends, and the convergence of the calculations should be checked with respect to increasing the range.

Energy Level Plots and Molecular Focusing /Deceleration

To predict the behavior of individual energy levels in a field, the Levels window allows energy levels to be plotted as a function of field, in addition to the normal plot as a function of angular momentum.

For a plot as a function of the field check the Field box and select the field to sweep and the required range.
If the Field box is not checked an energy level plot as a function of quantum number is obtained, in the presence of any field(s) set in the Simulation object.
The State window provides a more detailed listing of states in the presence of any fields set in the Simulation object. Note that if fields are present, M is selected instead of J and the symmetry is ignored.