Fourier path integrals

Path integrals can be expressed directly in Cartesian coordinates [1, 2] or can be transformed to Fourier variables [1,2,20,45]. A Fourier path integral method will be used here [20]. The major reason for doing this is that length scales are directly built 

In what follows, a one-dimensional picture will mainly be employed to avoid a notational proliferation. We will also consider a single atomic solute. The methods discussed are easily generalized to three dimensions and many particles the x, y, and z coordinates of the cyclic path for each atom are represented by a set of Fourier coefficients. 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation 

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20] 

5 This coordinate change is analogous to changing integration variables x, y, z to spherical polar coordinates r, 0, 4 - 

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Ortiz, V. Lopez, G. E., Fourier path integral Monte Carlo study of a two-dimensional model quantum monolayer, Mol. Phys. 2002,100, 1003-1009... 

Srinivisan, J. Volobuev, Y. L. Mielke, S. L. Truhlar, D. G., Parallel Fourier path-integral Monte Carlo calculations of absolute free energies and chemical equilibria, Comput. Phys. Commun. 2000, 128, 446-464... 

Doll, J. D. Beck, T. L. Freeman, D. L., Equilibrium and dynamical Fourier path integral... 

Mielke, S. L. Truhlar, D. G., A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical calculation of free energies, J. Chem. Phys. 2001,114, 621-630... 

IV. Fourier Path Integral Monte Carlo Methods... 

In Eq. (3.2) m is the mass of particle i at coordinate r,-, M is the total mass, and Rc is the constraining radius. We have chosen a continuous constraining potential because it is well suited to the Fourier path integral Monte Carlo calculations to be described in the next section. 

IV. FOURIER PATH INTEGRAL MONTE CARLO METHODS... 

We follow Doll et al. [42] and begin by writing a Fourier path-integral (FPI) formula for the ratio of the density matrix of the full system to that of a free-particle system (zero potential energy). The Jacobian and other prefactors cancel in the ratio, and we obtain... 

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. 

D. L. Freeman and J. D. Doll, j. Chem. Phys., 101, 848 (1984). Fourier Path Integral Methods for the Calculation of the Microcanonical Density of States. 

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