Herman-Wallis Factors Research Articles

Herman-Wallis factor for a molecule of type HCN

We suggest a novel theoretical approach to calculate Herman-Wallis factors in rovibrational spectra of non-centrosymmetric linear molecules possessing a permanent dipole. A fresh glance allowed to extend the theory recently formulated in [K.V. Kazakov and A.A. Vigasin, Phys. Rev. A, 104, 042805 (2021)]. Our obtained expressions for the first Herman-Wallis correction factors result in evaluation of the rovibrational interaction impact on the line intensities belonging to fundamental modes ν1,ν2, and ν3 for H12C14N, H13C14N, H12C15N, D12C14N, D13C14N, and D12C15N. As an auxiliary outcome, the sum rule for the Coriolis constants is generally derived in detail.

Theory of rovibrational line intensities in allowed and collision-induced absorption spectra of linear molecules

A theoretical approach is developed for the Herman-Wallis factors describing the variation of the line intensities caused by rovibrational interaction in a linear molecule. In the case of ${\ensuremath{\nu}}_{2}$ and ${\ensuremath{\nu}}_{3}$ fundamentals and the ${\ensuremath{\nu}}_{1}+{\ensuremath{\nu}}_{3}/2{\ensuremath{\nu}}_{2}+{\ensuremath{\nu}}_{3}$ resonance combination band in a ${\mathrm{CO}}_{2}$ molecule, our theory is capable of generating reliable numerical estimates for the linear Herman-Wallis coefficients without recourse to complicated algebra. On the assumption of nearly free rotating monomers, an analog of the Herman-Wallis factors is introduced for the dipole-forbidden ${\ensuremath{\nu}}_{2}+{\ensuremath{\nu}}_{3}$ band, which manifests itself in ${\mathrm{CO}}_{2}$ collision-induced absorption (CIA). The rovibrational perturbation is appropriate to explain a pronounced crossflow of intensity among rovibrational branches in a virtually rotationally unresolved CIA envelope.

Features of Measuring Low CO Concentrations in N2-Containing Mixtures at Different Temperatures Using Spontaneous Raman Spectroscopy.

Raman spectroscopy is a promising tool for combustion processes optimization, due to the possibility of rapid determination of the exhaust gases composition. An important gas component in this task is carbon monoxide whose emission limits vary from 100 to 200 parts per million (ppm), depending on the heat generator technology. However, for the correct determination of its concentration from the sample Raman spectrum, it is necessary to take into account the contribution of nitrogen lines intensity due to their mutual overlap. This paper discusses a technique for deriving carbon monoxide intensity based on fitting the nitrogen spectrum at various temperatures. It is shown that ignoring the Herman-Wallis factors in the fitting procedure lead to an additional measurement error, which increases with temperature and exceeds 350 ppm at T = 1800 K.

Frequency-comb-assisted absolute calibration and linestrength of H12C13CH ro-vibrational transitions in the 2ν3 band

We report on a precision spectroscopic study of room-temperature H12C13CH in the 6463–6520 cm−1 range, using a continuous-wave diode laser phase-locked to a near-infrared optical frequency comb synthesizer stabilized, in turn, against a GPS-disciplined rubidium clock. By tuning the comb repetition frequency, several ro-vibrational transitions in the P and R branches of the 2ν3 overtone band are recorded. Accuracy levels as low as 100 kHz, 5·10−10 in fractional terms, are obtained in the determination of absolute line-center frequencies. In addition, line intensities are measured for the first time, with a best fractional uncertainty of 1%, and used to retrieve the purely vibrational transition dipole moment in conjunction with the first-order Herman–Wallis factor. Finally, the pressure broadening coefficients γself are also derived for the studied transitions.

