Cornell University

Our research is centered around the following problems:

**Classical-Quantum correspondence and the development of semiclassical
methods for molecular problems**

What is the relation between the familiar classical description of molecules as "balls and springs", which underlies much of our intuitive understanding of chemical reaction dynamics, and the presumably more accurate but less easily interpreted quantum mechanical description? Can we forge qualitative (and preferably quantitative) links between, on the one hand, the trajectories of classical molecular dynamics, and, on the other, the energy levels and wavefunctions of quantum mechanics?

This question is clearly of great fundamental importance, as it explores the relation between
two basic theoretical descriptions of the world.
It has practical significance as well,
as many systems of chemical interest (condensed phase, biological molecules) are
simply too large to treat by the methods of exact quantum mechanics.
We are therefore interested in the development and application of
*semiclassical * theories, where we attempt to
compute accurate quantum mechanical properties using classical trajectories!

In earlier work we have applied semiclassical methods to the analysis of the spectrum of the doubly-excited He atom and to the interpretation of molecular vibrational spectra. (See our review of this early work.) Currently, we are interested in applications of semiclassical methods to computation of nonlinear response functions, and to the theory of molecules in intense external fields.

**Chaotic dynamics and the applicability of statistical theories of chemical reaction rates**

It has been known since the beginning of the 20th century that nonlinear classical
systems (such as ball and spring models for molecules) generally exhibit extremely
complex or *chaotic* motions. The motion of chaotic systems is by definition
effectively unpredictable ("sensitive dependence on initial conditions"). Such
unpredictability can be turned to advantage in theories of chemical kinetics
through the formulation of statistical theories of reaction rates. The Rice-Ramsperger-Kassel-Marcus theory of
unimolecular reaction rates is an important example of a statistical theory; transition state
theory is another.
We are interested in the fundamental question of the relation between the nature and extent
of intramolecular energy flow in polyatomic molecules and the applicability of
statistical theories of reaction rates. Statistical approaches essentially assume complete
scrambling of internal energy in a reactive molecule on a timescale rapid compared to
the reactive lifetime. Those instances where conventional statistical theories fail are of most interest,
of course, and
include important classes of ion-molecule reactions such as
proton transfer, and the
predissociation of weakly-bound van der Waals complexes.

**Assignment of molecular vibrational spectra and the phase space structure of multimode systems
**

In the simplified world of textbook vibration-rotation spectroscopy, the internal motions of molecules can be described in terms of effectively noninteracting harmonic normal modes vibrating about a rigidly rotating equilibrium structure. In the real world spectroscopy of polyatomic molecules containing a chemically significant amount of internal energy, vibrational modes interact with one another, leading to exchange of energy and the breakdown of the simple harmonic oscillator picture, while rotations interact with vibrations, leading to the breakdown of the rigid-rotor picture. These interactions lead to relaxation of the standard spectroscopic selection rules and to complicated energy level patterns.

We have a long-standing interest in the development of novel approaches to the analysis of rotation-vibration interaction, and to the assignment of vibrational states in strongly coupled multimode systems.

The problem of assigning vibration spectra and states in multi (*N > 2*) mode molecules
raises the fundamental question of understanding and visualizing the dynamics in such systems.
The phase space (see below) for multimode systems has high dimensionality, and over the years we (and many
other researchers) have grappled with the problem of gaining a useful and
intuitively satisfying visualization of the structure of multimode phase space.
In our own work, we pioneered the method of
local frequency analysis
of multidimensional systems, an approach that is undergoing a resurgence of interest.
Following the mathematical work of Wiggins, we made early attempts to
visualize multidimensional transition states
for vibration predissociation of van der Waals complexes.
We have also applied a classical template for assigning vibrational
states in multimode vibrational Hamiltonians.

As a result of recent mathematical advances and of improvements in computing power, there is renewed interest in the investigation of the structure of the phase space of multimode systems, and several of our current research projects lead us to again confront this fundamental problem.

**Dynamics of molecules in intense fields and the control of rotational motion**

Recent years have seen impressive progress made from both the theoretical
and (increasingly) the experimental side on the problem of the *control*
of molecular processes, such as branching ratios in chemical reactions and product state
distribution in photodissociation.

