Nonequilibrium statistical mechanics:

• As a member of Professor Todd Gingrich’s group, my research was focused on the understanding of complex microscopic behavior exhibited in systems that are driven away from equilibrium. Typically, the methods employed to elucidate details of such systems involve brute-force stochastic sampling algorithms such as Langevin dynamics and Monte Carlo methods, which tend to be inefficient especially when rare events qualitatively impact the overall macroscopic behavior. In my research, by modeling stochastic systems as Markov jump processes, I have been able to complement these costly sampling methods with powerful and cutting-edge numerical approaches such as tensor network methods. Thus, instead of sampling many trajectories, these alternative methods can directly evolve full distributions, and can much more efficiently and accurately account for rare but non-negligible fluctuations.
• My latest work involved the development of methodologies that would allow me to study the kinetics of chemical reaction networks, with potential applications to both chemical and biological systems. These methodologies were mainly based on transition path theory, whose goal is to enhance and facilitate the sampling of rare but crucial events within complex reaction networks, such as “reactive transitions” or transitions between metastable regions. I successfully identified sequences of reaction events responsible for triggering transitions from one metastable regime to another in a gene toggle switch (GTS). A publication of this work is forthcoming.
• In my most recently published research, I have successfully used tensor network methods, namely the density matrix renormalization group (DMRG) and time-dependent variational principle (TDVP) algorithms, to shed light on the dynamics of a multiparticle Brownian ratchet modeled as a Markov jump process on a grid. Ratchets are systems involving one or more particles which feel a spatially and temporally periodic force with local asymmetries, and as a result break time-reversal symmetry, allowing them to move in a preferred direction. When a single particle is involved, the jump process simply relies on traditional matrix operations such as diagonalization and exponentiation. As soon as more particles are added, however, dimensionality increases rapidly as the size of the relevant matrices grows exponentially with the lattice size and combinatorially with the number of particles. Tensor networks serve as an efficient and accurate way to combat this curse of dimensionality. Nonequilibrium steady-state currents of this multiparticle ratchet were successfully elucidated as a function of both the number of particles and the frequency of the periodic drive. This work is presented in two recent publications, in the form of a short communication focusing on the physics of the ratchet, and a longer article, which delves into the details of the tensor network methodology.
• My earlier published work is related to single-particle ratchets, and has mainly focused on current reversals, or the flipping of the sign of the current as various parameters, such as the frequency of the temporal driving and the particle density, are tuned (see here).

Electronic structure theory of heavy elements:

• During my first year at the PhD program at Northwestern, while working in Professor Toru Shiozaki’s lab, my research involved the electronic structure of heavy metals. A key aspect of this work is that the nature of such elements requires the inclusion of relativistic effects within relevant quantum chemistry algorithms. While full relativistic treatments (using 4-component methods with or without Gaunt/Breit interactions) are sometimes required, we found that often some accuracy could be sacrificed for a computationally less costly alteration of the non-relativistic Schrodinger equation.
• As part of this research, I contributed to the BAGEL (“Brilliantly Advanced General Electronic Structure Library”), a program package written in C++ from scratch by the group over the years. I successfully contributed to the BAGEL, by deriving and implementing analytical nuclear gradients (first derivatives with respect to nuclear displacements), applicable to energies obtained from any level of theory, incorporating scalar relativistic Douglas-Kroll-Hess (DKH) corrections. To achieve this, I derived a Lagrangian expression for the DKH energy, resulting in a mathematical relationship between the gradients and the atomic orbital integrals. One major disadvantage of previous DKH gradient strategies is that they involve the temporary storage of large matrices which render the computations highly memory intensive. The novelty of my approach is that it involves contractions of individual matrix elements on the fly, thus obviating the need to store matrices in memory. My DKH gradient approach conveniently couples with any level of theory, arbitrarily more advanced than Hartree-Fock. Our work resulted in a published article, which can be found here. Visit https://nubakery.org/for instructions on how to install and run BAGEL, as well as details about authors’ contributions. The latest version of BAGEL, which contains my full code, can be found here.

My codes within BAGEL:

• • src/grad/dkhgrad.cc
  • src/grad/dkhgrad.h
  • src/wfn/contractmat.h
  • src/wfn/contractmat.cc
  • src/util/math/diagmatrix.h