280
X. Liu et al.
36. Müser MH (2002) On new efficient algorithms for pimc and pimd. Comput Phys Commun
147(1–2):83–86
37. Drozdov AN, Talkner P (1998) Path integrals for fokker-planck dynamics with singular diffusion: accurate factorization for the time evolution operator. J Chem Phys 109(6):2080–2091
38. Tuckerman ME, Marx D, Klein ML, Parrinello M (1996) Efficient and general algorithms for
path integral car-parrinello molecular dynamics. J Chem Phys 104(14):5579–5588
39. Ceriotti M, Parrinello M, Markland TE, Manolopoulos DE (2010) Efficient stochastic
thermostatting of path integral molecular dynamics. J Chem Phys 133(12):124104
40. Liu X, Liu J (2018) Path integral molecular dynamics for exact quantum statistics of multielectronic-state systems. J Chem Phys 148(10):102319
41. Wang H, Liu X, Liu J (2018) Accurate calculation of equilibrium reduced density matrix for
the system-bath model: A multilayer multiconfiguration time-dependent hartree approach and
its comparison to a multi-electronic-state path integral molecular dynamics approach. Chin J
Chem Phys 31(4):446–456
42. Suzuki M (1985) Decomposition formulas of exponential operators and lie exponentials with
some applications to quantum mechanics and statistical physics. J Math Phys 26(4):601–612
43. Yoshida H (1990) Construction of higher order symplectic integrators. Phys Lett A 150(5):262–
268
44. Suzuki M (1991) General theory of fractal path integrals with applications to many-body
theories and statistical physics. J Math Phys 32(2):400–407
45. Ceperley DM (1995) Path integrals in the theory of condensed helium. Rev Mod Phys
67(2):279–355
46. Feynman RP (1953) Atomic theory of the lambda-transition in helium. Phys Rev 91(6):1291–
1301
47. Tuckerman ME, Berne BJ, Martyna GJ, Klein ML (1993) Efficient molecular-dynamics and
hybrid monte-carlo algorithms for path-integrals. J Chem Phys 99(4):2796–2808
48. Herman MF, Bruskin EJ, Berne BJ (1982) On path integral monte-carlo simulations. J Chem
Phys 76(10):5150–5155
49. Pollock EL, Ceperley DM (1984) Simulation of quantum many-body systems by path-integral
methods. Phys. Rev. B 30(5):2555–2568
50. Liu J, Li D, Liu X (2016) Supplementary material for the paper ‘a simple and
accurate algorithm for path integral molecular dynamics’. J Chem Phys 145:024103.
ftp://ftp.aip.org/epaps/journ_chem_phys/E-JCPSA6-145-007626
51. Feynman RP, Hibbs AR (1965) Quantum mechanics and path integrals. McGraw-Hill, New
York
52. Cao J, Berne BJ (1993) A Born-Oppenheimer approximation for path-integrals with an
application to electron solvation in polarizable fluids. J Chem Phys 99(4):2902–2916
53. Ryckaert J-P, Ciccotti G, Berendsen HJC (1977) Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes. J Comput Phys
23(3):327–341
54. Andersen HC (1983) Rattle: A “velocity” version of the shake algorithm for molecular
dynamics calculations. J Comput Phys 52(1):24–34
55. Hamelberg D, Mongan J, McCammon JA (2004) Accelerated molecular dynamics: A promising
and efficient simulation method for biomolecules. J Chem Phys 120(24):11919–11929
56. Leimkuhler B, Margul DT, Tuckerman ME (2013) Stochastic, resonance-free multiple timestep algorithm for molecular dynamics with very large time steps. Mol Phys 111(22–23):3579–
3594
57. Hasegawa T, Tanimura Y (2011) A polarizable water model for intramolecular and intermolecular vibrational spectroscopies. J Phys Chem B 115(18):5545–5553