Measurements and modeling of 16O12C17O spectroscopic parameters at 2 µm

The lack of spectroscopic measurements for rare CO2 isotopologues was the main motivation of this work. In our present study we report line intensity measurements for 16O12C17O made with a high resolution Fourier transform spectrometer (Bruker IFS-125HR) and a 21 m path cryogenic Herriott cell at Jet Propulsion Laboratory. For this, a 17O-enriched CO2 gas sample was used, which comes as a mixture of primary and several minor CO2 isotopologues. The mole fraction of the 16O12C17O isotopologue in the mixture was determined to be 0.3991 by mass spectrometry from a Stable Isotope Ratio Mass Spectrometer (SIRMS) under stochastic distribution assumption at thermal equilibrium. Since the collisional narrowing effect was observed, the Rautian molecular line shape profile was systematically adopted instead of the Voigt profile. Absolute line positions were also investigated by performing a wavenumber calibration based on CO, HCl and a few well-known 16O12C16O transitions. Finally, around 1000 transitions were studied between 4604 and 5126cm−1 involving 15 bands of the 16O12C17O isotopologue. All the measured line intensities were renormalized to be the values for 100% pure isotopologue sample. Transition dipole moments and Herman-Wallis factors were derived enabling a global comparison with theoretical calculations and predictions for the 15 bands of the 16O12C17O isotopologue. For the measured line positions, the absolute accuracy is around 2×10−4cm−1. The accuracies of retrieved line intensities are 2 – 5% for five cold and two hot bands, and 6–30% for eight other weaker hot bands. Results from this work were in a good agreement with HITRAN 2012 for positions, but showed rather significant discrepancies for line intensities. An extensive line list was generated from new experimental measurements in order to improve and validate spectroscopic knowledge of 12C16O17O isotopologue in support of atmospheric remote sensing for the Earth (e.g., OCO-2 mission), Mars and Venus.

Electric dipole moment function and line intensities for the ground state of carbon monxide**Project supported by the National Natural Science Foundation of China (Grant Nos. 11374217 and 11474207).

An accurate electric dipole moment function (EDMF) is obtained for the carbon monoxide (CO) molecule (X1Σ+) by fitting the experimental rovibrational transitional moments. Additionally, an accurate ab initio EDMF is found using the highly accurate, multi-reference averaged coupled-pair functional (ACPF) approach with the basis set, aug-cc-pV6Z, and a finite-field with ±0.005 a.u. (The unit a.u. is the abbreviation of atomic unit). This ab initio EDMF is very consistent with the fitted ones. The vibrational transition matrix moments and the Herman–Wallis factors, calculated with the Rydberg–Klein–Rees (RKR) potential and the fitted and ab initio EDMFs, are compared with experimental measurements. The consistency of these line intensities with the high-resolution transmission (HITRAN) molecular database demonstrates the improved accuracy of the fitted and ab initio EDMFs derived in this work.

The accurate transition dipole moment and line intensities of NO(X2П) based on averaged quadratic coupled-cluster calculations

For NO(X2П), the vibrational dipole moments calculated using previous accurate ab initio electric dipole moment functions (EDMFs) are not consistent with experiments, especially for high-overtone bands 3–0, 4–0, 21–17, 5–0, 6–0, and 7–0. Moreover, the band intensities of the current HITRAN database agree only roughly with recent experimental results for high-overtone bands 3–0, 4–0, and 5–0. Thus, further theoretical calculation is necessary. We adopt the highly accurate multi-reference averaged quadratic coupled-cluster (MR-AQCC) approach and four different basis sets to investigate the potential-energy curves (PECs) and EDMFs of NO(X2П). The PECs and EDMFs obtained using the aug-cc-pV6Z basis set agree well with Rydberg–Klein–Rees (RKR) potential and experimental EDMFs. They are used to deduce the vibrational dipole moments and Herman–Wallis factors, which are further be utilized to evaluate line intensities and band intensities. Both the calculated transition dipole moments and the calculated band intensities are in remarkably better agreement with the experimental results for high-overtone bands. The Herman–Wallis factors of band 7–0 are also fairly consistent between experiment and our result. We expect that these reliable and accurate theoretical results will be helpful in producing accurate theoretical predictions concerning the spectroscopic characteristics and the analysis of some experimentally measured properties of NO(X2П).

Accurate temperature measurements in flames with high spatial resolution using Stokes Raman scattering from nitrogen in a multiple-pass cell

A multiple-pass cell is aligned to focus light at two regions at the center of the cell. The two "points" are separated by 2.0 mm. Each probe region is 200 μm×300 μm. The cell is used to amplify spontaneous Raman scattering from a CH4-air laminar flame. The signal gain is 20, and the improvement in signal-to-noise ratio varies according to the number of laser pulses used for signal acquisition. The temperature is inferred by curve fitting high-resolution spectra of the Stokes signal from N2. The model accounts for details, such as the angular dependence of Raman scattering, the presence of a rare isotope of N2 in air, anharmonic oscillator terms in the vibrational polarizability matrix elements, and the dependence of Herman-Wallis factors on the vibrational level. The apparatus function is modeled using a new line shape function that is the convolution of a trapezoid function and a Lorentzian. The uncertainty in the value of temperature arising from noise, the uncertainty in the model input parameters, and various approximations in the theory have been characterized. We estimate that the uncertainty in our measurement of flame temperature in the least noisy data is ±9 K.

