Time-dependent Potential Energy Research Articles

Foundational Problems of Quantum Theory: Novel Approach to Temporal Probability Density

<p><span lang="EN-US">Present study focuses on some foundational problems of quantum theory specifically deals with the concept of probability density and relating introductory problems. In this sense, the work initially investigates the origins of the general probability theory and re-examines the concepts of spatial and temporal probability densities based on genuine epistemological and ontological arguments. In order to tackle the foundational problems, standard theory is primarily memorised and criticized scientifically and philosophically in terms of foundationally disappearing term of time dependent potential energy within the time and space dependent Schrödinger wave equation. Based on those arguments, the problematic inconsistency between the spatial and temporal probability density functions is underlined. Given the problem, an original approach previously suggested, is concisely described and extended to resolve the existing problem. The novel approach, based on a novel time dependent Schrödinger wave equation, resolves the discrepancy with the classical wave equation and also leads to time dependent temporal probability densities even for the time free potential energies. Novel temporal probability density function is also normalized and has a fluctuation period of around 10<sup>-16 </sup>s which is very short compared to the atomic time scales.</span></p>

In English

The symmetry conditions have been derived for the occurrence of the ratchet effect in Brownian photomotors. To this end, spatiotemporal symmetry operations in vector transformations, coordinate and time shifts, and in the overdamped regime were applied to the average photomotor velocity taken as a functional of the coordinate- and time-dependent potential energy. As established, individual Brownian particles (molecules) can move directionally only provided a symmetrically distributed charge fluctuates in them and they are placed on the substrates with an antisymmetric charge distribution or, vice versa, they are characterized by antisymmetrically distributed charge fluctuations and are placed on symmetric substrates. The collective directed motion of orientation-averaged particles is possible only in the former case. If a particle charge distribution is described by a time dependence with the universal type of symmetry (i.e., simultaneously symmetric, antisymmetric, and shift-symmetric), an additional symmetry constraint on the ratchet functioning arises: the ratchet effect is ruled out in the overdamped regime but allowed for inertial moving particles if the charge distributions in both the particle and the substrate are neither symmetric nor antisymmetric. The effect of the universal type of symmetry is exemplified by dipole photomotors derived from donor-acceptor conjugated organic molecules. With a specific type of molecular photoexcitation and a specific relationship of the dipole moments in the ground and excited states, the ratchet effect becomes symmetry-forbidden. The forbiddenness can be removed by molecular polarization effects, which in this case become the predominant factor governing the direction of the motion and average velocity of photomotors. The estimated velocities of polarization photomotors are an order of magnitude larger than for known motor proteins and dipole Brownian photomotors. These results can be helpful in the purposeful molecular design of dipole photomotors.

Case studies of the time-dependent potential energy surface for dynamics in cavities.

The exact time-dependent potential energy surface driving the nuclear dynamics was recently shown to be a useful tool to understand and interpret the coupling of nuclei, electrons, and photons in cavity settings. Here, we provide a detailed analysis of its structure for exactly solvable systems that model two phenomena: cavity-induced suppression of proton-coupled electron-transfer and its dependence on the initial state, and cavity-induced electronic excitation. We demonstrate the inadequacy of simply using a weighted average of polaritonic surfaces to determine the dynamics. Such a weighted average misses a crucial term that redistributes energy between the nuclear and the polaritonic systems, and this term can in fact become a predominant term in determining the nuclear dynamics when several polaritonic surfaces are involved. Evolving an ensemble of classical trajectories on the exact potential energy surface reproduces the nuclear wavepacket quite accurately, while evolving on the weighted polaritonic surface fails after a short period of time. The implications and prospects for application of mixed quantum-classical methods based on this surface are discussed.

Spin-Orbit Interactions in Ultrafast Molecular Processes.

We investigate spin-orbit interactions in ultrafast molecular processes employing the exact factorization of the electron-nuclear wave function. We revisit the original derivation by including spin-orbit coupling, and show how the dynamics driven by the time-dependent potential energy surface alleviates inconsistencies arising from different electronic representations. We propose a novel trajectory-based scheme to simulate spin-forbidden non-radiative processes, and we show its performance in the treatment of excited-state dynamics where spin-orbit effects couple different spin multiplets.

Exact Potential Energy Surface for Molecules in Cavities.

We find and analyze the exact time-dependent potential energy surface driving the proton motion for a model of cavity-induced suppression of proton-coupled electron transfer. We show how, in contrast to the polaritonic surfaces, its features directly correlate to the proton dynamics and we discuss cavity modifications of its structure responsible for the suppression. The results highlight the interplay between nonadiabatic effects from coupling to photons and coupling to electrons and suggest caution is needed when applying traditional dynamics methods based on polaritonic surfaces.

