Highlights of research

Ronnie Kosloff

Quantum molecular dynamics constitutes the thrust of my research. The goal is to gain insight into realistic elementary chemical encounters. This requires the development and application of a quantum description to molecular processes. In particular the emphasis is on time-dependent approaches which can follow naturally the stream of events.

The main topics of study are:

Coherent chemistry: light induced processes.

Coherent control and laser cooling.

Dynamical processes on surfaces.

Quantum thermodynamics.

Computational and teaching methods.

Coherent chemistry

Unraveling the quantum nature of elementary chemical encounters is difficult. Typically the quantum nature is masked by macroscopic averaging. For this reason inducing a chemical event by an ultrashort pulse, and following the coherent motion in time, is a primary source of insight into the elementary chemical process.

A non-perturbative quantum description of light-matter interaction is the key for a theoretical understanding. In condensed phases the phenomena of dissipation and relaxation which causes the loss of coherence has to be well understood.

The theoretical studies are closely linked to the experimental effort in ultrafats photodissociation processes in solution carried out in the laboratory of Sandy Ruhman.

Photodissociation dynamics of tri-halides in solution

The photodissociation of tri-halides in solution constitute an excellent case study exhibiting in a traceable fashion the primary events in solution chemistry. In a typical experiment a stong pulse photodissociates I3- to I2- + I. The product formed I2- shows a modulation in absorption of a weak probe pulse corresponding to the vibrational frequency. A probe at the reactant frequency reveals an induced modulation in the reactant frequency.

The theoretical study is aimed at unraveling the full dynamical picture from reactants to products as well as dissipation and relaxation phenomena induced by the solvent. Quantum time dependent wavepacket simulations have been constructed which are able to correlate the shape of the potential to the observed modulations 118 It has been found that breaking the symmetry of the molecule leads to enhanced modulations. Experimentally this can be done either by using a polar solvent such as methanol, or thermally exciting the anti-symmetric strech mode, or photodissociating I2Br-.

The dynamical "hole" induced by the ultrafast excitation on the ground electronic surface, is a primary source of insight into the relaxation phenomena. The decay of modulations is directly related to the vibrational dephasing time. A comprehensive quantum description of the process has been constructed based on the impulsive two-level-approximation. This description is supplemented by a quantum simulation based on the solution of the Liouville von Neumann equation 108,110,127.

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The evolution of the dynamical "hole" in phase space. The "hole" is created by a 60 fsec pulse of 305 nm in resonance to the I3- transition. The pulse excites the symmetric stretch of the molecule. The time is measured in periods of the vibrational motion. After four periods the "hole" is almost wiped out due to dephasing.

The insight gained on the dissipative dynamics of the "hole" is used to study the relaxation of the hot I2- created in the photo-reaction. A "push" pulse is employed to create a hole in the the nascent hot product. By following the decay of modulations as the product distribution approaches thermal equilibrium both relaxation times T1 and T2 are measured. The quantum simulation reveals the interplay between energy relaxation, dephasing, and dispersion due to anharmonicity of the potential 127. The following figure shows creation and dynamics of a "hole" induced on a nonequilibrium density.

Wigner distribution function of the simulated density of hot I2- after the photodissociation of I3-. After complete dephasing (top left frame), a "push" pulse is applied creating a new "hole" (top right), which rotates in phase space and dissipates (bottom).

Current theoretical efforts are devoted to the calculation of a potential energy surface and the study of polarization effects. Theoretical effort has been performed by Allon Bartana and Guy Ashkenazi. Experiments have been performed in the group of Sandy Ruhman by Uri Banin and Erez Gershgoren.

