Time-dependent Quantum Mechanics Research Articles

Vibronic Transitions and Quantum Dynamics in Molecular Oligomers: A Theoretical Analysis with an Application to Aggregates of Perylene Bisimides

Vibronic absorption spectra of molecular aggregates consisting of up to N = 9 monomer units are calculated employing methods of time-dependent quantum mechanics. Taking one vibrational degree of freedom for each monomer into account and treating one-exciton excited electronic states leads to a problem with N vibrations and N electronically coupled states. The demanding quantum propagation is carried out within the multiconfiguration time-depended Hartree method (MCTDH). Spectral features of and population transfer in the aggregates are analyzed as a function of the aggregate size and the strength of the electronic coupling. With a model for oligomers of perylene bisimides, it is shown how measured temperature-dependent absorption spectra correlate with the aggregate size. Furthermore, the exciton localization and dynamics in these aggregates are investigated.

Quasi-classical Trajectory Study of the Intramolecular Isotope Effect in the Reaction O(3P)+H2/HD

Theoretical studies of the dynamics of the reactions O(3P)+H2/HD(v = 0, j = 0)OH+H have been performed with quasi-classical trajectory method (QCT) on an ab initio potential surface for the lowest triplet electronic state of H2O(3A). The QCT-calculated integral cross sections are in good agreement with the earlier time-dependent quantum mechanics results. The state-resolved rotational distributions reveal that the product OH rotational distributions for O+HD have a preference for populating highly internally excited states compared with the O+H2 reaction. Distributions of differential cross sections show that directions of scattering are strongly dependent on the choice of quantum state. The polarization dependent generalized differential cross-sections and the distributions were calculated and a pronounced isotopic effect is revealed. The calculated results indicate that the product polarization is very sensitive to the mass factor.

Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics

For classical canonical transformations, one can, using the Wigner transfor- mation, pass from their representation in Hilbert space to a kernel in phase space. In this paper it will be discussed how the time-dependence of the uncertainties of the corresponding time-dependent quantum problems can be incorporated into this formalism.

An efficient and accurate method to obtain the energy-dependent Green function for general potentials

Time-dependent quantum mechanics provides an intuitive picture of particle propagation in external fields. Semiclassical methods link the classical trajectories of particles with their quantum mechanical propagation. Many analytical results and a variety of numerical methods have been developed to solve the time-dependent Schrödinger equation. The time-dependent methods work for nearly arbitrarily shaped potentials, including sources and sinks via complex-valued potentials. Many quantities are measured at fixed energy, which is seemingly not well suited for a time-dependent formulation. Very few methods exist to obtain the energy-dependent Green function for complicated potentials without resorting to ensemble averages or using certain lead-in arrangements. Here, we demonstrate in detail a time-dependent approach, which can accurately and effectively construct the energy-dependent Green function for very general potentials. The applications of the method are numerous, including chemical, mesoscopic, and atomic physics.

Electronic stopping in insulators: a simple model

A simple model is presented to explore, review and illustrate basic conceptsrelated to the electronic stopping power in large electronic band gap insulators.A projectile shooting through the solid is described by a local potentialV (r−vt). The flat band limit is assumed for both valence and conduction bands. Theenergy-transfer rate is calculated by solving the time-dependent quantum mechanics for thetimescale of electron excitation near the trajectory, taking the timescale for electronicenergy out-diffusion to be much larger. The threshold effect in the low projectile velocitylimit is characterized.

Influence of Reagent Rotation on (H-, D2) and (D-, H2) Collisions: A Quantum Mechanical Study

