We discover and characterize strong quantum scars, or eigenstates resembling classical periodic orbits, in two-dimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum near-degeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremize the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wave-packet recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.

%B Sci. Rep. %G eng %U https://arxiv.org/abs/1511.04198 %0 Journal Article %J arxiv %D 2017 %T Graphene Terahertz Absorption %A Yang, Yuan %A Grigory Kolesov %A Kocia, Lucas %A Heller, EricJ. %X The unique terahertz properties of graphene has been identified for novel optoelectronic applications. In a graphene sample with bias voltage added, there is an enhanced absorption in the far infrared region and a diminished absorption in the infrared region. The strength of enhancement(diminishment) increases with the gate voltage, and the enhancement compensates the diminishment. We find that it is the coherence length of electrons in graphene that allows pure electronic transitions between states differing by small momentums and makes intraband transition possible, is responsible for the far infrared enhancement. Phonon assisted processes are not necessary and would not in any case contribute to a sum rule. This naturally leads to results obeying the general sum-rule in optical absorptions. Our prediction of the strength of enhancement(diminishment) in terms of the bias agrees with experiments. This is the first direct calculation we are aware of, since the prior phonon assisted model for indirect transition should not obey a sum rule. %B arxiv %P arXiv:1705.06267 %G eng %U https://arxiv.org/abs/1705.06267 %0 Journal Article %J arxiv %D 2017 %T Reassessing Graphene Absorption and Emission Spectroscopy %A Yang, Yuan %A Grigory Kolesov %A Kocia, Lucas %A Heller, EricJ. %XWe present a new paradigm for understanding optical absorption and hot electron dynamics experiments in graphene. Our analysis pivots on assigning proper importance to phonon assisted indirect processes and bleaching of direct processes. We show indirect processes figure in the excess absorption in the UV region. Experiments which were thought to indicate ultrafast relaxation of electrons and holes, reaching a thermal distribution from an extremely non-thermal one in under 5-10 fs, instead are explained by the nascent electron and hole distributions produced by indirect transitions. These need no relaxation or ad-hoc energy removal to agree with the observed emission spectra and fast pulsed absorption spectra. The fast emission following pulsed absorption is dominated by phonon assisted processes, which vastly outnumber direct ones and are always available, connecting any electron with any hole any time. Calculations are given, including explicitly calculating the magnitude of indirect processes, supporting these views.

%B arxiv %P arXiv:1704.07500 %G eng %U https://arxiv.org/abs/1704.07500 %0 Journal Article %J arxiv preprint %D 2017 %T Characterizing time-irreversibility in disordered fermionic systems by the effect of local perturbations %A Shreya Vardhan %A Giuseppe De Tomasi %A Markus Heyl %A Heller, EricJ. %A Pollmann, Frank %XWe study the effects of local perturbations on the dynamics of disordered fermionic systems in order to characterize time-irreversibility. We focus on three different systems, the non-interacting Anderson and Aubry-Andr\'e-Harper (AAH-) models, and the interacting spinless disordered t-V chain. First, we consider the effect on the full many-body wave-functions by measuring the Loschmidt echo (LE). We show that in the extended/ergodic phase the LE decays exponentially fast with time, while in the localized phase the decay is algebraic. We demonstrate that the exponent of the decay of the LE in the localized phase diverges proportionally to the single-particle localization length as we approach the metal-insulator transition in the AAH model. Second, we probe different phases of disordered systems by studying the time expectation value of local observables evolved with two Hamiltonians that differ by a spatially local perturbation. Remarkably, we find that many-body localized systems could lose memory of the initial state in the long-time limit, in contrast to the non-interacting localized phase where some memory is always preserved.

%B arxiv preprint %P arXiv:1702.01622 %G eng %U https://arxiv.org/abs/1702.01622 %0 Journal Article %J J. Chem. Phys. %D 2016 %T Dynamical tunneling versus fast diffusion for a non-convex Hamiltonian %A S. M. Pittman %A E. Tannenbaum %A E. J. Heller %XThis paper attempts to resolve the issue of the nature of the 0.01-0.1 cm^{−1} peak splittings observed in high-resolution IR spectra of polyatomic molecules. One hypothesis is that these splittings are caused by dynamical tunneling, a quantum-mechanical phenomenon whereby energy flows between two disconnected regions of phase-space across dynamical barriers. However, a competing classical mechanism for energy flow is Arnol’d diffusion, which connects different regions of phase-space by a resonance network known as the Arnol’d web. The speed of diffusion is bounded by the Nekhoroshev theorem, which guarantees stability on exponentially long time scales if the Hamiltonian is steep. Here we consider a non-convex Hamiltonian that contains the characteristics of a molecular Hamiltonian, but does not satisfy the Nekhoroshev theorem. The diffusion along the Arnol’d web is expected to be fast for a non-convex Hamiltonian. While fast diffusion is an unlikely competitor for longtime energy flow in molecules, we show how dynamical tunneling dominates compared to fast diffusion in the nearly integrable regime for a non-convex Hamiltonian, as well as present a new kind of dynamical tunneling.

Raman spectroscopy plays a key role in studies of graphene and related carbon systems. Graphene is perhaps the most promising material of recent times for many novel applications, including electronics. In this paper, the traditional and well established Kramers-Heisenberg-Dirac (KHD) Raman scattering theory (1925-1927) is extended to crystalline graphene for the first time. It demands different phonon production mechanisms and phonon energies than does the popular "double resonance" Raman scattering model. The latter has never been compared to KHD. Within KHD, phonons are produced instantly along with electrons and holes, in what we term an electron-hole-phonon triplet, which does not suffer Pauli blocking. A new mechanism for double phonon production we name "transition sliding" explains the brightness of the 2D mode and other overtones, as a result of linear (Dirac cone) electron dispersion. Direct evidence for sliding resides in hole doping experiments performed in 2011 \cite{chenCrommie}. Whole ranges of electronic transitions are permitted and may even constructively interfere for the same laser energy and phonon q, explaining the dispersion, bandwidth, and strength of many two phonon Raman bands. Graphene's entire Raman spectrum, including dispersive and fixed bands, missing bands not forbidden by symmetries, weak bands, overtone bands, Stokes anti-Stokes anomalies, individual bandwidths, trends with doping, and D-2D band spacing anomalies emerge naturally and directly in KHD theory.

%B ACS Nano %V 10 %P 2803–2818 %G eng %U http://dx.doi.org/10.1021/acsnano.5b07676 %N 2 %R 10.1021/acsnano.5b07676 %0 Journal Article %J arXiv preprint arXiv:1511.04198 %D 2015 %T Strong quantum scarring by local impurities %A Luukko, Perttu JJ %A Drury, Byron %A Klales, Anna %A Kaplan, Lev %A Heller, EricJ. %A Esa Räsänen %B arXiv preprint arXiv:1511.04198 %G eng %U http://arxiv.org/abs/1511.04198 %0 Journal Article %J The Journal of Physical Chemistry A %D 2015 %T The Degree of Ergodicity of Ortho-and Para-aminobenzonitrile in an Electric Field %A Baker, Suzanne M %A Heller, EricJ. %XWe study the dynamics of the two molecules *ortho*-aminobenzonitrile (OABN) and *para*-aminobenzonitrile (PABN). They are structural isomers, with differing asymmetries and dipole moments. In this paper, we show that the dynamics of the system strongly depends on the region of phase space of the initial rotational state, the asymmetry of the molecule, and on the direction of the dipole. We also show that the ergodicity of the system varies gradually with energy, except where the rotational energy of the initial state is much less than the Stark interaction. In this regime, the projection of the dipole along the lab-frame *z*-axis varies linearly with increasing energy and follows the microcanonical ergodic estimate. Both molecules are far from full chaos for total angular momentum quanta *J* ∈ [0,45]. However, the initial rotational states in OABN access much more of the available phase space than in PABN. We show that this is a likely cause for the experimental discrepancies in molecular beam deflection experiments.

Polyacetylene has been a paradigm conjugated organic conductor since well before other conjugated carbon systems such as nanotubes and graphene became front and center. It is widely acknowledged that Raman spectroscopy of these systems is extremely important to characterize them and understand their internal quantum behavior. Here we show, for the first time, what information the Raman spectrum of polyacetylene contains, by solving the 35-year-old mystery of its spectrum. Our methods have immediate and clear implications for other conjugated carbon systems. By relaxing the nearly universal approximation of ignoring the nuclear coordinate dependence of the transition moment (Condon approximation), we find the reasons for its unusual spectroscopic features. When the Kramers–Heisenberg–Dirac Raman scattering theory is fully applied, incorporating this nuclear coordinate dependence, and also the energy and momentum dependence of the electronic and phonon band structure, then unusual line shapes, growth, and dispersion of the bands are explained and very well matched by theory.

%B ACS Central Science %I ACS Publications %G eng %U https://pubs.acs.org/doi/abs/10.1021/acscentsci.5b00021 %0 Journal Article %J The Journal of chemical physics %D 2014 %T Collision dynamics of polyatomic molecules containing carbon rings at low temperatures %A Li, Zhiying %A Krems, Roman V %A Heller, EricJ. %XWe explore the collision dynamics of complex hydrocarbon molecules (benzene, coronene, adamantane, and anthracene) containing carbon rings in a cold buffer gas of ^{3}He. For benzene, we present a comparative analysis of the fully classical and fully quantum calculations of elastic and inelastic scattering cross sections at collision energies between 1 and 10 cm^{−1}. The quantum calculations are performed using the time-independent coupled channel approach and the coupled-states approximation. We show that the coupled-states approximation is accurate at collision energies between 1 and 20 cm^{−1}. For the classical dynamics calculations, we develop an approach exploiting the rigidity of the carbon rings and including low-energy vibrational modes without holonomic constraints. Our results illustrate the effect of the molecular shape and the vibrational degrees of freedom on the formation of long-lived resonance states that lead to low-temperature clustering.

The collective excitation of the conduction electrons in subwavelength structures gives rise to the Localized Surface Plasmon(LSP). The system consisting of two such LSPs, known as the dimer system,is of fundamental interest and is being actively investigated in the literature. Three regimes have been previously identified and they are the photonic regime, the strong coupling regime and the quantum tunneling regime. In this Letter, we propose a new regime for this intriguing systems, the intermediate regime. In this new regime, the quasistatic approximation, which is widely used to study such LSP systems, fails to capture the main physics: the multiple scattering of the electromagnetic waves between the two LSPs, which significantly modifies the properties of the resonant modes in the system. This intermediate regime provides a new route to explore in plasmonics, where controlling both the excited plasmon modes and the damping rates are of paramount significance.

%B arXiv preprint arXiv:1403.4310 %G eng %U https://arxiv.org/abs/1403.4310 %0 Journal Article %J arXiv preprint arXiv:1405.2807 %D 2014 %T Shaping Electromagnetic Fields %A Liu, Bo %A Heller, EricJ. %XThe ability to control electromagnetic fields on the subwavelength scale could open exciting new venues in many fields of science. Transformation optics provides one way to attain such control through the local variation of the permittivity and permeability of a material. Here, we demonstrate another way to shape electromagnetic fields, taking advantage of the enormous size of the configuration space in combinatorial problems and the resonant scattering properties of metallic nanoparticles. Our design does not require the engineering of a material's electromagnetic properties and has relevance to the design of more flexible platforms for probing light-matter interaction and many body physics.

%B arXiv preprint arXiv:1405.2807 %G eng %U https://arxiv.org/abs/1405.2807 %0 Journal Article %J Physical Review E %D 2013 %T Fractal dynamics in chaotic quantum transport %A Kotimäki, Ville %A Esa Räsänen %A Hennig, Holger %A Heller, EricJ. %XDespite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis—a widely used method in time-series analysis—and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.

%B Physical Review E %I APS %V 88 %P 022913 %G eng %U http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.022913 %0 Journal Article %J The European Physical Journal B %D 2013 %T Optimal control of quantum revival %A Esa Räsänen %A Heller, EricJ. %K Computational Methods %X

Increasing fidelity is the ultimate challenge of quantum information technology. In addition to decoherence and dissipation, fidelity is affected by internal imperfections such as impurities in the system. Here we show that the quality of quantum revival, i.e., periodic recurrence in the time evolution, can be restored almost completely by coupling the distorted system to an external field obtained from quantum optimal control theory. We demonstrate the procedure with wave-packet calculations in both one- and two-dimensional quantum wells, and analyze the required physical characteristics of the control field. Our results generally show that the inherent dynamics of a quantum system can be idealized at an extremely low cost.

