^{1,a)}, R. González-Férez

^{1}and P. Schmelcher

^{2}

### Abstract

We examine the impact of the combination of a static electric field and a non-resonant linearly polarized laser field on an asymmetric top molecule. Within the rigid rotor approximation, we analyze the symmetries of the Hamiltonian for all possible field configurations. For each irreducible representation, the Schrödinger equation is solved by a basis set expansion in terms of a linear combination of symmetric top eigenfunctions respecting the corresponding symmetries, which allows us to distinguish avoided crossings from genuine ones. Using the fluorobenzene and pyridazine molecules as prototypes, the rotational spectra and properties are analyzed for experimentally accessible static field strengths and laser intensities. Results for energy shifts, orientation, alignment, and hybridization of the angular motion are presented as the field parameters are varied. We demonstrate that a proper selection of the fields gives rise to a constrained rotational motion in three Euler angles, the wave function being oriented along the electrostatic field direction, and aligned in other two angles.

Financial support by the Spanish project FIS2008-02380 (MICINN) (Grant Nos. FQM-2445 and FQM-4643) (Junta de Andalucía) is gratefully appreciated. J.J.O. and R.G.F. belong to the Andalusian research group FQM-207. J.J.O. acknowledges the support of ME under the program FPU. We thank J. Küpper for fruitful discussions.

I. INTRODUCTION

II. HAMILTONIAN OF AN ASYMMETRIC TOP MOLECULE IN THE PRESENCE OF THE FIELDS

A. Symmetries

III. RESULTS

A. Impact of a linearly polarized laser field

B. Constant static electric field and increasing laser intensity

C. Constant laser intensity and increasing electric field strength

D. Orientation and 2-D alignment by means of perpendicular fields

E. Influence of the inclination of the fields

IV. CONCLUSIONS

### Key Topics

- Electrostatics
- 18.0
- Static electric fields
- 18.0
- Wave functions
- 17.0
- Electric dipole moments
- 14.0
- Polarizability
- 13.0

## Figures

Laboratory and molecular fixed coordinate frames and field configuration.

Laboratory and molecular fixed coordinate frames and field configuration.

Structure of the (a) fluorobenzene and (b) pyridazine molecules.

Structure of the (a) fluorobenzene and (b) pyridazine molecules.

Laser interaction term *H* _{ L }, see Eq. (4), in units of cm^{−1} with *I* = 10^{11} W cm^{−2}, for (a) χ = π/2 and (b) θ = π/2 (b) for fluorobenzene (solid) and pyridazine (dash).

Laser interaction term *H* _{ L }, see Eq. (4), in units of cm^{−1} with *I* = 10^{11} W cm^{−2}, for (a) χ = π/2 and (b) θ = π/2 (b) for fluorobenzene (solid) and pyridazine (dash).

(a) Ground state energy, (b) ⟨cos ^{2}θ⟩, (c) ⟨sin ^{2}χ⟩, and (d) ⟨*K* ^{2}⟩, as a function of the intensity of a linearly polarized laser field, for the fluorobenzene (solid) and pyridazine (dashed) molecules.

(a) Ground state energy, (b) ⟨cos ^{2}θ⟩, (c) ⟨sin ^{2}χ⟩, and (d) ⟨*K* ^{2}⟩, as a function of the intensity of a linearly polarized laser field, for the fluorobenzene (solid) and pyridazine (dashed) molecules.

(a)–(c) Energies and the expectation values, (d)–(f) ⟨cos θ_{ S }⟩, (g)–(i) ⟨cos ^{2}θ⟩, and (j)–(l) ⟨cos ^{2}χ⟩ for a constant field *E* _{ S } = 20 kV cm^{−1} as a function of the intensity of the laser field for for the first states with both *M* + *q* + *K* and *K* even for fluorobenzene. The states are 0_{00}0 (solid black), 1_{01}1 (solid green), 1_{01}0 (dashed–dotted black), 2_{02}2 (dashed–dotted green), 2_{02}1 (dashed black), and 3_{03}2 (dashed green). The spectrum (a, b, c) contains also highly excited states (very thin lines).

(a)–(c) Energies and the expectation values, (d)–(f) ⟨cos θ_{ S }⟩, (g)–(i) ⟨cos ^{2}θ⟩, and (j)–(l) ⟨cos ^{2}χ⟩ for a constant field *E* _{ S } = 20 kV cm^{−1} as a function of the intensity of the laser field for for the first states with both *M* + *q* + *K* and *K* even for fluorobenzene. The states are 0_{00}0 (solid black), 1_{01}1 (solid green), 1_{01}0 (dashed–dotted black), 2_{02}2 (dashed–dotted green), 2_{02}1 (dashed black), and 3_{03}2 (dashed green). The spectrum (a, b, c) contains also highly excited states (very thin lines).

(a)–(c) Energies and the expectation values, (d)–(f) ⟨cos θ_{ S }⟩, (g)–(i) ⟨cos ^{2}θ⟩, and (j)–(l) ⟨cos ^{2}χ⟩ for a constant field *E* _{ S } = 20 kV cm^{−1} as a function of the intensity of the laser field for for the first states with both *M* + *q* + *K* and *K* even for pyridazine. The states are 0_{00}0 (solid black), 1_{01}1 (solid green), 2_{02}2 (dashed–dotted black), 1_{01}0 (dash–dotted green), 2_{02}1 (dashed black), and 2_{21}2 (dashed green). The spectrum (a, b, c) contains also highly excited states (very thin lines).