When the exact factorization meets conical intersections...

Capturing nuclear dynamics through conical intersections is pivotal to understand the fate of photoexcited molecules. The concept of a conical intersection, however, belongs to a specific definition of the electronic states, within a Born–Huang representation of the molecular wavefunction. How would these ultrafast funneling processes be translated if an exact factorization of the molecular wavefunction were to be used? In this article, we build upon our recent analysis [B.F.E. Curchod, F. Agostini, J. Phys. Chem. Lett. 8, 831 (2017)] and address this question in a broader perspective by studying the dynamics of a nuclear wavepacket through two types of conical intersections, differing by the strength of their underlying diabatic coupling. Our results generalize our previous findings by (i) showing that the time-dependent potential energy surface smoothly varies, both in time and in position, between the corresponding diabatic and adiabatic potentials, with sometimes more complex features if interferences are observed, (ii) highlighting the non-trivial behavior of the time-dependent vector potential and the fact that it cannot be gauged away in general, and (iii) justifying some approximations employed in the derivation of a mixed quantum/classical scheme based on the exact factorization.

On the Dynamics through a Conical Intersection.

Conical intersections represent critical topological features of potential energy surfaces and open ultrafast nonradiative deactivation channels for photoexcited molecules. In the following, we investigate how this funneling picture is transposed in the eyes of the exact factorization formalism for a 2D model system. The exact factorization of the total molecular wave function leads to the fundamental concept of time-dependent potential energy surface and time-dependent vector potential, whose behavior during a dynamics through a conical intersection has up to now remained unexplored. Despite the fact that these quantities might be viewed as time-dependent generalizations of the adiabatic potential energy surfaces and the nonadiabatic coupling vectors, characteristic quantities appearing in the Born-Oppenheimer framework, we observe that they do not exhibit particular topological features in the region of conical intersection but still reflect the complex dynamics of the nuclear wavepacket.

An exact factorization perspective on quantum interferences in nonadiabatic dynamics.

Nonadiabatic quantum interferences emerge whenever nuclear wavefunctions in different electronic states meet and interact in a nonadiabatic region. In this work, we analyze how nonadiabatic quantum interferences translate in the context of the exact factorization of the molecular wavefunction. In particular, we focus our attention on the shape of the time-dependent potential energy surface-the exact surface on which the nuclear dynamics takes place. We use a one-dimensional exactly solvable model to reproduce different conditions for quantum interferences, whose characteristic features already appear in one-dimension. The time-dependent potential energy surface develops complex features when strong interferences are present, in clear contrast to the observed behavior in simple nonadiabatic crossing cases. Nevertheless, independent classical trajectories propagated on the exact time-dependent potential energy surface reasonably conserve a distribution in configuration space that mimics one of the exact nuclear probability densities.

Laser-induced electron localization in H₂⁺: mixed quantum-classical dynamics based on the exact time-dependent potential energy surface.

We study the exact nuclear time-dependent potential energy surface (TDPES) for laser-induced electron localization with a view to eventually developing a mixed quantum-classical dynamics method for strong-field processes. The TDPES is defined within the framework of the exact factorization [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett., 2010, 105, 123002] and contains the exact effect of the couplings to the electronic subsystem and to any external fields within a scalar potential. We compare its features with those of the quasistatic potential energy surfaces (QSPES) often used to analyse strong-field processes. We show that the gauge-independent component of the TDPES has a mean-field-like character very close to the density-weighted average of the QSPESs. Oscillations in this component are smoothened out by the gauge-dependent component, and both components are needed to yield the correct force on the nuclei. Once the localization begins to set in, the gradient of the exact TDPES tracks one QSPES and then switches to the other, similar to the description provided by surface-hopping between QSPESs. We show that evolving an ensemble of classical nuclear trajectories on the exact TDPES accurately reproduces the exact dynamics. This study suggests that the mixed quantum-classical dynamics scheme based on evolving multiple classical nuclear trajectories on the exact TDPES will be a novel and useful method to simulate strong field processes.

Electronic Schrödinger equation with nonclassical nuclei

We present a rigorous reformulation of the quantum mechanical equations of motion for the coupled system of electrons and nuclei that focuses on the dynamics of the electronic subsystem. Usually the description of electron dynamics involves an electronic Schr\"odinger equation where the nuclear degrees of freedom appear as parameters or as classical trajectories. Here we derive the exact Schr\"odinger equation for the subsystem of electrons, staying within a full quantum treatment of the nuclei. This exact Schr\"odinger equation features a time-dependent potential energy surface for electrons (e-TDPES). We demonstrate that this exact e-TDPES differs significantly from the electrostatic potential produced by classical or quantum nuclei.