%B The European Physical Journal B %I Springer-Verlag %V 86 %G eng %U http://dx.doi.org/10.1140/epjb/e2012-30921-4 %R 10.1140/epjb/e2012-30921-4 %0 Journal Article %J EPL (Europhysics Letters) %D 2013 %T Quantum flux and reverse engineering of quantum wave functions %A Mason, Douglas J %A Mario F. Borunda %A Heller, EricJ. %X

An interpretation of the probability flux is given, based on a derivation of its eigenstates and relating them to coherent-state projections on a quantum wave function. An extended definition of the flux operator is obtained using coherent states. We present a "processed Husimi" representation, which makes decisions using many Husimi projections at each location. The processed Husimi representation reverse engineers or deconstructs the wave function, yielding the underlying classical ray structure. Our approach makes possible interpreting the dynamics of systems where the probability flux is uniformly zero or strongly misleading. The new technique is demonstrated by the calculation of particle flow maps of the classical dynamics underlying a quantum wave function in simple model systems such as a circular billiard with and without a magnetic field.

%B EPL (Europhysics Letters) %I IOP Publishing %V 102 %P 60005 %G eng %U http://dx.doi.org/10.1209/0295-5075/102/60005 %0 Journal Article %J Physical Review Letters %D 2013 %T Stability of Branched Flow from a Quantum Point Contact %A Liu, Bo %A Eric J. Heller %X In classically chaotic systems, small differences in initial conditions are exponentially magnified over time. However, it was observed experimentally that the (necessarily quantum) ‘‘branched flow’’ pattern of electron flux from a quantum point contact (QPC) traveling over a random background potential in two- dimensional electron gases remains substantially invariant to large changes in initial conditions. Since such a potential is classically chaotic and unstable to changes in initial conditions, it was conjectured that the origin of the observed stability is purely quantum mechanical, with no classical analog. In this Letter, we show that the observed stability is a result of the physics of the quantum point contact and the nature of the experiment. We show that the same stability can indeed be reproduced classically, or quantum mechanically. In addition, we explore the stability of the branched flow with regards to changes in the eigenmodes of the quantum point contact. %B Physical Review Letters %V 111 %P 236804 %G eng %U http://prl.aps.org/abstract/PRL/v111/i23/e236804 %0 Journal Article %J Physical Review B %D 2013 %T Ballistic versus diffusive transport in graphene %A Mario F. Borunda %A Hennig, H %A Heller, EricJ. %XWe investigate the transport of electrons in disordered and pristine graphene devices. Fano shot noise, a standard metric to assess the mechanism for electronic transport in mesoscopic devices, has been shown to produce almost the same magnitude (

The dephasing relation (DR), a linearization of semiclassical fidelity, is generalized to include the overlap of “off-diagonal” elements. The accuracy of the formulation is tested in integrable and chaotic systems and its scaling with dimensionality is studied in a Caldeira-Leggett model with many degrees of freedom. It is shown that the DR is often in very good agreement with numerically analytic quantum results and frequently outperforms an alternative semiclassical treatment. Most importantly, since there is no computationally expensive prefactor, and Monte Carlo Metropolis sampling is used to facilitate the calculation, the DR is found to scale remarkably well with increasing dimension. We further demonstrate that a propagator based on the DR can include more quantum coherence and outperform other popular linearized semiclassical methods, such as forward-backward semiclassical dynamics (FBSD) and the linearized semiclassical initial value representation (LSC-IVR).

%B The Journal of Chemical Physics %V 139 %P 124110 %G eng %U http://scitation.aip.org/content/aip/journal/jcp/139/12/10.1063/1.4820880 %0 Journal Article %J Physical Review B %D 2013 %T Optimal local control of coherent dynamics in custom-made nanostructures %A Thomas Blasi %A Mario F. Borunda %A Esa Räsänen %A Heller, EricJ. %XWe apply quantum optimal control theory to establish a local voltage-control scheme that operates in conjunction with the numerically exact solution of the time-dependent Schrödinger equation. The scheme is demonstrated for high-fidelity coherent control of electronic charge in semiconductor double quantum dots. We find tailored gate voltages in the viable gigahertz regime that drive the system to a desired charge configuration with

We present a method for bridging the gap between the Dirac effective field theory and atomistic simulations in graphene based on the Husimi projection, allowing us to depict phenomena in graphene at arbitrary scales. This technique takes the atomistic wave function as an input, and produces semiclassical pictures of quasiparticles in the two Dirac valleys. We use the Husimi technique to produce maps of the scattering behavior of boundaries, giving insight into the properties of wave functions at energies both close to and far from the Dirac point. Boundary conditions play a significant role to the rise of Fano resonances, which we examine using the processed Husimi map to deepen our understanding of bond currents near resonance.

%B Physical Review B %I APS %V 88 %P 165421 %G eng %U http://journals.aps.org/prb/abstract/10.1103/PhysRevB.88.165421 %0 Journal Article %J Phys. Rev. B %D 2012 %T Optical control of entangled states in semiconductor quantum wells %A E. Räsänen %A T. Blasi %A M. F. Borunda %A E. J. Heller %XWe present theory and calculations for coherent high-fidelity quantum control of many-particle states in semiconductor quantum wells. We show that coupling a two-electron double quantum dot to a terahertz optical source enables targeted excitations that are one to two orders of magnitude faster and significantly more accurate than those obtained with electric gates. The optical fields subject to physical constraints are obtained through quantum optimal control theory that we apply in conjunction with the numerically exact solution of the time-dependent Schrödinger equation. Our ability to coherently control arbitrary two-electron states, and to maximize the entanglement, opens up further perspectives in solid-state quantum information.

%B Phys. Rev. B %I American Physical Society %V 86 %P 205308 %8 Nov %G eng %U http://link.aps.org/doi/10.1103/PhysRevB.86.205308 %R 10.1103/PhysRevB.86.205308 %0 Journal Article %J The Journal of chemical physics %D 2012 %T Cold collisions of complex polyatomic molecules %A Li, Zhiying %A Heller, EricJ. %X

%B The Journal of chemical physics %I AIP Publishing %V 136 %P 054306 %G eng %U http://dx.doi.org/10.1063/1.3682982 %0 Journal Article %J Physics Letters A %D 2011 %T The activation of classical vibro-rotational resonances in diatom molecules through slow collision processes %A Ruiz, Antonia %A Palao, José P %A Heller, EricJ. %X

Classical atom–diatom collisions at low velocities can be considered as a transient perturbation to the (integrable) diatomic system. We present an analysis that makes explicit the contributions of the terms of the Fourier expansion of the interaction potential to the changes in the molecular actions due to the collision process. Each term is associated with a resonance condition between the vibrational and rotational molecular frequencies, and leads to a vibrational, rotational or vibrotational contribution to the total action changes. The analysis is applied to the system Ne.

%B Physics Letters A %I Elsevier %V 375 %P 2555–2562 %G eng %U http://dx.doi.org/10.1016/j.physleta.2011.05.040 %0 Journal Article %J Physical Review B %D 2011 %T Algorithm for efficient elastic transport calculations for arbitrary device geometries %A Mason, Douglas J %A Prendergast, David %A Neaton, Jeffrey B %A Heller, EricJ. %XWith the growth in interest in graphene, controlled nanoscale device geometries with complex form factors are now being studied and characterized. There is a growing need to understand new techniques to handle efficient electronic transport calculations for these systems. We present an algorithm that dramatically reduces the computational time required to find the local density of states and transmission matrix for open systems regardless of their topology or boundary conditions. We argue that the algorithm, which generalizes the recursive Green's function method by incorporating the reverse Cuthill-McKee algorithm for connected graphs, is ideal for calculating transmission through devices with multiple leads of unknown orientation and becomes a computational necessity when the input and output leads overlap in real space. This last scenario takes the Landauer-Buttiker formalism to general scattering theory in a computational framework that makes it tractable to perform full-spectrum calculations of the quantum scattering matrix in mesoscopic systems. We demonstrate the efficacy of these approaches on graphene stadiums, a system of recent scientific interest, and contribute to a physical understanding of Fano resonances which appear in these systems.

%B Physical Review B %I APS %V 84 %P 155401 %G eng %U https://doi.org/10.1103/PhysRevB.84.155401 %0 Journal Article %J ACS nano %D 2011 %T Imaging Universal Conductance Fluctuations in Graphene %A Mario F. Borunda %A Jesse Berezovsky %A Robert M. Westervelt %A Heller, EricJ. %XWe study conductance fluctuations (CF) and the sensitivity of the conductance to the motion of a single scatterer in two-dimensional massless Dirac systems. Our extensive numerical study finds limits to the predicted universal value of CF. We find that CF are suppressed for ballistic systems near the Dirac point and approach the universal value at sufficiently strong disorder. The conductance of massless Dirac fermions is sensitive to the motion of a single scatterer. CF of order *e*^{2}/*h* result from the motion of a single impurity by a distance comparable to the Fermi wavelength. This result applies to graphene systems with a broad range of impurity strength and concentration while the dependence on the Fermi wavelength can be explored *via* gate voltages. Our prediction can be tested by comparing graphene samples with varying amounts of disorder and can be used to understand interference effects in mesoscopic graphene devices.

Motivated by recent experiments by the Westervelt group, which used a mobile tip to probe the electronic state of a segmented nanowire, we calculate shifts in Coulomb blockade peak positions, as a function of tip location, which we term “Coulomb blockade microscopy.” We show that if the tip can be brought sufficiently close to the nanowire, one can distinguish a high-density electronic liquid state from a Wigner-crystal state by microscopy with a weak-tip potential. In the opposite limit of a strongly negative tip potential, the potential depletes the electronic density under it and divides the quantum wire into two partitions. There the tip can push individual electrons from one partition to the other and the Coulomb blockade micrograph can clearly track such transitions. We show that this phenomenon can be used to qualitatively estimate the relative importance of the electron interaction compared to one-particle potential and kinetic energies. Finally, we propose that a weak-tip Coulomb blockade micrograph focusing on the transition between electron number

Graphene provides a fascinating testbed for new physics and exciting opportunities for future applications based on quantum phenomena. To understand the coherent flow of electrons through a graphene device, we employ a nanoscale probe that can access the relevant length scales—the tip of a liquid-He-cooled scanning probe microscope (SPM) capacitively couples to the graphene device below, creating a movable scatterer for electron waves. At sufficiently low temperatures and small size scales, the diffusive transport of electrons through graphene becomes coherent, leading to universal conductance fluctuations (UCF). By scanning the tip over a device, we map these conductance fluctuations versus scatterer position. We find that the conductance is highly sensitive to the tip position, producing δ*G* ~ *e*^{2}/*h* fluctuations when the tip is displaced by a distance comparable to half the Fermi wavelength. These measurements are in good agreement with detailed quantum simulations of the imaging experiment and demonstrate the value of a cooled SPM for probing coherent transport in graphene.

We study matter-wave scattering from an ultracold, many-body atomic system trapped in an optical lattice. The angular cross section of the target lattice for a matter wave is determined and is demonstrated to have a strong dependence on the many-body phase, superfluid, or Mott insulator. Analytical approaches are employed deep in the superfluid and Mott-insulator regimes, while intermediate points in the phase transition are treated numerically. Matter-wave scattering offers a convenient method for nondestructively probing the quantum many-body phase transition of atoms in an optical lattice.

%B Physical review letters %I APS %V 105 %P 035301 %G eng %U https://doi.org/10.1103/PhysRevLett.105.035301 %0 Journal Article %J Physical Review B %D 2010 %T Self-consistent calculation of electric potentials in Hall devices %A Kramer, Tobias %A Krueckl, Viktor %A Heller, EricJ. %A Parrott, Robert E %XUsing a first-principles classical many-body simulation of a Hall bar, we study the necessary conditions for the formation of the Hall potential: (i) Ohmic contacts with metallic reservoirs, (ii) electron-electron interactions, and (iii) confinement to a finite system. By propagating thousands of interacting electrons over million time-steps we capture the build-up of the self-consistent potential. The microscopic model sheds light on the current injection process and directly links the Hall effect to specific boundary conditions at the particle reservoirs.

%B Physical Review B %I APS %V 81 %P 205306 %G eng %U https://doi.org/10.1103/PhysRevB.81.205306 %0 Journal Article %J Physical Review B %D 2010 %T Theory of the quantum Hall effect in finite graphene devices %A Kramer, Tobias %A Kreisbeck, Christoph %A Krueckl, Viktor %A Heller, EricJ. %A Parrott, Robert E %A Liang, Chi-Te %XWe study the quantum Hall effect (QHE) in graphene based on the current injection model, which takes into account the finite rectangular geometry with source and drain electrodes. In our model, the presence of disorder, the edge-state picture, extended states, and localized states, which are believed to be indispensable ingredients in describing the QHE, do not play an important role. Instead the boundary conditions during the injection into the graphene sheet, which are enforced by the presence of the Ohmic contacts, determine the current-voltage characteristics.

