Gaussian integration, harmonic oscillators

The quantum-to-classical mapping for this system follows the same Feynman path integral procedure used in the first example. The harmonic oscillator degrees of freedom lead to Gaussian integrals and can thus be integrated out exactly. The resulting classical Hamiltonian reads... 

Figure 1 Two potential mutations of an isolated harmonic oscillator which illustrate the difference in the convergence properties of the perturbation and integration formuia. A represents a shift in the position,. ro, of the oscillator and B represents a shift in the offset, C, of the potential energy. The parabolic curves correspond to the potential energy, U x) (equation 25) and the Gaussian curves correspond to the probability distribution of the particle...

The remaining terms in the system-bath Hamiltonian correspond to linearly displaced harmonic oscillators which enter the path integral in a Gaussian fashion. Following the procedure of Feynman and Vernon, these Gaussian variables... 

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