Time-dependent Schroedinger Equation Research Articles

Anharmonicity and quantum nuclear effects in theoretical vibrational spectroscopy: a molecular tale of two cities

Anharmonic effects due to the shape of the molecular potential energy surface far from the equilibrium geometry are major responsible for the deviations of the actual frequencies of vibration from the harmonic estimates. However, anharmonic effects are not the solely responsible for this. Quantum nuclear effects also play a prominent role in theoretical vibrational spectroscopy as they contribute to drive away the molecular vibrational frequencies from their harmonic counterpart. The consequence of this is that anharmonicity and quantum effects may be difficult to separate spectroscopically and get often confused. In this work we show that anharmonicity can be detected by means of classical simulations, while quantum nuclear effects need to be identified by means of an approach originating from either the time independent or the time dependent Schroedinger equation of quantum mechanics. We show that classical methods are sensitive to the temperature or energy conditions under which they are undertaken. This leads to wrong frequency estimates, when dealing with few-Kelvin experiments, if one performs simulations simply matching the experimental temperature. Conversely, quantum approaches are not affected by this issue and they provide more and better information.

Measuring time-dependent induced quantum coherences via two-dimensional coherence spectroscopy

We propose a two-dimensional spectroscopic protocol for measuring the time-dependent coherences between the stationary states of a system induced by a time-dependent system-bath interaction. We also investigate the role of temporally-correlated noise on coherence dephasing. This protocol enables dynamical information about the system and its coupling to the environment to be determined. Our results are based on the quantum-trajectory method, and are obtained from both approximate, analytical and exact, numerical solutions of the time-dependent Schroedinger equation. As an example, we show how this protocol can be used to investigate exciton dynameds in conjugated polymers induced by the coupling of their torsional modes with the environment.

Magnetic dichroism in few-photon ionization of polarized atoms

We consider few-photon ionization of atomic lithium by linearly polarized femtosecond laser pulses, and demonstrate that asymmetries of the electron angular distribution can occur for initially polarized (2p, m=+1) target atoms. The dependence of the photoelectron emission angle relative to the electric field direction is investigated at different laser intensities and wavelengths. The experimental spectra show excellent agreement with numerical solutions of the time-dependent Schroedinger equation. In the perturbative picture, the angular shift is traced back to interferences between partial waves with mean magnetic quantum number <m>$\ne$0. This observation allows us to obtain quantum mechanical information on the electronic final state.

Scale-free Unique Continuation Estimates and Logvinenko–Sereda Theorems on the Torus

We study uncertainty principles for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier transform of the considered functions is allowed to be supported in a (finite number of) parallelepipeds. The estimates we obtain do not depend on the size of the torus and the position of the parallelepipeds, but only on their size and number, and the density and scale of the observability set. Our results are on the one hand closely related to unique continuation for linear combinations of eigenfunctions (aka spectral inequalities) which can be obtained by Carleman estimates, on the other hand to observability estimates for the time-dependent Schroedinger and for the heat equation, and finally to the Logvinenko & Sereda theorem. In fact, they are based on the methods developed by Kovrijkine to refine and generalize the results of Logvinenko & Sereda and Kacnel'son. Furthermore, relying on completely different techniques associated with the time-dependent Schroedinger equation, we prove a companion theorem where the energy of the considered functions is allowed to be in a spectral subspace of a Schroedinger operator.

