- 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.

Computational and teaching methods.

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*.

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.

i

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.

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.*

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*

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*}.

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.

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:

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 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.*

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.

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.

Calculations by *Ofra Citri and Gil Katz *

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.

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.

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*}.

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.

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*.

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.*

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*.

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*.

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.

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.

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*.

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.

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 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 *

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.

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.

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

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.