Time-independent quantum mechanical (TIQM) approach (helicity basis truncated at k = 2) has been used for computing differential and integral cross sections for the exchange reaction H- + D2 (v = 0, j = 0-4) --> HD + D- and D- + H2 (v = 0, j = 0-3) --> HD + H- in three dimensions on an accurate ab initio potential energy surface. It is shown that the j-weighted differential reaction cross section values are in good agreement with the experimental results reported by Zimmer and Linder at four different relative translational energies (Etrans = 0.55, 0.93, 1.16 and 1.48 eV) for (H-, D2) and at one relative translational energy (Etrans = 0.6 eV) by Haufler et al. for both (H-, D2) and (D-, H2) collisions. The j-weighted integral reaction cross section values are in good agreement with the crossed beam measurements by Zimmer and Linder in the Etrans range 0.5-1.5 eV and close to the guided ion beam results by Haufler et al. for (H-, D2) in the range 0.8-1.2 eV. Time-dependent quantum mechanical (TDQM) results obtained using centrifugal sudden approximation are reported in the form of integral reaction cross section values as a function of Etrans in the range 0.3-3.0 eV for both reactions in three dimensions on the same potential energy surface. The TDQM reaction cross section values decline more sharply than the TIQM results with increase in the initial rotational quantum number (j) for the D2 molecules in their ground vibrational state (v = 0) for (H-, D2) collisions. The computed j-weighted reaction cross section values are in good agreement with the experimental results reported by Zimmer and Linder for (H-, D2) collisions and guided ion beam results by Haufler et al. for both (H-, D2) and (D-, H2) collisions for energies below the threshold for electron detachment channel.

Overdamping of coherently driven quantum systems

The phenomenon of overdamping, when frictional forces overwhelm the restoring force of an harmonic oscillator, is well known in classical mechanics. Analogous phenomena occur in time-dependent quantum mechanics when probability loss rates (e.g. photoionization rates) become larger than the characteristic inverse times for coherent excitation (the Rabi frequencies). These results often seem surprising, even for simple two-state quantum systems. We discuss examples of this overdamping phenomenon—when an increase of the loss rate actually produces a slower rate of probability loss and even freezes population—for a number of quantum systems: steady and pulsed two-state excitation, time evolution with adiabatic passage, simple multistate chains, and branched chains of excitation linkage. In each situation there occurs for very large damping, often unexpectedly, phenomena akin to the overdamping of the classical harmonic oscillator.

On the geometry dependence of molecular dimer spectra with an application to aggregates of perylene bisimide

We study spectroscopic properties of molecular dimers coupled by dipole–dipole interactions within the framework of time-dependent quantum mechanics. A systematic variation of the dimer geometry allows to establish relationships between the latter and structures in the absorption spectrum. The theoretical model is constructed with the purpose to characterize the changes in absorption and emission properties arising upon aggregation of perylene bisimides. Measured and calculated spectra are compared, thereby addressing the question if a simple exciton model is capable to describe excited state properties of nanoaggregates of these molecules.

Perturbative approach to the adiabatic approximation

A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of ``adiabatic'' evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to not be widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.

Semiclassical treatment of fusion processes in collisions of weakly bound nuclei

We describe a semiclassical treatment of nuclear fusion reactions involving weakly bound nuclei. In this treatment, the projectile-target relative motion is approximated by a classical trajectory while the intrinsic dynamics is handled by time-dependent quantum mechanics. The complete fusion probabilities are approximated by products of two factors: A tunneling probability and the probability that the system is in its ground state at the strong absorption radius. We investigate the validity of the method in a schematic two-channel application, where the continuum is represented by a single effective state with finite lifetime. Coupling effects below the Coulomb barrier are included through an analytical continuation of the time. Comparisons with full coupled-channels calculations are performed. The agreement between semiclassical and fully quantal calculations is quite good, suggesting that the procedure may be extended to more sophisticated discretizations of the continuum.

On the role of second number-conserving functional derivatives

It is found that number-conserving second derivatives, of functional differentiation constrained to the domain of functional variables ρ ( x ) of a given norm ∫ ρ ( x ) d x , are not obtained via two successive number-conserving differentiations, contrary to the case of unrestricted second derivatives. Investigating the role of second number-conserving derivatives, with the density-functional formulation of time-dependent quantum mechanics in focus, it is shown how number-conserving differentiation handles the dual nature of the Kohn–Sham potential arising in the practical use of the theory. On the other hand, it is pointed out that number-conserving derivatives cannot resolve the causality paradox connected with the second derivative of the exchange-correlation part of the action density functional.

Time-dependent density functional theory.

