Research

Below is a "note" I wrote on Facebook to describe my research project on Fermion dynamics from a classical Hamiltonian. Those of you with a solid background in molecular electronics (or even quantum mechanics generally) may recognize that I oversimplified a few points. But the purpose of this was more to explain why I'd been excited by this work than to teach people subtleties of the field of molecular electronics.

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While he was on sabbatical in Berkeley, Prof. Eran Rabani approached Prof. Miller about the idea of developing a semiclassical approach to study molecular electronics. Together, we developed what we eventually called the Dynamics for Classically Mapped Fermions (DCMF) method, which generates classical model Hamiltonian that can describe the dynamics of fermionic systems. Fermion dynamics are particularly challenging for a classical model because of the Pauli exclusion principle and because of the anticommutivity of fermionic operators.

In an effort to reduce the number of trajectories required for semiclassical calculations, I tried including information about the time-evolved distribution in the Monte Carlo sampling for the double Herman-Kluk (DHK) semiclassical initial value representation (IVR). Historically, we have sampled using only the distribution at time zero. However, many trajectories initiated from those initial conditions end up unimportant at later times. Our idea was to reduce the contributing trajectories to those which are also important at the later time of interest.

While at the University of California, Berkeley, I developed a new method for calculating the monodromy matrix. The monodromy (or stability) matrix is a quantity of central importance in many semiclassical theories; it is required to calculate the semiclassical prefactor, which helps capture many quantum effects in semiclassical calculations. Calculating it is one of the most computationally difficult parts of many semiclassical calculations.

Under the direction of William H. Miller at The University of California, Berkeley, I worked on applying the semiclassical initial value representation [1] to systems with arbitrary holonomic constraints.

The semiclassical initial value representation (SC-IVR) was originally developed by W.H. Miller in the early 1970s. As computers have improved, it has re-emerged as a practical method to include approximate quantum effects in a wide range of dynamical molecular properties. P.-N. Roy has recently explored the idea of adding constraints to the SC-IVR [2], and I've been working on possible improvements to the framework he developed.

Under the direction of Dr. Cristian Predescu (at the time a post-doctoral scholar at the University of California, Berkeley) I used a new grid-based integration scheme to solve model electrostatic problems. We found that the method scaled very well for the interior space once accurate boundary conditions had been established.

The results of this project were presented at the Mini Statistical Mechanics Meeting held in Berkeley in January 2007. [Poster: 504K]

Under the direction of Prof. Clifford E. Dykstra of Indiana University-Purdue University Indianapolis, I used the Molecular Mechanics for Clusters [1] scheme to develop an approximate version of the H₂-benzene ab initio potential energy surface. Using previously developed parameters for the H₂-H₂ interaction, [2] I was able to show that our model represented clusters of H₂ around a benzene with good accuracy.