Experimental and theoretical studies of CO2 spectra for planetary atmosphere modelling: region 600–9650 cm−1 and pressures up to 60 atm

Extensive experimental studies of room-temperature carbon dioxide absorption coefficients are reported for a wide range of wavenumbers and pressures requested in atmospheric spectra modelling. The quality of measurements is optimised by the use of two complementary setups with long- and short-path optical cells for low and high gas densities, respectively. The recorded spectra provide a representative picture of band-shape evolutions from gaseous to nearly liquid phases of CO2 and enable a theoretical analysis of line-mixing effects. Various kinds of vibrational bands (Σ←Σ, Π←Σ as well as Π←Π transitions) are modelled using a specific, non-Markovian in the general case, approach of Energy-Corrected Sudden type which is based on the symmetric relaxation matrix and, in contrast to the standard ECS model used for infrared absorption calculations, ensures automatically the fundamental relations of detailed balance and double-sided sum rules. Moreover, this method properly accounts for the vibrational angular momenta of the initial and final molecular states and allows including Coriolis resonances via the usual Herman-Wallis factors in the dipole transition moments. With a set of ECS parameters previously obtained for isotropic and anisotropic Raman spectra modelling, completely neglected imaginary part of the relaxation operator and a simple change in the tensorial rank to get the dipole absorption case in the working formulae, the computed spectra reproduce quite correctly the vibrotational band shapes up to 20 amagat without any additional parameter. An empirical correction factor tentatively introduced to account globally for the Coriolis effects on the relaxation matrix leads to better matches with high-density band shapes but its role merits further studies with an accurately modelled imaginary part of the relaxation matrix.

On the sensitivity of rotational O2 CARS thermometry to the Herman–Wallis factor

Contributions of vibration–rotation interaction to line intensities of pure rotational coherent anti‐Stokes Raman scattering (CARS) of oxygen are here considered in the perspective of what was remarked by us recently for nitrogen. The interaction is conventionally evaluated by means of the so‐called Herman–Wallis (HW) factor, the need of which is conceptually clear but the theoretical expression is debatable, as shown in the literature of this field. In this work, the various solutions for the anisotropic HW factor of pure rotational Raman transitions are compared in light of the implications for the corresponding CARS spectral line intensities. The results show that neglecting the use of HW factors in rotational N2 and O2 CARS thermometry leads to systematic overestimation of temperatures; on the order of 1.5% in the temperature range 300–2100 K in comparison with the use of the most plausible HW factors. The results also suggest that, in general, oxygen is more sensitive than nitrogen to the HW correction, and in particular, different S‐branch HW expressions for oxygen are responsible for thermometric uncertainties between 0.5% and 1% at flame temperatures. Copyright © 2012 John Wiley & Sons, Ltd.

Herman–Wallis corrections in dual‐pump CARS intensities for combustion temperature and species

Interaction between vibration and rotation in light molecules is responsible for many spectroscopic corrections to the fundamental rigid‐rotor model. In this context, we discuss the interpretation of the coherent anti‐Stokes Raman scattering (CARS) intensity that is recurrently considered in combustion science to measure high temperatures. Intensity changes are quantified by the Herman–Wallis (HW) factor and, unlike other recent works appearing on the subject where the focus is on thermometry, the present work examines the consequences for the quantitative detection of chemical species. Indeed, the diagnostic potential of CARS spectra is not limited to gas‐phase thermometry in that multiple species detection is also possible. To this end, we describe an experiment based on a conventional set‐up designed according to the principles of dual‐pump CARS that, in our case, allows for the simultaneous measurements of temperatures and mole fractions of nitrogen and oxygen observed in a reference flame operated at equivalence ratios below or near stoichiometry. Two HW factors are compared with common reference to the experimental spectra and the results show that, beyond the confirmation of thermometric corrections of about 0.5% for nitrogen CARS, the choice between different vibrational HW models significantly affects the measurement of mole fractions. The effect reaches 1% for nitrogen at stoichiometric conditions, whereas the sensitivity of oxygen to HW models reaches 2.5%. Copyright © 2012 John Wiley & Sons, Ltd.

