^{1,a)}and K. Birgitta Whaley

^{2}

### Abstract

We present calculations of the rotational excitations of and in helium using correlated basis function theory for excited states of spherical top molecules, together with ground statehelium density distributions computed by diffusion Monte Carlo simulations. We derive the rotational self-energy for symmetric top molecules, generalizing the previous analysis for linear molecules. The analysis of the self-energy shows that in helium the symmetry of a rigid spherical rotor is lost. In particular, rotational levels with split into states of E and of symmetry. This splitting can be analyzed in terms of an effective tetrahedral distortion that is induced by coupling of the molecular rotation to density fluctuations of the helium. Additional splitting occurs within each symmetry group as a result of rotational coupling to the high density of states between the roton and maxon excitations of , which also results in broad bands in the corresponding rotational absorption spectra. Connecting these pure rotational dynamics of methane to experimental rovibrational spectra, our results imply that the R(1) line of is significantly broadened, while the P(2) is not broadened by rotational relaxation, which is consistent with experiment. Comparison of our results for and shows that the reduction in the moment of inertia in scales approximately quadratically with the gas phase moment of inertia, as has also been observed experimentally.

I. INTRODUCTION

II. THEORY

A. CBF theory and self-energy of symmetric and spherical tops

B. Tetrahedral distortion of solvating helium

C. Rotational absorptionspectrum

III. RESULTS FOR AND

A. Ground state

B. Excitations

1. Pure rotational spectra

2. Relation to rovibrational spectra

3. Excitation energies and distorted rotor parametrization

4. Isotope effect

5. Importance of zero-point motion

C. Mechanisms for increased coupling to helium

IV. DISCUSSION AND CONCLUSION

### Key Topics

- Methane
- 39.0
- Absorption spectra
- 38.0
- Rotation constants
- 32.0
- Liquid helium
- 28.0
- Ground states
- 23.0

## Figures

Pair distribution function in the molecular frame between and in bulk . Color maps of are shown in two perpendicular planar cuts. The top panel shows the DMC result with rotation correctly taken into account, while for the bottom panel the rotational motion was artificially frozen, i.e., was treated as a spherical rotor with infinite moment of inertia.

Pair distribution function in the molecular frame between and in bulk . Color maps of are shown in two perpendicular planar cuts. The top panel shows the DMC result with rotation correctly taken into account, while for the bottom panel the rotational motion was artificially frozen, i.e., was treated as a spherical rotor with infinite moment of inertia.

The tetrahedral expansion coefficients of the pair distribution function between and bulk (in the frame), as defined in the Appendix B. The spherical average of , i.e., , is clearly the largest contribution, but as noted in the text, this does not contribute to the rotational self-energy . The dominant contribution to is therefore .

The tetrahedral expansion coefficients of the pair distribution function between and bulk (in the frame), as defined in the Appendix B. The spherical average of , i.e., , is clearly the largest contribution, but as noted in the text, this does not contribute to the rotational self-energy . The dominant contribution to is therefore .

Spectra for dipole transition (bottom) and quadrupole transition (middle: E symmetry; top: symmetry) for . Transitions that are not homogeneously broadened due to rotational relaxation ( and the main peak of , E symmetry) appear as sharp lines. We have broadened these transitions by an artificial imaginary part (dashed lines), see text.

Spectra for dipole transition (bottom) and quadrupole transition (middle: E symmetry; top: symmetry) for . Transitions that are not homogeneously broadened due to rotational relaxation ( and the main peak of , E symmetry) appear as sharp lines. We have broadened these transitions by an artificial imaginary part (dashed lines), see text.

Spectra for dipole transition (bottom) and quadrupole transition (middle: E symmetry; top: symmetry) for . Transitions that are not homogeneously broadened due to rotational relaxation ( and the main peak of , E symmetry) appear as sharp lines. We have broadened these transitions by an artificial imaginary part (dashed lines), see text.

Left: schematic of the rovibrational energy levels in the ground and first excited vibrational states of the mode of together with the corresponding spectroscopic transitions. Right: schematic of the rotational energy levels and corresponding transitions in the ground vibrational state that derive from the and rotational levels calculated in this work.

Left: schematic of the rovibrational energy levels in the ground and first excited vibrational states of the mode of together with the corresponding spectroscopic transitions. Right: schematic of the rotational energy levels and corresponding transitions in the ground vibrational state that derive from the and rotational levels calculated in this work.

The tetrahedral expansion coefficients (top panel) and (bottom panel) for and in bulk for obtained by DMC simulation. , the dominant contribution to the reduction in rotational energies, is about twice as large for (dotted line) as for (solid line).

The tetrahedral expansion coefficients (top panel) and (bottom panel) for and in bulk for obtained by DMC simulation. , the dominant contribution to the reduction in rotational energies, is about twice as large for (dotted line) as for (solid line).

Reduction in lowest rotational excitation energy for and as a function of the relative effective mass , where this is treated as a variable parameter.

Reduction in lowest rotational excitation energy for and as a function of the relative effective mass , where this is treated as a variable parameter.

## Tables

Rotational energies , , and for and in obtained from CBF-DMC calculations. The square brackets indicate whether the MP4 or VB potential of Ref. 20 for the methane-helium interaction has been used. All energies are given in .

Rotational energies , , and for and in obtained from CBF-DMC calculations. The square brackets indicate whether the MP4 or VB potential of Ref. 20 for the methane-helium interaction has been used. All energies are given in .

Rotational constants , , and obtained from the rotational energies for of given in Table I. Also shown are the experimental values of rotational constant (vibrational ground state) and distortion constant from Refs. 22 and 23 and experimental values from Ref. 24. Square brackets indicate whether the MP4 or VB potential for the methane-helium interaction has been used in the calculations. All energies are given in . Reductions in relative to the gas phase value are shown in parentheses.

Rotational constants , , and obtained from the rotational energies for of given in Table I. Also shown are the experimental values of rotational constant (vibrational ground state) and distortion constant from Refs. 22 and 23 and experimental values from Ref. 24. Square brackets indicate whether the MP4 or VB potential for the methane-helium interaction has been used in the calculations. All energies are given in . Reductions in relative to the gas phase value are shown in parentheses.

For both methane-helium interaction potentials, the rigid rotor estimates are tabulated for and , as well as the changes with respect to the gas phase value (, (Ref. 17)), , and of the effective moment of inertia . We also tabulate the relative reduction for and , which scales roughly linearly with the respective gas phase moment of inertia . Energies are given in , moments of inertia are given in .

For both methane-helium interaction potentials, the rigid rotor estimates are tabulated for and , as well as the changes with respect to the gas phase value (, (Ref. 17)), , and of the effective moment of inertia . We also tabulate the relative reduction for and , which scales roughly linearly with the respective gas phase moment of inertia . Energies are given in , moments of inertia are given in .

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