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1.W. H. Miller, J. Phys. Chem. A 105, 2942 (2001).
2.T. J. H. Hele, M. J. Willatt, A. Muolo, and S. C. Althorpe, J. Chem. Phys. 142, 134103 (2015).
3.J. Cao and G. A. Voth, J. Chem. Phys. 100, 5106 (1994).
4.S. Jang and G. A. Voth, J. Chem. Phys. 111, 2371 (1999).
5.I. R. Craig and D. E. Manolopoulos, J. Chem. Phys. 121, 3368 (2004).
6.S. Habershon, D. E. Manolopoulos, T. E. Markland, and T. F. Miller III, Annu. Rev. Phys. Chem. 64, 387 (2013).
7. This article considers only dynamics on a single Born–Oppenheimer potential energy surface; it might be possible to extend the analysis to multi-surface methods using the Meyer-Miller representation;
7.see H. D. Meyer and W. H. Miller, J. Chem. Phys. 70, 3214 (1979);
7.N. Ananth, ibid. 139, 124102 (2013);
7.J. O. Richardson and M. Thoss, ibid. 139, 031102 (2013).
8. To reduce clutter, we define CAB(t) without the factor of 1/Z (where Z is the quantum partition function).
9. To simplify the algebra, we assume that the system is one-dimensional, and that and are functions of . Matsubara dynamics generalizes straightforwardly to multi-dimensions and to operators that are functions of ; see Ref. 2.
10. Note that this distribution corresponds to the Matsubara (i.e., smoothed) version of the ring-polymer distribution of Ref. 11 which converges in the large M limit to the same distribution as the more commonly used discrete form of Refs. 12 and 13.
11.D. L. Freeman and J. D. Doll, J. Chem. Phys. 80, 5709 (1984).
12.D. Chandler and P. G. Wolynes, J. Chem. Phys. 74, 4078 (1981).
13.M. Parrinello and A. Rahman, J. Chem. Phys. 80, 860 (1984).
14.See supplementary material at for details of the contour-integration and for a harmonic analysis of the fluctuations.[Supplementary Material]
15.R. L. C. Akkermans and W. J. Briels, J. Chem. Phys. 113, 6409 (2000).
16.See Q. Shi and E. Geva, J. Chem. Phys. 118, 8173 (2003), which obtains CMD by decoupling the centroid from LSC-IVR dynamics (which is equivalent to filtering out the non-Matsubara modes from LSC-IVR, then following the steps above).
17.A. Witt, S. D. Ivanov, M. Shiga, H. Forbert, and D. Marx, J. Chem. Phys. 130, 194510 (2009).
18. Strictly speaking, this is the Matsubara-smoothed version of RPMD, but it converges in the large M limit to the more familiar (and practical) discrete version of RPMD in Refs. 5 and 6; see also Ref. 10.
19. The only property that cannot be derived from Matsubara dynamics is that CMD is equivalent to a minimum-energy wave-packet treatment in the limit T → 0 (where, clearly, one does not expect a description based on quantum statistics and classical dynamics to be valid);
19.see R. Ramírez and T. López-Ciudad, J. Chem. Phys. 111, 3339 (1999).
20.B. J. Braams and D. E. Manolopoulos, J. Chem. Phys. 125, 124105 (2006).
21.S. Jang, A. V. Sinitskiy, and G. A. Voth, J. Chem. Phys. 140, 154103 (2014).
22. Note that two of the RPMD normal-mode frequencies become imaginary on cooling below the instanton cross-over temperature of βħ = 2π/|ω|;
22.see J. O. Richardson and S. C. Althorpe, J. Chem. Phys. 131, 214106 (2009).
23.T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 138, 084108 (2013);
23.T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 139, 084115 (2013);
23.T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 139, 084116 (2013).
24. This may explain the success of the recently developed thermostatted-RPMD method;
24.see M. Rossi, M. Ceriotti, and D. E. Manolopoulos, J. Chem. Phys. 140, 234116 (2014).
25. The RPMD, Matsubara, and exact quantum results would tend to the same t → ∞ limit in the case of an ergodic system (which this is not).

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We recently obtained a quantum-Boltzmann-conserving classical dynamics by making a single change to the derivation of the “Classical Wigner” approximation. Here, we show that the further approximation of this “Matsubara dynamics” gives rise to two popular heuristic methods for treating quantum Boltzmann time-correlation functions: centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD). We show that CMD is a mean-field approximation to Matsubara dynamics, obtained by discarding (classical) fluctuations around the centroid, and that RPMD is the result of discarding a term in the Matsubara Liouvillian which shifts the frequencies of these fluctuations. These findings are consistent with previous numerical results and give explicit formulae for the terms that CMD and RPMD leave out.


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