%B Physical Review B %I APS %V 81 %P 081410 %G eng %U https://doi.org/10.1103/PhysRevB.81.081410 %0 Journal Article %J Physical Review A %D 2009 %T Classical and quantum analysis of quasiresonance in grazing atom-surface collisions %A Ruiz, Antonia %A Palao, José P %A Heller, EricJ. %XQuasiresonance is a general effect that may arise from the coupling between approximately resonant degrees of freedom in a system perturbed by some transient interaction. In a process induced by a slowly switching on and off of the coupling interaction, quasiresonance is characterized by the existence of significant ranges of initial states in the perturbed system over which some very specific and efficient transfer of energy between the approximately resonant degrees of freedom occurs. This work presents a classical and quantum analysis of quasiresonant processes in grazing incident angle atom-surface collisions. The momentum transfer between the normal components to an index direction is investigated. For fast atoms with grazing angle of incidence there is an interval of azimuthal angles around the index directions, the quasiresonance region, in which the energy transfer can be very efficient. This effect is reflected in quantum diffraction patterns with large nonspecular peaks, associated with the parallel to the surface and normal to the index direction momentum component. We demonstrate the essentially classical underlying mechanism for the persistence of a pattern of diffraction peak intensities for incidence close to an index direction. The analysis also shows that the size of the quasiresonance region is approximately equal to the spectral width of the diffraction pattern.

%B Physical Review A %I APS %V 79 %P 052901 %G eng %U https://doi.org/10.1103/PhysRevA.79.052901 %0 Journal Article %J Physical Review A %D 2009 %T Coherent scattering from a free gas %A Sanders, Scott N %A Mintert, Florian %A Heller, EricJ. %XWe investigate decoherence in atom interferometry due to scattering from a background gas and show that the supposition that residual coherence is due to near-forward scattering is incorrect. In fact, the coherent part is completely unscattered, although it is phase shifted. This recoil-free process leaves both the atom and the gas in an unchanged state, but allows for the acquisition of a phase shift. This is essential to understanding decoherence in a separated-arm atom interferometer, where a gas of atoms forms a refractive medium for a matter wave. Our work elucidates the actual microscopic, many-body, quantum-mechanical scattering mechanism that gives rise to prior phenomenological results for the phase shift and decoherence.

%B Physical Review A %I APS %V 79 %P 023610 %G eng %U https://doi.org/10.1103/PhysRevA.79.023610 %0 Journal Article %J Physical Review E %D 2009 %T Nearly resonant multidimensional systems under a transient perturbative interaction %A Ruiz, Antonia %A Palao, José P %A Heller, EricJ. %XWe analyze the response of a classical system with

A new semiclassical method is proposed to obtain accurate ground-state energies for many-electron systems. The method borrows its semiclassical character from Thomas-Fermi (TF) theory, but improves upon it by including exchange-correlation effects, at least approximately. We illustrate our method (correlated TF method) on simple models of 1D-interacting electrons, showing that it yields dramatic improvements over TF theory, particularly in the strongly correlated regime.

%B Physical review letters %I APS %V 103 %P 066401 %G eng %U https://doi.org/10.1103/PhysRevLett.103.066401 %0 Journal Article %J EPL (Europhysics Letters) %D 2009 %T Simulation of open quantum systems %A Mintert, Florian %A Heller, EricJ. %XWe present an approach for the semiclassical treatment of open quantum systems. An expansion into localized states allows restriction of a simulation to a fraction of the environment that is located within a predefined vicinity of the system. Adding and dropping environmental particles during the simulation yields an effective reduction of the size of the system that is being treated.

%B EPL (Europhysics Letters) %I IOP Publishing %V 86 %P 50006 %G eng %U http://dx.doi.org/10.1209/0295-5075/86/50006 %0 Conference Paper %B Journal of Physics: Conference Series %D 2008 %T An efficient and accurate method to obtain the energy-dependent Green function for general potentials %A Kramer, Tobias %A Heller, EricJ. %A Parrott, Robert E %XTime-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.

%B Journal of Physics: Conference Series %I IOP Publishing %V 99 %P 012010 %G eng %U http://dx.doi.org/10.1088/1742-6596/99/1/012010 %0 Journal Article %J Physical Review A %D 2008 %T Investigating interaction-induced chaos using time-dependent density-functional theory %A Wasserman, Adam %A Maitra, Neepa T %A Heller, EricJ. %XSystems whose underlying classical dynamics are chaotic exhibit signatures of the chaos in their quantum mechanics. We investigate the possibility of using the linear response formalism of time-dependent density functional theory (TDDFT) to study the case when chaos is induced by electron-interaction alone. Nearest-neighbor level-spacing statistics are in principle exactly and directly accessible from TDDFT. We discuss how the TDDFT linear response procedure can reveal information about the mechanism of chaos induced by electron-interaction alone. A simple model of a two-electron quantum dot highlights the necessity to go beyond the adiabatic approximation in TDDFT.

%B Physical Review A %I APS %V 77 %P 042503 %G eng %U https://doi.org/10.1103/PhysRevA.77.042503 %0 Journal Article %J Physics Today %D 2008 %T Postmodern quantum mechanics %A Heller, EricJ. %A Steven Tomsovic %X

Postmodern movements are well known in the arts. After a major artistic revolution, and after the “modern” innovations have been assimilated, the threads of premodern thought are always reconsidered. Much of value may be rediscovered and put to new use. The modern context casts new light on premodern thought, which in turn shades perspectives on modernism.

Recent progress in semiclassical theory has overcome barriers posed by classical chaos and cast light on the correspondence principle. Semiclassical ideas have also become central to new experiments in atomic, molecular, microwave and mesoscopic physics.

%B Physics Today %I American Institute of Physics %V 46 %P 38–46 %G eng %U http://dx.doi.org/10.1063/1.881358 %0 Journal Article %J Nature Physics %D 2008 %T Surface physics: Electron wrangling in quantum corrals. %A Eric J. Heller %K Christopher %K CROMMIE %K Don %K EIGLER %K ELECTRIC waves %K LUTZ %K METALS – Surfaces %K Michael %K OPTICAL instruments %K Quantum Theory %X The article reports on the discovery of image electron waves' ability to propagate along a metal surface by Don Eigler, Christopher Lutz, and Michael Crommie at IBM Almaden in 1990. Eigler, Lutz, and Crommie noticed unexpectedly long and periodic undulations in the scanning tunnelling microscope (STM) conductance signal as a function of tip position. The undulations were recognized as the signature of surface confined electron waves whose de Broglie wavelength is bigger than the lattice. %B Nature Physics %V 4 %P 443 - 444 %G eng %U http://ezp-prod1.hul.harvard.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=32460505&site=ehost-live&scope=site %0 Journal Article %J Proceedings of the National Academy of Sciences of the United States of America %D 2008 %T Inflationary dynamics for matrix eigenvalue problems. %A Eric J. Heller %A Kaplan, Lev %A Pollmann, Frank %K Algorithms %K CONJUGATE gradient methods %K eigenpairs %K EIGENVALUES %K Electrons %K MATRICES %K sparse matrices %X Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the acoustic modes of a concert hall, or hundreds of other physical quantities. Often only the few eigenpairs with the lowest or highest frequency (extremal solutions) are needed. Methods that have been developed over the past 60 years to solve such problems include the Lanczos algorithm. Jacobi-Davidson techniques, and the conjugate gradient method. Here, we present a way to solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear classical mechanical system with a modified Lagrangian constraint. The constraint induces exponential inflationary growth of the desired extremal solutions. [ABSTRACT FROM AUTHOR] %B Proceedings of the National Academy of Sciences of the United States of America %V 105 %P 7631 - 7635 %G eng %U http://ezp-prod1.hul.harvard.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=32791015&site=ehost-live&scope=site %0 Journal Article %J Physical review letters %D 2007 %T Displacement echoes: classical decay and quantum freeze %A Petitjean, Cyril %A Bevilaqua, Diego V %A Heller, EricJ. %A Jacquod, Philippe %X

Motivated by neutron scattering experiments, we investigate the decay of the fidelity with which a wave packet is reconstructed by a perfect time-reversal operation performed after a phase-space displacement. In the semiclassical limit, we show that the decay rate is generically given by the Lyapunov exponent of the classical dynamics. For small displacements, we additionally show that, following a short-time Lyapunov decay, the decay freezes well above the ergodic value because of quantum effects. Our analytical results are corroborated by numerical simulations.

%B Physical review letters %I APS %V 98 %P 164101 %G eng %U https://doi.org/10.1103/PhysRevLett.98.164101 %0 Journal Article %J Nano letters %D 2007 %T Spin-orbit coupling induced interference in quantum corrals %A Walls, Jamie D %A Heller, EricJ. %XLack of inversion symmetry at a metallic surface can lead to an observable spin−orbit interaction. For certain metal surfaces, such as the Au(111) surface, the experimentally observed spin−orbit coupling results in spin rotation lengths on the order of tens of nanometers, which is the typical length scale associated with quantum corral structures formed on metal surfaces. In this work, multiple scattering theory is used to calculate the local density of states (LDOS) of quantum corral structures composed of nonmagnetic adatoms in the presence of spin−orbit coupling. Contrary to previous theoretical predictions, spin−orbit coupling induced modulations are observed in the theoretical LDOS, which should be observable using scanning tunneling microscopy.

%B Nano letters %I ACS Publications %V 7 %P 3377–3382 %G eng %U http://pubs.acs.org/doi/abs/10.1021/nl071711z %0 Journal Article %J Journal of Physics A: Mathematical & Theoretical %D 2007 %T Statistical properties of many particle eigenfunctions. %A Heller, EricJ. %A Brian R. Landry %K BESSEL functions %K CONSTRAINTS (Physics) %K CORRELATION (Statistics) %K DENSITY matrices %K EIGENFUNCTIONS %K GREEN'S functions %K STATISTICAL physics %K Stochastic Processes %K SYMMETRY (Physics) %K WAVE functions %XWavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N? ?. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries and collisions between particles can be included. [ABSTRACT FROM AUTHOR]

%B Journal of Physics A: Mathematical & Theoretical %V 40 %P 9259 - 9274 %G eng %U http://ezp-prod1.hul.harvard.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=44660056&site=ehost-live&scope=site %0 Book Section %B Extreme Events in Nature and Society %D 2006 %T Freak ocean waves and refraction of Gaussian seas %A Heller, EricJ. %XRogue or freak waves sink ships at an alarming rate — estimated at one large ship every few weeks worldwide. It is thought that vulnerable ships (light cargo ships) simply break in two when they plough into a 60 foot wave preceded by a 40 foot hole in the sea, as some sailors that have survived such experiences have called it. Wave refraction due to current eddies (which are ubiquitous in the oceans) has long been suspected to play a role in concentrating wave energy into rogue waves. Existing theories have been based on refraction of plane waves, not the stochastic Gaussian seas one finds in practice. Gaussian seas ruin the dramatic focal caustic concentration of energy, and this fact has discouraged further investigations. Although it was thought that chaos, or the extreme sensitivity to initial conditions displayed by individual ray trajectories would quickly wipe out all significant fluctuations, we show that this is incorrect, and the fluctuations are “structually stable” entities. Significant “lumps” of energy survive the averaging over wave directions and wavelengths. We furthermore demonstrate that the probability of freak waves increases dramatically in the presence of these lumps, even though most parameters, such as the significant wave height, are unchanged. We show here that a single dimensionless parameter determines the potential for freak waves; this is the “freak index” of the current eddies — a typical angular deflection in one focal distance, divided by the initial angular uncertainty of the incoming waveset. If the freak index is greater than 2 or so, truly spectacular enhancements of freak index waves can result, even though the caustics are washed out by the Gaussian nature of the impinging sea.

%B Extreme Events in Nature and Society %I Springer %P 189–210 %G eng %U http://link.springer.com/chapter/10.1007%2F3-540-28611-X_9 %0 Journal Article %J Physical Review B %D 2006 %T Multiple-scattering theory for two-dimensional electron gases in the presence of spin-orbit coupling %A Walls, Jamie D %A Huang, Jian %A Robert M. Westervelt %A Heller, EricJ. %XIn order to model the phase-coherent scattering of electrons in two-dimensional electron gases in the presence of Rashba spin-orbit coupling, a general partial-wave expansion is developed for scattering from a cylindrically symmetric potential. The theory is applied to possible electron flow imaging experiments using a moveable scanning probe microscope tip. In such experiments, it is demonstrated theoretically that the Rashba spin-orbit coupling can give rise to spin interference effects, even for unpolarized electrons at nonzero temperature and no magnetic field.

%B Physical Review B %I APS %V 73 %P 035325 %G eng %U https://doi.org/10.1103/PhysRevB.73.035325 %0 Journal Article %J Molecular Physics %D 2006 %T Quasiresonance %A Ruiz, Antonia %A Heller, EricJ. %XThe concept of quasiresonance was introduced in connection with inelastic collisions between one atom and a vibro-rotationally excited diatomic molecule. In its original form, the collisions induce *quasiresonant* transfer of energy between the internal degrees of freedom of the diatom: there is a surprisingly accurate low order rational value for the ratio of the changes in the vibrational and rotational classical actions, provided the vibrational and rotational frequencies of the diatom are approximately related by low order rational values, and the collision was longer than the rotational period of the molecule. In this paper, we show that quasiresonance can be extended to many other processes and systems, and that it may be understood in terms of the adiabatic invariance theory and the method of averaging.

