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.

As a simple example, consider the Stark effect in a symmetric top with B = 10 cm^-1, C = 6 cm^-1 and a dipole of 1.5 Debye, approximately those of NH₃. (Note that the inversion doubling makes the true picture in NH₃ more complicated - see NH3 pure rotation. To set up this calculation for the ground vibrational state (see Making a Data File for a Symmetric Top Molecule)

Set up simple symmetric top with File, New, Symmetric Top
Adjust the ground state by clicking on the "v=0" entry under "Ground" and set B = 10 , C = 6.
Set up the dipole moment in the ground state by right clicking on <Excited|mu|Ground> (the transition moments object) and selecting rename. Change the "Bra" entry in the rename dialog so that both entries read "Ground". (This converts the transition moment to act within the ground state, rather than connecting two different states.). Enter the dipole moment by clicking on <v=0|T(1)|v=0> and enter 1.45 for Strength.
Delete the upper state that was created by default - right click on "Excited" and select delete.
To avoid excessive calculation time, set Jmax at the species level to 5. (This should be fine for J < 3, unless the fields are very high.) For a physically reasonable spectrum, set the temperature to 10 K.
Bring up the Energy Level plot window (View, Levels) and click the Field check box. The default range of M starts at zero, but negative values need to be included to see the full pattern, so set "min" under M range to -1.
The plot below can be obtained by setting the maximum field to 10⁷ V/m, the vertical plot range to -2 .. 2, the labels position to "End" and the "shift mult" to 10. (The latter setting scales up the shifts from zero field by a factor of 10 for plotting.)

The resulting file is available as symstark.pgo.

Note that the J = 0 level show a small, quadratic (second order) effect, while the K = 1 levels show a strong linear (first order) splitting.

Two levels of detail are available from this window. The "List" button prints out the energy levels as a function of field, as for example:

Energy Level List for symstark.pgo
   M  Sym   # Field(V/m)       Energy    g Population Name J K (kl) Sym M 
   0    -   1 100             -0.0000    1 .675473722 v=0  0  0 A1  0     
   0    -   1 1000000         -0.0010    1 .675569751 v=0  0  0 A1  0     
   0    -   1 2000000         -0.0040    1 .675857881 v=0  0  0 A1  0     
   0    -   1 3000000         -0.0089    1 .676338232 v=0  0  0 A1  0     
   0    -   1 4000000         -0.0158    1 .677011004 v=0  0  0 A1  0     
   0    -   1 5000000         -0.0247    1 .677876479 v=0  0  0 A1  0     
   0    -   1 6000000         -0.0355    1 .678935021 v=0  0  0 A1  0     
   0    -   1 7000000         -0.0483    1 .680187072 v=0  0  0 A1  0     
   0    -   1 8000000         -0.0631    1 .681633159 v=0  0  0 A1  0     
   0    -   1 9000000         -0.0798    1 .683273889 v=0  0  0 A1  0     
   0    -   1 10000000        -0.0985    1 .685109954 v=0  0  0 A1  0     
Linear: -9.85038087317928E-009 cm-1/(Vm-1) -.0586611 Debye (15%)
Quadratic: -9.84803161106173E-016 cm-1/(Vm-1)^2 (.061%)

The values shown here are for the J = 0, K = 0, M = 0 level, as can be seen from the state label at the end of the line. To provide a simple model for the behavior, a least squares fit is performed of energy against field to the following three functions:

E(F) = E₀ + CF	Linear, corresponding to a first order stark effect
E(F) = E₀ + cF²	Quadratic, corresponding to a second order stark effect
E(F) = E₀ ± (Δ²/4 + C²F²)^½	Two level model giving intermediate behavior. It corresponds to two levels at E₀ ± ½Δ, mixed by a matrix element CF. The sign of the square root is taken to be the same sign as C.

