^{1}, S. Ramakrishna

^{1}and Tamar Seideman

^{1,a)}

### Abstract

We discuss several interesting phenomena in the dynamics of strong field-triggered torsional wavepackets, which carry implications for the problem of torsional alignment in nonrigid molecules. Our results point to the origin and consequences of the fundamental differences between rotational and torsional coherences. In addition, we provide design guidelines for torsional control experiments by illustrating the role played by the laser intensity, pulse width, temperature, and molecular parameters. Specifically, as an example of several classes of molecules expected to make suitable candidates for laboratory experiments, we explore the torsional control of 9-[2-(anthracen-9-yl)ethynyl]anthracene and contrast it with that of biphenyl. Finally, we propose several potential applications for coherent torsional control in chemistry, physics, and material science.

The authors thank Shane Parker and Andrew Rasmussen for enlightening discussions and assistance with numerical problems. The Department of Energy (Grant No. DE-FG02-04ER15612) and the Air Force Office of Scientific Research (Grant No. P.O.217178/01/FA9550-11-1-0001) are gratefully acknowledged for support.

I. INTRODUCTION

II. THEORY

A. Hamiltonian

B. Time evolution

III. RESULTS AND DISCUSSION

A. Model system

1. Energy spacing

B. Light-controlled torsional dynamics

1. Field parameters

2. Time evolution

3. High-intensity dynamics

IV. CONCLUSIONS

### Key Topics

- Polarizability
- 13.0
- Probability theory
- 7.0
- Charge transfer
- 6.0
- Excited states
- 6.0
- Molecular dynamics
- 6.0

## Figures

The potential energy curves (Eq. (2) ) of biphenyl (red, solid) and AAC (blue, dashed) vs the torsion angle. The structure of biphenyl is shown on the left, and that of AAC is shown on the right. The biphenyl torsional potential is taken from Ref. ^{ 52 } , and that of AAC was calculated within density functional theory with the BP86 functional and a TZVP basis set using Q-Chem, ^{ 58 } and fit to the double cosine form of Eq. (2) . Details of the calculation of the AAC potential energy are given in the supplementary material. ^{ 59 }

The potential energy curves (Eq. (2) ) of biphenyl (red, solid) and AAC (blue, dashed) vs the torsion angle. The structure of biphenyl is shown on the left, and that of AAC is shown on the right. The biphenyl torsional potential is taken from Ref. ^{ 52 } , and that of AAC was calculated within density functional theory with the BP86 functional and a TZVP basis set using Q-Chem, ^{ 58 } and fit to the double cosine form of Eq. (2) . Details of the calculation of the AAC potential energy are given in the supplementary material. ^{ 59 }

Torsional level spectra. (a) The first 26 torsional energy levels of biphenyl superimposed on its potential energy curve. (b) Energy level spacing between adjacent eigenstates for biphenyl (solid red, left ordinate) and AAC (dashed blue, right ordinate). Both molecules exhibit the same level spacing characteristics, differing only in the density of torsional states. Energy levels shown are for the *w* _{1} Whittaker-Hill solution symmetry set.

Torsional level spectra. (a) The first 26 torsional energy levels of biphenyl superimposed on its potential energy curve. (b) Energy level spacing between adjacent eigenstates for biphenyl (solid red, left ordinate) and AAC (dashed blue, right ordinate). Both molecules exhibit the same level spacing characteristics, differing only in the density of torsional states. Energy levels shown are for the *w* _{1} Whittaker-Hill solution symmetry set.

Maximum alignment achieved as a function of the dimensionless interaction parameter Ω (Eq. (10) ), where Ω = 1 corresponds to 16 TW/cm^{2} for biphenyl (solid red), and to 3 TW/cm^{2} for AAC (dashed blue). All simulations correspond to *T* = 0 K. The Ω = 0 limit corresponds to the field-free equilibrium configuration. ⟨cos ^{2}β⟩ →1 in the limit of coplanarity.

Maximum alignment achieved as a function of the dimensionless interaction parameter Ω (Eq. (10) ), where Ω = 1 corresponds to 16 TW/cm^{2} for biphenyl (solid red), and to 3 TW/cm^{2} for AAC (dashed blue). All simulations correspond to *T* = 0 K. The Ω = 0 limit corresponds to the field-free equilibrium configuration. ⟨cos ^{2}β⟩ →1 in the limit of coplanarity.

Time evolution of the expectation value ⟨cos ^{2}β⟩ (shown in red) for biphenyl in response to an Ω = 1 laser pulse with a FWHM of 175 fs (shown in black). The left-hand graph shows the dynamics in the presence of the field, whereas the right-hand graph shows the dynamics over 40 ps. The temperature is 0 K.

Time evolution of the expectation value ⟨cos ^{2}β⟩ (shown in red) for biphenyl in response to an Ω = 1 laser pulse with a FWHM of 175 fs (shown in black). The left-hand graph shows the dynamics in the presence of the field, whereas the right-hand graph shows the dynamics over 40 ps. The temperature is 0 K.

