Optimal control theory for a target state distributed in time: optimizing the probe-pulse signal of a pump-probe-scheme

Optimal control theory (OCT) is formulated for the case of a two-color pump-probe experiment. The approach allows to calculate the pump-pulse shape in such a way that the probe-pulse absorption signal is maximized. Since the latter quantity is given by the time-averaged expectation value of a time dependent operator (the probe-pulse field-strength times the dipole operator) a version of OCT has to be used where the target state is distributed in time. The method is applied to a molecular three-level system with the pump-pulse driving the transition from the electronic ground state into the first-excited electronic state and the probe-pulse connecting the first-excited state with a higher lying electronic state. Depending on the probe-pulse duration, the vibrational wave packet becomes localized or at least highly concentrated in the Franck-Condon window for the transition into the higher-excited state. The dependence on the probe-pulse duration and on the delay time between the optimized pump-pulse and the probe-pulse is discussed in detail. The whole study demonstrates the feasibility of laser pulse induced temporal wave packet localization and the use of spectroscopic quantities as target states in experiments on femtosecond laser pulse control.     

Isotope selective photoionization of NaK by optimal control: theory and experiment

We present a joint theoretical and experimental study of the maximization of the isotopomer ratio (23)Na(39)K(23)Na(41)K using tailored phase-only as well as amplitude and phase modulated femtosecond laser fields obtained in the framework of optimal control theory and closed loop learning (CLL) technique. A good agreement between theoretically and experimentally optimized pulse shapes is achieved which allows to assign the optimized processes directly to the pulse shapes obtained by the experimental isotopomer selective CLL approach. By analyzing the dynamics induced by the optimized pulses we show that the mechanism involving the dephasing of the wave packets between the isotopomers (23)Na (39)K and (23)Na (41)K on the first excited state is responsible for high isotope selective ionization. Amplitude and phase modulated pulses, moreover, allow to establish the connection between the spectral components of the pulse and corresponding occupied vibronic states. It will be also shown that the leading features of the theoretically shaped pulses are independent from the initial conditions. Since the underlying processes can be assigned to the individual features of the shaped pulses, we show that optimal control can be used as a tool for analysis.     

An application of RPA theory to conjugated systems in the excited states

RPA theory has been applied to the calculations of the electronic spectra of some conjugated systems, considering only -electrons explicitly. Electron repulsion integrals have been calculated using Slater AO's with appropriate orbital exponents. The present calculation shows that the correlation effect between -electrons is rather small for the lowest — ^* transitions. Good results have been obtained, when we calculate the electron repulsion integrals using orbital exponents evaluated by Slater rule. Effective electron interaction in the excited states has been discussed, using the calculated results. The calculated oscillator strengths were considerably improved by RPA,

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Zusammenfassung Die RPA wurde auf die Berechnung der UV-Spektren des -Elektronensystems konjugierter Moleküle angewendet, wobei die Coulombintegrale für Slaterorbitate mit Exponenten gemäß der Slaterregeln ermittelt wurden. Es zeigt sich, daß die Korrelationseffekte für die tieferen Übergänge ziemlich gering sind und daß sich insbesondere die Oszillatorenstärken sehr gut ergeben. Schließlich wurden eine effektive Elektronenwechselwirkung in angeregten Zuständen diskutiert.

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Résumé La théorie RPA fondée sur l'équation du mouvement a été appliquée à des calculs de la structure électronique et des spectres de systèmes conjugués en ne considérant explicitement que les électrons . Les intégrales de répulsion électronique ont été calculées en utilisant des OA de Slater avec des exposants orbitaux appropriés. Nos calculs montrent que les effets de corrélation entre électrons sont plutôt faibles pour les plus basses transitions — ^*. De bons résultats ont été obtenus en calculant les intégrales de répulsion électronique en utilisant les exposants orbitaux évalués par la règle de Slater. L'interaction électronique effective dans les états excités est discutée sur la base des résultats obtenus.

     

An application of RPA theory to conjugated systems in the excited states

In order to obtain a relationship between the molecular dimension and the correlation effect, RPA method has been applied to the calculation of electronic transition energies of linear polyenes. It has been found that the effect of electron correlation on the excitation energy decreases with increasing the size of molecule. The calculated oscillator strengths are remarkably improved by RPA calculation.     

