^{1}, Gergely Gidofalvi

^{1}and David A. Mazziotti

^{1,a)}

### Abstract

Understanding and modeling the interaction between light and matter is essential to the theory of optical molecular control. While the effect of the electric field on a molecule’s electronic structure is often not included in control theory, it can be modeled in an optimal control algorithm by a set or toolkit of potential energy surfaces indexed by discrete values of the electric field strength where the surfaces are generated by Born-Oppenheimer electronic structure calculations that directly include the electric field. Using a new optimal control algorithm with a trigonometric mapping to limit the maximum field strength explicitly, we apply the surface-toolkit method to control the hydrogen fluoride molecule. Potential energy surfaces in the presence and absence of the electric field are created with two-electron reduced-density-matrix techniques. The population dynamics show that adjusting for changes in the electronic structure of the molecule beyond the static dipole approximation can be significant for designing a field that drives a realistic quantum system to its target observable.

One of the authors (D.A.M.) gratefully acknowledges the NSF, the Henry-Camille Dreyfus Foundation, the Alfred P. Sloan Foundation, and the David-Lucile Packard Foundation for support. Another author (G.G.) acknowledges the NSF for its generous support.

I. INTRODUCTION

II. THEORY

A. Molecular control theory

B. Toolkit of potential energy surfaces

C. Optimal control algorithm

III. RESULTS

IV. DISCUSSION

### Key Topics

- Potential energy surfaces
- 38.0
- Electric fields
- 19.0
- Electronic structure
- 17.0
- Excited states
- 12.0
- Control theory
- 10.0

## Figures

The set or toolkit of potential surfaces corresponding to a discrete set of electric-field values between and . The surfaces were calculated using 2-RDM techniques.

The set or toolkit of potential surfaces corresponding to a discrete set of electric-field values between and . The surfaces were calculated using 2-RDM techniques.

Three dipole moment surfaces corresponding to electric-field values of , 0, and are shown. Potential energy surfaces were generated with the 2-RDM method with the electric field included, and then the dipole surfaces were calculated by centered finite differences. In the dipole approximation the set of potential energy surfaces, generated from a single dipole surface in the absence of the electric field, does not include field-induced polarization effects. The figure shows that the dipole surfaces at the minimum and maximum electric-field values are qualitatively different. These qualitative differences cannot be corrected simply with the addition of a polarization term.

Three dipole moment surfaces corresponding to electric-field values of , 0, and are shown. Potential energy surfaces were generated with the 2-RDM method with the electric field included, and then the dipole surfaces were calculated by centered finite differences. In the dipole approximation the set of potential energy surfaces, generated from a single dipole surface in the absence of the electric field, does not include field-induced polarization effects. The figure shows that the dipole surfaces at the minimum and maximum electric-field values are qualitatively different. These qualitative differences cannot be corrected simply with the addition of a polarization term.

Population dynamics of the HF molecule under the influence of an optimized electric field where the field is optimized (a) with potential surfaces from the dipole approximation and (b) with a set of precalculated Born-Oppenheimer potential energy curves corresponding to a set of electric-field amplitudes. The initial state of the system is the ground state, and the target state is the fifth vibrational level.

Population dynamics of the HF molecule under the influence of an optimized electric field where the field is optimized (a) with potential surfaces from the dipole approximation and (b) with a set of precalculated Born-Oppenheimer potential energy curves corresponding to a set of electric-field amplitudes. The initial state of the system is the ground state, and the target state is the fifth vibrational level.

The magnitude of the optimized control field that drives the population of HF from its ground vibrational state to its fifth state is shown. Only the last 10% of the time interval is shown, but the remainder of the electric field exhibits similar behavior. The field was optimized with the toolkit of precalculated Born-Oppenheimer potential energy curves corresponding to a set of electric-field amplitudes.

The magnitude of the optimized control field that drives the population of HF from its ground vibrational state to its fifth state is shown. Only the last 10% of the time interval is shown, but the remainder of the electric field exhibits similar behavior. The field was optimized with the toolkit of precalculated Born-Oppenheimer potential energy curves corresponding to a set of electric-field amplitudes.

The power spectrum of the optimized control field that drives the population of HF from its ground vibrational state to its fifth state is shown. The frequencies of the control field are an order of magnitude smaller than those corresponding to the lowest electronic transition. The field was optimized with a toolkit of precalculated Born-Oppenheimer potential energy curves corresponding to a set of electric-field amplitudes.

The power spectrum of the optimized control field that drives the population of HF from its ground vibrational state to its fifth state is shown. The frequencies of the control field are an order of magnitude smaller than those corresponding to the lowest electronic transition. The field was optimized with a toolkit of precalculated Born-Oppenheimer potential energy curves corresponding to a set of electric-field amplitudes.

Population dynamics of the HF molecule modeled by a set of potential surfaces beyond the dipole approximation but subjected to an optimal field computed by surfaces within the dipole approximation. The field from the dipole approximation is unable to drive an appreciable amount of the population from the ground state into the target fifth-excited state and scarcely fills the third excited state.

Population dynamics of the HF molecule modeled by a set of potential surfaces beyond the dipole approximation but subjected to an optimal field computed by surfaces within the dipole approximation. The field from the dipole approximation is unable to drive an appreciable amount of the population from the ground state into the target fifth-excited state and scarcely fills the third excited state.

Population dynamics from a control calculation using a toolkit of Born-Oppenheimer potential energy surfaces precalculated by full configuration interaction. The basis set and parameters are the same as those in Fig. 3(b) where the toolkit of Born-Oppenheimer surfaces is computed with the variational 2-RDM method. The exhibited mechanism is similar to that of the 2-RDM calculation, but the populations have different timings and peak heights.

Population dynamics from a control calculation using a toolkit of Born-Oppenheimer potential energy surfaces precalculated by full configuration interaction. The basis set and parameters are the same as those in Fig. 3(b) where the toolkit of Born-Oppenheimer surfaces is computed with the variational 2-RDM method. The exhibited mechanism is similar to that of the 2-RDM calculation, but the populations have different timings and peak heights.

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