^{1,a)}, Fabien Gatti

^{1,b)}, Christophe Iung

^{1,c)}and Hans-Dieter Meyer

^{2,d)}

### Abstract

A theoretical study of the vibrational spectrum of the molecule is carried out with the aid of the multiconfiguration time-dependent Hartree (MCTDH) algorithm. In order to obtain the eigenvalues and the eigenstates, recent developments in the MCTDH *improved relaxation* method in a block form are exploited. Around 80 eigenvalues are reported, which are converged with a very high accuracy. The results obtained with our study are compared with those of a previous work using the wave operator sorting algorithm approach. The present investigation exemplifies the robustness and the accuracy of the improved relaxation method.

This work was supported by ANR of the French Centre National de la Recherche Scientifique (CNRS) under Project No. NT05-3_42315. Financial support by the Deutsche Forschungsgemeinschaft is also gratefully acknowledged. The authors thank Yohann Scribano for providing the vibrational fundamental transitions converged with the VMCFI method.

I. INTRODUCTION

II. THEORY

A. Hamiltonian

B. MCTDH and improved relaxation

III. COMPUTATIONAL RESULTS AND DISCUSSION

A. Vibrational spectrum

B. Eigenstate analysis of the region

IV. CONCLUSION

### Key Topics

- Eigenvalues
- 13.0
- Excitation energies
- 8.0
- Wave functions
- 8.0
- Mean field theory
- 5.0
- Excited states
- 4.0

## Tables

Vibrational modes of . Here, the notation in Ref. 74 is used.

Vibrational modes of . Here, the notation in Ref. 74 is used.

Parameters for the primitive basis sets employed. HO denotes a harmonic oscillator (Hermite) DVR. denotes the number of grid points. For each degree of freedom, we have used the frequency given in Table I to define the frequency parameter of the harmonic oscillator DVR.

Parameters for the primitive basis sets employed. HO denotes a harmonic oscillator (Hermite) DVR. denotes the number of grid points. For each degree of freedom, we have used the frequency given in Table I to define the frequency parameter of the harmonic oscillator DVR.

Vibrational levels obtained with the improved relaxation method (second column), the WOSA method (third column), and by experiment (fourth column). Energies, given in , are relative to the vibrational ground state. The energy of the ground state is . The numbers denote the average values of the quantum numbers associated with the zeroth-order nondegenerate or degenerate harmonic oscillator states (see text). The symbol indicates that the state is degenerate in harmonic approximation.

Vibrational levels obtained with the improved relaxation method (second column), the WOSA method (third column), and by experiment (fourth column). Energies, given in , are relative to the vibrational ground state. The energy of the ground state is . The numbers denote the average values of the quantum numbers associated with the zeroth-order nondegenerate or degenerate harmonic oscillator states (see text). The symbol indicates that the state is degenerate in harmonic approximation.

Vibrational levels obtained with the improved relaxation method (second column), the WOSA method (third column), and by experiment (fourth column). Energies, given in , are relative to the vibrational ground state. The energy of the ground state is . The numbers denote the average values of the quantum numbers associated with the zeroth-order nondegenerate or degenerate harmonic oscillator states (see text). The symbol indicates that the state is degenerate in harmonic approximation.

Energies associated to eigenstates whose projection on are larger than 0.05. The two last values correspond to the so-called “satellite lines.” Ovlp denotes the overlap squared between the eigenstates of the harmonic and full Hamiltonian. The harmonic eigenstates are characterized by their quantum numbers and the relaxed eigenstates by their average quantum numbers (see text).

Energies associated to eigenstates whose projection on are larger than 0.05. The two last values correspond to the so-called “satellite lines.” Ovlp denotes the overlap squared between the eigenstates of the harmonic and full Hamiltonian. The harmonic eigenstates are characterized by their quantum numbers and the relaxed eigenstates by their average quantum numbers (see text).

Comparison of the energies (in ) of the vibrational fundamental transitions obtained by improved relaxation with those computed with the VMFCI method Ref. 87).

Comparison of the energies (in ) of the vibrational fundamental transitions obtained by improved relaxation with those computed with the VMFCI method Ref. 87).

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