Relaxation and dephasing in solution

The dissipation and energy relaxation of a molecule embedded in a solvent is of primary importance for chemical encounters in solution. A quantum analysis requires a reduced description which divides the system to the primary part and the bath. The implication of this description and its relation to the quantum semigroups formalism is the subject of study. A phase space picture of the dynamics of the primary system has been found to be useful in visualizing the distinction between energy relaxation and dephasing. A phase space picture in action angle coordinates is an important addition 127. These methods have been applied to the dissipative dynamics of I3- in solution. Work by Guy Ashkenazi

Relaxation and dephasing of an harmonic oscillator. Left: succession of snapshots of the density in position-momentum phase space. Right: action-angle phase space. Top energy relaxation. Bottom: energy relaxation and dephasing.

Coherent control

The ultimate theme of coherent control is to use light to determine the evolution of matter. The main idea is to employ the coherent properties of light to induce interference of matter waves of the molecular constitute. By controlling the interference properties active and passive control on the outcome of observables is obtained. The route followed is an active control in time domain. Optimal control theory has been formulated in Liouville space for strong fields enabling the study of control under dissipative conditions. The phase relation between the instantaneous dipole and the field is found to determine the direction of flow of energy and population 94,k121. The study is in close collaboration with David Tannor and Stuart Rice}.

Laser cooling of internal degrees of freedom

The methods of optimal control has been applied to study the possibility of Laser cooling of molecular internal degrees of freedom. The emphasis has been on general concepts and universal constraints. A distinction between intensive and extensive considerations has been used to analyse the cooling process in open systems.

The limitations on cooling processes imposed by Hamiltonian generated unitary transformations have been analyzed. For a single mode system with a ground and excited electronic surfaces driven by an external field, it is impossible to increase the ground state population beyond its initial value. A numerical example based on optimal control theory demonstrates this result. For this model only intensive cooling is possible which can be classified as evaporative cooling. To overcome this constraint, a single bath degree of freedom is added to the model. This allows a heat pump mechanism in which entropy is pumped by the radiation from the primary degree of freedom to the bath mode, resulting in extensive cooling 129.

Heat pump cooling for the two-mode model. The x mode is the primary mode. The y mode is the entropy sink. The light induces population and energy transitions from the ground to the excited surface and vise versa. Energy can flow from one mode to the other through the excited electronic surface. The system can be partitioned to the ground and excited electronic manifolds and/or the x and y vibrational modes. The reduced description of the primary and bath modes is shown as the projections on the perpendicular one dimensional planes).

Extensive cooling can be visualized by the shrinking of the phase space distribution. With the use of optimal control the system can be cooled close to its ground state:

Coordinate - momentum phase space Wigner distribution of the reduced density on x primary mode showing the shrinking of the distribution . a) Initial state, and b) final state. These results were obtained within ten periods of the primary oscillator. Initially the two modes had the same entropy.

Entropy measures reveal the same heat pump mechanism where the a bath mode serves as the entropy sink of the primary mode. This study has been carried out by Allon Bartana.

Inversion of spectroscopic data

Inversion of spectroscopic data to obtain the underlying potential is closely related to the theory of optimal control. A scheme designed to invert ultrafast pump-probe spectroscopic data has been developed 107. Optimal control methods can be applied to optimize the harvesting of information on the system. The idea beeing that information is more valuable than materials. Work by Roi Baer.

Dynamical processes on surfaces

Primary dynamical process on surfaces exhibit a profound quantum nature. The focus is on primary catalytic mechanism where quantum phenomena dominate. The outcome of these process is determined by tunneling and nonadiabatic phenomena. Nitrogen dissociation on transition metals which is the rate determining step in ammonia synthesis, has been the subject of extensive study in conjunction with the experimental effort in the laboratory of Micha Asscher. Hydrogen transport in nickel is responsible for the catalytic hydrogenation reactions on nickel. A quantum first principle understanding of these processes has been the subject of study.