Images of a single-electron quantum dot were obtained in the Coulomb blockade regime at liquid He temperatures using a cooled scanning probe microscope (SPM). The charged SPM tip shifts the lowest energy level in the dot and creates a ring in the image corresponding to a peak in the Coulomb-blockade conductance. Fits to the line shape of the ring determine the tip-induced shift of the energy of the electron state in the dot. SPM manipulation of electrons in quantum dots promises to be useful in understanding, building, and manipulating circuits for quantum information processing.

%B Nano letters %I ACS Publications %V 5 %P 223–226 %G eng %U http://pubs.acs.org/doi/abs/10.1021/nl048405v %0 Journal Article %J Journal of Physics A: Mathematical and General %D 2004 %T Consolidating boundary methods for finding the eigenstates of billiards %A Cohen, Doron %A Lepore, Natasha %A Heller, EricJ. %XThe plane-wave decomposition method, a widely used means of numerically finding eigenstates of the Helmholtz equation in billiard systems is described as a variant of the mathematically well-established boundary integral method (BIM). A new unified framework encompassing the two methods is discussed. Furthermore, a third numerical method, which we call the gauge freedom method is derived from the BIM equations. This opens the way to further improvements in eigenstate search techniques.

%B Journal of Physics A: Mathematical and General %I IOP Publishing %V 37 %P 2139 %G eng %U http://dx.doi.org/10.1088/0305-4470/37/6/013 %0 Journal Article %J Physics Today %D 2004 %T Imaging Electron Flow %A M. A. Topinka, R. M. Westervelt, %A E. J. Heller %B Physics Today %P 47 %G eng %0 Journal Article %J Physica E: Low-dimensional Systems and Nanostructures %D 2004 %T Imaging electron waves %A R. M. Westervelt %A M. A. Topinka, R. M. Westervelt, %A LeRoy, B.J %A A.C Bleszynski %A K Aidala %A Shaw, S.E.J %A E. J. Heller %A K. D. Maranowski %A A. C. Gossard %K Quantum point contact %XOne can image the coherent flow of electron waves through a quantum point contact (QPC) into a two-dimensional electron gas by using scanning probe microscopy. A negatively charged tip depletes the electron gas below, backscatters electron waves, and reduces the QPC conductance. By raster scanning the tip over the sample, an image of electron flow is obtained. Images at liquid He temperatures show the individual quantum modes of the QPC. At greater distances, the electron flow forms narrow branches caused by small-angle scattering. Interference fringes in the images demonstrate the coherence of electron flow. An electron interferometer that acts as a quantum phase shifter was constructed by adding a gate to reflect electron waves back to the QPC, producing a V-shaped path for interfering electron waves with the apex at the QPC. When the length of one leg of the V is altered by changing the reflector gate voltage, the fringes at the other end of the V, under the tip, shift by the same distance. The interferometer is sensitive to transit time differences as small as ∼0.1 ps between the two electron paths. These observations are in good agreement with theoretical simulations of electron flow.

%B Physica E: Low-dimensional Systems and Nanostructures %V 24 %P 63 - 69 %G eng %U http://www.sciencedirect.com/science/article/pii/S1386947704001869 %R 10.1016/j.physe.2004.04.025 %0 Journal Article %J Reviews of Modern Physics %D 2003 %T Colloquium: Theory of quantum corrals and quantum mirages %A Fiete, Gregory A %A Heller, EricJ. %XQuantum corrals are two-dimensional structures built atom by atom on an atomically clean metallic surface using a scanning tunneling microscope (STM). These two-dimensional structures “corral” electrons in the surface states of noble metals, leading to standing-wave patterns in the electron density inside the quantum corral. The authors review the physics of quantum corrals and relate the signal of the STM to the scattering properties of substrate electrons from atomic impurities supported on the surface. The theory includes the effects of incoherent surface-state electron scattering at the impurities and quantitively describes nearly all of the current STM data on quantum corrals, including the recent quantum mirage experiments with Kondo effect. The physics underlying the recent mirage experiments is discussed, as are some of the outstanding questions regarding the Kondo effect from impurities in nanoscale structures on metallic surfaces. The authors also summarize recent work on variations of “quantum” corrals: Optical corrals and acoustical corrals.

%B Reviews of Modern Physics %I APS %V 75 %P 933 %G eng %U https://doi.org/10.1103/RevModPhys.75.933 %0 Journal Article %J Physical Review E %D 2003 %T Comment on “Ehrenfest times for classically chaotic systems” %A Steven Tomsovic %A Heller, EricJ. %XIn a recent Rapid Communication [P. G. Silvestrov and C. W. J. Beenakker, Phys. Rev. E **65**, 035208(R) (2002)], the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time

We show that for a general system of *N* *s*-wave point scatterers, there are always *N* eigenmodes. These eigenmodes or eigenchannels play the same role as spherical harmonics for a spherically symmetric target—they give a phase shift only. In other words, the *T* matrix of the system is of rank *N*, and the eigenmodes are eigenvectors corresponding to nonzero eigenvalues of the *T* matrix. The eigenmode expansion approach can give insight to the total scattering cross section; the position, width, and superradiant or subradiant nature of resonance peaks; the unsymmetric Fano line shape of sharp proximity resonance peaks based on the high-energy tail of a broadband; and other properties. Off-resonant eigenmodes for identical proximate scatterers are approximately angular-momentum eigenstates.

We present a numerically feasible semiclassical (SC) method to evaluate quantum fidelity decay (Loschmidt echo) in a classically chaotic system. It was thought that such evaluation would be intractable, but instead we show that a uniform SC expression not only is tractable but it also gives remarkably accurate numerical results for the standard map in both the Fermi-golden-rule and Lyapunov regimes. Because it allows Monte Carlo evaluation, the uniform expression is accurate at times when there are *a posteriori* comparison with numerical results. We explain in more detail the extended validity of the classical perturbation approximation and show that within this approximation, the so-called “diagonal approximation” is automatic and does not require ensemble averaging.

A general semiclassical approach to quantum systems with system-bath interactions is developed. We study system decoherence in detail using a coherent-state semiclassical wave-packet method which avoids singularity issues arising in the usual Green’s function approach. We discuss the general conditions under which it is approximately correct to discuss quantum decoherence in terms of a “dephasing” picture and we derive semiclassical expressions for the phase and phase distribution. Remarkably, an effective system wavefunction emerges whose norm measures the decoherence and is equivalent to a density-matrix formulation.

%B Physical Review A %I APS %V 68 %P 022112 %G eng %U https://doi.org/10.1103/PhysRevA.68.022112 %0 Journal Article %J The Journal of chemical physics %D 2003 %T Similarity transformed semiclassical dynamics %A Van Voorhis, Troy %A Heller, EricJ. %X

%B The Journal of chemical physics %I AIP Publishing %V 119 %P 12153–12162 %G eng %U http://dx.doi.org/10.1063/1.1626621 %0 Journal Article %J Physical Review E %D 2003 %T Uniform semiclassical wave function for coherent two-dimensional electron flow %A Van{\'ıček, Jiř{\'ı %A Heller, EricJ. %X

We find a uniform semiclassical (SC) wave function describing coherent branched flow through a two-dimensional electron gas (2DEG), a phenomenon recently discovered by direct imaging of the current using scanned probed microscopy [M.A. Topinka, B.J. LeRoy, S.E.J. Shaw, E.J. Heller, R.M. Westervelt, K.D. Maranowski, and A.C. Gossard, Science

Recent experimental work in the Westervelt laboratory at Harvard has succeeded in directly imaging electron flow in two degree of freedom electron gasses formed in semiconductor microstructures. Here, we give a brief account of the unexpected high resolution of the resulting images, the surprising branching of the flow which was observed, and the survival of quantum fringing beyond where it was thought to have been obliterated by thermal effects.

%B International Journal of Modern Physics B %I World Scientific %V 17 %P 3977-3987 %G eng %U http://dx.doi.org/10.1142/S0217979203021964 %0 Journal Article %J Physical Review A %D 2002 %T Nearly real trajectories in complex semiclassical dynamics %A Van Voorhis, Troy %A Heller, EricJ. %X

We introduce a very general approximation to the quantum propagator that is based on the assumption that the most important contributions to the complex semiclassical propagator evolve from real classical trajectories that *almost* satisfy the desired boundary conditions. Our results for two systems — the autocorrelation function for the quartic anharmonic oscillator and the photodissociation spectrum of

This paper presents a perturbative model for the vibrational predissociation dynamics of inert gas hydrogen halide (RgHX) complexes. The predissociation is modeled as a Fermi Golden Rule (FGR) process from a bound state residing on a two-dimensional potential energy surface (PES) obtained by averaging over the HX vibrational state of interest to a series of one-dimensional exit channels obtained by averaging over the HX rovibrational state of interest. This model is applied to ArHF, for which a high-quality ab initio interaction potential is available. In particular, we focus on the *v* → *v* − 1 transition for the bound states (1000), (2000), (2110), (3000), and (3110). We confirm the experimental observation that the product HF tends to come off at the highest accessible *j* state, which is *j* = 13 for this system. This results from a strong angular anisotropy in the ArHF interaction potential that couples low-*j* and high-*j* HF states. The basic mechanism for this high-*j* preference is determined to be the suppression of the low-*j* exit channels arising from highly oscillatory low-*j* outgoing wave functions. We also observed that the tails of the bound-state wave functions, in the inner wall region of the interaction potential, gave the main contribution to the predissociation rate, indicating that the vibrational predissociation process is due to tunneling and is therefore a purely quantum effect. The calculations also confirm the strong *v*-dependence of the predissociation rates, as well as the stabilization of the complex that occurs when energy is placed into the HF bending mode. For the (1000) and (2110) states, we obtain rates well below 1600 s^{-1}, which is consistent with the observation by Miller that the vibrational predissociation rates are too slow to be measured. The (2000) state does give a measurable rate, with a computed decay into the *j* = 13 exit channel of 14 000 s^{-1}. The (3000) state gives a corresponding rate of 200 000 s^{-1}, in good agreement with the overall dissociation rate of 250 000 s^{-1}. The (3110) rate is slower, with a value of 12 000 s^{-1}. Though this rate seems somewhat small, given that the lifetime of the (3110) state was measured to only be twice as long as that of the (3000) state, this rate and all our rates are within an order of magnitude of the measured rates.

We provide a general and nonperturbative theoretical basis for quantum reflection of an ultracold atom incident on a cold or warm surface. Sticking is identified with the formation of a long-lived resonance, from which it emerges that the physical reason for not sticking is that the many internal degrees of freedom of the target serve to decohere the incident one body wave function, thereby upsetting the delicate interference process necessary to form a resonance. We then explore the transition to the post-threshold behavior, when sticking prevails at higher incident energies. Studying the WKB wave functions of the atom provides a quick understanding of our results even in the ultracold regime where WKB is not applicable. Explicit examples are examined in detail and we predict the temperatures required to reach the various regimes.

%B Physical Review B %I APS %V 64 %P 085418 %G eng %U https://doi.org/10.1103/PhysRevB.64.085418 %0 Journal Article %J Physical Review E %D 2001 %T Parametric evolution for a deformed cavity %A Cohen, Doron %A Barnett, Alex %A Heller, EricJ. %XWe consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x controls a deformation of the boundary. The quantum eigenstates of the system are \|n(x)>. We describe how the parametric kernel P(n\|m)=\|<n(x)\|m(x_{0})>\|^{2}, also known as the local density of states, evolves as a function of δx=x-x_{0}. We illuminate the nonunitary nature of this parametric evolution, the emergence of nonperturbative features, the final nonuniversal saturation, and the limitations of random-wave considerations. The parametric evolution is demonstrated numerically for two distinct representative deformation processes.

Quantum tunnelling breaks the rules of classical physics — and leads to ghost-like transfer of matter through barriers. Demonstrations of a new type of quantum tunnelling have the ghosts taking new liberties.

%B Nature %I Nature Publishing Group %V 412 %P 33–34 %G eng %U http://dx.doi.org/10.1038/35083672 %0 Journal Article %J Journal of Physics A: Mathematical and General %D 2001 %T Rate of energy absorption for a driven chaotic cavity %A Barnett, Alex %A Cohen, Doron %A Heller, EricJ. %XWe consider the response of a chaotic cavity in *d* dimensions to periodic driving. We are motivated by older studies of one-body dissipation in nuclei, and also by anticipated mesoscopic applications. For calculating the rate of energy absorption due to time-dependent deformation of the confining potential, we introduce an improved version of the wall formula. Our formulation takes into account that a special class of deformations causes no heating in the zero-frequency limit. We also derive a mesoscopic version of the Drude formula, and explain that it can be regarded as a special example of our calculations. Specifically we consider a quantum dot driven by an electro-motive force which is induced by a time-dependent homogeneous magnetic field.

