Path integral molecular dynamics

Path integral molecular dynamics (PIMD) is a method of incorporating quantum mechanics into molecular dynamics simulations using Feynman path integrals. In PIMD, one uses the Born–Oppenheimer approximation to separate the wavefunction into a nuclear part and an electronic part. The nuclei are treated quantum mechanically by mapping each quantum nucleus onto a classical system of several fictitious particles connected by springs (harmonic potentials) governed by an effective Hamiltonian, which is derived from Feynman's path integral. The resulting classical system, although complex, can be solved relatively quickly. There are now a number of commonly used condensed matter computer simulation techniques that make use of the path integral formulation including Centroid Molecular Dynamics (CMD),^{[1][2][3][4][5]} Ring Polymer Molecular Dynamics (RPMD),^[6][7] and the Feynman-Kleinert Quasi-Classical Wigner (FK-QCW) method.^[8][9] The same techniques are also used in path integral Monte Carlo (PIMC).^{[10][11][12][13][14]}

Applications

The technique has been used to calculate time correlation functions.^[15]

References

1. Cao, J.; Voth, G. A. (1994). "The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties". The Journal of Chemical Physics 100 (7): 5093. Bibcode:1994JChPh.100.5093C. doi:10.1063/1.467175.
2. Cao, J.; Voth, G. A. (1994). "The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamical properties". The Journal of Chemical Physics 100 (7): 5106. Bibcode:1994JChPh.100.5106C. doi:10.1063/1.467176.
3. Jang, S.; Voth, G. A. (1999). "A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables". The Journal of Chemical Physics 111 (6): 2371. Bibcode:1999JChPh.111.2371J. doi:10.1063/1.479515.
4. RamíRez, R.; LóPez-Ciudad, T. (1999). "The Schrödinger formulation of the Feynman path centroid density". The Journal of Chemical Physics 111 (8): 3339. arXiv:cond-mat/9906318. Bibcode:1999JChPh.111.3339R. doi:10.1063/1.479666.
5. Polyakov, E. A.; Lyubartsev, A. P.; Vorontsov-Velyaminov, P. N. (2010). "Centroid molecular dynamics: Comparison with exact results for model systems". The Journal of Chemical Physics 133 (19): 194103. Bibcode:2010JChPh.133s4103P. doi:10.1063/1.3484490. PMID 21090850.
6. Craig, I. R.; Manolopoulos, D. E. (2004). "Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics". The Journal of Chemical Physics 121 (8): 3368–3373. Bibcode:2004JChPh.121.3368C. doi:10.1063/1.1777575. PMID 15303899.
7. Braams, B. J.; Manolopoulos, D. E. (2006). "On the short-time limit of ring polymer molecular dynamics". The Journal of Chemical Physics 125 (12): 124105. Bibcode:2006JChPh.125l4105B. doi:10.1063/1.2357599. PMID 17014164.
8. Smith, Kyle K. G.; Poulsen, Jens Aage; Nyman, Gunnar; Rossky, Peter J. (2015-06-28). "A new class of ensemble conserving algorithms for approximate quantum dynamics: Theoretical formulation and model problems". The Journal of Chemical Physics 142 (24): 244112. doi:10.1063/1.4922887. ISSN 0021-9606.
9. Smith, Kyle K. G.; Poulsen, Jens Aage; Nyman, Gunnar; Cunsolo, Alessandro; Rossky, Peter J. (2015-06-28). "Application of a new ensemble conserving quantum dynamics simulation algorithm to liquid para-hydrogen and ortho-deuterium". The Journal of Chemical Physics 142 (24): 244113. doi:10.1063/1.4922888. ISSN 0021-9606.
10. Berne, B. J.; Thirumalai, D. (1986). "On the Simulation of Quantum Systems: Path Integral Methods". Annual Review of Physical Chemistry 37: 401. Bibcode:1986ARPC...37..401B. doi:10.1146/annurev.pc.37.100186.002153.
11. Gillan, M. J. (1990). "The path-integral simulation of quantum systems, Section 2.4". In C. R. A. Catlow, S. C. Parker and M. P. Allen. Computer Modelling of Fluids Polymers and Solids. NATO ASI Series C 293. pp. 155–188. ISBN 978-0-7923-0549-1.
12. Trotter, H. F. (1959). "On the Product of Semi-Groups of Operators". Proceedings of the American Mathematical Society 10 (4): 545–551. doi:10.1090/S0002-9939-1959-0108732-6. JSTOR 2033649.
13. Chandler, D. (1981). "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids". The Journal of Chemical Physics 74 (7): 4078–4071. Bibcode:1981JChPh..74.4078C. doi:10.1063/1.441588.
14. Marx, D.; Müser, M. H. (1999). "Path integral simulations of rotors: Theory and applications". Journal of Physics: Condensed Matter 11 (11): R117. doi:10.1088/0953-8984/11/11/003.
15. Cao, J.; Voth, G. A. (1996). "Semiclassical approximations to quantum dynamical time correlation functions". The Journal of Chemical Physics 104: 273. Bibcode:1996JChPh.104..273C. doi:10.1063/1.470898.

Further reading

• Feynman, R. P. (1972). "Chapter 3". Statistical Mechanics. Reading, Massachusetts: Benjamin. ISBN 0-201-36076-4.
• Morita, T. (1973). "Solution of the Bloch Equation for Many-Particle Systems in Terms of the Path Integral". Journal of the Physical Society of Japan 35 (4): 980–984. doi:10.1143/JPSJ.35.980.
• Wiegel, F. W. (1975). "Path integral methods in statistical mechanics". Physics Reports 16 (2): 57–37. doi:10.1016/0370-1573(75)90030-7.
• Barker, J. A. (1979). "A quantum-statistical Monte Carlo method; path integrals with boundary conditions". The Journal of Chemical Physics 70 (6): 2914–2911. Bibcode:1979JChPh..70.2914B. doi:10.1063/1.437829.
• Ceperley, D. M. (1995). "Path integrals in the theory of condensed helium". Reviews of Modern Physics 67 (2): 279–355. Bibcode:1995RvMP...67..279C. doi:10.1103/RevModPhys.67.279.
• Chakravarty, C. (1997). "Path integral simulations of atomic and molecular systems". International Reviews in Physical Chemistry 16 (4): 421–444. doi:10.1080/014423597230190.

External links

• "Density matrices and path integrals" (computer code). SMAC-wiki.
• John Shumway; Matthew Gilbert (2008). "Path Integral Monte Carlo Simulation".