Vibration–rotation interactions in H₂, HD and D₂ : centrifugal distortion factors and the derivatives of polarisability invariants

ABSTRACT

We report the correction factors for centrifugal distortion in Raman intensities for pure rotation (O0- and S0-branch) and vibration–rotation (O1- and S1-branch) transitions in the ground electronic state of H₂, HD and ${\mathrm{D}}_2$. These factors are presented for 52 selected excitation wavelengths and for the initial rotational states, $J=2-21$. This data is useful in applications of intensity calibration of spectrometers and the spectroscopy of flames. The classical treatment of centrifugal distortion involved the expansion of polarisability anisotropy (γ) over the internuclear distance, while assuming the diatomic molecule behaves as a harmonic oscillator. Here, this approximation of polarisability invariants as a Taylor series expansion is tested, revealing that truncation up to the second-order derivatives of mean polarisability ($\overline{\alpha }$) and polarisability anisotropy (γ) gives faithful representations, yielding accurate expectation values with error $<0.2\mathrm{\%}$, for the ground rovibrational state and for the fundamental transition.

GRAPHICAL ABSTRACT

Keywords:

• Polarisability
• hydrogen molecule
• polarisability invariants
• Raman spectroscopy

7. Supplementary material and data availability

Supplementary material includes the following content. Section S1 : Tabulation of correction factors to Raman intensities for pure rotation (O0- and S0-branch) in ${\mathrm{H}}_2$, HD and ${\mathrm{D}}_2$; Section S2 : Tabulation of correction factors to Raman intensities for vibration–rotation transitions (O1- and S1-branch) in ${\mathrm{H}}_2$, HD and ${\mathrm{D}}_2$; Sections S3 and S4 : 2D plots of the correction factors for HD and ${\mathrm{D}}_2$; Section S5 : List of band frequencies for transitions covered in this work; Section S6 : Fraction of molecules populating ro-vibrational states at different temperatures; and Section S7: Details on the computation of the derivatives of polarisability and the matrix elements of the Taylor series expansions. Program written in Python for computing the derivatives of polarisability invariants and their matrix elements for ${\mathrm{H}}_2$, HD and ${\mathrm{D}}_2$ have been made available on GitHub [34] indexed with doi:10.5281/zenodo.2576016.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 S0-branch consists of the $\mathrm{\Delta }v=0,\mathrm{\Delta }J=+2$ transitions appearing on the Stokes side. The O0-branch consists of the corresponding anti-Stokes bands.

2 S1-branch includes $\mathrm{\Delta }v=1,\mathrm{\Delta }J=+2$ transitions appearing on the higher frequency side (in Ramanshift, $\mathrm{\Delta }\nu$), while the O1-branch consists of the $\mathrm{\Delta }v=1,\mathrm{\Delta }J=-2$ transitions appearing on the lower frequency side.

3 Several symbols have been used for the correction factors like the Herman–Wallis factor as $F_{v,v^{\mathrm{\prime }}}\left(m\right)$ [1], and $Z_{v,v^{\mathrm{\prime }}}^2(J,\nu )$ given by Cheung et al. [19]. In this work, we use $F_{v^{\mathrm{\prime }},v}^{\gamma }\left(J\right)$ for the correction factors following Buldakov et al. [9]. This notation explicitly specifies the involved polarisability invariant(s). The notation of the involved vibrational states is shown as $v^{\mathrm{\prime }},v$ for consistency with the notation for the rovibrational matrix elements discussed in later sections of this work.

4 This section discusses the approximation made in earlier classical works on the vibration–rotation interaction. These works involved the Taylor series expansion centred at some internuclear distance, and included first and second derivatives to discuss the change in polarisability over internuclear distance, thus accounting for vibration–rotation interaction. This section focuses on verifying the accuracy of classical approach when studying vibration–rotation interaction, and we use ab initio polarisability and wavefunctions (from numerical solution of the radial nuclear equation) for this purpose.

Additional information

Funding

This work was financially supported by Ministry of Science and Technology of Taiwan (Grant Nos. MOST105-2923-M-009-001-MY3, MOST105-2113-M-009-018-MY3 and MOST103-2113-M-009-001) and the Center for Emergent Functional Matter Science of National Chiao Tung University from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE), Taiwan.