#### Keynote Talk V

**Path Integrals for Nonadiabatic Processes: Quantum Dynamics from Classical Trajectories**

The challenge of designing new materials for efficient charge and energy transfer requires new theoretical methods that can accurately capture quantum effects in the simulation of complex systems such as organic photovoltaics for solar energy harvesting and transition metal catalysts for water splitting. Methods based on path integrals are particularly promising as they provide a classically isomorphic representation of quantum mechanics. In the first half of this talk, we will discuss ‘imaginary-time’ path integrals that allow us to employ classical molecular dynamics to calculate quantum equilibrium properties for complex systems. We then intro- duce approximate, path-integral based, quantum dynamic methods for the simulation of charge transfer in the condensed phase. We numerically demonstrate the efficiency and the accuracy of these methods in atomistic simulations of thermal electron transfer reactions in transition metal complexes as shown in Fig. 1A.

**Figure 1:** (A) Classical ring polymers that exactly describe the properties of a quantum system at thermal equilibrium. To a good approximations, the classical dynamics of these ring polymers accurately describe electron transfer in transition metal complex like Cobalt Hexammine self-exchange reaction shown here. (B) A cartoon of a novel classical-like ring polymers that represents multi-state quantum systems and exactly describes their equilibrium properties. Approximate dynamics based on these ring polymers can be used to model photo-initiated excited state dynamics. For instance, we can simulate the dynamics of singlet fission, a phenomenon exhibited by certain organic molecules that can significantly increase the efficiency of photovoltaic devices.

In the second half, we discuss path integrals for multi-state systems where coupled electron-nuclear motion (nonadiabatic effects) play an important role. We map discrete electronic states to continuous classical-like variables and derive a path integral formulation that is exact for quantum equilibrium properties as shown in Fig. 1B. We extend these ideas to develop an approximate dynamic methods for the simulation of photo-initiated excited state dynamics, and we conclude with a discussion of the applications, limitations, and future prospects of this class of methods.

**Recommended Reading:**

1. *The Path Integral Representation of Quantum Mechanics:*

(i) Quantum Mechanics and Path Integrals, R. P. Feynman and A. R. Hibbs, McGraw-Hill (1965).

(ii) Statistical Mechanics: Theory and Molecular Simulation, M. E. Tuckerman, Oxford University Press (2010).

2. *Ring Polymers for Thermal Electron Transfer:*

(i) S. Habershon, D. E. Manolopoulos, T. E. Markland, and T. F. Miller III, Annu. Rev. Phys. Chem., 64, 387 (2013). (ii)A. Menzeleev, N. Ananth and T. F. Miller, III, J. Chem. Phys., 135, 074106 (2011). (iii) R. L. Kenion and N. Ananth, Phys. Chem. Chem. Phys., 18, 26117 (2016).

3. *Excited State Dynamics with Ring Polymers:*

(i) N. Ananth, J. Chem. Phys., 139, 124102 (2013). (ii) J. R. Duke and N. Ananth, J. Phys. Chem. Lett., 6, 4219 (2015).