Before attempting to fit the data set to a global rotational-torsional model, it is important to check for the correctness of the assignments. Mistaken assignments can lead to a subtle shift away from the global minimum of the Hamiltonian model that can be difficult to detect when working with very large data sets as is the case here. This step is critically important for transitions that play central roles in the model, such as those connecting different K stacks, and that are sparsely represented in the data set. Several methods are available to check if
10000 5000 0 -5000 -10000 Intensity (A.U.) 880 860 840 820 800 Frequency (GHz) 500 0 -500 Intensity (A.U.) 847.4 847.2 847.0 846.8 846.6 Frequency (GHz)
Figure 2.1: A scan of the methanol rotational spectrum in the 800-880 GHz region. These graphs show both the density and the large dynamic range of the molecular signals. The settings in the spectrum in the top plot were chosen such that many of the strongest transitions are saturated. This was done to allow measurement of transitions at the weakest levels given the available data acquisition capabilities of the instrument. The bottom plots illustrate the dynamic range of the spectrum, including the expansion of a section with “weaker” features that reveals them to still have significant signal-to-noise ratios. Many still weaker features can be seen at or above the noise level/standing wave level achieved.
an assignment is correct. The first step is paying close attention to the initial assignment, taking into account the location of both the assigned and alternative features compared to a prediction, as well as the intensity of the transition. As mentioned, initial assignments were done here using a prediction from a previous high-resolution study that extended to moderately high frequencies, and covering states up to J = 25 and K = 8 [46]. For quantum numbers outside of this range, the prediction was generally still close to a spectral feature up to about J ∼30-35 and K ∼ 11 and paying close attention to the intensity progression of the transitions, assignments were made with reasonable confidence. Beyond these states, the energy levels from the methanol atlas [45] proved invaluable. Even though the spectra in that work were of significantly lower resolution then the data analyzed here, the extremely large data set (∼19000 transitions were included) allowed the fitting of individual levels to many transitions, thus increasing the accuracy. Predictions from this atlas were generally within a few MHz of the spectral feature assigned using the multiplier chain THz data acquired here. Furthermore, it is possible to trace the pattern within an individual branch to extend assignments to higher J levels. In particular, with Q-branches that are closely spaced in specific parts of the spectrum we were able to look at the progression of the spacing and intensity of individual lines and use this to extend assignments to high J thanks to the wide coverage and good absolute intensity calibration of the JPL spectrometer.
After the initial identification of a transition using these methods, the assignments were further checked by using basic models and calculations. It is possible to break the global fitting problem down into smaller sub-models, fitting the individual K-stacks as vibrational states separated by an energy term. This method is not particularly useful in the global description of the problem as these models cannot simply be tied together, but does provide a valuable means of checking the assignments within K-stacks. The A state levels were
treated as a symmetric top and E state levels as a linear molecule, which for many states permitted fitting to high J (but with a few exceptions). In the A state, the K = 9 levels diverge above J∼25 due to a perturbative interaction that occurs with K = 5 in νt = 1,
and in both the A and E state low values of K diverge at the highest J’s due to several interactions between these states. The strengths of these repulsive interactions are inversely proportional to the ∆K between the states, with a single unique interaction term describing each ∆K interaction. The mapping of these level crossings is thus important in the global modeling of the spectrum as it allows prediction and fitting of further interactions of similar ∆K that are currently not part of the measured data set.
Ultimately, the only way to determine if a spectral assignment is absolutely correct is by calculating the sum over closed combination-difference loops, in which the individual transitions in the loop have all been measured. Several examples of such loops are shown in Fig. 2.2. If the lines in the loop have equal measurement uncertainties, ∆νmeas, one would
expect the loop to sum to zero within a factor of ∆νmeas
√
N. Since it is highly unlikely that all lines in the loop have been assigned incorrectly such that the errors cancel, especially as many connected loops are calculated, this technique can generally be considered the final word on the correctness of an assignment. Calculating the sums over many loops of N = 3,4,5,6 lines, and in a few select cases up to N= 10 lines, we managed to include ∼ 63% of all ground state transitions in loops. The various aspects in assembling and checking the database of assigned lines can be thought of as different levels of description of the energy level landscape with increasing levels of accuracy. The loop-sum calculations are the final aspect of fully weaving together the landscape in great detail. Thus, even though not all transitions were included in loops, the large fraction of included lines in addition to the other methods gives a high level of confidence in the set and that the few incorrect
Figure 2.2: Graphs explaining the principles of loop calculations. The colored lines are the ro- tational transitions that connect the different energy levels in the diagrams. In the calculation, a closed loop must be formed by a series of transitions, and the error in the sum of the transition frequencies should not surpass the sum in the expected uncertainties. Panel A) represents the case when no asymmetry splitting is present, and examples of 3-, 4- and 5-line loops are shown. Panel B) shows the situation in the presence of asymmetry splitting, in which case symmetry selection rules must be obeyed. Two examples of 6-line loops are shown.
assignments left in the set will be identified by the computerized fitting procedure.
In work related to that presented here for the ground state, the THz and microwave data set for the ground state were combined with similar data sets for the first and second excited torsional states to create a global data set for the first three torsional states (νt = 0,1,2).
Where appropriate, high resolution infrared data were also included, resulting in a global model up to J ≤ 30, K ≤ 14. This work was published recently [47], and will be briefly described here. The program used in the fitting procedure was obtained from I. Kleiner, available on the web [48]. The general approach involves fitting a rotational Hamiltonian to which the large amplitude internal rotational motion of the vibrational motion has been
transferred. The zeroth-order Hamiltonian has the following form:
H =F(Pγ+ρPa)2+ (1/2)V3(1−cos3γ) +APa2+BPb2+CPc2+Dab(PaPb+PbPa) (2.1)
where γ is the angle of internal rotation, Pγ is its conjugate momentum, V3 is the barrier
to internal rotation, and Pa, Pb and Pc are the rotational angular momentum operators.
Higher-order terms are generated by taking products of a power series of the momenta
PγpPaqPbrPcswith terms (1 - cos3γ) in the Fourier expansion of the internal rotation potential
function. Group theory and time reversal is then used to eliminate symmetry-forbidden terms. The program was modified in some ways to take maximum advantage of the present fit, the details of which can be found in [47]. The approximately 25000 total methanol lines were fit to 119 parameters, and the fit reaches the experimental uncertainty of the infrared line frequencies and about twice the experimental uncertainty for the microwave and THz lines. The Hamiltonian include rotational, torsional and rotational-torsional cross terms up to 10th order, whereas previous fits only extended to 6th order.