Partition function Fourier coefficients

To obtain the partition function, we simply take the trace of the density matrix by setting x = x and integrating over all x. Realizing that for practical work we will generally truncate all of the Fourier series at some maximum number of Fourier coefficients ( ), we obtain the result Q = Urn -, QK such that... 

We can then evaluate the ratio of partition functions by a Monte Carlo calculation, drawing random samples uniformly from configuration space (this inefficient scheme will be improved on later) and nonuniformly in the Fourier coefficient space according to P(a). We then simply multiply the average over many Monte Carlo samples by which is an analytic function, to obtain Q Ki. This result is then divided by <7, the symmetry number of the molecule, to finally obtain an approximation to the properly sym-... 

Figure 5. Plot of the partition function Q as a function of the hyperradius R of the great sphere enclosing the molecules H 0 and D20. A mass-weighted Jacobi coordinate system is used in each case. One Fourier coefficient is used per degree of freedom, and 10s Monte Carlo samples were used for each calculation. The Monte Carlo samples were drawn uniformly on the interior of the great sphere in coordinate space and from P(a) in Fourier coefficient space.

Vibration-rotation partition function for HC1 obtained via Fourier path-integral AOSS-U Monte Carlo calculations from Topper et al. [46]. Error bars are given at 95% confidence level (2w ). Unless otherwise noted, all calculations used = 128 Fourier coefficients per degree of freedom and n = 100000 Monte Carlo samples. 

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