^{1,a)}, David Lauvergnat

^{2,b)}, Fabien Gatti

^{3,c)}and Hans-Dieter Meyer

^{1,d)}

### Abstract

Generalized curvilinear coordinates, as, e.g., polyspherical coordinates, are in general better adapted to the resolution of the nuclear Schrödinger equation than rectilinear ones like the normal mode coordinates. However, analytical expressions of the kinetic energy operators (KEOs) for molecular systems in polyspherical coordinates may be prohibitively complicated for large systems. In this paper we propose a method to generate a KEO numerically and bring it to a form practicable for dynamical calculations. To examine the new method we calculated vibrational spectra and eigenenergies for nitrous acid (HONO) and compare it with results obtained with an exact analytical KEO derived previously [F. Richter, P. Rosmus, F. Gatti, and H.-D. Meyer, J. Chem. Phys.120, 6072 (2004)]10.1063/1.1651051. In a second example we calculated π → π* photoabsorption spectrum and eigenenergies of ethene (C_{2}H_{4}) and compared it with previous work [M. R. Brill, F. Gatti, D. Lauvergnat, and H.-D. Meyer, Chem. Phys.338, 186 (2007)]10.1016/j.chemphys.2007.04.002. In this ethene study the dimensionality was reduced from 12 to 6 by freezing six internal coordinates. Results for both molecules show that the proposed method for obtaining an approximate KEO is reliable for dynamical calculations. The error in eigenenergies was found to be below 1 cm^{−1} for most states calculated.

K.S. is grateful to Dr. D. Peláez and M. Eroms for fruitful discussions on MCTDH. K.S. thanks IMPRS Quantum Dynamics in Physics, Chemistry and Biology for support. Financial support by the German and French science foundations through contract DFG/ANR-09-BLA-0417 is gratefully acknowledged.

I. INTRODUCTION

II. THE MULTI-CONFIGURATION TIME-DEPENDENT HARTREE METHOD

A. The MCTDH wavefunction

B. Wavepacket propagation

C. Improved relaxation

III. NUMERIC KINETIC ENERGY OPERATOR

A. Kinetic energy operators

B. Kinetic energy operator with Tnum

C. n-Mode representation

D. Polyspherical coordinates

IV. RESULTS

A. HONO

B. Ethene

V. CONCLUSION

### Key Topics

- Wave functions
- 9.0
- Ground states
- 8.0
- Tensor methods
- 6.0
- Acids
- 4.0
- Mean field theory
- 4.0

## Figures

Shown is HONO in *trans*-geometry. The *cis*-geometry is reached by increasing the dihedral angle φ by π, i.e., by rotating *R* _{1} with respect to *R* _{2} around *R* _{3}.

Shown is HONO in *trans*-geometry. The *cis*-geometry is reached by increasing the dihedral angle φ by π, i.e., by rotating *R* _{1} with respect to *R* _{2} around *R* _{3}.

Spectrum of HONO. (a) Spectrum of HONO calculated by propagation of an initial wavepacket (see text). Displayed are the Fourier transforms of autocorrelation functions obtained by using analytical (continuous line) and cluster expansion (dotted line) KEOs. The two spectra lie on top of each other and their difference can hardly be seen in this figure. (b) Enlarged view on the high energy part of the above spectrum, tiny differences can now be observed.

Spectrum of HONO. (a) Spectrum of HONO calculated by propagation of an initial wavepacket (see text). Displayed are the Fourier transforms of autocorrelation functions obtained by using analytical (continuous line) and cluster expansion (dotted line) KEOs. The two spectra lie on top of each other and their difference can hardly be seen in this figure. (b) Enlarged view on the high energy part of the above spectrum, tiny differences can now be observed.

Polyspherical coordinates for C_{2}H_{4}. G is the center of mass of the system. denotes the center of mass of the H_{2} subsystem. Reprinted with permission from M. R. Brill, F. Gatti, D. Lauvergnat, and H.-D. Meyer, Chem. Phys.338, 186 (2007)10.1016/j.chemphys.2007.04.002. Copyright 2007, Elsevier B.V.