Nonadiabatic dissociation mechanism of N$_2$ on transition metals

The framework of study assumes a direct dissociation paradigm, based on a universal quantum nonadiabatic picture where a gas phase nitrogen molecule approaches on the physisorption potential surface and crosses to the chemisorption surface where the molecule dissociates. In energies below the nonadiabatic crossing seam, the reaction mechanism can be classified as a tunneling event. The experiments were simulating by solving the time dependent Schrodinger equation using threes degrees of freedom: the nitrogen-nitrogen distance r, the nitrogen-surface distance z, and the surface recoil coordinate x. Three metals were analyzed: iron rhenium and ruthenium. A universal behavior of an increase in dissociation probability of orders of magnitude upon increasing the incident kinetic energy, in agreement with experiment, is found. The crystal temperature effect show differences between th metals where iron has a negative temperature effect ruthenium is neutral and rhenium has a positive temperature effect 117. These observations are due to the influence of the metal mass on the nonadiabatic transition.

Dissociation probability as a function of incident kinetic energy for Fe and Re. The lower line for Fe corresponds to T=500K and the higher one to 100K. For Re the order is reversed. The graphs were shifted to fit the same high energy asymptote. Notice the orders of magnitude increase in dissociation probability.

Calculations by Ofra Citri and Gil Katz

Hydrogen dynamics in nickel

The dynamics of hydrogen on a nickel surface is dominated by transport phenomena where hydrogen moves from one metastable site to another. The transport of bulk hydrogen to a surface site is an activated process. Due to the light mass of the hydrogen the primary reaction route at low temperature occurs via tunneling 124.

A critical study of the influence of lattice motion on tunneling has been carried out. Representing the lattice motion by a single oscillator led to an enhancement of the tunneling rate. A time scale separation for this model failed, the conclusion being that the primary flux in tunneling highly correlates the two degrees of freedom. A multi-mode bath description of the lattice was developed, based on a short time finite representation of the bath dynamics. This model shows that if the bath is primarily coupled to the barrier height then it's overall effect is to enhance tunneling.

Top. Views of the nickel atoms at their lattice points (right) and of an open nickel triangle (left). Bellow: The corresponding reaction path potentials for the tunneling hydrogen, These paths have been adiabatically corrected for the perpendicular mode zero point energies. The open configuration was obtained from an MD simulation at 90$^o$K corresponding to the lowest barrier in a 10 psec simulation period.

However, if the bath influence is restricted to the metastable subsurface well, the crossing rate is suppressed. It was found that for hydrogen on nickel the two mechanisms approximately cancel each other. The multi-mode treatment is based on a spectral density calculated using a molecular dynamics simulation. It was also demonstrated that the nonadiabatic interactions of the hydrogen with the electron-hole-pairs in the metal has a relatively small hindering effect on the tunneling. From this analysis it has been concluded that hydrogen tunneling is extremely sensitive to the multi-mode nature of the lattice vibrations.

The resurfacing rate of subsurface hydrogen in nickel as a function of inverse temperature. Different approximations for the influence of the metal are shown. Notice the crossover temperature from the tunneling regime to the activated one.

Theses studies were used to analyze the possibility of recombination of a bulk and surface hydrogen and to compare to experimental observations. Study by Roi Baer and Yehuda Zeiri}.

Dissociation of oxygen on metals

The dissociation dynamics of oxygen on silver surfaces is a primary example of nonadiabatic effects. A universal functional form for the potential energy surfaces has been employed. The diabatic potentials describing the sequence of events leading to dissociation begin from the physisorption potential crossing over to a charged molecular chemisorption potential and crossing over again to the dissociated atomic-surface potential.

The three diabatic potential energy surfaces representing the oxidation states of oxygen on a silver surface.

Dynamical time dependent calculations on these potentials have shown that oxygen is captured by the molecular chemisorption well for a considerable length of time, long enough for thermalization.