There has also been great interest in the control of molecular orientational (rotational) variables: in the production of anisotropic states localized in rotational angle, or in the production of highly excited rotational states. Beams of oriented molecules have been used for some time to explore the stereochemistry of reaction dynamics.

Our recent work in this area is concerned with understanding the dynamics of small molecules in
intense fields. For example, we have studied the
classical and quantum dynamics of diatomic molecules
subject to both a strong static electric field and a nonresonant infrared laser pulse. (This combination
of fields was predicted by Friedrich and Herschbach to be a good way to prepare oriented
molecules.) When the two field polarization vectors are *tilted* with respect to one
another, the dynamics becomes complex and particularly interesting.

We have recently investigated the phenomenon of quantum monodromy for diatomic molecules subject to electrostatic and pulsed nonresonant laser fields. We have also successfully applied semiclassical methods to compute the rotational dynamics of molecules subject to strong laser pulses.

Recently we have investigated the use of ultrashort laser pulses to "reconstruct" the quantum state (in general, the density matrix) of molecular rotors.

**Semiclassical theory of nonlinear response functions**

There is great current interest in nonlinear optical phenomena such as the two-pulse vibrational echo, which is analogous to the familiar spin echo of magnetic resonance spectroscopy. Vibrational echo experiments are in principle capable of revealing detailed nuclear dynamical information that cannot be extracted from conventional linear vibrational spectroscopy.

The observable quantities in vibrational echo experiments are expressed in terms of fundamental
quantities known as *nonlinear response functions*,
and the computation of such response functions presents a major theoretical challenge.

Nonlinear response functions are difficult to compute using either classical or quantum methods, and the significance of quantum effects is not well understood at present. In collaboration with R. Loring, we have applied semiclassical approaches to the calculation of the response function relevant for the vibrational photon echo. We have developed a semiclassical theory for optical response functions, and have applied the theory with success to both the linear response and nonlinear response (photon echo) for a thermal ensemble of non-interacting Morse oscillators. Current work is concerned with extension of this approach to systems of coupled oscillators.

**Geometry of Hamiltonian and non-Hamiltonian dynamics**

Consider a system (a model of a polyatomic molecule, for example) treated according to the laws
of classical mechanics, as in a typical molecular dynamics simulation. In order to give a
complete description of such a system, we need to know both the values of all the relevant
coordinates *and* the associated velocities (strictly, momenta). This coordinate/momentum space is
*phase space*, which is the arena in which classical dynamics operates.

If the molecule is isolated from its surroundings, then the dynamics is
*Hamiltonian*. Being Hamiltonian implies, amongst other things, that the phase space volume is conserved by
the time evolution (this is Liouville's theorem, a cornerstone of traditional statistical mechanics).

If the molecule is not isolated from its surroundings, so that is in contact with a heat bath,
for example, then the dynamics of the individual moleule itself is no longer Hamiltonian. In order to describe
the motions of individual molecules in contact with their environment, without having to describe
completely the motion of every atom in the surroundings, it is necessary to resort to
a *non-Hamiltonian* description of the molecular motion. Such a description can simulate a molecule
constrained to be at constant temperature *T*, for example.
For non-Hamiltonian systems, the familiar form of Liouville's theorem usually no longer applies, so
that phase space volume is not necessarily conserved by the system evolution.

The phase space for both Hamiltonian and non-Hamiltonian systems is a non-metric space in which there is no unique definition of distance. The appropriate mathematical apparatus for the description of geometry and dynamics in such spaces is the theory of differential forms on manifolds. We have applied this theory to give a general formulation of response theory in non-Hamiltonian systems, and to discuss general aspects of the statistical mechanics of non-Hamiltonian systems.

In recent work, we have shown that the mathematical formalism can be turned to practical use in the design of efficient methods for the integration of non-Hamiltonian dynamics. These geometric integration methods respect the underlying non-Hamiltonian structure of the phase space, and are therefore expected to be (and are indeed found to be) very accurate.

Ludwig Wittgenstein

Link to complete list of publications, including downloadable PDF versions of papers.

G.S. Ezra Faculty Page.

Cornell Chemistry Faculty Page.

Cornell Chemistry Department Home Page.

gse1@cornell.edu