We present a semiclassical technique that relies on replacing complicated classical manifold structure with simpler manifolds, which are then evaluated by the usual semiclassical rules. Under circumstances where the original manifold structure gives poor or useless results semiclassically the replacement manifolds can yield remarkable accuracy. We give several working examples to illustrate the theory presented here.

%B Physical Review E %I APS %V 64 %P 026215 %G eng %U https://dx.doi.org/10.1103/PhysRevE.64.026215 %0 Journal Article %J Physical review letters %D 2001 %T Scattering theory of Kondo mirages and observation of single Kondo atom phase shift %A Fiete, Gregory A %A Hersch, Jesse S %A Heller, EricJ. %A Manoharan, HC %A Lutz, CP %A Eigler, DM %XWe explain the origin of the Kondo mirage seen in recent quantum corral Scanning Tunneling Microscope (STM) experiments with a scattering theory of electrons on the surfaces of metals. Our theory combined with experimental data provides the first direct observation of a single Kondo atom phase shift. The Kondo mirage observed at the empty focus of an elliptical quantum corral is shown to arise from multiple electron bounces off the corral wall adatoms in a manner analagous to the formation of a real image in optics. We demonstrate our theory with direct quantitive comparision to experimental data. *This research was supported by the National Science Foundation under Grant No. CHE9610501 and by ITAMP.

%B Physical review letters %I APS %V 86 %P 2392 %G eng %0 Journal Article %J The Journal of Physical Chemistry A %D 2001 %T Semiclassical Quantization Using Invariant Tori: A Gradient-Descent Approach %A Tannenbaum, Emmanuel %A Heller, EricJ. %XThis paper presents a PDE-based, gradient-descent approach (GDA) to the EBK quantization of nearly separable Hamiltonians in the quasi-integrable regime. The method does this by finding an optimal semiclassical basis of invariant tori which minimizes the angular dependence of the Hamiltonian. This representation of the Hamiltonian is termed an intrinsic resonance representation (IRR), and it gives the smallest possible basis obtainable from classical mechanics. Because our method is PDE-based, we believe it to be significantly faster than previous IRR algorithms, making it possible to EBK quantize systems of higher degrees of freedom than previously studied. In this paper we demonstrate our method by reproducing results from a two-degree-of-freedom system used to demonstrate the previous Carioli, Heller, and Moller (CHM) implementation of the IRR approach. We then go on to show that our method can be applied to higher dimensional Hamiltonians than previously studied by using it to EBK quantize a four- and a six-degree-of-freedom system.

%B The Journal of Physical Chemistry A %I ACS Publications %V 105 %P 2803–2813 %G eng %U http://pubs.acs.org/doi/abs/10.1021/jp004371d?journalCode=jpcafh %0 Journal Article %J Nature %D 2001 %T Coherent branched flow in a two-dimensional electron gas. %A Topinka, M.A. %A LeRoy, B.J %A Westervelt, R. M. %A Shaw, S.E.J %A Fleischmann, R. %A E. J. Heller %A Maranowski, K.D. %A Gossard, A.C. %K Electron Transport %K Nanostructures %K Nanotechnology %K Physics %X Presents a study to observe electron flow through a narrow constriction in a semiconductor nanostructure. Methods; Results; Conclusion that a branching of current flux is due to focusing of the electron paths by ripples in the background potential. %B Nature %V 410 %P 183 %G eng %U http://ezp-prod1.hul.harvard.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=4185454&site=ehost-live&scope=site %0 Journal Article %J Science %D 2000 %T Imaging Coherent Electron Flow from a Quantum Point Contact %A M. A. Topinka, R. M. Westervelt, %A B. J. LeRoy %A S. E. J. Shaw %A E. J. Heller %A R. M. Westervelt %A K. D. Maranowski %A A. C. Gossard %XScanning a charged tip above the two-dimensional electron gas inside a gallium arsenide/aluminum gallium arsenide nanostructure allows the coherent electron flow from the lowest quantized modes of a quantum point contact at liquid helium temperatures to be imaged. As the width of the quantum point contact is increased, its electrical conductance increases in quantized steps of 2*e* ^{2}/*h*, where *e* is the electron charge and *h* is Planck's constant. The angular dependence of the electron flow on each step agrees with theory, and fringes separated by half the electron wavelength are observed. Placing the tip so that it interrupts the flow from particular modes of the quantum point contact causes a reduction in the conductance of those particular conduction channels below 2*e* ^{2}/*h* without affecting other channels.

We consider chaotic billiards in

We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where x is a constant parameter. Specifically, we discuss a gas particle inside a cavity, where x controls a deformation of the boundary or the position of a "piston." The quantum eigenstates of the system are |n(x)>. We describe how the parametric kernel P(nmid R:m) = |<n(x)mid R:m(x(0))>|(2) evolves as a function of deltax = x-x(0). We explore both the perturbative and the nonperturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as random waves and random-matrix-theory considerations.

%B Physical review letters %I APS %V 84 %P 2841 %G eng %U https://dx.doi.org/10.1103/PhysRevLett.84.2841 %0 Journal Article %J The Physics and Chemistry of Wave Packets; Yeazel, J.; Uzer, T. eds, Wiley: New York %D 1999 %T Semiclassical wave packets %A Eric J. Heller %B The Physics and Chemistry of Wave Packets; Yeazel, J.; Uzer, T. eds, Wiley: New York %G eng %0 Journal Article %J Physical review letters %D 1999 %T Quasiresonant energy transfer in ultracold atom-diatom collisions %A Forrey, Robert C %A Balakrishnan, N %A Dalgarno, A %A Haggerty, Michael R %A Heller, EricJ. %XQuantum and classical quasiresonant vibration-rotation energy transfer is investigated for ultracold

A time domain approach employing the semiclassical approximation to the quantum mechanical propagator, as applied to Gaussian wavepackets, is used to study the barrier penetration problem. We have observed that qualitative agreement with the exact quantum calculations for the correlation function and the transmission probability can be achieved by considering only classically allowed trajectories. The results slowly tend to the classical step function at the barrier top as a function of the wavepacket center parameter, however. We suggest that a full semiclassical calculation, including nonclassical trajectories with complex energies, would improve the results.

%B Chemical physics letters %I Elsevier %V 241 %P 45–50 %G eng %U http://dx.doi.org/10.1016/0009-2614(95)00605-4 %0 Journal Article %J Physical Review A %D 1994 %T Ionization of hydrogen by positron impact near the fragmentation threshold %A Rost, Jan M %A Heller, EricJ. %XThe semiclassical approximation of Feynman’s path integral is used to calculate the *S* matrix for the positron-impact ionization of hydrogen. The formulation provides a full scattering amplitude, and more importantly does not require knowledge of the asymptotic three-body Coulomb state in the continuum. In the limit of vanishing excess energy, the results confirm Wannier’s classical model for fragmentation [Phys. Rev. *90*, 817 (1953)]. The experimentally observable ratio of fragmentation versus total ionization (including positronium formation) is predicted.

The semiclassical techniques developed in the previous paper are applied to the understanding of the hierarchical structure underlying the spectra. This organization, as analyzed by Davis with statistical models, is revealed by continuously changing the energy resolution of the spectra and noting the branching pattern of the peaks. We argue that the greater part of this hierarchical organization can be understood with classical events in the time domain.

%B The Journal of chemical physics %I AIP Publishing %V 101 %P 8016–8027 %G eng %U http://dx.doi.org/10.1063/1.468228 %0 Journal Article %J The Journal of chemical physics %D 1994 %T Semiclassical calculation and analysis of dynamical systems with mixed phase space %A Sepúlveda, Miguel Angel %A Heller, EricJ. %XA semiclassical method for the propagation of arbitrary wave packets in a multidimensional Hamiltonian is presented. The method is shown to be valid for treating Hamiltonian systems whose classical phase space is a combination of chaotic and quasiperiodic motion (mixed dynamics). The propagation can be carried out long enough for the nonlinearities of the system to be important. The nonlinear dynamics is reflected in spectra and correlation functions. We suggest this new semiclassical method can be a tool for analyzing the nonlinear aspects of the vibrational spectra.

%B The Journal of chemical physics %I AIP Publishing %V 101 %P 8004–8015 %G eng %U http://dx.doi.org/10.1063/1.468227 %0 Journal Article %J Physical Review E %D 1993 %T Long-time semiclassical dynamics of chaos: The stadium billiard %A Steven Tomsovic %A Heller, EricJ. %XIn a recent Letter [Phys. Rev. Lett. *67*, 664 (1991)] we found semiclassical propagation to be remarkably accurate in the chaotic stadium billiard long after classical fine structure had developed on a scale much smaller than ħ. We give a complete account of that work and derive an approximate time scale for the validity of the semiclassical approximation as a function of ħ.

It has been a long-standing problem to understand the eigenfunctions of a system whose classical analog is strongly chaotic. We show that in some cases the eigenfunctions can be constructed by purely semiclassical calculations.

%B Physical review letters %I APS %V 70 %P 1405 %G eng %U http://dx.doi.org/10.1103/PhysRevLett.70.1405 %0 Journal Article %J Journal of statistical physics %D 1992 %T Accuracy of semiclassical dynamics in the presence of chaos %A Patrick W. O'Connor %A Steven Tomsovic %A Heller, EricJ. %XWe review some of the issues facing semiclassical methods in classically chaotic systems, then demonstrate the long-time accuracy of semiclassical propagation of a nonstationary wave packet using the quantum baker's map of Balazs and Voros. We show why some of the standard arguments against the efficacy of semiclassical dynamics for long-time chaotic motion are incorrect.

%B Journal of statistical physics %I Springer %V 68 %P 131–152 %G eng %U http://link.springer.com/article/10.1007/BF01048839 %0 Journal Article %J Physica D: Nonlinear Phenomena %D 1992 %T Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier %A Patrick W. O'Connor %A Steven Tomsovic %A Heller, EricJ. %XWe investigate the behavior of the quantum baker's transformation, a system whose classical analogue is completely chaotic, for time scales where the classical mechanics generates phase space structures on a scale smaller than Planck's constant (i.e., past the log time *t*^{∗} ≈ ln ħ^{-1}). Surprisingly, we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime.

A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time‐dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.

%B The Journal of chemical physics %I AIP Publishing %V 94 %P 2723–2729 %G eng %U http://dx.doi.org/10.1063/1.459848 %0 Journal Article %J Physical review letters %D 1991 %T Semiclassical dynamics of chaotic motion: unexpected long-time accuracy %A Steven Tomsovic %A Heller, EricJ. %XChaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green’s function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell (*h*). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.

A new semiclassical propagator based on a local expansion of the potential up the wavepacket is proposed. Formulas for the propagator are derived and the implementation using grid and fast Fourier transform (FFT) methods is discussed. The semiclassical propagator can be improved up to the exact quantum mechanical limit by including anharmonic corrections using a split operator approach. Preliminary applications to the CH_{3}I photodissociation problem show the applicability and accuracy of the proposed method.

The effect of Born–Oppenheimer potential energy surface crossings on energy transfer in polyatomic molecules is investigated, classically and quantum mechanically. The hopping from one energy surface to another is enough to cause classical chaos, and strong mixing of the levels quantum mechanically. The *r* *a* *t* *e* of classical mixing determines the *e* *x* *t* *e* *n* *t* of quantum mixing, even though classical mixing is complete at long times.

We recently published a new method for the calculation of the time evolution of a wave function. We used an accurate approximate method to calculate the time propagator for a finite time Δ*t*. Numerical calculations showed that this scheme works quite accurately, but that it is not more efficient than conventional methods. In this paper we propose to use a very fast and simple, but less accurate semiclassical method for the calculation of the time propagator. The approximation consists in the replacement of the Hamiltonian by a quadratic approximation around the center of the evolving wave packet called thawed Gaussian dynamics. We show by numerical examples in one and two dimensions that, despite this crude approximation, we achieve nearly the same accuracy as in the foregoing paper, but with an efficiency that is typically more than an order of magnitude better. We further show that the method is able to describe tunneling and long time dynamics (e.g., 1000 vibrational periods).

We present an approach to quantum dynamics, based entirely on Cartesian coordinates, which covers vibrational as well as rotational motion. The initial state is represented in terms of multidimensional Gaussian wave packets. Rotational adaptation to angular momentum eigenstates is done by using angular momentum projection operators. This gives an initial state represented as a weighted superposition of Gaussians with different average orientation in space. It is shown that the subsequent dynamics can be determined from the dynamics of Gaussians corresponding to just *o* *n* *e* of these orientations. An application to the 3*D*photodissociationdynamics of ICN is presented. All six degrees of freedom which describe the internal motion of the triatomic are included, the only approximation introduced in the present calculation being the thawed Gaussian wave packet approximation for the dynamics. The total absorptionspectrum out of vibrational–rotational eigenstates of ICN as well as fully resolved final product distributions are calculated.