Thus the calculation is split into two parts: the calculation of "direct" dissociation probability and the calculation of nonadiabatic dissociative tunneling rate from the thermalize chemisorbed molecular state. For the direct probabilities, the Fourier method with a Chebychev-polynomial expansion of the evolution operator has been used to solve the time dependent Schrödinger equation. For the tunneling rate calculation, a similar expansion of the Green's operator has been developed. The output of the direct-reaction calculation is the dissociation probability as a function of the initial energy content, while the tunneling calculation yields the dissociation rate. The dependence of the direct dissociation probability on the initial kinetic energy is found to be non-monotonic. A strong isotope effect has been found, favoring the dissociation of the light species 120. Work performed by Roi Baer and Ofra Citri.

Photodesorption through electronic excitation

Based on the numerical solution of the Liouville-von Neumann equation for dissipative systems, the photodesorption dynamics of the NO/Pt(111) system has been studied. The Redhead Gomer Menzel (RGM) model was employed where vie scattering hot electrons the molecule is promoted to the excited surface. The nuclear dynamics on the excited surface after quenching to the ground surface leads to desorption. Dissipative terms were used to describe the quenching of electronically excited states on the metal as well as electronic dephasing, and the indirect (hot-electron mediated) excitation processes in the DIMET and DIET limits. Norm and energy flow, desorption probabilities, and density time-of-flight spectra were computed 123. Work in collaboration with Peter Saalfrank.

Dissipative processes on surfaces

A new approach to describe dissipative dynamics of an adsorbates near a metal surface has been developed. The formulation has been based on replacing the infinite system-bath Hamiltonian by a finite surrogate Hamiltonian. This finite representation has been designed to generate the true short time dynamics of a primary system coupled to a bath. A detailed wavepacket description is employed for the primary system while the bath is represented by an array of two-level-systems. The number of bath modes determines the period the surrogate Hamiltonian reproduces the dynamics of the primary system.

The convergence of this construction has been studied for the dissipating Harmonic oscillator and the double-well tunneling problem. Converged results are obtained for a finite duration by a bath consisting of 4-11 modes.

The formalism has been extended to treat dissipation caused by electron-hole-pair excitations. The stopping power for a slow moving proton was studied showing deviations from the frictional limit at low velocities. Vibrational lineshapes of hydrogen and deuterium on nickel were calculated. In the bulk the lineshape was found to be influenced by nonadiabatic effects. The interplay between two bathes was studied for low temperature tunneling between two surface sits of hydrogen on nickel. A distinction between lattice modes that enhance the tunneling and ones that suppress it was found. Work performed by Roi Baer.

Photoinduced diffraction on insulator surfaces

Photolysis of an HCl adsorbate on a rigid MgO surface can lead to quantum diffraction phenomena. Quantum calculations have shown a strong oscillatory structure in the angular distribution of the photo-fragmented hydrogen as well as in the absorption spectrum. It is caused by resonances and is quantitatively related to the initial perpendicular adsorption geometry. Corrugation of the surface potential leads to a significant modification of these interference patterns, which exist even for a flat surface.

Within a mixed quantum/classical time-dependent self-consistent field (Q/C TDSCF) propagation the influence of additional degrees of freedom on the interference pattern are investigated. Thermal motion of the surface and inelastic collisions of the hydrogen atom with the surface and the chlorine atom lead to a smearing of the peak structure. The angular and energy resolved spectra nevertheless still show clearly distinguishable peaks, which are related to adsorption geometry and surface potential 128.

The photolysis of hydrogen containing molecules at surfaces and following the diffraction pattern can serve as a new surface probe. Work by Michael Hintender, Franck Robentrost and R G. Gerber.

Quantum thermodynamics

The manifestations of the three laws of thermodynamics has been explored in a model of an irreversible quantum heat engine. The purpose is to explore the quantum origins of the thermodynamical laws. An engine composed of a three-level system simultaneously coupled to hot and cold heat baths, and driven by an oscillating external field has been studied. General quantum heat baths are considered, which are weakly coupled to the three-level system. The work reservoir is modeled by a semi-classical electro-magnetic driving field of arbitrary intensity, which is coupled to the three-level system. The first law of thermodynamics is related to the rate of change of energy obtained from the quantum master equation in the Heisenberg picture. The fluxes of the thermodynamic heat and work are then directly related with the expectation values of quantum observables.

