A high speed fitting program for rotational spectroscopy
Graphical abstract
Introduction
The relentless improvement of high-speed electronics driven principally by the telecommunications industry has resulted in the widespread availability of extremely high bandwidth frequency sources, digitizers, and high power amplifiers[1]. The advent of chirped pulse Fourier transform microwave (CP-FTMW) spectroscopy [2] successfully leveraged these advances, with numerous instruments based on this approach now in operation throughout the radio band [3], [4], [5], [6], [7]. These spectrometers routinely supply large, information-rich data sets [8], [9] at a rate far faster than the data can be analyzed, resulting in spectral assignment limiting both the rate and completeness of analysis.
Broadband rotational spectra present a particularly difficult set of challenges for spectral assignment. Among these are the non-linear scaling of potential solutions with data set size, the large potential parameter space and sparse solution space, and the limitations inherent to experimental data, e.g., finite frequency coverage, intensity variations, etc. A further complication is that a spectrum almost invariably contains features from more than one carrier or variant; the high-sensitivity of CP-FTMW spectrometers allows for species of low abundance, for example isotopologues, to be readily observed. Taken together, an enormous number of potential solutions may exist for any data set. Confidently decomposing experimental spectra into individual molecular components is therefore a formidable undertaking.
This situation is further exacerbated for open-ended searches where few constraints may be available in advance. Because little information is encoded in a single rotational feature, as a practical matter it is nearly impossible to make definitive assignments without a priori rotational constants. However, the magnitude of observed rotational constants can span several orders of magnitude depending on the precursors and the type of experiment. As illustrated in Fig. 1, even for a rigid rotor, there are numerous local minima in search space, a phenomenon that often precludes use of standard least-squares fitting approaches. Furthermore, because solution space is often extremely narrow, different minima may be found depending on the search resolution, with the global minimum only obvious at high resolution.
As an example, for a spectrum containing 100 unknown features, there are 1.6 105 possible sets of three transitions, the minimum needed to fit all three rotational constants. The number of unique sets grows exponentially with the number of features. In practice using reasonable exclusion criteria far fewer sets will be plausible, however it should be noted that for larger data sets worst case scenarios can still reach several million sets [10]. This requires an input of initial rotational constants. When these are not known, a search of rotational constants scales with the cube of the search range and inverse cube of the search resolution. Practical values give millions or more unique sets of rotational constants. Together these two factors alone can easily result in several trillion unique sets of constants and features to be checked.
To address this multitude of factors, numerous computational and experimental methods have been developed to aid in rotational assignment[11], [12], [10], [13], [14], [15]. None of these methods are complete solutions alone, and may struggle in various situations. Regardless of the specific approach, a constraint for many is simply the time required to calculate a rotational spectrum. As such, strongly nonlinear scaling can make it difficult or impossible to fully explore parameter space when potentially millions to billions of spectra must be considered. Alternative solutions such as machine learning may be able to address these issues, but at present struggle with real world data sets [13]. An attractive alternative is to simply to increase the sheer speed with which rotational spectra are calculated.
To that end, we have developed a method for high-speed calculation of rotational spectra of asymmetric top rigid rotors[16]. An archived version of the code discussed can be found with the digital object identifier doi:10.5281/zenodo.39505431. By pre-calculating rigid rotor eigenvalues, the need for expensive matrix diagonalization is eliminated. This method is not only inherently much faster, but also eliminates the need for many extraneous calculations, thereby allowing us to realize a several of order of magnitude speed-up relative to programs commonly used by the microwave community. This method can be used as the basis for building search and fitting algorithms that previously would have been computationally intractable. This simple brute force algorithm has already been used with considerable success in studies to determine rotational constants for entirely new molecules based on a very small number of double resonance linkages[17], [9]. The following sections describe the implementation and performance of this program.
Section snippets
Implementation
Our program effectively functions as a lookup table. On startup the program loads three items into memory: first is a pre-calculated set of eigenvalues for all states and geometries to be used, as described in Section 2.1; second, a catalog listing transitions as connectivity between states, and an associated dipole axis; third, a dictionary containing quantum numbers associated with each state. For speed and simplicity the archived version of the catalog contains only first order ( = 0,
Discussion
This program is potentially applicable to numerous problems in microwave spectroscopy. Its restriction to rigid rotors certainly precludes its use in more complex problems, for which programs such as CALPGM are far more suitable. However, in cases where rigid rotor approximations are sufficient it may be highly effective provided it meets two criteria.
Conclusion
We have built and demonstrated a program capable of predicting and fitting rigid rotor rotational spectra at very high speed. The program substantially outperforms existing programs as measured by clock speed, while retaining sufficient accuracy for use with high precision rotational data. The potential uses for this program are varied. Simple use cases employing double resonance data have been demonstrated [17], [9], although far more complex and varied methods can almost certainly be
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
M.C.M., K.L.K.L., and P.B.C. acknowledge financial support from NSF grant AST-1908576 and NASA grant 80NSSC18K0396. P.B.C. is funded by the Simons Collaboration on the Origins of Life. We thank Zachary Nicolaou for helpful discussions.
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