Mixed quantum-classical dynamics on the exact time-dependent potential energy surface: a fresh look at non-adiabatic processes

The exact nuclear time-dependent potential energy surface arises from the exact decomposition of the electronic and nuclear motions, recently presented in [A. Abedi, N.T. Maitra, and E.K.U. Gross, Phys. Rev. Lett. 105, 123002 (2010)]. Such time-dependent potential drives nuclear motion and fully accounts for the coupling to the electronic subsystem. We investigate the features of the potential in the context of electronic non-adiabatic processes and employ it to study the performance of the classical approximation on nuclear dynamics. We observe that the potential, after the nuclear wave packet splits at an avoided crossing, develops dynamical steps connecting different regions, along the nuclear coordinate, in which it has the same slope as one or the other adiabatic surface. A detailed analysis of these steps is presented for systems with different non-adiabatic coupling strength. The exact factorisation of the electron-nuclear wave function is at the basis of the decomposition. In particular, the nuclear part is the true nuclear wave function, solution of a time-dependent Schrödinger equation and leading to the exact many-body density and current density. As a consequence, the Ehrenfest theorem can be extended to the nuclear subsystem and Hamiltonian, as discussed here with an analytical derivation and numerical results.

Dynamical Steps that Bridge Piecewise Adiabatic Shapes in the Exact Time-Dependent Potential Energy Surface

We study the exact time-dependent potential energy surface (TDPES) in the presence of strong nonadiabatic coupling between the electronic and nuclear motion. The concept of the TDPES emerges from the exact factorization of the full electron-nuclear wave function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)]. Employing a one-dimensional model system, we show that the TDPES exhibits a dynamical step that bridges between piecewise adiabatic shapes. We analytically investigate the position of the steps and the nature of the switching between the adiabatic pieces of the TDPES.

Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction

It was recently shown [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] that the complete wavefunction for a system of electrons and nuclei evolving in a time-dependent external potential can be exactly factorized into an electronic wavefunction and a nuclear wavefunction. The concepts of an exact time-dependent potential energy surface (TDPES) and exact time-dependent vector potential emerge naturally from the formalism. Here, we present a detailed description of the formalism, including a full derivation of the equations that the electronic and nuclear wavefunctions satisfy. We demonstrate the relationship of this exact factorization to the traditional Born-Oppenheimer expansion. A one-dimensional model of the H(2)(+) molecule in a laser field shows the usefulness of the exact TDPES in interpreting coupled electron-nuclear dynamics: we show how features of its structure indicate the mechanism of dissociation. We compare the exact TDPES with potential energy surfaces from the time-dependent Hartree-approach, and also compare traditional Ehrenfest dynamics with Ehrenfest dynamics on the exact TDPES.

Quantum Dynamics of H2+ in Intense Laser Fields on Time-Dependent Potential Energy Surfaces

We have exploited the fully time-dependent Born-Oppenheimer approximation to develop time-dependent potential energy surfaces for the lowest two states of H(2)(+) in the presence of intense, time-varying, few-cycle laser fields of 2-8 fs duration. Quantum dynamics are explored on these field-dressed, time-dependent potentials. Our results show that the potential well in the lowest-energy state of H(2)(+) (i) collapses as the laser pulse reaches its peak amplitude and (ii) regains its form on the trailing edge of the pulse, and (iii) the trapped nuclear wavepacket has a higher probability of leaking out from the well in the case of longer laser pulses. The carrier envelope phase is found to have negligible effect on the nuclear dynamics.

A full-dimensional wave packet dynamics study of the photodetachment spectra of ${\rm FCH}_{4}^{-}$ FCH 4−

The low-resolution photodetachment spectrum of FCH(4)(-) is studied in full dimensionality employing the multi-configurational time-dependent Hartree approach and potential energy surfaces recently developed by Bowman and co-workers. The computed spectrum qualitatively agrees with the low-resolution spectrum measured by Neumark and co-workers. It displays two peaks which can be assigned to different vibrational states of methane in the quasi-bound F·CH(4) van der Waals complex. The first intense peak correlates to methane in its vibrational ground state while the second much smaller peak results from methane where one of the bending modes is excited. The present simulations consider only a single potential energy surface for the neutral FCH(4) system and thus do not include spectral contributions arising from transitions to excited electronic states correlating to the F((2)P) + CH(4) asymptote. Considering the quantitative differences between the computed and the experimental spectra, one cannot decide whether beside the vibrational excitation of the methane fragment also electronic excitation of FCH(4) contributes to the second peak in the experimental photodetachment spectrum.