A three level system operated as a heat engine when radiation is amplified. Heat pump operation requires radiation as input and is used to cool the cold bath.

An analysis of the standard quantum master equation for the amplifier, first introduced by Lamb, are shown to be thermodynamically inconsistent when strong driving fields are used. A generalized master equation is rigorously derived, starting from the underlying quantum dynamics, which includes relaxation terms that explicitly depend upon the field. For weak fields the generalized Master equation reduce to the standard equations. In very intense fields, the amplifier splits into two heat engines. One engine operates accelerates as the field intensifies, while the other slows down and eventually switches direction to become a heat pump. The relative weight of the slower engine increases with the field intensity, which leads to a maximum in power as a function of the field intensity. The amplifier is shown to go through four "phases" in the post-saturation region, throughout all of which the second law of thermodynamics is generally satisfied. One phase corresponds to a "refrigeration window" which allows for the extraction of heat out of a cold bath of temperatures down to the absolute zero. This window disappears at absolute zero, which is conjectured to be a dynamical manifestation of the third law of thermodynamics 122.

This quantum model of a laser based on a three level system can be operated as a heat pump. Thermodynamic currents of power and heat are derived, showing strictly positive entropy production thus showing consistency with the second law of thermodynamics.

The cooling rate and entropy production as a function of inverse temperature close to the absolute temperature. The insert shows that the cooling rate vanishes linearly as the absolute temperature is reached.

When operated at ultra-cold conditions the maximum cooling rate vanishes linearly with temperature maintaining constant rate of entropy production. This phenomena is the a generalization of the third law of thermodynamics. Work by Eitan Geva.

A simple example of a four-stroke engine operated in finite-time has been analyzed. The working medium consists of non-interacting two-level systems or harmonic oscillators. The cycle of operation is analogous to a four-stroke Otto cycle. The engine is shown to settle to a stable limit cycle for given contact periods with the hot and cold baths. A maximization of the power with respect to the cycle time leads to a finite optimal cycling frequency 116. Work by Tova Feldman and Eitan Geva.

Computational and teaching methods

The study of quantum molecular dynamics requires an intensive effort in developing new algorithms and methods of visualization. Quantum mechanical calculations scale exponentially with the number of degrees of freedom. For this reason only very efficient methods are able to simulate realistic encounters. The computation and visualization techniques employed are ideal for use as a teaching aid.

Representation theory

The representation of a quantum system by an evenly spaced Fourier grid is the most common method. This grid faithfully represents wave functions whose projection is contained in a rectangular phase space. This is mathematically equivalent to a band limited function with finite support. In general, wave packets decay exponentially in classically forbidden regions of phase space. This idea is then used first to optimize the rectangular shape of the Fourier grid, leading to exponential convergence 126. Nevertheless, in most cases the representation is suboptimal. The representation efficiency can then be extremely enhanced by mapping the coordinates. The mapping procedure reshapes the wave function to fit into the rectangular Fourier shape such that the wasted phase space area is minimal. It is shown that canonical transformations, which re-scale the coordinates, improve the representation dramatically. A specific scaling transformation enables the representation of the notoriously difficult Coulomb potentials.

The Wigner phase space distribution of the 63 eigenvalue of the Morse potential for I$_2$. The area represented in phase space corresponds to a Fourier grid of 128 points. Notice the exponential decay of the wavefunction in the classical forbidden regions. The left panel shows the uniform grid and the right panel the mapped grid.

This scaling transformation can bridge the gap between quantum chemistry and quantum molecular dynamics by enabling the treatment of electronic problems in the vicinity of Coulomb potentials by grid methods developed for molecular dynamics 119. Work by Eyal Fatal and Roi Baer.

