Correlation between normal character and Raman intensities of isotopologues of carbon dioxide

The gist: This contribution establishes correlations between the degree of normality and Raman transition intensities, mean polarizability derivatives, and variational Raman shift errors across the complete series of carbon dioxide isotopologues at room temperature.

Introduction to Normal Modes and Internal Coordinates

The concept of normal modes is crucial in vibrational spectroscopy because they provide the frequency associated with collective motion of atoms specified by normal coordinates The direct approach to obtain the normal modes relies on establishing the Lagrangian in terms of Cartesian coordinates, which allows for calculation using standard programs like GAUSSIAN and MOLPRO However, using Cartesian coordinates prevents a direct physical interpretation of force constants and avoids access to chemical information A crucial step involves introducing internal coordinates to link force constants to the concept of chemical bond in valence bond theory This framework allows force constants estimated for one molecule to be carried over into another similar molecule. Furthermore, internal coordinates are essential for establishing the concept of local modes, which were coined from overtone spectroscopy studies of molecules with X–H bonds. Molecules with terminal bonds involving large mass differences exhibit local mode behavior at high energies, experimentally proven by the existence of doublets of energy.

Defining the Degree of Normality

A new criterion to establish a normal mode degree is proposed based on internal coordinates, relying on the connection between normal and local coordinates using a second quantization scheme. This parameter, denoted as δ−, is obtained by averaging the coefficients associated with the annihilation operators in an expansion of the creation bosonic operator belonging to a given normal mode. The important property of this expansion is that in the pure local limit, the annihilation contributions vanish and consequently, the parameter δ− measures the degree of normality of the system. For the isotopologue series of CO2, this degree of normality ranges from 0.0338 to 0.0418, which represents a remarkably large value compared to that of local-mode molecules such as H2O. This parameter represents an alternative to the Child and Halonen’s parameter used to measure the degree of locality.

Model for Raman Spectra Simulation

The present model simulates Raman spectra by generalizing a scheme devised for the principal CO2 isotopologue to asymmetric triatomic linear molecules. The standard approach for computing spectral intensities is based on the double harmonic approximation, where the potential energy surface is truncated at second order and the polarizability surface is restricted to linear terms. This study goes beyond this by employing anharmonic vibrational wave functions obtained from a higher-order PES expansion, evaluating transition moments in an interacting anharmonic oscillator model. The Raman line intensities for the isotropic contribution are obtained with the trace of the differential cross section, which is given by Eq. (24). The matrix elements Mi j in Eq. (24) are calculated by expanding the mean polarizability around molecular equilibrium in terms of curvilinear symmetry coordinates S up to cubic terms.

Correlation Results and Physical Interpretation

The study found a correlation for asymmetric isotopologues between the first order derivatives (∂α¯ /∂S1)0 and the coefficients c(1,1)− as well as between the derivatives (∂α¯ /∂S3)0 and the coefficients c(3,1)+. This feature also appears in the correlation between the parameter δ− and the Fermi interaction. The ratio of principal intensities for both symmetries Σ+ and Π are strongly correlated with the parameter δ−, which can be interpreted as arising from the increasing strength of the Fermi resonance with δ−. The largest deviations from experimental results appear in the extremes of normal mode behavior centered at the principal isotopologue, with minimum deviation.

Conclusion

Both the parameter δ− and the local coefficients c(ι, j)± provide tools to make predictions for a given series of isotopologues. They represent descriptors of a molecule, encoding information of the molecule that manifests in fundamental observables in spectroscopy, i.e., the Raman intensities. The normal mode criterion is general to any molecular system but, in this work, it is particularly relevant in the analysis of a series of isotopologues. This approach will be employed in a forthcoming work for the study of samples containing a mixture of carbon dioxide isotopologues.

Acknowledgments

The authors acknowledge useful discussions with Marisol Bermúdez-Montaña, José M. Fernández and Guzmán Tejeda

Funding

This work is partially supported by DGAPA-UNAM, México, under project IN-212224, and project BiMiAb H2 at the Federal Institute for Geosciences and Natural Resources, Germany. This project has also received funding from the European Union’s Horizon 2020 research and innovation program under Marie Sklodowska-Curie grant agreement No. 872081, and grant PID2022-136228NBC21 (M.C.) funded by MCIN/AEI/10.13039/501100011033, and, as appropriate, by ERDF A way of making Europe, the European Union, or the European Union NextGenerationEU/PRTR.

Supplementary Material

In the supplementary material the justification of analyzing the Raman spectra in involving only vibrational degrees of freedom is presented. In addition, fits to the transition moments and experimental Raman intensities are included.

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1. Improved AI can predict Raman transition moments for any carbon dioxide isotopologue by utilizing an algebraic approach based on an effective Hamiltonian and fitting 42 experimental transition moments, as described in Section 5 and Table 4, thereby enhancing predictive accuracy beyond the harmonic approximation.

2. Improved AI can quantify the degree of normality of a molecule by calculating the parameter δ−, which is defined as a parameter δ− measuring the degree of normality, and correlating it with experimental observables like Raman intensities to identify isotopologues through Raman spectra analysis.

3. Improved AI can assess deviations from Born-Oppenheimer approximations by calculating Raman frequency shift errors using an algebraic approach, comparing them against variational methods to determine which theoretical description is most accurate for a given isotopologue series.

4. Improved AI can explore the influence of rotational structure on spectral correlations by analyzing the difference between vibrationally dependent rotational constants, ∆Bν = Bνf − Bνi, and the degree of normality δ−, as shown in Figure 4, to refine simulations that include rotational degrees of freedom.

Previous studies on the vibrational structure of carbon dioxide isotopologues have revealed a correlation between the strength of Fermi interactions and a recently defined normal-mode parameter. In this contribution, we investigate how the Raman intensities of the most intense bands evolve across the complete series of carbon dioxide isotopologues at room temperature. To this end, we introduce a general scheme for simulating Raman intensities in linear triatomic molecules, based on the computation of the mean polarizability surface up to cubic terms for all twelve stable species. For the first time, we establish correlations between the degree of normality and the Raman transition intensities, mean polarizability derivatives, and variational Raman shift errors across the CO 2 isotopologue series. The observed trends are further analyzed by considering rotational contributions, which may partially account for the deviations observed in the correlations. Our study is supported by experimental Raman spectra of isotopically enriched samples and theoretical simulations. Although this study focuses on carbon dioxide, similar behavior is expected for other series of isotopologues.