^{1}, Loïc Joubert-Doriol

^{2}, Hans-Dieter Meyer

^{3}, André Nauts

^{1,4}, Fabien Gatti

^{2}and David Lauvergnat

^{1,a)}

### Abstract

We develop a new general code to automatically derive exact analytical kinetic energy operators in terms of polyspherical coordinates. Computer procedures based on symbolic calculations are implemented. Sets of orthogonal or non-orthogonal vectors are used to parametrize the molecular systems in space. For each set of vectors, and whatever the size of the system, the exact analytical kinetic energy operator (including the overall rotation and the Coriolis coupling) can be derived by the program. The correctness of the implementation is tested for different sets of vectors and for several systems of various sizes.

Financial support by the German and French science foundations through contact DFG/ANR-09-BLA-0417 is gratefully acknowledged.

I. INTRODUCTION

II. OVERVIEW OF THE POLYSPHERICAL APPROACH

A. Standard parametrization: Without subsystems

B. Generalized parametrization: Separation into subsystems

1. Notation: Subsystems, vectors, and frames

2. Definition of vectors and frames

3. KEO in the case of uncoupled subsystems

III. OUTLINE OF THE IMPLEMENTATION

IV. APPLICATION

A. Tetra-atomic systems

B. Systems with more than four atoms

V. CONCLUSIONS AND PERSPECTIVES

### Key Topics

- Coriolis effects
- 7.0
- Angular momentum
- 6.0
- Tensor methods
- 6.0
- Symbolic computation
- 3.0
- Carbon
- 2.0

## Figures

Representation scheme of a system, *S* _{BF} = *S* _{1}, containing two subsystems, *S* _{ j, 1} ⊂ *S* _{1}, *j* = 1, 2. According to the notation discussed below, F^{(1)} is the global BF frame attached to the system, *S* _{1}. It is oriented with respect to the space-fixed frame, SF. We have assumed that F^{(1)} is defined such that the -axis and the half plane are parallel to the vectors, and , respectively.

Representation scheme of a system, *S* _{BF} = *S* _{1}, containing two subsystems, *S* _{ j, 1} ⊂ *S* _{1}, *j* = 1, 2. According to the notation discussed below, F^{(1)} is the global BF frame attached to the system, *S* _{1}. It is oriented with respect to the space-fixed frame, SF. We have assumed that F^{(1)} is defined such that the -axis and the half plane are parallel to the vectors, and , respectively.

Case 1: . Since is not involved in the definition of the F^{(r)}-frame, the orientation F^{(j, r)} with respect to F^{(r)}-frame is characterized by two Euler angles.

Case 1: . Since is not involved in the definition of the F^{(r)}-frame, the orientation F^{(j, r)} with respect to F^{(r)}-frame is characterized by two Euler angles.

Case 2: The vectors and are chosen to define the F^{(j, r)}-frame. Its orientation with respect to the F^{(r)}-frame is characterized by the three Euler angles.

Case 2: The vectors and are chosen to define the F^{(j, r)}-frame. Its orientation with respect to the F^{(r)}-frame is characterized by the three Euler angles.

Case 3: and the vector is involved in the definition of the *F* ^{(r)}-frame. In such situation, the orientation is characterized by one Euler angle.

Case 3: and the vector is involved in the definition of the *F* ^{(r)}-frame. In such situation, the orientation is characterized by one Euler angle.

(Case 4: Orientation of the F^{(1, r)}-frame with respect to the F^{(r)}-frame when the vector is involved in the definition of the F^{(r)}-frame and . The orientation is characterized by the two Euler angles.

(Case 4: Orientation of the F^{(1, r)}-frame with respect to the F^{(r)}-frame when the vector is involved in the definition of the F^{(r)}-frame and . The orientation is characterized by the two Euler angles.

Characterization by the spherical angles of a vector which is not involved in the definition of the F^{(j, r)}-frame as described by Case 4.

Characterization by the spherical angles of a vector which is not involved in the definition of the F^{(j, r)}-frame as described by Case 4.

Representation scheme of the mass-matrix **M** associated with parametrization shown in Fig. 1.

Representation scheme of the mass-matrix **M** associated with parametrization shown in Fig. 1.

Computation scheme of for a given subsystem *S* _{ j, r }. It is assumed that all are already computed by the same procedure. The total KEO is computed from the recursion equation, Eq. (20). The mass-matrix **M** is computed from TNUM.^{18}

Computation scheme of for a given subsystem *S* _{ j, r }. It is assumed that all are already computed by the same procedure. The total KEO is computed from the recursion equation, Eq. (20). The mass-matrix **M** is computed from TNUM.^{18}

Examples of parametrization of the HONO molecule with Jacobi vectors (a) and with valence vectors (b).

Examples of parametrization of the HONO molecule with Jacobi vectors (a) and with valence vectors (b).

Parametrization of H_{2}CCH_{2} molecule with Valence vectors.

Parametrization of H_{2}CCH_{2} molecule with Valence vectors.

Parametrization of the benzopyran molecule in terms of subsystems (upper graph) and in terms of vectors (lower graph).

Parametrization of the benzopyran molecule in terms of subsystems (upper graph) and in terms of vectors (lower graph).

Mixed Jacobi/Valence vectors to parametrize the water protonated dimmer.

Mixed Jacobi/Valence vectors to parametrize the water protonated dimmer.

## Tables

Numerical values of the coefficients *C* _{ i }, *i* = 1, …, 5, of the analytical expression given in Eq. (30).

Numerical values of the coefficients *C* _{ i }, *i* = 1, …, 5, of the analytical expression given in Eq. (30).

Numerical values of the coefficients *C* _{ i }, *i* = 1, …, 9, of the analytical form shown in Eq. (31).

Numerical values of the coefficients *C* _{ i }, *i* = 1, …, 9, of the analytical form shown in Eq. (31).

Numerical values of the coefficients *C* _{ i } of Eq. (B1) that corresponds to the analytical expression of the KEO of the HONO molecule in a non-orthogonal parametrization.

Numerical values of the coefficients *C* _{ i } of Eq. (B1) that corresponds to the analytical expression of the KEO of the HONO molecule in a non-orthogonal parametrization.

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