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time‐dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.

%B J. Chem. Phys. %V 89 %P 2003 %G eng %U http://dx.doi.org/10.1063/1.455714 %R 10.1063/1.455714 %0 Journal Article %J J. Chem. Phys. %D 1988 %T Ring torsional dynamics and spectroscopy of benzophenone: A new twist %A John H. Frederick %A Heller, EricJ. %A Judy L. Ozment %A David W. Pratt %XThe low energy portion of the high resolution *S* _{1}←*S* _{0}fluorescence excitation spectrum of benzophenone recently reported by Holtzclaw and Pratt [J. Chem. Phys. **8** **4**, 4713 (1986)] is modeled here using a simple two‐degree‐of‐freedom vibrational Hamiltonian. The Hamiltonian features a 1:1 nonlinear resonance between the two low frequency ring torsional modes of the molecule in its *S* _{1} state. Line positions and intensities of the two major spectral progressions are well reproduced using parameters similar to those derived from earlier matrix diagonalizations. The comparison of the theory and experiment results in a determination of the displacement of the *S* _{1}surface relative to the ground electronic state along the symmetric torsional coordinate and permits a calculation of the excitation spectra of various isotopically substituted molecules not yet measured in the laboratory. A clear picture of the relationship between the dynamics on the *S* _{1}surface and the spectroscopy of benzophenone is revealed by comparing a time domain analysis of the experimental data with wave packet dynamics on the model *S* _{1}surface. This comparison provides new insight into energy flow in the isolated molecule and permits a qualitative simulation of the effects of collisional quenching on the fluorescencespectrum. We also discuss, using a classical trajectory analysis, the resonance dynamics of the torsional modes and note the existence of heretofore undetected local modes in the high resolution spectrum.

We study the quantum mechanics of a Hamiltonian system which is classically chaotic: the stadium billiard. We have examined many of the eigenstates of the stadium, up to about the 10,000th. Complex periodic orbits play an active role in shaping the eigenstates.

%B AIP Conference Proceedings %C Crystal City, VA, USA %V 162 %P 43–59 %G eng %U http://dx.doi.org/10.1063/1.36892 %R 10.1063/1.36892 %0 Journal Article %J J. Chem. Phys. %D 1987 %T Multidimensional quantum eigenstates from the semiclassical dynamical basis set %A John H. Frederick %A Heller, EricJ. %X

A new method for obtaining molecular vibrational eigenstates using an efficient basis set made up of semiclassical eigenstates is presented. Basis functions are constructed from a ‘‘primitive’’ basis of Gaussian wave packets distributed uniformly on the phase space manifold defined by a single quasiperiodic classical trajectory (an invariant *N*‐torus). A uniform distribution is constructed by mapping a grid of points in the Hamilton–Jacobi angle variables, which parametrize the surface of the *N*‐torus, onto phase space by means of a careful Fourier analysis of the classical dynamics. These primitive Gaussians are contracted to form the semiclassical eigenstates via Fourier transform in a manner similar to that introduced by De Leon and Heller [J. Chem. Phys. **8** **1**, 5957 (1984)]. Since the semiclassical eigenstates represent an extremely good approximation of the quantum eigenstates, small matrix diagonalizations are sufficient to obtain eigenvalues ‘‘converged’’ to 4–5 significant figures. Such small diagonalizations need not include the ground vibrational state and thus can be used to find accurate eigenstates in select regions of the eigenvalue spectrum. Results for several multidimensional model Hamiltonians are presented.

%B J. Chem. Phys. %V 87 %P 6592 %G eng %U http://dx.doi.org/10.1063/1.453444 %R 10.1063/1.453444 %0 Journal Article %J J. Chem. Phys. %D 1987 %T Semiclassical wave packet treatment of the rigid asymmetric rotor %A Daniel Huber %A Heller, EricJ. %A William G. Harter %X

We discuss a semiclassical treatment of the rigid asymmetric rotor that delivers eigenenergies as well as eigenstates. We give possibilities to improve the semiclassical wave functions to any accuracy required. The method is devised so that inclusion of vibrations is possible. As no information about energetically lower states is included in the procedure, the calculation of highly excited states is easier than with conventional quantum methods. Calculation of quantum splitting from semiclassical eigenstates is treated. We give numerical examples for every procedure developed, so that the performance of the theory can be judged.

%B J. Chem. Phys. %V 87 %P 1116 %G eng %U http://dx.doi.org/10.1063/1.453344 %R 10.1063/1.453344 %0 Journal Article %J J. Chem. Phys. %D 1986 %T Classical and semiclassical aspects of dissipative molecular processes: Radiative and nonradiative vibrational relaxation in polyatomic molecules %A Robert P. Parson %A Heller, EricJ. %XThe time dependence of spontaneous emission from isolated, highly vibrationally excited molecules is studied from the point of view of the classical and semiclassical mechanics of dissipatively perturbed Hamiltonian systems. A correlation function expression for the time‐dependent emission spectrum serves as a starting point for quasiclassical approximations. These in turn lead to an investigation of the classical dynamics of model molecular systems in which small, energy‐nonconserving terms have been added to Hamilton’s equations. Numerical calculations show rich dynamical behavior which can be qualitatively understood in terms of the resonance structure of the unperturbed system. For example, trajectories tend to be captured by zones of nonlinear resonance. This ‘‘mode locking’’ produces a characteristic cleanup of an emission spectrum that otherwise is rather congested at these energies. The close connection between spectra and dynamics suggests that the classical dynamics of dissipatively perturbed systems may provide a useful language for interpreting radiative and condensed‐phase vibrational relaxation, and possibly some types of intramolecular relaxation phenomena as well.

%B J. Chem. Phys. %V 85 %P 2581 %G eng %U http://dx.doi.org/10.1063/1.451065 %R 10.1063/1.451065 %0 Book Section %B Quantum Chaos and Statistical Nuclear Physics: Proceedings of the 2nd International Conference on Quantum Chaos and the 4th International Colloquium on Statistical Nuclear Physics, Held at Cuernavaca, México, January 6–10, 1986 %D 1986 %T Qualitative properties of eigenfunctions of classically chaotic Hamiltonian systems %A Heller, EricJ. %XWe begin by discussing the properties expected of eigenfunctions of a classically chaotic Hamiltonian system, using simple Correspondence Principle arguments. The properties involve nodal surfaces, coordinate and momentum space amplitude distribution, and phase space distribution. The eigenfunctions of the stadium billiard are examined, and it is found that the periodic orbits of shortest periods and smallest stability parameter profoundly affect the eigenfunctions: “scars” of higher wavefunction density surround the special periodic orbits. Finally a theory is presented for the scars, showing that they must exist, and relating them directly to the special periodic orbits. These same periodic orbits cause level density fluctuations.

%B Quantum Chaos and Statistical Nuclear Physics: Proceedings of the 2nd International Conference on Quantum Chaos and the 4th International Colloquium on Statistical Nuclear Physics, Held at Cuernavaca, México, January 6–10, 1986 %I Springer Berlin Heidelberg %P 162–181 %G eng %U http://link.springer.com/chapter/10.1007%2F3-540-17171-1_15 %R 10.1007/3-540-17171-1_15 %0 Book Section %B Dynamics of Wave Packets in Molecular and Nuclear Physics: Proceedings of the International Meeting Held in Priorij Corsendonck, Belgium July 2–4, 1985 %D 1986 %T Recent Progress in Semiclassical Wavepacket Methods for Chemical Physics Applications %A E. J. Heller %A J. R. Reimers %B Dynamics of Wave Packets in Molecular and Nuclear Physics: Proceedings of the International Meeting Held in Priorij Corsendonck, Belgium July 2–4, 1985 %I Springer-Verlag Berlin Heidelberg %P 79-81 %G eng %U http://link.springer.com/book/10.1007/3-540-16772-2 %R 10.1007/3-540-16772-2_10 %0 Journal Article %J J. Chem. Phys. %D 1986 %T Semiclassical mechanics of dissipative molecular processes: The correspondence principle and population relaxation %A Robert P. Parson %A Heller, EricJ. %XThe relaxation of a primary system coupled weakly to a bath of environmental modes is examined from the standpoint of recent developments in the semiclassical theory of molecular bound states. Emphasis is placed upon highly excited, strongly nonlinear (but quasiperiodic) primary systems and zero temperature baths. The starting point for the analysis is a master equation for the populations of the eigenstates of the primary system. The correspondence principle provides semiclassical approximations to the transition rates, allowing quantum state populations to be calculated from classical trajectories. A second semiclassical approximation leads to an equation of motion for a probability density in the classical action variables. As *h*→0, this density agrees with the density generated by running an ensemble of damped classical trajectories and averaging out the angle variables; retention of terms of order *h* provides smoothed quantum corrections. Numerical examples of both semiclassical approximations are presented.

The combination of *a* *b* *i* *n* *i* *t* *i* *o* calculation of electronic wave functions with a wave packet calculation of the nuclear motion is used, within the Born–Oppenheimer approximation to compute the vibrational and electronic absorption of a polyatomic molecule. A particular virtue of this approach is that high as well as low temperature spectra are both calculable. This method is applied to C_{2}H, for which the complete active space self‐consistent field (CASSCF) method is used to determine full Born–Oppenheimer potential surfaces. Using the assumption that the *A*(^{2}Π) ← *X*(^{2}Σ^{+}) absorption can be written as the sum of the *A*(^{2} *A*’) ← *X* and *A*(^{2} *A*‘) ← *X*absorptions, the spectra are determined to 60 cm^{−} ^{1} resolution at a temperature of 3000 K. As a result of the large thermal bending amplitude at 3000 K, the calculated spectra are broad and have little resolved structure. Two bands are resolvable, one is due to the *A*(^{2} *A*‘) ← *X*absorption and is centered at 5500 cm^{−} ^{1}, while the other is due to *A*(^{2} *A*’) ← *X*absorption and is centered at 9500 cm^{−} ^{1}. The dramatic blue shift of the *A*(^{2} *A*’) ← *X* band results from the combination of the large *X* state thermal bending amplitude and high bending frequency of the *A*(^{2} *A*’) state. We also determine the *X* state pure vibrational absorptionspectrum and show it to be of much lower intensity than the pure electronic spectrum.

Exact eigenfunctions for a two‐dimensional rigid rotor are obtained using Gaussian wave packet dynamics. The wave functions are obtained by propagating, without approximation, an infinite set of Gaussian wave packets that collectively have the correct periodicity, being coherent states appropriate to this rotational problem. This result leads to a numerical method for the semiclassical calculation of rovibrational, molecular eignestates. Also, a simple, almost classical, approximation to full wave packet dynamics is shown to give exact results: this leads to an *a* *p* *o* *s* *t* *e* *r* *i* *o* *r* *i* justification of the De Leon–Heller spectral quantization method.

The exact thermal rotational spectrum of a two‐dimensional rigid rotor is obtained using Gaussian wave packet dynamics. The spectrum is obtained by propagating, without approximation, infinite sets of Gaussian wave packets. These sets are constructed so that collectively they have the correct periodicity, and indeed, are coherent states appropriate to this problem. Also, simple, almost classical, approximations to full wave packet dynamics are shown to give results which are either exact or very nearly exact. Advantages of the use of Gaussian wave packet dynamics over conventional linear response theory are discussed.

%B J. Chem. Phys. %V 83 %G eng %U http://dx.doi.org/10.1063/1.449515 %N 516 %R 10.1063/1.449515 %0 Book Section %B Chaotic Behavior in Quantum Systems %D 1985 %T Quantum Ergodicity and Intensity Fluctuations %A E. J. Heller %A R. L. Sundberg %X

Flow in phase space is intimately related to the concept of ergodicity. An ergodic system will sample all regions of phase space democratically in the time average, subject only to known a priori constants of the motion. For convenience, we shall refer to the accessible phase space as the “energy shell.” This shell may have some thickness if a distribution of energies are initially populated An energy envelope is defined by the initial distribution of energy and the envelope plays a role both in classical and quantum phase Space flow, as discussed before and as will become evident in the theory and examples to follow.

