^{1}, Soo-Ying Lee

^{1}, Minghui Yang

^{2,a)}and Yunpeng Lu

^{1,a)}

### Abstract

Full-dimensional quantum calculations of the vibrational states of have been performed on the accurate potential energy surface developed by Xie et al. [J. Chem. Phys.122, 224307 (Year: 2005)10.1063/1.1927529]. The zero point energies of , H_{4}D^{+}, D_{4}H^{+}, and and their ground-state geometries are presented and compared with earlier theoretical results. The first 10 low-lying excited states of are assigned to the fundamental, overtone, and combination of the H_{2}– stretch, the shared proton hopping and the out-of-plane torsion. The ground-state torsional tunneling splitting, the fundamental of the photon hopping mode and the first overtone of the torsion mode are 87.3 , 354.4 , and 444.0 , respectively. All of these values agree well with the diffusion Monte Carlo and multi-configuration time-dependent Hartree results where available.

H. Song, S.-Y. Lee, and M. Lu were supported by a Ministry of Education, Singapore, Grant No. MOE2011-T2-2-087. M. Yang was supported by the National Science Foundation of China (Project Nos. 21221064 and 21073229). The authors would like to thank Professor Joel M. Bowman for helpful discussion and Dr. Yimin Wang for sending us the PES used in this work.

I. INTRODUCTION

II. THEORY

III. RESULTS

A. Numerical parameters and convergence

B. Ground state properties

C. Assignment of vibrational states

IV. CONCLUSIONS

### Key Topics

- Excited states
- 17.0
- Protons
- 17.0
- Wave functions
- 13.0
- Ground states
- 12.0
- Angular momentum
- 8.0

## Figures

The Jacobi coordinates for AB-ECD system. *R* is the distance from the center-of-mass of AB to the center-of-mass of ECD, *r* _{1} is the bond distance of AB, *r* _{2} is the distance between E and the center-of-mass of CD, *r* _{3} is the bond distance of CD, *A* _{1} is the bending angle between *R* and *r* _{1}, *A* _{2} is the bending angle between *R* and *r* _{2}, *A* _{3} is the bending angle between *r* _{2} and *r* _{3}, *B* _{1} is the torsion angle around *R* and *B* _{2} is the torsion angle around *r* _{2}.

The Jacobi coordinates for AB-ECD system. *R* is the distance from the center-of-mass of AB to the center-of-mass of ECD, *r* _{1} is the bond distance of AB, *r* _{2} is the distance between E and the center-of-mass of CD, *r* _{3} is the bond distance of CD, *A* _{1} is the bending angle between *R* and *r* _{1}, *A* _{2} is the bending angle between *R* and *r* _{2}, *A* _{3} is the bending angle between *r* _{2} and *r* _{3}, *B* _{1} is the torsion angle around *R* and *B* _{2} is the torsion angle around *r* _{2}.

One-dimensional probability distribution functions of the ground state wave function of along R, r_{2}, and B_{1}.

One-dimensional probability distribution functions of the ground state wave function of along R, r_{2}, and B_{1}.

Cuts through the wave functions of the first five vibrationally excited states of the ion. Other coordinates are fixed at DVR points with maximum amplitude of the corresponding one-dimensional wave function. Contours are from 20% to 80% of the maximum amplitude with an interval of 20%. The red (blue) curves enclose regions of positive (negative) amplitude.

Cuts through the wave functions of the first five vibrationally excited states of the ion. Other coordinates are fixed at DVR points with maximum amplitude of the corresponding one-dimensional wave function. Contours are from 20% to 80% of the maximum amplitude with an interval of 20%. The red (blue) curves enclose regions of positive (negative) amplitude.

Cuts through the wave functions of the sixth to tenth vibrationally excited states of the ion. Other coordinates are fixed at DVR points with maximum amplitude of the corresponding one-dimensional wave function. Contours are from 20% to 80% of the maximum amplitude with an interval of 20%. The red (blue) curves enclose regions of positive (negative) amplitude.

Cuts through the wave functions of the sixth to tenth vibrationally excited states of the ion. Other coordinates are fixed at DVR points with maximum amplitude of the corresponding one-dimensional wave function. Contours are from 20% to 80% of the maximum amplitude with an interval of 20%. The red (blue) curves enclose regions of positive (negative) amplitude.

## Tables

Numerical parameters used in the full-dimensional quantum bound state calculations of and its isotopic variants. The range of *B* _{1} can be reduced from [0, 2π] to [0, π] according to the space inversion symmetry of the whole system. (Atomic units are used unless stated otherwise.)

Numerical parameters used in the full-dimensional quantum bound state calculations of and its isotopic variants. The range of *B* _{1} can be reduced from [0, 2π] to [0, π] according to the space inversion symmetry of the whole system. (Atomic units are used unless stated otherwise.)

Convergence of the vibrational energies (in cm^{−1}) of with respect to basis/grid number. The angular basis number are first tested in a five-dimensional (5D) model, in which all the radial coordinates are fixed at the global minimum of the PES. Then, the radial basis number are tested in the full-dimensional (FD) model.

Convergence of the vibrational energies (in cm^{−1}) of with respect to basis/grid number. The angular basis number are first tested in a five-dimensional (5D) model, in which all the radial coordinates are fixed at the global minimum of the PES. Then, the radial basis number are tested in the full-dimensional (FD) model.

Convergence of the ZPE values (in cm^{−1}) of with respect to basis/grid number.

Convergence of the ZPE values (in cm^{−1}) of with respect to basis/grid number.

Comparison of zero point energy (ZPE) values of and its isotopic variants with other theoretical results. (The ZPE values are in cm^{−1} and relative to the potential minimum. In H_{4}D^{+} and D_{4}H^{+}, the D and H atom, respectively, is in between the two diatoms.)

Comparison of zero point energy (ZPE) values of and its isotopic variants with other theoretical results. (The ZPE values are in cm^{−1} and relative to the potential minimum. In H_{4}D^{+} and D_{4}H^{+}, the D and H atom, respectively, is in between the two diatoms.)

Vibrational energies of the first 10 excited states of with total angular momentum *J* = 0 with the corresponding ZPE as a reference (in cm^{−1}). The intermolecular vibrational quantum numbers (*n* _{ str }, *n* _{ hop }, *n* _{ tor }) denote the H_{2}– stretch, the shared-proton hopping, and the out-of-plane torsion, respectively.

Vibrational energies of the first 10 excited states of with total angular momentum *J* = 0 with the corresponding ZPE as a reference (in cm^{−1}). The intermolecular vibrational quantum numbers (*n* _{ str }, *n* _{ hop }, *n* _{ tor }) denote the H_{2}– stretch, the shared-proton hopping, and the out-of-plane torsion, respectively.

Comparison of the ground-state torsional splitting, the fundamental of , and the first overtone of the torsion mode with other theoretical results (in cm^{−1}).

Comparison of the ground-state torsional splitting, the fundamental of , and the first overtone of the torsion mode with other theoretical results (in cm^{−1}).

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