Vibration–rotation interaction in time‐resolved coherent anti‐Stokes Raman scattering for gas‐phase thermometry

Different approaches to the vibration–rotation (VR) interaction in time‐resolved intensities of coherent anti‐Stokes Raman scattering (CARS) of light homonuclear diatomics (namely H2, N2 and O2) are here compared in view of the diagnostic refinement of the technique that has an acknowledged thermometric potential for studies in combustion science. From this perspective, VR interaction is treated in agreement with past works about conventional or frequency‐resolved CARS, and although the current theoretical context refers to optical nonlinearities excited by ultrashort laser pulses, the results confirm the strong sensitivity of H2 CARS to the VR interaction quantified by means of the Herman–Wallis (HW) factor. Indeed, depending on the model of the interaction, time‐resolved H2 signals can be subject to indeterminations on the order of 10% leading to relevant thermometric inaccuracies. Heavier diatomics are less sensitive, but the choice of an appropriate HW factor can contribute to improve the confidence in CARS thermometry with ultrashort laser pulses. Copyright © 2012 John Wiley & Sons, Ltd.

Absolute line intensities and self-broadening coefficients in the ν3 − ν1 band of carbon dioxide

Using a tunable diode-laser spectrometer, we have measured the self-broadening coefficients and strengths of 26 absorption lines in the ν3−ν1 band of 12CO2 and 13CO2 at room temperature. These lines, ranging from P(34) to R(40), are located around 960.9 and 913.4cm−1, respectively for the 12CO2 and 13CO2 molecules. The collisional widths and the intensities were obtained by fitting Voigt and Rautian and Sobel’man profiles to the measured shapes of the lines. From the individual line intensities and using a least-squares method, we have determined the vibrational band strength as well as the Herman–Wallis factors for the ν3−ν1 band of 12CO2 and 13CO2.

On the sensitivity of rotational CARS N2 thermometry to the Herman–Wallis factor

AbstractPurely rotational spectral signals of coherent anti‐Stokes Raman scattering (CARS) from nitrogen molecules are studied as a function of the vibration–rotation interaction that weakens the rigid rotor approximation under which the dominant terms of the Raman cross section are calculated. The effect of the vibration–rotation interaction is quantified by means of the Herman–Wallis (HW) factor, and different approaches to its determination are evaluated in terms of their relative contribution to the CARS intensity and thermometric measurements made in a fuel‐rich hydrocarbon flame. Known HW factors are contrasted with more complete expressions of recent derivation, and it is found that relative line strength adjustments amount to about a few percent. Such differences result in temperature corrections of less than 1%. This value should be considered for the definition of the ideal thermometric accuracy of the technique but it is of minor importance in comparison with other sources of uncertainty (e.g. Raman line widths) that emerge from the complexity typical of reactive gas mixtures. Copyright © 2011 John Wiley & Sons, Ltd.

Forbidden vibrational-rotational transitions and Herman-Wallis factors in fundamental IR bands of symmetric-top molecules

Within the theory of coupled schemes of ordering of vibrational-rotational interactions, the operator of the effective dipole moment of single-quantum vibrational transitions is represented in the form of an infinite series in vibrational (normal coordinates and conjugate momenta) or rotational variables (components of the total angular momentum). Mechanisms of activation of infrared-inactive totally symmetric vibrations in molecules of the D 2a , D 3h , C 3h , D n (n ≥ 3), S 4, T, T a , and O symmetries and forbidden vibrational-rotational transitions in IR bands of active vibrations have been studied. The group-theoretic analysis of tensor parameters in higher-order effective dipole moments of single-quantum vibrational transitions in axially symmetric molecules has been performed. The strengths of allowed transitions and forbidden transitions in fundamental and hot IR bands of axially symmetric molecules are calculated with allowance for the Herman-Wallis factors. For effective dipole moments of multiquantum transitions in molecules, models are developed in the form of infinite series in rotational variables and in the form of Pade approximants.

Line strength measurements of carbonyl sulfide ( 16O 12C 32S) in the 2 v3, v1+2 v2+ v3, and 4 v2+ v3 bands