%B Chaotic Behavior in Quantum Systems %I Springer US %C Boston, MA %P 255-292 %G eng %U http://dx.doi.org/10.1007/978-1-4613-2443-0_18 %0 Journal Article %J Journal of Chemical Physics %D 1985 %T The exact thermal rotational spectrum of a two-dimensional rigid rotor obtained using Gaussian wave packet dynamics. %A Reimers, Jeffrey R. %A Eric J. Heller %K Quantum Theory %K WAVE packets %X The exact thermal rotational spectrum of a two-dimensional rigid rotor is obtained using Gaussian wave packet dynamics. The spectrum is obtained by propagating, without approximation, infinite sets of Gaussian wave packets. These sets are constructed so that collectively they have the correct periodicity, and indeed, are coherent states appropriate to this problem. Also, simple, almost classical, approximations to full wave packet dynamics are shown to give results which are either exact or very nearly exact. Advantages of the use of Gaussian wave packet dynamics over conventional linear response theory are discussed. [ABSTRACT FROM AUTHOR] %B Journal of Chemical Physics %V 83 %P 516 %G eng %U http://ezp-prod1.hul.harvard.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=7633254&site=ehost-live&scope=site %0 Journal Article %J J. Chem. Phys. %D 1984 %T Comparisons of classical and quantum dynamics for initially localized states %A Michael J. Davis %A Heller, EricJ. %XWe compare the dynamics of quantum wave packets with the dynamics of classical trajectory ensembles. The wave packets are Gaussian with expectation values of position and momenta which centers them in phase space. The classical trajectory ensembles are generated directly from the quantum wave packets via the Wigner transform. Quantum and classical dynamics are then compared using several quantum measures and the analogous classical ones derived from the Wigner equivalent formalism. Comparisons are made for several model potentials and it is found that there is generally excellent classical–quantum correspondence except for certain specific cases of tunneling and interference. In general, this correspondence is also very good in regions of phase space where there is classical chaos.

%B J. Chem. Phys. %V 80 %P 5036 %G eng %U http://dx.doi.org/10.1063/1.446571 %R 10.1063/1.446571 %0 Journal Article %J J. Chem. Phys. %D 1984 %T Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems %A Charles W. Eaker %A George C. Schatz %A N. De Leon %A E. J. Heller %XTwo methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie–Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon–Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates *Q* _{1}±*i* *Q* _{2}. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three‐quarters of the dissociation energy).

We investigate the consequences of quasiperiodic and ergodic classical dynamics on predissociation rates from bound vibrational states into a continuum. We find a strong correlation between the quasiperiodic/ergodic motion and rate constants for certain locations of the predissociationsurface. Other locations for the predissociative surface are insensitive to the difference between classical motion which is confined to small regions of phase space and motion which samples a large portion of its available phase space.

%B J. Chem. Phys. %V 80 %P 3680 %G eng %U http://dx.doi.org/10.1063/1.447190 %R 10.1063/1.447190 %0 Journal Article %J J. Chem. Phys. %D 1984 %T Quantum manifestations of classical resonance zones %A N. De Leon %A M. J. Davis %A E. J. Heller %XWe examine the concept of nodal breakup of wave functions as a criterion for quantum mechanical ergodicity. We find that complex nodal structure of wave functions is *n* *o* *t* sufficient to determine quantum mechanical ergodicity. The influence of classical resonances [which manifest themselves as classical resonance zones (CRZ)] may also be responsible for the seeming complexity of nodal structure. We quantify this by reexamining one of the two systems studied by Stratt, Handy, and Miller [J. Chem. Phys. **7** **1**, 3311 (1974)] from both a quantum mechanical and classical point of view. We conclude that quasiperiodic classical motion can account for highly distorted quantum eigenstates. One should always keep this in mind when addressing questions regarding quantum mechanical ergodicity.

Semiclassical quantization of the quasiperiodic vibrational motion of molecules is usually based on Einstein–Brillouin–Keller (EBK) conditions for the quantization of the classical actions. Explicit use of the EBK conditions for molecular systems of *K* degrees of freedom requires *K* quantization conditions. Therefore, explicit use of the EBK conditions becomes increasingly difficult if not impossible for polyatomic systems of three or more degrees of freedom. In this paper we propose a semiclassical quantization method which makes explicit use of phase coherence of the de Broglie wave associated with the trajectory rather than the EBK conditions. We show that taking advantage of phase coherence reduces the *K* quantization conditions to a single quantum condition—regardless of the number of degrees of freedom. For reasons that will become obvious we call this method ‘‘spectral quantization.’’ Polyatomic vibrational wave functions and energy eigenvalues are generated from quasiperiodic classical trajectories. The spectral method is applied to an ABA linear triatomic molecule with two degrees of freedom and to an anharmonic model of the molecule cyanoacetylene. The usefulness of the technique is demonstrated in this latter calculation since the cyanoacetylene model will have four coupled vibrational degrees of freedom.

A time dependent wave packet method is presented for the rapid calculation of the properties of systems in thermal equilibrium and is applied, as an illustration, to electronic spectra. The thawed Gaussian approximation to quantum wave packet dynamics combined with evaluation of the density matrix operator by imaginary time propagation is shown to give exact electronic spectra for harmonic potentials and excellent results for both a Morse potential and for the band contours of the three transitions of the visible electronic absorptionspectrum of the iodine molecule. The method, in principle, can be extended to many atoms (e.g., condensed phases) and to other properties (e.g., infrared and Raman spectra and thermodynamic variables).

%B J. Chem. Phys. %V 79 %P 4749 %G eng %U http://dx.doi.org/10.1063/1.445618 %R 10.1063/1.445618 %0 Journal Article %J J. Phys. Chem. %D 1983 %T Electronic spectra from molecular dynamics: A simple approach %A J. P. Bergsma %A P. H. Berens %A K. R. Wilson %A D. R. Fredkin %A E. J. Heller %B J. Phys. Chem. %V 88 %P 612-619 %G eng %U http://pubs.acs.org/doi/abs/10.1021/j150647a055 %N 3 %0 Journal Article %J J. Chem. Phys. %D 1983 %T Radiationless transitions in a new light %A Heller, EricJ. %A Robert C. Brown %XRadiationless transitions in polyatomic molecules prove to be quite amendable to a semiclassical treatment both below and above crossings between the potential surfaces involved in the transition. Below such crossings, tunneling integrals are easily performed which give good estimates of the dependence of the nonradiative rate on the energy gap and excess energy in the electronic state. Above the surface crossing, the transitions become classically allowed and a Tully–Preston surface hopping model suffices. We find that a nonlinear dependence of ln(*k* _{ n } _{ r }) vs *E* plots is the rule rather than the exception. The ln(*k* _{ n } _{ r }) vs *E* plots tend to flatten out with increasing energy. This effect can occur below surface crossings, but is most dramatic when a surface crossing is reached. The recent beam results of Smalley and co‐workers on pyrazine and pyrimidine are seen to be a possible case of this simple behavior.

We present in this paper a coordinate independent semiclassical quantization method. We demonstrate that in order to extract accurate eigenvalues and eigenfunctions the trajectory does not necessarily have to reside on the quantizing torus, rather, one can use information obtained on arbitrary tori. Because the method is coordinate independent, no difficulty is encountered in quantizing within classical resonance zones. Furthermore, nearby eigenstates and eigenvalues (nearby in action space) may be extracted from the same trajectory—this is especially convenient when the density of states becomes large.

%B J. Chem. Phys. %V 78 %P 4005 %G eng %U http://dx.doi.org/10.1063/1.445126 %R 10.1063/1.445126 %0 Journal Article %J J. Chem. Phys. %D 1982 %T Excited state geometry changes from preresonance Raman intensities: Isoprene and hexatriene %A Anne B. Myers %A Richard A. Mathies %A David J. Tannor %A Heller, EricJ. %XA method is presented for using a single preresonance Raman spectrum and an absorptionspectrum to obtain changes in equilibrium geometry upon electronic excitation. The relative displacements along each of the vibrational normal coordinates are obtained from the Raman intensities, while the overall scaling of the displacements is determined by the absorption band shape. The absorption spectra, as well as Raman excitation profiles, are calculated using either a sum over vibronic states or a formally equivalent time‐dependent method [S.‐Y. Lee and E. J. Heller, J. Chem. Phys. 71, 4777 (1979)]. The time‐dependent method is computationally much faster than the vibronic sum for large multidimensional systems. Our analysis, which assumes isolated molecules and separable, harmonic surfaces, yields a good fit to the vapor phase absorptionspectrum of *t* *r* *a* *n* *s*‐hexatriene with a Lorentzian linewidth of 175 cm^{−1}. However, the diffuse absorptionspectrum of isoprene cannot be adequately reproduced using Lorentzian line shapes, even when all 33 normal modes are included. Finite temperature and excited state frequency changes are also found to have little effect on the calculated band shapes. These results suggest that inhomogeneous broadening may be a major factor, but calculations using Gaussian broadening fail to accurately reproduce the experimental spectrum.

While liquid solution reactions are much more important in chemistry, gas phase reactions are much better understood. Given the central importance of solution reactions to inorganic, organic, industrial and biochemistry, it is rather surprising that, as yet, there is not a single such reaction whose molecular dynamics are understood in detail. Theoretical and experimental evidence already makes clear that much of the important molecular dynamic action in solution reactions occurs on the picosecond and subpicosecond time scales. The dihalogen photodissociation and recombination reactions, *X* _{2} + *hv*→*X* + *X*→*X* _{2}, involving the simplest possible molecular reactants and products, diatomics, and in rare gas solution involving only two elements, seem excellent candidates for study.

Lee and Heller’s time‐dependent theory of resonance Raman scattering is reviewed. This theory is formally identical to the traditional Kramer–Heisenberg–Dirac (KHD) theory but, in its wave packet interpretation, the time‐dependent theory has distinct calculational and conceptual advantages over the KHD sum‐over‐states method. For polyatomics with large Franck–Condon displacements and Duschinsky rotations, where typically the KHD sum is over 10^{10} states with complicated Franck–Condon factors, these advantages are most pronounced. Wave packet propagation on general harmonic potential surfaces (Franck–Condon displacement, frequency shifts, and Duschinsky rotation) is exact. Formulas for the propagated wave packet are given for various levels of harmonic sophistication. The role of symmetry in the wave packet dynamics is discussed and explicit formulas are derived for the overlap of the moving wave packet ‖φ_{ i }(*t*)〉 with the final state of interest ‖φ_{ f }〉. The half Fourier transform of this overlap gives the Raman amplitude α. The transform method of Tonks and Page, relating absorption and Raman excitation profiles, is shown to arise naturally in our approach. We show excitation profiles calculated by the time‐dependent theory for multidimensional harmonic potential surfaces with and without Duschinsky rotation. For the no‐Duschinsky cases, we compare our results with the profiles of Myers and Mathies and of Champion and Albrecht, which were calculated by a sum‐over‐states; we then discuss some discrepancies between the latter’s results and ours.

A time‐dependent semiclassical method for generating energy‐dependent photodissociation partial cross sections is presented. The method is based on the Wigner equivalent formulation of quantum mechanics with the semiclassical limit arising from one dynamical approximation: the replacement of the quantum Liouville operator by its h/→0 limit. The results of the present scheme for the collinear dissociation of ICN on a single dissociativesurface are compared to those obtained from a distorted‐wave analysis and a semiclassical wave packet propagation. A model calculation employing standard trajectory techniques indicates that the present method has several distinct advantages over the traditional quasiclassical approach.

%B J. Chem. Phys. %V 75 %P 186 %G eng %U http://dx.doi.org/10.1063/1.441822 %R 10.1063/1.441822 %0 Journal Article %J J. Chem. Phys. %D 1981 %T Errors in the Wigner approach to quantum dynamics %A Heller, EricJ. %A Robert C. Brown %XIt is shown that the recent results and error trends of Lee and Scully and Brown and Heller are explicable in terms of the ’’dangerous cross term’’ analysis. Extended quantum state suffer larger errors in the Wigner method.

%B J. Chem. Phys. %V 75 %P 1048 %G eng %U http://dx.doi.org/10.1063/1.442056 %R 10.1063/1.442056 %0 Journal Article %J J. Chem. Phys. %D 1981 %T Frozen Gaussians: A very simple semiclassical approximation %A Heller, EricJ. %XA new and convenient semiclassical method is proposed. It relies only upon classical trajectories and Gaussian integrals. It seems to work very well for the model molecular vibrational spectra investigated here. It should be applicable to a wide variety of processes and can be variationally improved if necessary.

%B J. Chem. Phys. %V 75 %P 2923 %G eng %U http://dx.doi.org/10.1063/1.442382 %R 10.1063/1.442382 %0 Journal Article %J J. Chem. Phys. %D 1981 %T Multidimensional wave functions from classical trajectories %A Michael J. Davis %A Heller, EricJ. %XA new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov‐type correction. Wave functionsgenerated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generatingwave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.

%B J. Chem. Phys. %V 75 %P 3916 %G eng %U http://dx.doi.org/10.1063/1.442548 %R 10.1063/1.442548 %0 Journal Article %J J. Chem. Phys. %D 1981 %T Quantum dynamical tunneling in bound states %A Michael J. Davis %A Heller, EricJ. %XTunneling involves an allowed quantum event which fails to take place classically. Dynamical tunneling is the subset of such events which do not involve a classically insurmountable potential barrier. In this paper, we present unambiguous evidence for dynamical tunneling in bound state quantum systems.The tunneling occurs between two distinct regions of classically trapped quasiperiodic motion. Close analogies are shown to exist between this situation and ordinary barrier penetration in a double minimum potential. In the cases we study, tunneling occurs between equivalent or nearly equivalent local mode motions, which have arisen out of a resonance between the symmetric and nonsymmetric stretch.