Transition states in modern valence-bond theory: application to the Cope rearrangement

Modern valence-bond theory, in its spin-coupled form, is used to study the electronic structure of the transition structures in the Cope rearrangement. It is found that the transition structure described by a “chair” geometry with a “6-in-6” CASSCF/6-31G^* wave function is clearly aromatic while the CASSCF/6-31G^*“boat” transition structure corresponds more closely to two weakly interacting allyl radicals. Moreover, there is a striking resemblance between the CASSCF chair transition structure and the benzene molecule, arising from the modern valence-bond analysis in terms of Rumer spin functions. In agreement with previous works, dynamical correlated wave functions show shorter interallylic distances in the optimized transitions structures. The use of spin-coupled wave functions on the latter geometries results in diradical and aromatic character for the chair and boat transition structures, respectively. Received: 13 October 1998 / Accepted: 30 December 1998 / Published online: 7 June 1999     

An application of second-order n-electron valence state perturbation theory to the calculation of excited states

n–electron valence state perturbation theory (NEVPT) is a form of multireference perturbation theory where all the zero-order wave functions are of multireference nature, being generated as eigenfunctions of a two–electron model Hamiltonian. The absence of intruder states makes NEVPT an interesting choice for the calculation of electronically excited states. Test calculations have been performed on several valence and Rydberg transitions for the formaldehyde and acetone molecules; the results are in good accordance with the best calculations and with the existing experimental data.Contribution to the Jacopo Tomasi Honorary Issue     

Limitations of density functional theory in application to degenerate states

It is shown that the claims that density functional theory (DFT) can handle orbitally degenerate states are ungrounded. The constraint search formulation of DFT allows one to determine a set of densities and eigenvalues for the degenerate term that, however, are neither observables, nor can they be used to solve the system of coupled equations for the nuclear motions to obtain observables, as in the wave function presentation. A striking example of the failure of the existing versions of DFT to describe degenerate states is provided by the Berry phase problem: the strong dependence of the results on the phase properties of the electronic wave function that are smeared out in the density formulation. The solution of the Jahn-Teller E-e problem illustrates these statements. For nondegenerate states with the full wave function taken in the adiabatic approximation as a product of the electronic and nuclear parts, the formulation of DFT is rigorous if and only if the dependence of the electronic wave function on nuclear coordinates is ignored. This lowers the accuracy of the results, in general, and may lead to erroneous presentation as in the case of molecular systems in strong magnetic fields. © 1997 by John Wiley & Sons, Inc.     

Optimal control of quantum non-Markovian dissipation: reduced Liouville-space theory

An optimal control theory for open quantum systems is constructed containing non-Markovian dissipation manipulated by an external control field. The control theory is developed based on a novel quantum dissipation formulation that treats both the initial canonical ensemble and the subsequent reduced control dynamics. An associated scheme of backward propagation is presented, allowing the efficient evaluation of general optimal control problems. As an illustration, the control theory is applied to the vibration of the hydrogen fluoride molecule embedded in a non-Markovian dissipative medium. The importance of control-dissipation correlation is evident in the results.     

Partial third-order quasiparticle theory: An application to the photoelectron spectrum of S-tetrazine

An alternative choice of reference state averages in electron propagator theory and retention of all self-energy terms through third-order gives rise to the partial third-order self-energy approximation, P3. Quasiparticle P3 calculations avoid the chief computational bottlenecks of third-order theory and the outer valence Green's function (OVGF). P3 requires no multiplicative factors for scaling self-energy terms and produces better accuracy than OVGF. An application to the photoelectron spectrum of s-tetrazine illustrates the ability of the P3 method to predict correct final state orderings. © 1997 John Wiley & Sons, Inc.     

An application of minimax robust optimal control theory for selective vibrational excitation in molecules

In recent investigations control theory was applied to design electromagnetic fields capable of producing vibrational excitation in molecular systems. This approach has been applied to linear or non-linear classical approximations of molecular systems or to quantal systems using distributed cost functionals. Practical computations of molecular optimal control theory for large molecules especially with anharmonic potentials become difficult due to the increased dimensionality and the mixed nature of the boundary conditions. This paper proposes to approach the control design for such systems by treating a portion of the molecule containing the target and dipole bonds in full detail while the effect of the remainder of the system is modelled as a disturbance of limited energy. The optimal fieldminimizes the cost functional which is simultaneouslymaximized with respect to the disturbance. Such assumptions give rise to a robust controller akin to theH theory of robust estimation. We investigate the various field designs for truncated harmonic systems associated with different disturbance energies and demonstrate that the existence of the solution to the associated Ricatti equation ensures the existence of the equilibrium game point. In addition, in the range of physically reasonable disturbance energy the optimal field could be accurately predicted from an asymptotic expansion involving only the undisturbed reference case. As an application we show the optimal field design for a 20 atom truncated molecular chain containing both the target bond (the 5th bond) and the dipole bonds (1st and 9th) where the disturbance only affects the end bond of the system attached to the remainder of the chain. In an effort to improve on the efficiency of the bond energy deposition we investigate shortened target times and also a 40 atom truncated chain. This approach presents very conservative estimates of possible disturbances but provides insight into the sensitivity of different configurations with respect to external disturbances. The minimax approach can be generalized to non-linear molecular systems by modelling the original system as a linear system plus an energy constrained disturbance.     