(a)–(c) Energies and the expectation values, (d)–(f) ⟨cos θ_{ S }⟩, (g)–(i) ⟨cos ^{2}θ⟩, and (j)–(l) ⟨cos ^{2}χ⟩ for a constant field *E* _{ S } = 20 kV cm^{−1} as a function of the intensity of the laser field for for the first states with both *M* + *q* + *K* and *K* even for pyridazine. The states are 0_{00}0 (solid black), 1_{01}1 (solid green), 2_{02}2 (dashed–dotted black), 1_{01}0 (dash–dotted green), 2_{02}1 (dashed black), and 2_{21}2 (dashed green). The spectrum (a, b, c) contains also highly excited states (very thin lines).

(a)–(c) Energies and the expectation values, (d)–(f) ⟨cos θ_{ S }⟩, and (g)–(i) ⟨cos ^{2}θ⟩ for a constant *I* = 10^{10} W cm^{−2} as a function of the strength of the static field for for the energetically lowest states with both *M* + *q* + *K* and *K* even for fluorobenzene. The states are 0_{00}0 (solid black), 1_{01}0 (solid green), 1_{01}1 (dashed–dotted black), 2_{02}0 (dashed–dotted green), 2_{02}1 (dashed black), and 2_{02}2 (dashed green). The spectrum (a, b, c) contains also highly excited states (very thin lines).

(a)–(c) Energies and the expectation values, (d)–(f) ⟨cos θ_{ S }⟩, and (g)–(i) ⟨cos ^{2}θ⟩ for a constant *I* = 10^{10} W cm^{−2} as a function of the strength of the static field for for the energetically lowest states with both *M* + *q* + *K* and *K* even for fluorobenzene. The states are 0_{00}0 (solid black), 1_{01}0 (solid green), 1_{01}1 (dashed–dotted black), 2_{02}0 (dashed–dotted green), 2_{02}1 (dashed black), and 2_{02}2 (dashed green). The spectrum (a, b, c) contains also highly excited states (very thin lines).

Same as Fig. 7 but for pyridazine.

Same as Fig. 7 but for pyridazine.

One-dimensional probability density distribution in each Euler angle for the ground state of pyridazine for (a) *E* _{ S } = 20 kV cm^{−1} and *I* = 10^{11} W cm^{−2} (solid), (b) *E* _{ S } = 50 kV cm^{−1} and *I* = 10^{11} W cm^{−2} (dashed–dotted), and (c) *E* _{ S } = 20 kV cm^{−1} and *I* = 5 × 10^{11} W cm^{−2} (dashed).

One-dimensional probability density distribution in each Euler angle for the ground state of pyridazine for (a) *E* _{ S } = 20 kV cm^{−1} and *I* = 10^{11} W cm^{−2} (solid), (b) *E* _{ S } = 50 kV cm^{−1} and *I* = 10^{11} W cm^{−2} (dashed–dotted), and (c) *E* _{ S } = 20 kV cm^{−1} and *I* = 5 × 10^{11} W cm^{−2} (dashed).

(a) Energies and expectation values, (b) ⟨cos θ_{ S }⟩, (c) ⟨cos ^{2}θ⟩, and (d) ⟨*M* ^{2}⟩ for pyridazine in the presence of a static field *E* _{ S } = 20 kV cm^{−1} and a laser field *I* = 10^{11} W cm^{−2} as a function of β for the energetically lowest states with both *M* + *q* + *K* an *K* even. The states are 0_{00}0 (solid black), 1_{01}1 (solid green), 2_{02}2 (dashed–dotted black), 1_{01}0 (dashed–dotted green), 2_{02}1 (dashed black), and 2_{21}2 (dashed green).

(a) Energies and expectation values, (b) ⟨cos θ_{ S }⟩, (c) ⟨cos ^{2}θ⟩, and (d) ⟨*M* ^{2}⟩ for pyridazine in the presence of a static field *E* _{ S } = 20 kV cm^{−1} and a laser field *I* = 10^{11} W cm^{−2} as a function of β for the energetically lowest states with both *M* + *q* + *K* an *K* even. The states are 0_{00}0 (solid black), 1_{01}1 (solid green), 2_{02}2 (dashed–dotted black), 1_{01}0 (dashed–dotted green), 2_{02}1 (dashed black), and 2_{21}2 (dashed green).

Width Δ*E* (a) and electrostatic field strength *E* _{ S } (b) at the avoided crossing taking place between the states 2_{02}0 and 2_{02}1 for pyridazine, for *I* = 10^{10} W cm^{−2} and different inclination angles β.

Width Δ*E* (a) and electrostatic field strength *E* _{ S } (b) at the avoided crossing taking place between the states 2_{02}0 and 2_{02}1 for pyridazine, for *I* = 10^{10} W cm^{−2} and different inclination angles β.

## Tables

Action of the symmetry operations on the Euler angles.

Action of the symmetry operations on the Euler angles.

Relevant data for fluorobenzene^{50–52} and pyridazine.^{53,54}

Relevant data for fluorobenzene^{50–52} and pyridazine.^{53,54}

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