Propagation methods

Propagation algorithms are tha basic tool which allows to extract dynamical information. The basic idea is to apply recursively the Hamiltonian operator to an initial state. Both time dependent and time independent information can be obtained simultaneously. The method is a much more efficient algorithm than traditional diagonalization methods or methods based on linear equation solvers.

A new method to calculate resonances has been developed. The method is based on a dual filter both in the energy as well as in the time domain. Extreme accuracy has been demonstrated 124.

The flux of an evolving wavepacket is the definite time integral of it's probability current density. A new method for calculating the flux, based on a Chebychev polynomial expansion of the quantum evolution operator has been developed. The central point of the development is that the time integration of the current density is performed analytically, resulting in a scheme which eliminates additional numerical errors. Using this method, one benefits from both the time-dependent and time-independent frameworks of the dynamics. Furthermore, the method requires only a small modification to the existing Chebychev-Polynomial evolution code \cite{k115}. Work by Gil Katz and Roi Baer

Dissipative dynamics

The idea of partitioning the system into primary and bath modes has been the key element in the quantum theory of dissipative dynamics. Starting from the work of Bloch, reduced equations of motion for the primary system have been derived. The reduction is obtained by performing a partial trace over the bath degrees of freedom resulting in a Liouville description of the primary mode. The most well studied derivation is based on the assumption of weak coupling between the system and bath leading to a differential equation describing the systems dynamics. In this derivation, commonly called Redfield dynamics, the influence of the bath is described by it's correlation functions. This basic derivation has been supplemented by the requirement that the reduced equations of motion have the semigroup form, meaning that they preserve the complete positivity of the density operator.

A complementary approach to dissipative dynamics is to axiomatically require a semigroup form. This leads to a general form for the reduced evolution equations. These equations allow a consistent study of different dissipative models, but require an empirical treatment when a particular system is studied. A continuous effort is devoted to the development of ne algorithms for solving the Liouville von Neumann equation 118,127 Such an approach has been used for modeling the photodesorption of NO from a nickel surface, where the influence of the metal electrons was imposed empirically by using the semigroup form 112,123.

The practical disadvantage of both the semigroup and the Redfield theories is that they are formulated in Liouville space where the state of the system is represented by a density operator. This fact squares the number of required representation points in comparison to a wavefunction description. Although powerful numerical techniques have been developed to solve the dynamics in Liouville space it still is extremely taxing to treat these problems, limiting the scope of systems that can be studied. For this reason the alternative surrogate Hamiltonian approach was developed.

New teaching aids

A computer assisted instruction modular package for quantum mechanics is in the process of development. The instructional strategy is based on a three stage process. The fundamental novel aspects of quantum mechanics are addressed first. Interactive computational and graphical tools, demonstrating the fundamentals of quantum mechanics in an intuitive way, will be used at this stage to bypass the mathematical difficulties of quantum mechanics.

At a second stage the mathematical foundations of quantum mechanics will be studied with the use of specially designed computerized tools. The final stage will be dedicated to applications of quantum mechanics, such as spectroscopy, chemical bonding, reactive scattering, and more, through the use of computational and graphical tools. The process and the product of this development may serve as a model for further development for upper devision courses in the physical sciences.

A teaching aid constructed to demonstrate the composition of a wavepacket. Left panel shows the amplitude of the wavepaket in coordinate space. The color is coded by the phase. The middle panel shows the wavepacket in the complex plane as a function of coordinate. The right panel allows the student to control the composintion of the wavepacket. The upper part controls the amplitude in momentum space, the lower part controles the phase.

Work by Guy Ashkenazi, Nava Ben-Zvi and Michael Bermann.

Overview

Quantum molecular dynamics constitutes a rich field of study. The scope covers fundamental questions, such as the relation between quantum mechanics and thermodynamics to practical ones, such as the tunneling characteristics of ammonia synthesis. In particular this research serves as an excellent discipline for teaching graduate students.