Atomic-scale structural evolution from disorder to order in an amorphous metal

In this paper, we performed molecular dynamics simulations to study the atomic-scale structural evolution from disorder to order during the isothermal annealing of an amorphous Ni. Three plateaus in the time dependent potential energy and mean square displacement (MSD) curves were observed, indicating that the atomic ordering process from amorphous to nanocrystalline Ni undergoes three distinct stages. The structural analyses reveal that the atomic structural evolution is associated with these three stages: Disordered atoms adjust their relative positions to form a one-dimensional (1D) periodic structure at the first stage, then form a 2D periodic structure at the second stage, and finally form a 3D periodic nanocrystal. Further analyses of potential energy and MSD difference and dynamics demonstrate that the structural change from the 2D to 3D structure is more difficult than that from the 1D to 2D structure, because both the 1D and 2D quasi-ordered structures belong to transition states and have similar...

Dynamical model of surrogate reactions

A new dynamical model is developed to describe the whole process of surrogate reactions; transfer of several nucleons at an initial stage, thermal equilibration of residues leading to washing out of shell effects and decay of populated compound nuclei are treated in a unified framework. Multi-dimensional Langevin equations are employed to describe time-evolution of collective coordinates with a time-dependent potential energy surface corresponding to different stages of surrogate reactions. The new model is capable of calculating spin distributions of the compound nuclei, one of the most important quantity in the surrogate technique. Furthermore, various observables of surrogate reactions can be calculated, e.g., energy and angular distribution of ejectile, and mass distributions of fission fragments. These features are important to assess validity of the proposed model itself, to understand mechanisms of the surrogate reactions and to determine unknown parameters of the model. It is found that spin distributions of compound nuclei produced in $^{18}$O+$^{238}$U $\rightarrow ^{16}$O+$^{240*}$U and $^{18}$O+$^{236}$U $\rightarrow ^{16}$O+$^{238*}$U reactions are equivalent and much less than 10$\hbar$, therefore satisfy conditions proposed by Chiba and Iwamoto (PRC 81, 044604(2010)) if they are used as a pair in the surrogate ratio method.

Exact Factorization of the Time-Dependent Electron-Nuclear Wave Function

We present an exact decomposition of the complete wave function for a system of nuclei and electrons evolving in a time-dependent external potential. We derive formally exact equations for the nuclear and electronic wave functions that lead to rigorous definitions of a time-dependent potential energy surface (TDPES) and a time-dependent geometric phase. For the H(2)(+) molecular ion exposed to a laser field, the TDPES proves to be a useful interpretive tool to identify different mechanisms of dissociation.

Quantum dynamical study of the amplitude collapse and revival of coherent A 1g phonons in bismuth: a classical phenomenon?

We parameterize the potential energy surface of bismuth after intense laser excitation using accurate full-potential linearized augmented plane wave calculations. Anharmonic contributions up to the fifth power in the A 1g phonon coordinate are given as a function of the absorbed laser energy. Using a previously described model including effects of electron–phonon coupling and carrier diffusion due to Johnson et al., we obtain the time-dependent potential energy surface for any given laser pulse shape and duration. On the basis of this parameterization we perform quantum dynamical simulations to study the experimentally observed amplitude collapse and revival of coherent A 1g phonons in bismuth considering work of Misochko et al. Our results strongly indicate that the observed beatings are not related to quantum effects and are most probably of classical origin.

High-temperature brownian motors: Deterministic and stochastic fluctuations of a periodic potential

The high-temperature unidirectional motion of a Brownian particle with time-dependent potential energy described by a spatially asymmetric periodic function is considered. A general formula derived for the mean velocity ν of such a motion is specified for dichotomic deterministic and Markovian stochastic processes. In both cases, ν increases linearly for low-frequencies γ of potential-energy fluctuations and reaches maxima for γ about the inverse time of diffusion by the spatial period of the potential. The behaviors of ν for large γ values are different in these cases: ν ∝ γ−2 and ν ts γ−1 for the deterministic and stochastic processes, respectively. It is shown that the direction of the motor motion depends on the relative lifetimes of each of the dichotomic-process states if the amplitude of the potential-energy fluctuations is fairly large in comparison with the mean value.