Time-dependent density functional theory (TDDFT) can be viewed as an exact reformulation of time-dependent quantum mechanics, where the fundamental variable is no longer the many-body wave function but the density. This time-dependent density is determined by solving an auxiliary set of noninteracting Schrodinger equations, the Kohn-Sham equations. The nontrivial part of the many-body interaction is contained in the so-called exchange-correlation potential, for which reasonably good approximations exist. Within TDDFT two regimes can be distinguished: (a) If the external time-dependent potential is "small," the complete numerical solution of the time-dependent Kohn-Sham equations can be avoided by the use of linear response theory. This is the case, e.g., for the calculation of photoabsorption spectra. (b) For a "strong" external potential, a full solution of the time-dependent Kohn-Sham equations is in order. This situation is encountered, for instance, when matter interacts with intense laser fields. In this review we give an overview of TDDFT from its theoretical foundations to several applications both in the linear and in the nonlinear regime.

Spectral difference Lanczos method for efficient time propagation in quantum control theory.

Spectral difference methods represent the real-space Hamiltonian of a quantum system as a banded matrix which possesses the accuracy of the discrete variable representation (DVR) and the efficiency of finite differences. When applied to time-dependent quantum mechanics, spectral differences enhance the efficiency of propagation methods for evolving the Schrodinger equation. We develop a spectral difference Lanczos method which is computationally more economical than the sinc-DVR Lanczos method, the split-operator technique, and even the fast-Fourier-Transform Lanczos method. Application of fast propagation is made to quantum control theory where chirped laser pulses are designed to dissociate both diatomic and polyatomic molecules. The specificity of the chirped laser fields is also tested as a possible method for molecular identification and discrimination.

Vibrational Echoes: Dephasing, Rephasing, and the Stability of Classical Trajectories

The experimental observables in coherent, multiple pulse infrared spectroscopic measurements can be calculated from the nonlinear response functions describing the nuclear dynamics of molecular and condensed phase systems. Within classical mechanics, these nonlinear response functions can be expressed in terms of the monodromy matrices that quantify the stability of classical trajectories. We use an ensemble of noninteracting, anharmonic oscillators to examine the effects of the divergence in time of the classical stability matrix on the analytic properties of the third-order response function, relevant to vibrational echo spectroscopy. The twopulse echo measurement is designed to rephase a macroscopic variable, that is, to reverse the effects of destructive interference among the dynamics of microscopic systems characterized by a static distribution of energies. Within classical mechanics, this rephasing is shown to preserve the growth with time of the nonlinear response function that is the signature of the divergence of nearby trajectories. For systems with nearly classical nuclear motions, the vibrational echo measurement may then be interpreted as a probe of the stability of atomic trajectories. I. Introduction Observables in nonlinear infrared and Raman measurements 1-7 of nuclear motions may be computed from nonlinear response functions characterizing the material system. The challenges posed by computing nonlinear response functions for large anharmonic systems with time-dependent quantum mechanics has motivated the analysis of these quantities within classical mechanics. 8-17 Nonlinear response functions of a classical mechanical system may not be calculated directly from a conventional, equilibrium molecular dynamics simulation, because their computation requires knowledge of stability matrices, 8 which quantify the effects of small deviations in initial conditions on classical trajectories. An alternative to simulating nonlinear response functions, which obviates the need to compute stability matrices, is to perform a nonequilibrium molecular dynamics simulation of the material system in the presence of an electromagnetic field and to evaluate numerically the appropriate low-field limit. 15-17 The relevance of stability matrices to nonlinear optical measurements is intuitively clear.

Quantum and classical studies of the O(3P) + H2(v = 0-3,j = 0) --> OH + H reaction using benchmark potential surfaces.

We present results of time-dependent quantum mechanics (TDQM) and quasiclassical trajectory (QCT) studies of the excitation function for O(3P) + H2(v = 0-3,j = 0) --> OH + H from threshold to 30 kcal/mol collision energy using benchmark potential energy surfaces [Rogers et al., J. Phys. Chem. A 104, 2308 (2000)]. For H2(v = 0) there is excellent agreement between quantum and classical results. The TDQM results show that the reactive threshold drops from 10 kcal/mol for v = 0 to 6 for v = 1, 5 for v = 2 and 4 for v = 3, suggesting a much slower increase in rate constant with vibrational excitation above v = 1 than below. For H2(v > 0), the classical results are larger than the quantum results by a factor approximately 2 near threshold, but the agreement monotonically improves until they are within approximately 10% near 30 kcal/mol collision energy. We believe these differences arise from stronger vibrational adiabaticity in the quantum dynamics, an effect examined before for this system at lower energies. We have also computed QCT OH(v',j') state-resolved cross sections and angular distributions. The QCT state-resolved OH(v') cross sections peak at the same vibrational quantum number as the H2 reagent. The OH rotational distributions are also quite hot and tend to cluster around high rotational quantum numbers. However, the dynamics seem to dictate a cutoff in the energy going into OH rotation indicating an angular momentum constraint. The state-resolved OH distributions were fit to probability functions based on conventional information theory extended to include an energy gap law for product vibrations.

