Scaling of the Zero-order Hamiltonian

5 Scaling of the Zero-order Hamiltonian. - A particularly simple modification of the zero-order hamiltonian is obtained by scaling. Let us recall that if, following Feenberg, we modify the zero-order hamiltonian operator by multiplying it by an arbitrary scalar, p, say, the zero-order hamiltonian is then 

SO that the full hamiltonian is recovered when these two modified operators are added. It can be easily demonstrated that the second-order energy is then given by 

However, this simple modification shows that there is some choice of zero-order hamiltonian which yields the exact correlation in second order. In such a case the remainder term is zero, 3R. 

Values of /i can be obtained from perturbation theory calculations taken to higher-order. The second-order energy provides a useful approximation to the total correlation energy 

In the past, a number of prescriptions for the determination of p have been investigated. The most popular is to set the modified third-order energy coefficient to zero 

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Rg. 14J. Convergence of the perturbation expansion for different values of the gap-shift parameter y using a = 1 and /S = 2 in the zero-order Hamiltonian (14.5.18) and 3 = in the pertuibation (14.5.19). For each order of the expansion, the error has been plotted on a logarithmic scale. 

Different approaches have been or are being investigated. Path integral approaches scale favorably with the number of degrees of freedom. However, an efficient real time path integral treatment requires the introduction of a simple reference Hamiltonian. The propagator associated with the reference Hamiltonian has to be known in closed form and the reference Hamiltonian has to yield a resonable zero order approximation for the total dynamics. 

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