Roles played by the system and field parameters. The left portion of each panel shows the dynamics in the presence of the laser field (*t* = 1.5 − 3.0 ps) along with the pulse envelope (black curve), whereas the right portion shows the long time, field-free evolution. (a) Biphenyl at *T* = 0 K, for a pulse duration of 175 fs FWHM, and interaction strengths of Ω = 1 (16 TW/cm^{2}) (red), Ω = 3 (48 TW/cm^{2}) (blue). (b) Biphenyl for an interaction strength of Ω = 1, pulse duration of 175 fs FWHM, and temperatures *T* = 0 K (red), *T* = 300 K (blue). (c) AAC for an interaction strength of Ω = 1 (3 TW/cm^{2}) and temperatures *T* = 0 K (red), *T* = 300 K (blue). (d) Both molecules for Ω = 1, pulse duration of 353 fs FWHM and *T* = 0 K. The biphenyl response is shown in red, and that of AAC in blue.

Roles played by the system and field parameters. The left portion of each panel shows the dynamics in the presence of the laser field (*t* = 1.5 − 3.0 ps) along with the pulse envelope (black curve), whereas the right portion shows the long time, field-free evolution. (a) Biphenyl at *T* = 0 K, for a pulse duration of 175 fs FWHM, and interaction strengths of Ω = 1 (16 TW/cm^{2}) (red), Ω = 3 (48 TW/cm^{2}) (blue). (b) Biphenyl for an interaction strength of Ω = 1, pulse duration of 175 fs FWHM, and temperatures *T* = 0 K (red), *T* = 300 K (blue). (c) AAC for an interaction strength of Ω = 1 (3 TW/cm^{2}) and temperatures *T* = 0 K (red), *T* = 300 K (blue). (d) Both molecules for Ω = 1, pulse duration of 353 fs FWHM and *T* = 0 K. The biphenyl response is shown in red, and that of AAC in blue.

The probability density associated with a biphenyl torsional wavepacket vs time and the torsion angle for the parameters of Figure 4 . The periodicity of the potential energy curve leads to two equivalent wavepackets, each centered around one of the two minima (only one is shown).

The probability density associated with a biphenyl torsional wavepacket vs time and the torsion angle for the parameters of Figure 4 . The periodicity of the potential energy curve leads to two equivalent wavepackets, each centered around one of the two minima (only one is shown).

(a) Time evolution of the torsional alignment of AAC for Ω = 7 (corresponding to an intensity of 21 TW/cm^{2}), *T* = 0 K, and a pulse duration of 615 fs FWHM. Shown is the total expectation value (black, dashed) along with its population ⟨cos ^{2}β⟩_{ p } (red, solid) and coherence ⟨cos ^{2}β⟩_{ c } (blue, dotted) projections along with the pulse profile (green, dashed). (b) The probability densities associated with the torsional eigenfunctions of *w* _{1} symmetry for AAC.

(a) Time evolution of the torsional alignment of AAC for Ω = 7 (corresponding to an intensity of 21 TW/cm^{2}), *T* = 0 K, and a pulse duration of 615 fs FWHM. Shown is the total expectation value (black, dashed) along with its population ⟨cos ^{2}β⟩_{ p } (red, solid) and coherence ⟨cos ^{2}β⟩_{ c } (blue, dotted) projections along with the pulse profile (green, dashed). (b) The probability densities associated with the torsional eigenfunctions of *w* _{1} symmetry for AAC.

The absolute value of density matrix elements for biphenyl after the pulse turnoff for the WH solutions of the 1st symmetry set at *T* = 0 K. (a) Ω = 1, (b) Ω = 3, (the conditions of Figure 5(a) ). Although torsional excitation is dominated by single-quantum processes, matrix elements distant from the diagonal (corresponding to high order torsional coherences) have comparable magnitudes to first order coherences.

The absolute value of density matrix elements for biphenyl after the pulse turnoff for the WH solutions of the 1st symmetry set at *T* = 0 K. (a) Ω = 1, (b) Ω = 3, (the conditions of Figure 5(a) ). Although torsional excitation is dominated by single-quantum processes, matrix elements distant from the diagonal (corresponding to high order torsional coherences) have comparable magnitudes to first order coherences.

## Tables

Parameters used to describe biphenyl and AAC. *I* _{rel} and the polarizability anisotropy Δα are taken from Ref. ^{ 45 } , *V* _{2} and *V* _{4} for biphenyl from Ref. ^{ 52 } . Our electronic structure calculations of the AAC potential are summarized in the caption of Figure 1 .

Parameters used to describe biphenyl and AAC. *I* _{rel} and the polarizability anisotropy Δα are taken from Ref. ^{ 45 } , *V* _{2} and *V* _{4} for biphenyl from Ref. ^{ 52 } . Our electronic structure calculations of the AAC potential are summarized in the caption of Figure 1 .

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