The Kleinert Variational Perturbation Theory

The path-integral (PI) representation of the quantum canonical partition function Qqm for a quantized particle can be written in terms of the effective centroid potential IT as a classical configuration integral  

Details of the derivation for Kleinert s variational perturbation (KP) theory can be found elsewhere [21], The nth-order KP approximation W, (r) to the centroid potential W (f) is given by 

The remaining terms in Eq. (4-24) are the nth-order corrections to approximate the real system, in which the expectation value ( c is called cumulant, which can be written in terms of the standard expectation value ( by cumulant expansion in terms of Gaussian smearing convolution integrals  

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as  

The computational procedure for obtaining the first and second order KP approximations to the centroid potential is summarized below  

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