Schematic illustration of the complex-time plane. The disk bounded by the dotted circle shows the complex-time points for which . The dashed line indicates the contour along which the symmetrized correlation function is obtained at .
Expectation value of the (a) total and (b) kinetic energy per atom as a function of time at . The solid circles show the results of the pair-product approximation, Eq. (5.2). The hollow triangles show results obtained using the Trotter formula for the complex-time propagators. The solid square shows the result of the PIMC calculation.
Radial distribution function as a function of time for liquid para-hydrogen at . Solid circles: . Hollow squares: . Up triangles: . Down triangles: . Hollow circles: . The solid line shows the results obtained from a numerically exact PIMC simulation. The dotted line shows the results of the single-step Trotter approximation.
The partition function as a function of time obtained from Eq. (5.9).
The velocity autocorrelation function for liquid para-hydrogen at . Solid circles: quantum mechanical results obtained from Eq. (5.10). Hollow triangles: results obtained using the single-bead Trotter approximation to the complex-time propagator. The arrow shows the converged results of a PIMC calculation at using 30 beads.
Comparison of the accurate quantum mechanical results for the symmetrized velocity autocorrelation function of liquid para-hydrogen at to those obtained using the FBSD approximation. Black circles: quantum PP propagator results obtained from Eq. (5.10). Solid line: transformed FBSD results. Dashed line: scaled transformed FBSD results.
The normalized self-intermediate scattering function, Eq. (5.15), for liquid para-hydrogen at . Hollow triangles: . Hollow diamonds: . Hollow circles: . Solid squares: .
Numerical (points) and fitted (line) forms of the normalized self-intermediate scattering function at .
The normalized dynamic structure factor, Eq. (5.15), for liquid para-hydrogen for at . Solid line: PPP results. Dashed line: results using the FBSD velocity autocorrelation function in the Gaussian approximation.
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