This is the golden age of theoretical chemistry. Advances in computer hardware and software sometimes allow calculation of experimental results faster and cheaper than actually doing the experiments (and without hazardous waste to dispose of). We use a range of computers from large supercomputer clusters to personal computers to predict or explain the results of complex experiments.

My research has evolved over the years. Previous interests include:

1. Vibrational and Rotational Dynamics. Because molecules can easily have too many quantum states to perform traditional quantum calculations, we developed ways of getting quantum information out of classical mechanics. Using numerically integrated trajectories of molecular vibration and rotation, we applied the Einstein-Brillouin-Keller (EBK) quantization condition to give us approximations to quantum states in classical system. We developed a new Fourier transform methods for studying the quantum mechanical meaning of classically chaotic motion to understand the nature of quantum states and the spectra of vibrational and rotational systems (J. Chem. Phys. 101, 7763 (1994)).

2. Infrared Multiple Photon Absorption. Molecules can concertedly absorb several photons of the same or different frequencies to reach a state of high excitation. We developed an innovative geometrical way to understand how the relative quantum phases between states interact to make multiphoton absorption possible. We also investigated infrared multiphoton absorption using rapidly pulsed light. This absorption process can be thought of as the absorption of several different photons of different frequencies, and it can both be very rapid and reach highly excited vibrational states in a single mode (J. Chem. Phys. 94, 3451 (1991)).

3. Molecular Recognition. We studied ion specific binding by oxygen-containing ligands using quantum mechanical calculations and force fields methods. We showed that binding of ions to polar molecules cannot be accurately modeled using traditional forcefields which neglect atomic polarizability and the ability of partial atomic charges to redistribute within a molecule. (J. Inclusion. Phenom. 20, 297 (1995)).

My current research is in the statistical mechanics of nanoscale systems. As experimenters work on smaller systems, finite-size effects (differences from bulk behavior) become increasing important. One important size effect is that the Boltzmann distribution for one-particle energies is no longer accurate. Some of our recent results include:

• Exact, microcanonical, classical, one-particle energy distributions for arbitrary particle number and total energy. These can deviate significantly from the Boltzmann distribution at high energies.

• Approximate quantum one-particle energy distributions that show behavior similar to the classical distributions (J. Chem. Phys. 117, 5564 (2002)).

• Corrections to hard-sphere virial coefficients depending on both the number of particles and the size of the particle compared to the size of the container.

• Corrections to the one-particle energy and velocity distributions in computer simulations using periodic boundary conditions. Imposition of these, commonly used boundary conditions alter the usual relationships between the temperature and the average relative velocity, for example. In particular, for a mixture of heavy and light particles, the two sets of particles will actually have slightly different temperatures. (J. Chem.. Phys. 125, 164102 (2006), violating the equipartition principle.

All of these finite-system distributions limit to the standard ones for bulk systems, but corrections are required when doing theoretical simulations of finite systems or when interpreting nanoscale experiments.

The plot below shows the logarithm of probability versus energy in the simulation of 20 two-dimensional hard spheres in a reflecting box. The average energy is 25 (in arbitrary units). The black line is the Boltzmann distribution. The red diamonds are the numerical results from the simulation. The blue line is our theoretical distribution. It agrees with the simulation results within statistical errors. Note that the actual probability of having a large energy is considerably less than the value predicted by the Boltzmann distribution. This might be of considerable importance, for example, in predicting the rate of a unimolecular reaction that depends on the probability of the reactive degree of freedom having an energy above the activation energy. The actual reaction rate may be orders of magnitude less than that predicted using a Boltzmann distribution.

The following plot shows the breakdown of the equipartition principle in a simulation of ten hard-spheres with mass 1 and ten with mass 4 using periodic boundary conditions. The theoretically predicted equilibrium energy distributions and their averages are correctly predicted (1.0316 and 0.9684) within statistical errors (±0.0004), but the averages, and thus the temperatures are different by over 6%.