Polyspherical coordinates for C_{2}H_{4}. G is the center of mass of the system. denotes the center of mass of the H_{2} subsystem. Reprinted with permission from M. R. Brill, F. Gatti, D. Lauvergnat, and H.-D. Meyer, Chem. Phys.338, 186 (2007)10.1016/j.chemphys.2007.04.002. Copyright 2007, Elsevier B.V.

Absorption spectrum of π → π* of ethene. The lower curve is obtained with the first-order cluster expansion of the KEO and the upper curve is the spectrum of Ref. 15, i.e., calculated using the analytical main term and zeroth order cluster for the correction term Eq. (26).

Absorption spectrum of π → π* of ethene. The lower curve is obtained with the first-order cluster expansion of the KEO and the upper curve is the spectrum of Ref. 15, i.e., calculated using the analytical main term and zeroth order cluster for the correction term Eq. (26).

## Tables

DVR type and size for the various degrees of freedom. The distances are given in atomic units. HO denotes a harmonic oscillator (hermite) DVR and exp denotes an exponential DVR. N is the number of grid points.

DVR type and size for the various degrees of freedom. The distances are given in atomic units. HO denotes a harmonic oscillator (hermite) DVR and exp denotes an exponential DVR. N is the number of grid points.

Vibrational levels *A* ^{′} (even quanta for ν_{6}) and *A* ^{″} (odd quanta for ν_{6}) for *trans*-HONO. ν_{1} is OH stretching, ν_{2} is NO stretching, ν_{3} is NOH bending, ν_{4} is ONO bending, ν_{5} is ON stretching, and ν_{6} is torsional mode. The first column in the table shows the assignment of the state. The second column displays the eigenvalues obtained with analytical KEO, the third column shows the population of , the fourth column is energy above ground state, fifth column shows the eigenvalues obtained by using cluster expanded numerical KEO, and the last column gives their difference. The last row shows the energies of the OH-stretch fundamental. All energies are in cm^{−1}.

Vibrational levels *A* ^{′} (even quanta for ν_{6}) and *A* ^{″} (odd quanta for ν_{6}) for *trans*-HONO. ν_{1} is OH stretching, ν_{2} is NO stretching, ν_{3} is NOH bending, ν_{4} is ONO bending, ν_{5} is ON stretching, and ν_{6} is torsional mode. The first column in the table shows the assignment of the state. The second column displays the eigenvalues obtained with analytical KEO, the third column shows the population of , the fourth column is energy above ground state, fifth column shows the eigenvalues obtained by using cluster expanded numerical KEO, and the last column gives their difference. The last row shows the energies of the OH-stretch fundamental. All energies are in cm^{−1}.

Vibrational levels *A* ^{′} and *A* ^{″} for *cis*-HONO. See caption of Table II.

Vibrational levels *A* ^{′} and *A* ^{″} for *cis*-HONO. See caption of Table II.

DVR type and size for the various degrees of freedom. Distances are given in atomic units. sin denotes *sine* DVR. Compare with Table I for further explanation.

DVR type and size for the various degrees of freedom. Distances are given in atomic units. sin denotes *sine* DVR. Compare with Table I for further explanation.

Single particle function basis sets used for propagation and eigenenergy calculations.

Single particle function basis sets used for propagation and eigenenergy calculations.

Energy states for ethene. The first column shows the eigenenergies for Hamiltonian with analytic main term KEO, the second column gives excitation energies, i.e., energies of the first column minus the zero point energy. The third column shows the eigenenergies for Hamiltonian with numerical KEO, and the fourth column displays the differences. All energies are in cm^{−1}.

Energy states for ethene. The first column shows the eigenenergies for Hamiltonian with analytic main term KEO, the second column gives excitation energies, i.e., energies of the first column minus the zero point energy. The third column shows the eigenenergies for Hamiltonian with numerical KEO, and the fourth column displays the differences. All energies are in cm^{−1}.

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