Optimal designs for pairwise calculation: An application to free energy perturbation in minimizing prediction variability

Pairwise-based methods such as the free energy perturbation (FEP) method have been widely deployed to compute the binding free energy differences between two similar host–guest complexes. The calculated pairwise free energy difference is either directly adopted or transformed to absolute binding free energy for molecule rank ordering. We investigated, through both analytic derivations and simulations, how the selection of pairs in the experiment could impact the overall prediction precision. Our studies showed that (1) the estimated absolute binding free energy () derived from calculated pairwise differences (ΔΔG) through weighted least squares fitting is more precise in prediction than the pairwise difference values when the number of pairs is more than the number of ligands and (2) prediction precision is influenced by both the total number of pairs and the specifically selected pairs, the latter being critically important when the number of calculated pairs is limited. Furthermore, we applied optimal experimental design in pair selection and found that the optimally selected pairs can outperform randomly selected pairs in prediction precision. In an illustrative example, we showed that, upon weighing ligand structure similarity into design optimization, the weighted optimal designs are more efficient than the literature reported designs. This work provides a new approach to assess retrospective pairwise-based prediction results, and a method to design new prospective pairwise-based experiments for molecular lead optimization. © 2019 Wiley Periodicals, Inc.     

Degenerate symmetry-adapted perturbation theory of weak interactions between closed- and open-shell monomers: application to Rydberg states of helium hydride

Symmetry-adapted perturbation theory is extended to the (quasi) degenerate, open-shell case. The new formalism is tested in calculations of the interaction energies for a helium atom in the ground state interacting with an excited hydrogen atom. It is shown that the method gives satisfactory results if the coupling with higher Rydberg states of the dimer is small, as is the case for the A²Σ⁺,B²Π,E²Π,3²Π, and 1²Δ states of HeH. For the C²Σ⁺ state convergence of the method is very slow, but it can be improved by including the n=3 states in the model space. Received: 3 June 1998 / Accepted: 9 September 1998 / Published online: 7 December 1998     

Complete quantum control of the population transfer branching ratio between two degenerate target states

A five-level four-pulse phase-sensitive extended stimulated Raman adiabatic passage scheme is proposed to realize complete control of the population transfer branching ratio between two degenerate target states. The control is achieved via a three-node null eigenstate that can be correlated with an arbitrary superposition of the target states. Our results suggest that complete suppression of the yield of one of two degenerate product states, and therefore absolute selectivity in photochemistry, is achievable and predictable, even without studying the properties of the unwanted product state beforehand.     

Theory of ultrafast nonresonant multiphoton transitions in polyatomic molecules: basics and application to optimal control theory

A systematic approach is presented to describe nonresonant multiphoton transitions, i.e., transitions between two electronic states without the presence of additional intermediate states resonant with the single-photon energy. The method is well suited to describe femtosecond spectroscopic experiments and, in particular, attempts to achieve laser pulse control of molecular dynamics. The obtained effective time-dependent Schrodinger equation includes effective couplings to the radiation field which combine powers of the field strength and effective transition dipole operators between the initial and final states. To arrive at time-local equations our derivation combines the well-known rotating wave approximation with the approximation of slowly varying amplitudes. Under these terms, the optimal control formalism can be readily extended to also account for nonresonant multiphoton events. Exemplary, nonresonant two- and three-photon processes, similar to those occurring in the recent femtosecond pulse-shaping experiments on CpMn(CO)(3), are treated using related ab initio potential energy surfaces.     

An application of hypergeometric distribution theory to competitive processes at deteriorating electrode surfaces

The hypergeometric distribution of probability theory is employed to predict the effect of surface deterioration on electrode behaviour in the presence of two competitive processes. The approach, carrying numerical illustrations, assumes that only the total number of deteriorating active centre clusters is known, but not their fractions supporting individual processes. The temporal variation of the computed probability of process-prevalence, independent of the deterioration mechanism, maps the history of surface efficiency, if the kinetics of deterioration is known.     

Optimal quantum control with multi-polarization fields

The dynamical advantages for employing up to three independently shaped polarization fields are explored in the optimal control of quantum systems. The analysis compares multi-polarization optimal control with what may be achieved using linearly polarized (1-D) control. Simulations on model systems show how multi-polarization (2-D, 3-D) optimally shaped pulses can overcome symmetry forbidden transitions and improve control quality by better accessibility to the target state.     

Phase-only shaped laser pulses in optimal control theory: application to indirect photofragmentation dynamics in the weak-field limit

We implement phase-only shaped laser pulses within quantum optimal control theory for laser-molecule interaction. This approach is applied to the indirect photofragmentation dynamics of NaI in the weak-field limit. It is shown that optimized phase-modulated pulses with a fixed frequency distribution can substantially modify transient dissociation probabilities as well as the momentum distribution associated with the relative motion of Na and I.