ABSTRACT
This is an important chapter because it introduces the temporal degree of freedom. Time evolution opens up opportunities to develop several powerful exact methods that circumvent the basis set exponential growth bottleneck. After introducing the time-dependent Schro¨dinger equation, the reader is guided to experience the time evolution using the most elementary model of quantum mechanics, the particle in a box. Then, we move on to the matrix representation of the time evolution operator, with the formal solution to the coupled equations and their first order Magnus expansion propagator. These methods are used to tackle the time evolution of quantum systems even when the Hamiltonian is time dependent and the equation is no longer separable. We explore the development of real-time path integrals. A separate chapter is dedicated to the development of computation methods for imaginary time path integrals. In this chapter, we develop two important stochastic ground state methods in imaginary time that are complementary to the path integral of statistical mechanics: the diffusion Monte Carlo (DMC) method and the variational Monte Carlo (VMC) method. The latter approach provides the approximate importance sampling function for the guided DMC algorithms. Two Importance Sampling Diffusion Monte Carlo (IS-DMC) methods are presented.
