A promising approach to the quantum dynamics of condensed-phase systems is Feynman’s path integral formulation of time-dependent quantum mechanics. In this, the quantum mechanical amplitude is expressed as the sum of amplitudes along all paths that connect the initial and final points or states. The path integral formulation avoids the unrealistic memory demands of the Schrodinger equation at the cost of introducing auxiliary integration variables (often referred to as “beads”). Monte Carlo techniques are ideally suited for evaluating the path integral in imaginary time, leading to the powerful Path Integral Monte Carlo (PIMC) method for simulating the equilibrium properties of many-body quantum systems. However, the real-time path integral involves a rapidly oscillatory phase (the essence of quantum interference), which leads to Monte Carlo methods to converge very poorly. The difficulty associated with the so-called “sign problem” increases exponentially with the number of integration variables. At the same time, the number of quantum paths grows exponentially with propagation time, quickly reaching astronomical values.
Our group has pioneered the development of path integral methods which allow efficient evaluation of the path sum without ad hoc assumptions. These methods are applicable to specific types of Hamiltonians with broad relevance to processes encountered in chemical reactions, biological processes, and low-temperature physics.
- The QuAPI Method for System-Bath Dynamics
- Small-Matrix Path Integral (SMatPI) Decomposition
- The Blip Decomposition
- Path Integral Methods for Anharmonic Environments
- Quantum-Semiclassical Dynamics
- Quantum-Classical Path Integral
- The Modular Path Integral