In the above, F is the field, E(F) is the energy as a function of field, E₀ is the energy at zero field (except in the last case where it is E₀ ± ½Δ) and c, C and Δ are constants. PGOPHER performs a fit to all three functions for each level, though the results for the two level model are not displayed if the fit gives poor results, typically when the state is close to pure quadratic behavior. The %ages indicate the maximum error in energy from the fitted function as a percentage of the total energy change over the sweep. For the values given above, the error in the quadratic fit is thus 0.061% of 0.0985 = 0.00006 cm-1, while the maximum error in the linear fit is 15% of 0.0985 = 0.015 cm-1, so the level clearly has a second order Stark effect.

The J = 1, K= 1, M = 1 level is a classic example of a linear Stark effect:

   1    -   1 100             16.0000    1 .067584303 v=0  1  1 E  1      
   1    -   1 1000000         15.8780    1 .068780609 v=0  1  1 E  1      
   1    -   1 2000000         15.7556    1 .070002665 v=0  1  1 E  1      
   1    -   1 3000000         15.6328    1 .071250937 v=0  1  1 E  1      
   1    -   1 4000000         15.5095    1 .072526022 v=0  1  1 E  1      
   1    -   1 5000000         15.3858    1 .073828535 v=0  1  1 E  1      
   1    -   1 6000000         15.2616    1 .075159102 v=0  1  1 E  1      
   1    -   1 7000000         15.1371    1 .076518368 v=0  1  1 E  1      
   1    -   1 8000000         15.0121    1 .07790699  v=0  1  1 E  1      
   1    -   1 9000000         14.8866    1 .079325643 v=0  1  1 E  1      
   1    -   1 10000000        14.7608    1 .080775017 v=0  1  1 E  1      
Linear: -1.23923698142084E-007 cm-1/(Vm-1) -.7379921 Debye (.262%)
Quadratic: -1.15112688486556E-014 cm-1/(Vm-1)^2 (17.2%)
Two Level:  16.004875 cm-1  .00993999 cm-1 -1.24153225238959E-007 cm-1/(Vm-1)  -.739359  Debye (.209%)

The linear fit is clearly better than quadratic here, though a small amount of second order behavior is also suggested by the slightly better fit to a two level model. For the linear and two level cases C can be considered as an effective (state dependent dipole moment); the textbook formula for the linear Stark shift in a symmetric top is MK/(J(J+1)) · μF giving an effective dipole of MK/(J(J+1)) · μ = 1.1/(1.2) · 1.45 = 0.725 Debye, with the 0.738 value resulting from higher order effects.

The "Summary" button prints out a summary of the fits for each level:

   M  Sym   #    g Population Name J K (kl) Sym M         Energy Linear         Dipole     Err  Quadratic       Err  Two Level Delta     C              Dipole2    Err
  -1    -   1    1 .067584122 v=0  1  1 E -1            16.0000 1.1947733e-7   .71151304 .304% 1.11179096e-14 18.5%
  -1    -   2    1 .038010557 v=0  1  0 A2 -1           20.0000 -2.96324193e-9 -.0176467 15.8% -2.963163e-16  .008%
   0    -   1    1 .675473722 v=0  0  0 A1  0           -0.0000 -9.85124183e-9 -.0586663 15.8% -9.8490399e-16 .071%
   0    -   2    1 .067584185 v=0  1  1 E  0            16.0000 -2.96324193e-9 -.0176467 15.8% -2.963163e-16  .008%
   0    -   3    1 .038010557 v=0  1  0 A2  0           20.0000 5.898223888e-9 .03512518 15.8% 5.89597675e-16 .121%
   1    -   1    1 .067584247 v=0  1  1 E  1            16.0000 -1.23924255e-7 -.7379954 .276% -1.1562606e-14 17.8% 16.004337 .00918344 -1.24055462e-7 -.7387768 .244%
   1    -   2    1 .038010557 v=0  1  0 A2  1           20.0000 -2.96324193e-9 -.0176467 15.8% -2.963163e-16  .008%

See The pure rotational spectrum of the ground state of NH₃ and the Stark effect for an example that is between first and second order.