58. Torrie GM, Valleau JP (1977) Nonphysical sampling distributions in monte carlo free-energy
estimation: Umbrella sampling. J Comput Phys 23(2):187–199
59. Laio A, Parrinello M (2002) Escaping free-energy minima. Proc Natl Acad Sci 99(20):12562
60. Gao YQ (2008) An integrate-over-temperature approach for enhanced sampling. J Chem Phys
128(6):064105
X. Liu et al.
36. Müser MH (2002) On new efficient algorithms for pimc and pimd. Comput Phys Commun
147(1–2):83–86
37. Drozdov AN, Talkner P (1998) Path integrals for fokker-planck dynamics with singular diffusion: accurate factorization for the time evolution operator. J Chem Phys 109(6):2080–2091
38. Tuckerman ME, Marx D, Klein ML, Parrinello M (1996) Efficient and general algorithms for
path integral car-parrinello molecular dynamics. J Chem Phys 104(14):5579–5588
39. Ceriotti M, Parrinello M, Markland TE, Manolopoulos DE (2010) Efficient stochastic
thermostatting of path integral molecular dynamics. J Chem Phys 133(12):124104
40. Liu X, Liu J (2018) Path integral molecular dynamics for exact quantum statistics of multielectronic-state systems. J Chem Phys 148(10):102319
41. Wang H, Liu X, Liu J (2018) Accurate calculation of equilibrium reduced density matrix for
the system-bath model: A multilayer multiconfiguration time-dependent hartree approach and
its comparison to a multi-electronic-state path integral molecular dynamics approach. Chin J
Chem Phys 31(4):446–456
42. Suzuki M (1985) Decomposition formulas of exponential operators and lie exponentials with
some applications to quantum mechanics and statistical physics. J Math Phys 26(4):601–612
43. Yoshida H (1990) Construction of higher order symplectic integrators. Phys Lett A 150(5):262–
268
44. Suzuki M (1991) General theory of fractal path integrals with applications to many-body
theories and statistical physics. J Math Phys 32(2):400–407
45. Ceperley DM (1995) Path integrals in the theory of condensed helium. Rev Mod Phys
67(2):279–355
46. Feynman RP (1953) Atomic theory of the lambda-transition in helium. Phys Rev 91(6):1291–
1301
47. Tuckerman ME, Berne BJ, Martyna GJ, Klein ML (1993) Efficient molecular-dynamics and
hybrid monte-carlo algorithms for path-integrals. J Chem Phys 99(4):2796–2808
48. Herman MF, Bruskin EJ, Berne BJ (1982) On path integral monte-carlo simulations. J Chem
Phys 76(10):5150–5155
49. Pollock EL, Ceperley DM (1984) Simulation of quantum many-body systems by path-integral
methods. Phys. Rev. B 30(5):2555–2568
50. Liu J, Li D, Liu X (2016) Supplementary material for the paper ‘a simple and
accurate algorithm for path integral molecular dynamics’. J Chem Phys 145:024103.
ftp://ftp.aip.org/epaps/journ_chem_phys/E-JCPSA6-145-007626
51. Feynman RP, Hibbs AR (1965) Quantum mechanics and path integrals. McGraw-Hill, New
York
52. Cao J, Berne BJ (1993) A Born-Oppenheimer approximation for path-integrals with an
application to electron solvation in polarizable fluids. J Chem Phys 99(4):2902–2916
53. Ryckaert J-P, Ciccotti G, Berendsen HJC (1977) Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes. J Comput Phys
23(3):327–341
54. Andersen HC (1983) Rattle: A “velocity” version of the shake algorithm for molecular
dynamics calculations. J Comput Phys 52(1):24–34
55. Hamelberg D, Mongan J, McCammon JA (2004) Accelerated molecular dynamics: A promising
and efficient simulation method for biomolecules. J Chem Phys 120(24):11919–11929
56. Leimkuhler B, Margul DT, Tuckerman ME (2013) Stochastic, resonance-free multiple timestep algorithm for molecular dynamics with very large time steps. Mol Phys 111(22–23):3579–
3594
57. Hasegawa T, Tanimura Y (2011) A polarizable water model for intramolecular and intermolecular vibrational spectroscopies. J Phys Chem B 115(18):5545–5553
58. Torrie GM, Valleau JP (1977) Nonphysical sampling distributions in monte carlo free-energy
estimation: Umbrella sampling. J Comput Phys 23(2):187–199
59. Laio A, Parrinello M (2002) Escaping free-energy minima. Proc Natl Acad Sci 99(20):12562
60. Gao YQ (2008) An integrate-over-temperature approach for enhanced sampling. J Chem Phys
128(6):064105