%B J. Chem. Phys. %V 75 %P 246 %G eng %U http://dx.doi.org/10.1063/1.441832 %R 10.1063/1.441832 %0 Journal Article %J J. Chem. Phys. %D 1980 %T Molecular spectra, Fermi resonances, and classical motion %A Heller, EricJ. %A Ellen B. Stechel %A Michael J. Davis %XClassically periodic molecular vibration (such as a totally symmetric stretch) can be unstable against the addition of small components of other modes, depending on anharmonic coupling strengths, near resonance of fundamental frequencies, and the total energy. We report here on some very strong correspondances between classical stability of the motion and quantum spectral features, wave functions, and energy transfer. The usual concept of a vibrational Fermi resonance turns out to apply best to the case where the transition to classical instability occurs at an energy below the first resonant quantum levels (this is the case for the famous Fermi resonance in CO_{2}). In the (probably more common) event that resonant classical instability should set in above several quanta of energy in the mode of interest, the quantum spectrum shows tell‐tale pre‐ and post‐resonant signatures which include *a* *t* *t* *r* *a* *c* *t* *i* *o* *n* of quantum levels (rather than the usual Fermi repulsion) and other features not normally associated with Fermi resonances. Evidence is presented which suggests that certain types of periodic motion in anharmonic molecules act as ’’traps’’, and are resistant to energy exchange with other types of motion. Numerical evidence linking the classical and quantum behavior, together with a new semiclassical theory presented here provides a very explicit connection between quantum and classical anharmonic motion.

We define a simple, purely local mode model vibrational Hamiltonian which gives rise to an apparent normal modespectrum under conditions resembling a symmetric stretch Franck–Condon transition. The model clearly distinguishes the question of the intrinsic separability of the Hamiltonian from the nature of the initial conditions, or "pluck", implied by the transition moment.

%V 73 %P 626 %G eng %U http://dx.doi.org/10.1063/1.440163 %R 10.1063/1.440163 %0 Journal Article %J J. Chem. Phys. %D 1980 %T Quantum intramolecular dynamics: Criteria for stochastic and nonstochastic flow %A Heller, EricJ. %XThis paper proposes new criteria by which to gauge the extent of quantum intramolecular randomization in isolated molecules. Several hallmarks of stochastic and nonstochastic behavior are identified, some of which are readily available from spectral data. We find that it is very important to tailor the criteria to the specific experimental situation, with the consequence that a given molecule can be labeled both stochastic and nonstochastic, even in the same general energy regime, depending on the experiment. This unsettling feature arises as a quantum analog of the necessity, in classical mechanics, of specifying the *a* *p* *r* *i* *o* *r* *i* known integrals of the motion before ergodic or stochastic behavior can be defined. In quantum mechanics, it is not possible to have flow or measure local properties (analog of trajectories and phase space measure) without some uncertainty in the integrals of the motion (most often the energy). This paper addresses the problems this creates for the definition of stochastic flow. Several systems are discussed which show significant differences in their quantum vs classical stochastic properties.

The effective Hamiltonian is quantized and the Frank–Condon spectra calculated for CO_{2}.

We present theory and numerical results for a new method for obtaining eigenfunctions and eigenvalues of molecular vibrational wave functions. The method combines aspects of the semiclassical nature of vibrational motion and variational, *a* *b* *i* *n* *i* *t* *i* *o* techniques. Localized complex Gaussian wave functions, whose parameters are chosen according to classical phase space criteria are employed in standard numerical basis set diagonalization routines. The Gaussians are extremely convenient as regards construction of Hamiltonian matrix elements, computation of derived properties such as Franck–Condon factors, and interpretation of results in terms of classical motion. The basis set is not tied to any zeroth order Hamiltonian and is readily adaptable to arbitrary smooth potentials of any dimension.

A recent time dependent formulation of total photodissociation cross sections is exploited to give a qualitative explanation of line shapes and absorption envelopes for symmetric triatomic (XY_{2}) vacuum uvspectra. Attention is given to the dependence of the cross section on potential surface parameters and on the nature of the initial vibrational wavefunctions. The symmetric triatomic case treated here is illustrative of techniques which can be applied to more complicated, unsymmetric polyatomic situations.

An unconventional time dependent formula for total photodissociation cross sections shows the importance of short time dynamics in direct photofragmentation. This is exploited to provide a systematic expansion in powers of h/ for the cross section. The lowest order term is a classical cross section which is shown to be an improvement upon the venerable reflection approximation. Terms to higher order in h/ lead to even greater improvements in accuracy as shown by simple numerical examples. Our formulas are directly applicable to polyatomic photofragmentation, and as a spinoff we derive the polyatomic generalization of the (diatomic) reflection method.

%B J. Chem. Phys. %V 68 %P 2066 %G eng %U http://dx.doi.org/10.1063/1.436029 %R 10.1063/1.436029 %0 Journal Article %J J. Chem. Phys. %D 1977 %T Generalized theory of semiclassical amplitudes %A Heller, EricJ. %XThe theory of quantum amplitudes in the semiclassical, h/→0 limit is considerably extended by including overlaps between Gaussian coherent states (wave packets) in the primitive ’’semiclassical algebra.’’ Traditionally, semiclassical formulations have exclusively involved overlaps between eigenstates of Hermitian operators; these are delocalized in most representations. Paradoxically, the states which are most localized and particlelike in the classical limit (wave packets) are not eigenfunctions of Hermitian observables, and have been omitted from asymptotic, formal theories of the semiclassical limit. When incorporated into a generalized theory, the Gaussian wave packet states apppear to alleviate many of the practical difficulties encountered in implementation of the Hermitian, delocalized state formulation. We present the generalized theory here, and show how it contains and reduces to the Hermitian semiclassical theory in various limits.

%B J. Chem. Phys. %V 66 %P 5777 %G eng %U http://dx.doi.org/10.1063/1.433853 %R 10.1063/1.433853 %0 Journal Article %J J. Chem. Phys. %D 1977 %T Phase space interpretation of semiclassical theory %A Heller, EricJ. %X

A useful and intuitive phase space picture of common semiclassical approximations and procedures is proposed and developed. Underlying the intuitive pictures is the semiclassical, *C*→0 limit of the Wigner distribution, which is discussed in an Appendix. Simple classical analog phase space diagrams are suggested which represent quantum amplitudes, completeness relations, stationary phase points and integrations, caustic singularities, uniformizations, and the interesting short and long time behavior of the semiclassical propagator. Semiclassical wave packet amplitudes, now known to be on equal footing with the more usual amplitudes between delocalized eigenstates of Hermitian operators, are included in the phase space pictures. It becomes apparent why the primitive semiclassical wave packet amplitudes are to some extent uniformizing, and numerical results are presented to support this conclusion. The power of the phase space picture as an aid in formulating new approximations is illustrated.

%B J. Chem. Phys. %V 67 %P 3339 %G eng %U http://dx.doi.org/10.1063/1.435296 %R 10.1063/1.435296 %0 Journal Article %J J. Chem. Phys. %D 1976 %T Classical S-matrix limit of wave packet dynamics %A Heller, EricJ. %X

The asymptotic (h/→0) equivalence of the Miller–Marcus classical *S*‐matrix theory and Gaussian wave packet dynamics is shown. This result is not suprising, but the analysis yields considerable insight into both methods. Both approaches are seen to rely upon a linear response of dynamical variables against small changes in initial conditions. However, the two theories ’’back off’’ the h/→0 limit in a very different manner. Wave packets emerge as a kind of *a* *p* *r* *i* *o* *r* *i* uniformization procedure as opposed to the *a* *p* *o* *s* *t* *e* *r* *i* *o* *r* *i* uniformizations of classical *S*‐matrix theory. In certain contexts the wave packets are shown to provide a parabolic cylinder function uniformization of the primitive semiclassical result. Wave packets in generalized coordinates are discussed. Analogy with classical *S*‐matrix theory suggests new procedures for through‐barrier tunneling of individual wave packets.

Explicitly time dependent methods for semiclassical dynamics are explored using variational principles. The Dirac–Frenkel–McLachlan variational principle for the time dependent Schrödinger equation and a variational correction procedure for wavefunctions and transition amplitudes are reviewed. These variational methods are shown to be promising tools for the solution of semiclassical problems where the correspondence principle, classical intuition, or experience suggest reasonable trial forms for the time dependent wavefunction. Specific trial functions are discussed for several applications, including the curve crossing problem. The useful semiclassical content of the time dependent Hartree approximation is discussed. Procedures for the variational propagation of density matrices are also derived.

%B J. Chem. Phys. %V 64 %P 63 %G eng %U http://dx.doi.org/10.1063/1.431911 %R 10.1063/1.431911 %0 Journal Article %J J. Chem. Phys. %D 1976 %T Wigner phase space method: Analysis for semiclassical applications %A Heller, EricJ. %XWe investigate the suitability of the Wigner method as a tool for semiclassical dynamics. In spite of appearances, the dynamical time evolution of Wigner phase space densities is found *n* *o* *t* to reduce to classical dynamics in most circumstances, even as *h*→0. In certain applications involving highly ’coherent’ density matrices, this precludes direct *h*‐expansion treatment of quantum corrections. However, by selective resummation of terms in the Wigner–Moyal series for the quantum phase space propagation it is possible to arrive at a revised or renormalized classicallike dynamics which solves the difficulties of the direct approach. In this paper, we review the Wigner method, qualitatively introduce the difficulties encountered in certain semiclassical applications, and derive quantitative means of surmounting these difficulties. Possible practical applications are discussed.

In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H_{2} system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

The “quantum trajectory” wavepacket approach to semiclassical collision dynamics is generalized to include effects which cause distortion of initially gaussian wavepackets. The generalization takes the form of a discrete phase space path “integral” or sum. A complete set of gaussian phase space localized basis functions is proposed, and in the semiclassical limit, each basis function time develops in a simple classical-like fashion. Thus, the formulation conveniently builds in the correct limiting semiclassical behavior. The phase space path sum is formally exact, numerically it appears convenient, and is apparently equally at home in quantum and semiclassical regimes.

%B Chemical Physics Letters %V 34 %P 321-325 %8 Jul 1975 %G eng %U http://dx.doi.org/10.1016/0009-2614(75)85284-5 %N 2 %0 Journal Article %J J. Chem. Phys. %D 1974 %T Random coupling model for molecular dissociation %A Heller, EricJ. %A Stuart A. Rice %XIn this paper, we propose and solve sequential coupling models for molecular dissociation of the Rice‐McLaughlin‐Jortner (RMJ) type in which the usual assumption of constant coupling among the states is replaced by an assumption of random coupling. The counter‐intuitive nonsequential branching behavior found previously for constant coupling is eliminated and we find completely sequential time dependence which obeys the phenomenological rate equations. We isolate the features of constant vs random coupling which give rise to the branching vs sequential behavior in terms of simple physical models and considerations of the coherence properties of the wavefunction. It is concluded that constant coupling is inappropriate for most molecules, and that the random coupling assumption has the effect of validating the use of a random phase approximation which in turn causes the molecule to decay as if each quasibound molecular level is coupled to its own continuum. Our conclusions do not change when we solve an extended model with many continua, with each molecular level coupled to each continuum.

%B J. Chem. Phys. %V 61 %P 936 %G eng %U http://dx.doi.org/10.1063/1.1682039 %R 10.1063/1.1682039 %0 Thesis %B Harvard University %D 1973 %T Discrete Methods in Quantum-Scattering Theory %A Eric Johnson Heller %B Harvard University %G eng %N 34-09 Section B Page 4559 %9 PhD thesis %0 Conference Paper %B 8th International Conference on the Physics of Electronic and Atomic Collisions, VIII ICPEAC %D 1973 %T a New Approach to L2 Expansion Scattering Calculations %A E. J. Heller %A H. A. Yamani %B 8th International Conference on the Physics of Electronic and Atomic Collisions, VIII ICPEAC %C Beograd, Yugoslavia %V I %P 5 %G eng %0 Journal Article %J Chemical Physics Letters %D 1973 %T Wigner R-Matrix Approach to Inelastic Collinear Collisions %A E. J. Heller %XThe quantum mechanical collinear atom-diatom collision problem is treated via a discrete expansion of the wave-function in terms of uncoupled, distorte wave states, using the Wigner-Eisenbud *R*-matrix formalism. Previously unreported narrow Feshbach resonances, which reduce the inelastic transition probability, are easily found and examined. Off resonant transition probabilities are in agreement with the work of Diestler and Feuer using the same model.