x ≡ {x a } and p ≡ {p a } are the mapping variables associated with the forward propagator, and x and p are for the backward propagator, both of which evolving according to the classical mapping Hamiltonian H(R, x, p) in Eq. 51. Further, G = exp -1 2 a (p a 2 + x a 2 ) is the Gaussian distribution for the initial forward mapping variables, with an analogous expression G for backward mapping variables. In addition,