Core features

i-PI includes a large number of advanced molecular dynamics features, with an obvious focus on path integral molecular dynamics, but also several methods for sampling classical trajectories. This is an (incomplete) list of some of the main features.

If you implement a major new feature, this is the place to briefly outline what it does and what are the papers to cite. Please follow as closely as possible the template.

Bosonic and Fermionic Path Integral Molecular Dynamics

PIMD simulations of bosonic particles with a polynomial scaling algorithm. Supports mixtures of distinguishable and bosonic particles. Fermionic statistics can be obtained by a reweigthing procedure by post processing the simulation (see ref. 2 below).

Main contributors: Yotam Feldman, Barak Hirshberg

Theory and Implementation:

Y.M.Y. Feldman and B. Hirshberg “Quadratic scaling bosonic path integral molecular dynamics simulations”, J. Chem. Phys. 159, 154107 (2023) DOI: 10.1063/5.0173749 — BibTeX: fetch

Theory:

B. Hirshberg, V. Rizzi and M. Parrinello, “Path integral molecular dynamics for bosons”, Proc. Natl. Acad. Sci. U.S.A. 116 (43) 21445-21449 (2019) DOI: 10.1073/pnas.1913365116 — BibTeX: fetch

B. Hirshberg, M. Invernizzi and M. Parrinello, “Path integral molecular dynamics for fermions: Alleviating the sign problem with the Bogoliubov inequality”, J. Chem. Phys. 152, 171102 (2020) DOI: 10.1063/5.0008720 — BibTeX: fetch

C. W. Myung, B. Hirshberg and M. Parrinello, “Prediction of a Supersolid Phase in High-Pressure Deuterium”, Phys. Rev. Lett., 128 045301 (2022) DOI: 10.1103/PhysRevLett.128.045301 — BibTeX: fetch

Cavity Molecular Dynamics for Polaritonics

This initial implementation provides an efficient cavity molecular dynamics (CavMD) scheme for simulating strong light-matter interactions between molecules and an optical cavity mode, particularly in the vibrational strong coupling regime. At present, the nuclear partial charges are assumed to be fixed during the simulation. CavMD is implemented with a new force evaluator: ffcavphsocket, which is operated similarly to the original ffsocket evaluator, but with additional parameters for controlling cavity photons. Hence, with this implementation, users can study different aspects of vibrational strong coupling with many sophisticated methods supported in i-pi.

Main contributors: Tao E. Li

Implementation and theory:

T. E. Li, J. E. Subotnik, and A. Nitzan, “Cavity molecular dynamics simulations of liquid water under vibrational ultrastrong coupling”, Proc. Natl. Acad. Sci. 117(31), 18324–18331. (2020)

DOI: 10.1073/pnas.2009272117 — BibTeX: fetch

T. E. Li, A. Nitzan, S. Hammes-Schiffer, and J. E. Subotnik, “Quantum simulations of vibrational strong coupling via path integrals”, J. Phys. Chem. Lett. 13(17), 3890–3895. (2022)

DOI: 10.1021/acs.jpclett.2c00613 — BibTeX: fetch

Committee models

Uncertainty estimation for machine-learning potentials based on committee models. Multiple potentials are used simultaneously (they should have been fitted with randomized training sets). The mean is used to drive the system, the spread is used as an estimate of the uncertainty. This uncertainty can be used to select high-error structures for active learning, to estimate the propagation of the error to thermodynamic averages, or to build a weighted baseline model that falls back to a safe baseline potential when the ML models fail.

Main contributors: Giulio Imbalzano, Venkat Kapil, Yongbin Zhuang, Federico Grasselli, Michele Ceriotti

Implementation and theory:

G. Imbalzano, Y. Zhuang, V. Kapil, K. Rossi, E. A. Engel, F. Grasselli, and M. Ceriotti, “Uncertainty estimation for molecular dynamics and sampling”, J. Chem. Phys. 154(7), 074102 (2021)

DOI: 10.1063/5.0036522 — BibTeX: fetch

F. Musil, M. J. Willatt, M. A. Langovoy, and M. Ceriotti, “Fast and Accurate Uncertainty Estimation in Chemical Machine Learning”, Journal of Chemical Theory and Computation 15(2), 906–915 (2019)

DOI: 10.1021/acs.jctc.8b00959 — BibTeX: fetch

Replica Exchange MD

Accelerated convergence of averages by performing Monte Carlo exchanges of configurations between parallel calculations.

Main contributors: Riccardo Petraglia, Robert Meissner, Michele Ceriotti

Implementation: R. Petraglia, A. Nicolaï, M. M. D. Wodrich, M. Ceriotti, C. Corminboeuf, “Beyond static structures: Putting forth remd as a tool to solve problems in computational organic chemistry”, J. Comput. Chem. 37(1), 83-92 (2016)

DOI: 10.1002/jcc.24025 — BibTeX: fetch

Theory:

Y. Sugita, Y. Okamoto, “Replica-exchange molecular dynamics method for protein folding”, Chem. Phys. