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Partial averaging approach to Fourier coefficient path integration
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19.Actually, we have not proved that terms with but not an integer do not occur. However, an appeal to the Wigner‐Kirkwood expansion enables us to leave out such terms with confidence. In any case, the important point for present purposes is simply that on the right‐hand side of Eq. (6.2).
20.It is somewhat unusual to propagate wave packets under the imaginary time Schrüdinger equation. However, it is not unheard of. For example, the calculation of finite temperature correlation functions (infrared and two‐surface) can be implemented by propagating localized wave packets alternately in real and imaginary time. [Cf. J. R. Reimers, K. R. Wilson, and E. J. Heller, J. Chem. Phys. 79, 4749 (1983).]
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31.By “linear exponentials” we mean functions of the form where x is the d dimensional Cartesian coordinate vector as defined in Sec. II and γ is a vector of scale constants. We wish to distinguish such functions from radial exponentials with r the three‐dimensional distance between two particles. Radial exponential functions, which arise in the study of quantum fluids, are not strictly Gaussian transformable.
32.It is important to stress that the partial averaging procedure does not completely eliminate this difficulty. It would, however, appear to lessen the problem since fewer integration variables suffice to obtain converged results. One can then afford to sample these variables more vigorously in order to reduce statistical error bars to a tolerable size.
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