Raman signal from a hindered hydrogen rotor

We present a method for calculation of Raman modes of the quantum solid phase I hydrogen and deuterium. We use the mean-field assumption that the quantized excitations are localized on one molecule. This is done by explicit solution of the time-dependent Schroedinger equation in an angle-dependent potential, and direct calculation of the polarization. We show that in the free rotor limit, the ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ frequencies differ by a factor of 2, which evolves toward $\sqrt{2}$ as the modes acquire librational character due to stronger interactions. The ratio overshoots $\sqrt{2}$ if anharmonic terms weaken the harmonic potential. We also use density functional theory and molecular dynamics to calculate the ${\phantom{\rule{4pt}{0ex}}E}_{{2}_{g}}$ optical phonon frequency and the Raman linewidths. The molecular dynamics shows that the molecules are not free rotors except at very low pressure and high temperature, and become like oscillators as phase II is approached. We fit the interaction strengths to experimental frequencies, but good agreement for intensities requires us to also include strong preferred orientation and stimulated Raman effects between ${\phantom{\rule{4pt}{0ex}}S}_{0}$(1) and ${\phantom{\rule{4pt}{0ex}}S}_{0}$(0) contributions. The experimental Raman spectrum for phase II cannot be reproduced, suggesting that the mean-field assumption is invalid in that case.

A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk.

Free-free matrix-elements for two-photon opacity

Recent opacity measurements have inspired a close study of the two-photon contributions to the opacity of hot plasmas. The absorption and emission of radiation is controlled by dipole matrix-elements of electrons in an atom or ion. This paper describes two independent methods to calculate matrix-elements needed for the two-photon opacity and tests the results by the f-sum rule. The usual f-sum rule is extended to a matrix f-sum that offers a rigorous test for bound-bound, bound-free and free-free transitions. An additional higher-order sum-rule for the two-photon transition amplitudes is described. We obtain a simple parametric representation of a key plasma density effect on the matrix-elements.The perturbation theory calculation of two-photon cross-sections is compared to an independent method based on the solution of the time-dependent Schroedinger equation for an atom or ion in a high-frequency electromagnetic field. This is described as a high frequency Stark effect or AC Stark effect. Two-photon cross sections calculated with the AC Stark code agree with perturbation theory to within about 5%. In addition to this cross check, the AC Stark code is well suited to evaluating important questions such as the variation of two-photon opacity for different elements.

Large-Scale Quantum Dynamics with Matrix Product States.

Dynamical electronic- and vibrational-structure theories have received a growing interest in the past few years due to their ability to simulate spectra recorded with ultrafast experimental techniques. The exact time evolution of a molecular system can, in principle, be obtained from the time-dependent version of full configuration interaction. Such an approach is, however, limited to few-atom systems due to the exponential increase of its cost with the system dimension. In the present work, we overcome this unfavorable scaling by employing the time-dependent density matrix renormalization group (TD-DMRG) which parametrizes the time-dependent wave function as a matrix product state. The time-dependent Schrödinger equation is then integrated with a sweep-based algorithm, as in standard time-independent DMRG. Unlike other TD-DMRG approaches, the one presented here leads to a set of coupled equations that can be integrated exactly. The resulting theory enables us to study real- and imaginary-time evolutions of Hamiltonians comprising more than 20 degrees of freedom that are challenging for current state-of-the-art quantum dynamics algorithms. We apply our algorithm to the simulation of quantum dynamics of models of increasing complexity, ranging from simple excitonic Hamiltonians to more complex ab initio vibronic ones.

Modelling vibrational relaxation in complex molecular systems.

In this paper we show how it is possible to treat the quantum vibrational relaxation of a chromophore, embedded in a complex atomic-molecular environment, via the explicit solution of the time-dependent Schroedinger equation once using a proper separation between quantum and semiclassical degrees of freedom. The rigorous theoretical framework derived, based on first principles and making use of well defined approximations/assumptions, is utilized to construct a general model for the kinetics of the vibrational relaxation as obtained by the direct evaluation of the density matrix for all the relevant quantum state transitions. Application to (deuterated) N-methylacetamide (the typical benchmark used as a model for the amino acids) shows that the obtained theoretical-computational approach captures the essential features of the experimental process, unveiling the basic relaxation mechanism involving several vibrational state transitions.