Bohmian versus semiclassical description of interference phenomena

The origin of quantum interference characteristic of bound nonlinear systems is investigated within the Bohmian formulation of time-dependent quantum mechanics. By contrast to time-dependent semiclassical theory, whereby interference is a consequence of phase mismatch between distinct classical trajectories, the Bohmian, fully quantum mechanical expression for expectation values has a quasiclassical appearance that does not involve phase factors or cross terms. Numerical calculations reveal that quantum interference in the Bohmian formulation manifests itself directly as sharp spatial/temporal variations of the density surrounding kinky trajectories. These effects are most dramatic in regions where the underlying classical motion exhibits focal points or caustics, and crossing of the Bohmian trajectories is prevented through extremely strong and rapidly varying quantum mechanical forces. These features of Bohmian dynamics, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations.

Ground and excited states of one‐dimensional self‐interacting nonlinear oscillators through time‐dependent quantum mechanics

AbstractEnergy eigenvalues, probability densities, and 〈x2r〉 values (r = 1 to 6) of one‐dimensional self‐interacting nonlinear oscillators have been obtained by evolving the time‐dependent Schrödinger equation in imaginary time, coupled with the minimization of energy expectation values. For excited states, the orthogonality constraint with lower states is maintained. Probability density plots for the ground and first three excited states are presented. A comparison of energy eigenvalues and probability densities plots is also made between oscillators with and without self‐interaction. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Reaction probabilities and reaction cross sections for three-dimensional He+H2+(v) collisions: A time-dependent quantum mechanical study

Vibrational (v) state-selected reaction probabilities (PvjJ) have been calculated for the reaction He+H2+(v=0,1,2,3)→HeH++H, for H2+ in its ground rotational state (j=0) for a range of total angular momentum (J) values, for total energy (E) in the range 0.94–1.4 eV, using time-dependent quantum mechanical (TDQM) approach and centrifugal sudden approximation. The number of oscillations in Pv0J(E) decreases with increase in J. In addition, there is a noticeable increase in the threshold energy (Eth) with increase in J. The magnitude of Pv0J decreases with increase in J resulting in converged cross section values by the time J becomes 35–45, depending upon v and E under investigation. The resulting (converged) reaction cross section values are in excellent agreement with the experimental results at E=1.14 eV, both in terms of the magnitude and in terms of the observed vibrational enhancement. Interestingly, there is a noticeable discrepancy between our TDQM results and earlier time-independent quantum mechanical studies at higher energies, for v=1 and 2. For v=3, the difference between the two theoretical studies is the largest near the threshold. Our results for v=2 and 3 suggest that the oscillations in the vibrational state-selected reaction cross section values are amenable to experimental observation.

Time-dependent wave packet calculations on parallel computers: A new and efficient algorithm for evaluating [formula omitted

A general parallel method for evaluating the action of the Hamiltonian operator on the wave function for use in time-dependent quantum mechanics is presented. In this algorithm the wave function is distributed over the processors of a parallel machine according to two different degrees-of-freedom, so that each processor holds two different slices of the same wave function. This leads to very efficient communication and computation characteristics of the algorithm. The algorithm is implemented for a gas–surface reactive system for two different communication protocols. Timings are presented and the scaling of the algorithm is discussed for each of the implementations. Finally, extensions of the algorithm are discussed.

Quantum dressed classical mechanics: application to non-adiabatic processes

A newly formulated theory for time-dependent molecular quantum mechanics is used to study processes involving more than one potential energy surface. Good agreement with exact numbers is obtained using one trajectory and just two grid points.