To support planetary studies of the Venus atmosphere, we measured line strengths of the 2 v 3, v 1+2 v 2+ v 3, and 4 v 2+ v 3 bands of the primary isotopologue of carbonyl sulfide ( 16O 12C 32S), whose band centers are located at 4101.387, 3937.427, and 4141.212 cm −1, respectively. For this, infrared absorption spectra in normal carbonyl sulfide (OCS) sample gas were recorded at an unapodized resolution of 0.0033 cm −1 at ambient room temperatures using a Bruker Fourier transform spectrometer (FTS) at the Jet Propulsion Laboratory. The FTS instrumental line shape (ILS) function was investigated, which revealed no significant instrumental line broadening or distortions. Various custom-made short cells and a multi-pass White cell were employed to achieve optical densities sufficient to observe the strong 2 v 3 and the weaker bands in the region. Gas sample impurities and the isotopic abundances were determined from mass spectrum analysis. Line strengths were retrieved spectrum by spectrum using a non-linear curve fitting algorithm adopting a standard Voigt line profile, from which Herman–Wallis factors were derived for the three bands. The band strengths of 2 v 3, v 1+2 v 2+ v 3, and 4 v 2+ v 3 of 16O 12C 32S (normalized at 100% of isotopologue) are observed to be 6.315(13)×10 −19, 1.570(2)×10 −20, and 7.949(20)×10 −21 cm −1/molecule cm −2, respectively, at 296 K. These results are compared with earlier measurements and the HITRAN 2004 database.

Calculation of higher-order approximations of the coefficients of the Herman-Wallis factor. Test for hydrogen halides

The higher-order approximations of the perturbation theory for the Herman-Wallis factor, which describes the effect of the vibrational-rotational interaction on the matrix elements of dipole moments of diatomic molecules, are calculated within the formalism of quantum number polynomials. An algorithm for taking into account higher orders of the perturbation theory for calculating random coefficients is found; the algorithm has a better convergence than previous theoretical methods. As applied to hydrogen halides, the results obtained agree well with current experimental data.

Comparative analysis of Herman‐Wallis factors for uses in coherent anti‐Stokes Raman spectra of light molecules

AbstractThe cross section of Raman processes involving light molecules takes into account the unavoidable coupling between rotational and vibrational modes. This coupling is quantified by the Herman‐Wallis (HW) factor and its use was recently criticized in view of the known thermometric application of vibrational spectra of coherent antiStokes Raman scattering (CARS) of hydrogen. Following this line of research, this paper addresses the physical limitations of the HW factors employed ordinarily in Q‐branch CARS measurements. Furthermore, new determinations of the vibration–rotation coupling are considered. These are based on a more realistic scenario consisting of anharmonic oscillations and polarizability variations during vibration. Numerical examples for fundamental diatomics, namely H2 and N2, are given throughout the text. It is found that, as expected, the Raman cross section of H2 is strongly dependent on the HW factor and the consequent correction to CARS line intensities is of the order of 10% or greater. Surprisingly, the analysis suggests that N2 spectral data are also affected, to a certain extent, by the new HW factors. Usually, this correction is completely disregarded, but under the new understanding of the vibration–rotation coupling, Q‐branch CARS lines of N2 are predicted to undergo variations between 1 and 5% for rotational levels that are sufficiently populated at temperatures of interest for CARS studies. Oxygen is also prone to vibration–rotation corrections and is finally shown as an example between the two limits of hydrogen and nitrogen. Copyright © 2009 John Wiley & Sons, Ltd.

Herman–Wallis factor to improve thermometric accuracy of vibrational coherent anti-Stokes Raman spectra of H2

The Herman–Wallis factor is a molecular parameter that measures the influence of centrifugal force on the intensity of spectral lines. Understandably, the effect is significant for very light molecules that necessarily have large couplings between vibrational and rotational degrees of freedom. Although known, the conceptual basis of the Herman–Wallis factor are nevertheless not clearly established in the literature. Over the years, different approaches have been proposed to explain the corrections to spectral line-strengths and, recently, an experimental study has demonstrated that Q-branch Raman transitions of H2 are highly sensitive to the theoretical model employed to determine the Herman–Wallis factor. In this paper, this fact is used to analyze the consequences on thermometry based on coherent anti-Stokes Raman scattering (CARS) designed to probe H2 molecules in combustion studies. It is found that the different Herman–Wallis factors lead to relative thermometric disagreements from several tens up to hundreds of degrees. This analysis could explain why H2 CARS thermometry has been considered less reliable than thermometric predictions based on CARS of more common molecules such as N2, O2 and others. In particular, it is remarked that unreliable expressions of Herman–Wallis factors have been used so far to interpret Q-branch H2 CARS experiments.

Reliability of Herman–Wallis factors for Raman spectroscopy of Q-branch molecular transitions

The centrifugal force in a rotating molecule influences the intensity of spectral lines. This effect is quantified by the so-called Herman–Wallis factor, but its calculation is rather involved and the result is uncertain for Q-branch Raman transitions. This Letter discusses the reliability of various predictions by means of an experimental proof based on coherent anti-Stokes Raman scattering from H 2 molecules.