The coupled transfer of electrons and protons is a central feature of biological and molecular catalysis, yet fundamental aspects of these reactions remain poorly understood. In this study, we extend the ring polymer molecular dynamics (RPMD) method to enable direct simulation of proton-coupled electron transfer (PCET) reactions across a wide range of physically relevant regimes. In a system-ba...

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes...

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general timedependent one-dimensional harmonic oscillator, whose Newton equation q̈+ω(t)q = 0 cannot be solved in general. We follow the time-evolution of an initial e...

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL2(K) representation variety, where K = C or R, answering a question of W. Goldman. We also prove the existence of faithful representations into PU(2, 1) with certain nonintegral Toledo invariants.

The crepant resolution conjecture states that the Gromov–Witten invariants of an orbifold X should be determined in a precise way by the Gromov–Witten invariants of a crepant resolution of its coarse moduli space. We compute the Gromov–Witten invariants of the stack symmetric square of P 2 and compare them with the Gromov– Witten invariants of its crepant resolution, Hilb 2 P 2 (which were comp...

We examine the effects of frequency variation along the reaction path in intramolecular or lattice modes on the rate constant for light atom transfer in the condensed phase. Accurate quantum mechanical calculations using the quasi-adiabatic propagator path integral representation of the flux–flux auto-correlation function reveal significant variations in the reaction rate compared with the cons...