Role of intermediate continuum states in exterior complex scaling calculations of two-photon ionization cross sections

The calculation of partial two-photon ionization cross sections in the above-threshold energy region is discussed in the framework of exterior complex scaling. It is shown that with a minor modification of the usual procedure, which is based on the calculation of the outgoing partial waves of the second-order scattering wave function, reliable partial ionization amplitudes can be obtained. The modified procedure relies on a few-term least-squares fit of radial functions pertaining to different partial waves. To test the procedure, partial and total two-photon ionization cross sections of the helium atom have been calculated for a broad range of incident photon energies. The calculated cross sections may be seen to agree well with the results found in the literature. Furthermore, it is shown that, using a similar approach, partial photoionization cross sections of an atom in an autoionizing (resonance) state may be calculated in a relatively straightforward way. Such photoionization cross sections may find their use in enhanced few-parameter models describing the atom-light interaction in cases where a direct solution of the time-dependent Schroedinger equation becomes too resource-intensive.

Direct reactions of weakly-bound nuclei within a one dimensional model

A line of research has been developed to describe structure and dynamics of weakly-bound systems with one or more valence particles. To simplify the problem we are assuming particles moving in one dimension and, despite the drastic assumption, the model encompasses many characteristics observed in experiments. Within this model we can describe, for example, one- and two-particle breakup and one- and two-particle transfer processes. We concentrate here in models involving weakly-bound nuclei with just one valence particle. Exact solutions obtained by directly solving the time-dependent Schroedinger equation can be compared with the results obtained with different approximation schemes (coupled-channels formalism, continuum discretization, etc). Our goal is to investigate the limitations of the models based on approximations, and in particular to understand the role of continuum in the reaction mechanism.

Charge separation at an organic/inorganic nano-hybrid interface: atomistic simulations of a para-sexiphenyl ZnO system.

A prototypical organic/inorganic interface is considered which is formed by vertical stacking of 20 para-sexiphenyl molecules physisorbed on a ZnO nano-cluster of 3903 atoms. Charge separation kinetics at the interface are investigated for their dependence on ultrafast optical excitation. In order to analyze the spatio-temporal evolution of the Frenkel exciton in the organic part and the formation of charge separated states a first principles parameterized Hamiltonian is introduced and the related time-dependent Schroedinger equation is solved. By determining the interface absorption spectrum the optically addressable states can be uncovered. The work continues our previous studies of J. Phys. Chem. Lett., 2018, 9, 209, but with a changed type of surface passivation. This prevents trapping of electrons close to the surface. Charge separated states are formed by direct optical excitation and also by exciton decay at the interface. Electron migration away from the interface into bulk regions becomes possible. The hole stays close to the interface for all excitation scenarios. Finally, it is demonstrated that energetic disorder is of minor influence.

An Egorov Theorem for Avoided Crossings of Eigenvalue Surfaces

We study nuclear propagation through avoided crossings of electron energy levels. We construct a surface hopping semigroup, which gives an Egorov-type description of the dynamics. The underlying time-dependent Schroedinger equation has a two-by-two matrix-valued potential, whose eigenvalue surfaces have an avoided crossing. Using microlocal normal forms reminiscent of the Landau-Zener problem, we prove convergence to the true solution in the semi-classical limit.

Periodically driven small polarons

Small lattice polarons driven by strong external electric fields are considered. A time-dependent Schroedinger equation is integrated directly in time domain. The field agitates ions both directly and through modulation of carrier density. It is found that when the field is in resonance with local ion oscillations, the polaron is liberated from its self-induced trap and the tunneling frequency increases exponentially.

Strong field approximation within a Faddeev-like formalism for laser-matter interactions

We consider the interaction of atomic hydrogen with an intense laser field within the strong-field approximation. By using a Faddeev-like formalism, we introduce a new perturbative series in the binding potential of the atom. As a first test of this new approach, we calculate the electron energy spectrum in the very simple case of a photon energy higher than the ionisation potential. We show that by contrast to the standard perturbative series in the binding potential obtained within the strong field approximation, the first terms of the new series converge rapidly towards the results we get by solving the corresponding time-dependent Schroedinger equation.

