We introduce two non-adiabatic semiclassical methods that employ two coupled Gaussian wavepackets, each one traveling on a separate diabatic potential energy surface. The wavepackets take the form of thawed Gaussians and are driven by classical equations of motion which account for the diabatic coupling. The classical equations of motion are derived in one case by enforcing the thawed Gaussian ansatz, while in the other the time-dependent variational principles to the thawed Gaussian ansatz. After a sanity check where both approximations reproduce Rabi oscillations, the methods are applied to two non-adiabatic potential energy scenarios. The first one involves two coupled displaced harmonic oscillators, as in a typical electron transfer reaction. The second one comprises a Morse potential coupled to an upper dissociative state, modeling a photo-dissociation process. In both scenarios, the variational thawed Gaussian approach is quite accurate, while the standard thawed Gaussian one fails to fully capture the non-adiabatic effects. Ultimately, non-adiabatic molecular dynamics is reproduced by means of two classical trajectories without introducing any artificial jump or other ad-hoc non-classical effects.
We introduce two non-adiabatic semiclassical methods that employ two coupled Gaussian wavepackets, each one traveling on a separate diabatic potential energy surface. The wavepackets take the form of thawed Gaussians and are driven by classical equations of motion which account for the diabatic coupling. The classical equations of motion are derived in one case by enforcing the thawed Gaussian ansatz, while in the other the time-dependent variational principles to the thawed Gaussian ansatz. After a sanity check where both approximations reproduce Rabi oscillations, the methods are applied to two non-adiabatic potential energy scenarios. The first one involves two coupled displaced harmonic oscillators, as in a typical electron transfer reaction. The second one comprises a Morse potential coupled to an upper dissociative state, modeling a photo-dissociation process. In both scenarios, the variational thawed Gaussian approach is quite accurate, while the standard thawed Gaussian one fails to fully capture the non-adiabatic effects. Ultimately, non-adiabatic molecular dynamics is reproduced by means of two classical trajectories without introducing any artificial jump or other ad-hoc non-classical effects.
Ehrenfest dynamics is a widely used mixed quantum--classical approach for nonadiabatic molecular dynamics, whereas thawed Gaussian wavepacket dynamics provides an efficient semiclassical description of adiabatic nuclear quantum dynamics. Here we describe thawed Gaussian Ehrenfest dynamics (TGED), which unifies and generalizes these two methods to capture both electronic nonadiabaticity and nuclear quantum effects within a single framework. The fully variational formulation of TGED is derived by applying the time-dependent variational principle to a Hartree product of electronic and Gaussian nuclear wavepackets. Replacing the effective locally quadratic molecular potential obtained from this variational treatment by alternative effective locally quadratic potentials yields an infinite family of TGED methods, of which we present several members. We analyze the limiting cases of the general formalism and show, in particular, that it reduces to conventional Ehrenfest dynamics in the classical limit for the nuclei and to thawed Gaussian wavepacket dynamics in the absence of electronic coupling. Finally, we present explicit geometric integrators for the entire family of methods and identify the conditions under which the different approximations become exact.
We consider a semiclassical approximation, first derived by Heller and coworkers, for the time evolution of an originally Gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed Gaussian approximation. The other approximations are non-Gaussian and may involve several trajectories determined by mixed initial–final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive Gaussian potential and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-Gaussian approximations with real trajectories work well in some cases, whereas the thawed Gaussian works only in very simple situations.
Gaussian wavepacket (GWP) methods are prevalent means of solving the nuclear time-dependent Schrodinger equation (TDSE) on classical computers. They consist of representing the nuclear wave function as a superposition of Gaussians. Here, we present a variational hybrid continuous-variable discrete-variable (CV-DV) algorithm for simulating adiabatic nuclear dynamics on qubit-oscillator quantum hardware. It encodes a superposition of frozen Gaussians (FGs) onto the qubit-oscillator register and evolves it according to the time-dependent variational principle (VP). We test two variants of the approach, one that evolves a single FG and another that evolves a superposition of FGs, on a set of prototypical 1D harmonic and anharmonic systems. We simulate the photoexcited state dynamics of SO2 using the single FG variant of the algorithm and compute its autocorrelation function on physical quantum hardware. For harmonic and Morse potentials, the variant evolving the superposition of FGs converges to the numerically exact solution as the number of Gaussians increases. In the case of the double-well potential, the algorithm simulates wavepacket bifurcation and recurrence correctly, and the ensuing irregular dynamics moderately well. Overall, this work establishes a pathway for simulating molecular dynamics on noisy intermediate-scale quantum devices.
Simulation and analysis of multidimensional dynamics of a quantum non-Hmeritian system is a challenging problem. Gaussian wavepacket dynamics has proven to be an intuitive semiclassical approach to approximately solving the dynamics of quantum systems. A Gaussian wavepacket approach is proposed for a continuous space extension to the Hatano-Nelson model that enables transparent analysis of the dynamics in terms of complex classical trajectories. We demonstrate certain cases where the configuration space trajectory can be made fully real by transforming the initial conditions to account for the non-Hermiticity appropriately through the momentum coordinates. However, in general the complex phase space is unavoidable. For the cases where the trajectory is real, the effective force can be decomposed into that due to the potential energy surface and that due to the imaginary vector potential. The impact of the vector potential on the trajectory of the wavepacket is directly proportional to both the strength of the vector potential and the width of the wavepacket.
A non-adiabatic nuclear wavepacket dynamics simulation of the H2O+ de-excitation process is performed based on electronic structure calculations using the variational quantum eigensolver. The adiabatic potential energy surfaces and non-adiabatic coupling vectors are computed with algorithms for noisy intermediate-scale quantum devices, and time propagation is simulated with conventional methods for classical computers. The results of non-adiabatic transition dynamics from the D2 state to the D1 state reproduce the trend reported in previous studies, which suggests that this quantum–classical hybrid scheme may be a useful application for noisy intermediate-scale quantum devices.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.