Semiclassical optimal control

As demonstrated above, bond-selective dissociation can be achieved with high efficiency by using an initial displaced-position and/or a directed-momentum wavepacket. The latter wavepacket can be prepared via the sequence of quadratically chirped pulses or by using semiclassical optimal control theory [34,35],... 

By combining the control of electronic transitions of wavepackets using quadratically chirped laser pulses with semiclassical optimal control [34,35] on a single adiabatic surface, we should be able to establish an effective methodology for controlling the dynamics of large-dimensional chemical and biological systems. 

Summary. An efficient semiclassical optimal control theory for controlling wave-packet dynamics on a single adiabatic potential energy surface applicable to systems with many degrees of freedom is discussed in detail. The approach combines the advantages of various formulations of the optimal control theory quantum and classical on the one hand and global and local on the other. The efficiency and reliability of the method are demonstrated, using systems with two and four dimensions as examples. 

Since the classical treatment has its restrictions and the applicability of the quantum OCT is limited to low-dimensional systems due to its formidable computational cost, it would be very desirable to incorporate the semiclassical method of wavepacket propagation like the Herman-Kluk method [20,21] into the OCT. Recently, semiclassical bichromatic coherent control has been demonstrated for a large molecule [22] by directly calculating the percent reactant as a function of laser parameters. This approach, however, is not an optimal control. 

Recently, a semiclassical formulation of the optimal control theory has been derived [23, 24] by combining the conjugate gradient search method... 

As this approach deals with a set of classical trajectories, its numerical cost remains reasonable for multidimensional systems. Contrary to the classical approach, which controls only the averaged classical quantities, the present semiclassical method can control the quantum motion itself. This makes it possible to reproduce almost all quantum effects at a computational cost that does not grow too rapidly as the dimensionality of the system increases. The new approach therefore combines the advantages of the quantum and classical formulations of the optimal control theory. 

The approach has been tested by controlling nuclear wavepacket motion in a two-dimensional model system [23], The relative simplicity of the system makes it possible to compare the semiclassical results with exact quantum ones. Numerical applications to the control of HCN-CNH isomerization [24] demonstrates that the new semiclassical formulation of optimal control theory provides an effective and powerful tool for controlling molecular dynamics with many degrees of freedom. 

The remainder of this paper is organized as follows the global optimization procedure used in the formulation is discussed in Sect. 6.2. The semiclassical expression of the correlation function is derived in Sect. 6.3, and the properties of the semiclassical correlation function are discussed in Sect. 6.4. In Sect. 6.5 we introduce the idea of guided optimal control. The full control algorithm is provided. In Sect. 6.6 we provide three numerical examples i) the control of wavepacket motion where a two-dimensional model of H2O is used as an example, ii) the control of the H + OD —> HO + D reaction using a two-dimensional model of HOD, and iii) the control of the 4-D model of HCN-CNH isomerization (i.e., isomerization in a plane). Future perspectives from the authors point of view are summarized in Sect. 6.7. 

In order to demonstrate the efficiency and accuracy of the semiclassical formulation of optimal control theory, let us consider the control of two elementary types of motion (a) a shift of the position of the ground-state wavepacket in the two-dimensional model system of H20 and (b) an acceleration of the ground-state wavepacket at the same position in the same model. 

The controlling field and its spectra, calculated both semiclassically (thin and dash lines) and quantum mechanically (bold fines), are shown in Fig. 6.5. The frequencies of the main components of the optimal field spectra are the same for all three cases. However, the optimal field obtained quantum mechanically and semiclassically with formula (6.34) contains second harmonics. This means that additional quantum effects are taken into account by the correction from the dipole moment gradient included in (6.34). 

Fig. 6.9. Optimal field (a-b) and its spectra (c-d) calculated for the control of the H + OD —> HO + D reaction. The bold line shows the exact quantum result. The dashed line shows the semiclassical result obtained with the simple formula (6.35) used for the correlation function...

Therefore, we present here our semiclassical Field-Induced Surface Hopping (FISH) method [59] for the simulation and control of the laser-driven coupled electron-nuclear dynamics in complex molecular systems including all degrees of freedom. It is based on the combination of quantum electronic state population dynamics with classical nuclear dynamics carried out on the fly . The idea of the method is to propagate independent trajectories in the manifold of adiabatic electronic states and allow them to switch between the states under the influence of the laser field. The switching probabilities are calculated fully quantum mechanically. The application of our FISH method will be illustrated in Sect. 17.6 on the example of optimal dynamic discrimination (ODD) of two almost identical flavin molecules. 

---