Modelling laser-atom interactions in the strong field regime

We consider the ionisation of atomic hydrogen by a strong infrared field. We extend and study in more depth an existing semi-analytical model. Starting from the time-dependent Schroedinger equation in momentum space and in the velocity gauge we substitute the kernel of the non-local Coulomb potential by a sum of N separable potentials, each of them supporting one hydrogen bound state. This leads to a set of N coupled one-dimensional linear Volterra integral equations to solve. We analyze the gauge problem for the model, the different ways of generating the separable potentials and establish a clear link with the strong field approximation which turns out to be a limiting case of the present model. We calculate electron energy spectra as well as the time evolution of electron wave packets in momentum space. We compare and discuss the results obtained with the model and with the strong field approximation and examine in this context, the role of excited states.

From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity.

Using shortcuts to adiabaticity, we solve the time-dependent Schrödinger equation that is reduced to a classical nonlinear integrable equation. For a given time-dependent Hamiltonian, the counterdiabatic term is introduced to prevent nonadiabatic transitions. Using the fact that the equation for the dynamical invariant is equivalent to the Lax equation in nonlinear integrable systems, we obtain the counterdiabatic term exactly. The counterdiabatic term is available when the corresponding Lax pair exists and the solvable systems are classified in a unified and systematic way. Multisoliton potentials obtained from the Korteweg-de Vries equation and isotropic XY spin chains from the Toda equations are studied in detail.

Above-threshold ionization and photoelectron spectra in atomic systems driven by strong laser fields

Above-threshold ionization (ATI) results from strong field laser-matter interaction and it is one of the fundamental processes that may be used to extract electron structural and dynamical information about the atomic or molecular target. Moreover, it can also be used to characterize the laser field itself. Here, we develop an analytical description of ATI, which extends the theoretical Strong Field Approximation (SFA), for both the direct and re-scattering transition amplitudes in atoms. From a non-local, but separable potential, the bound-free dipole and the re-scattering transition matrix elements are analytically computed. In comparison with the standard approaches to the ATI process, our analytical derivation of the re-scattering matrix elements allows us to study directly how the re-scattering process depends on the atomic target and laser pulse features -we can turn on and off contributions having different physical origins or corresponding to different physical mechanisms. We compare SFA results with the full numerical solutions of the time-dependent Schroedinger equation (TDSE) within the few-cycle pulse regime. Good agreement between our SFA and TDSE model is found for the ATI spectrum. Our model captures also the strong dependence of the photoelectron spectra on the carrier envelope phase of the laser field.

Ultrafast dynamics in the bifunctional PENNA neutral and cation molecule induced by ultrashort photoexcitation

SynopsisWe report on a theoretical investigation on ultrafast dynamics induced in the neutral and cationic states of the PENNA molecule by a short, strong optical pulse. The ultrafast, short time, electronic dynamics is computed by solving the time-dependent Schroedinger equation for the time-dependent Hamiltonian that includes the coupling to the strong field. The partitioning technique is implemented to take into account the coupling to the ionization continua. Preliminary results on the onset of nuclear dynamics that follows the excitation by the pump pulse will be discussed.

Structure and dynamics of weakly-bound systems: a one-dimensional model

A line of research has been developed to describe the structure and the dynamics of weakly-bound systems with one or more valence particles. To simplify the problem we are assuming particles moving in one dimension. Within this model one can describe, for example, one- and two-particle break-up and one- and two-particle transfer processes. In the case of reactions involving two particles one can study in particular the proper reaction mechanism in connection with the role of the pairing interaction. Exact solutions obtained by directly solving the many-body time-dependent Schroedinger equation can be compared with the results obtained with different approximation schemes (first-order, coupled channels, continuum discretization, etc). In this particular contribution we concentrate on one-particle transfer processes.