Abstract
Conventionally, Path Integral Monte Carlo (PIMC) calculations are performed with ‘classical beads’ (beads interacting via a classical potential) by using the primitive approximation for the thermal density matrix. Higher order propagators of the thermal density matrix have been proven to achieve faster convergence and better precision in quantum calculations than using just the primitive approximation. Use of different propagators in PIMC leads to methods equivalent to performing PIMC with ‘semi-classical beads’ (beads interacting via a semi-classical potential). We examine the Takahashi-Imada (TI) propagator as well as an ad hoc semi-classical potential in PIMC calculations for computing the quantum second virial coefficient for helium-4. We compare the performance of the two approaches based on semi-classical beads against values computed from PIMC using conventional classical beads. We find that while the TI propagator has the same or marginally better precision compared to the classical case, it has the best convergence rate (with respect to number of path-integral beads) among the three approaches. The convergence rate of the ad hoc potential is marginally better than its classical counterpart, and its precision is approximately the same as the classical case.
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References
Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals, 1st edn. McGraw-Hill Companies, Inc., New York (1965). Emended by Daniel F. Styer
Ceperley, D.: Path integrals in the theory of condensed helium. Rev. Mod. Phys. 67, 279 (1995)
Cui, T., Cheng, E., Alder, B., Whaley, K.: Rotational ordering in solid deuterium and hydrogen: a path integral Monte Carlo study. Phys. Rev. B 55, 12253 (1997)
Takahashi, M., Imada, M.: Monte Carlo calculation of quantum systems. II. higher order correction. J. Phys. Soc. Jpn. 53, 3765–3769 (1984)
Schenter, G.K.: The development of effective classical potentials and the quantum statistical mechanical second virial coefficient of water. J. Chem. Phys. 117, 6573 (2002)
Janke, W., Sauer, T.: Properties of higher-order Trotter formulas. Phys. Lett. A 165, 199–205 (1992)
Suzuki, M.: Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations. Phys. Lett. A 201, 425–428 (1995)
Yamamoto, T.M.: Path-integral virial estimator based on the scaling of fluctuation coordinates: application to quantum clusters with fourth-order propagators. J. Chem. Phys. 123, 104101 (2005)
Garberoglio, G., Harvey, A.H.: First-principles calculation of the third virial coefficient of helium. J. Res. Natl. Inst. Stand. Technol. 114, 249 (2009)
Fellmuth, B., Gaiser, C., Fischer, J.: Determination of the Boltzmann constant—status and prospects. Meas. Sci. Technol. 17, R145–R159 (2006)
Schmidt, J.W., Gavioso, R.M., May, E.F., Moldover, M.R.: Polarizability of helium and gas metrology. Phys. Rev. Lett. 98, 254504 (2007)
Pitre, L., Moldover, M.R., Tew, W.L.: Acoustic thermometry: new results from 273 K to 77 K and progress towards 4K. Metrologia 43, 142–162 (2006)
Moldover, M.R., McLinden, M.O.: Using ab initio data to accurately determine the fourth density virial coefficient of helium. J. Chem. Thermodyn. 42, 1193–1203 (2010)
Aziz, R.A., Janzen, A.R., Moldover, M.R.: Ab initio calculations for helium: a standard for transport property measurements. Phys. Rev. Lett. 74, 1586–1589 (1995)
Shaul, K.R.S., Schultz, A.J., Kofke, D.A., Moldover, M.R.: Semiclassical fifth virial coefficients for improved ab initio helium-4 standards. Chem. Phys. Lett. 531, 11–17 (2012)
Garberoglio, G., Harvey, A.H.: Path-integral calculation of the third virial coefficient of quantum gases at low temperatures. J. Chem. Phys. 134, 134106 (2011)
Garberoglio, G., Moldover, M.R., Harvey, A.H.: Improved first-principles calculation of the third virial coefficient of helium. J. Res. Natl. Inst. Stand. Technol. 116, 729–742 (2011)
Shaul, K.R., Schultz, A.J., Kofke, D.A.: Path-integral Mayer-sampling calculations of the quantum Boltzmann contribution to virial coefficients of helium-4. J. Chem. Phys. 137, 184101 (2012)
Tester, J.W., Modell, M.: Thermodynamics and Its applications, 3rd edn. Prentice Hall Inc, New Jersey (1997)
Masters, A.J.: Virial expansions. J. Phys.: Condens. Matter 20, 283102 (2008)
Hansen, J.P., McDonald, I.R.: Theory of Simple Liquids, 3rd edn. Academic Press (2006)
Przybytek, M., Cencek, W., Komasa, J., Łach, G., Jeziorski, B., Szalewicz, K.: Relativistic and quantum electrodynamics effects in the helium pair potential. Phys. Rev. Lett. 104, 183003 (2010)
Singh, J.K., Kofke, D.A.: Mayer sampling: calculation of cluster integrals using free-energy perturbation methods. Phys. Rev. Lett. 92, 220601 (2004)
Schultz, A.J., Kofke, D.A.: Sixth, seventh and eighth virial coefficients of the Lennard-Jones model. Mol. Phys. 107, 2309 (2009)
Percus, J.K., Yevick, G.J.: Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 110, 1–13 (1958)
Shaul, K.R.S., Schultz, A.J., Perera, A., Kofke, D.A.: Integral-equation theories and Mayer-sampling Monte Carlo: a tandem approach for computing virial coefficients of simple fluids. Mol. Phys. 109, 2395–2406 (2011)
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This work is supported by the U.S. National Science Foundation (CHE-1027963).
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Subramanian, R., Schultz, A.J., Kofke, D.A. (2016). Quantum Virial Coefficients via Path Integral Monte Carlo with Semi-classical Beads. In: Snurr, R., Adjiman, C., Kofke, D. (eds) Foundations of Molecular Modeling and Simulation. Molecular Modeling and Simulation. Springer, Singapore. https://doi.org/10.1007/978-981-10-1128-3_6
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DOI: https://doi.org/10.1007/978-981-10-1128-3_6
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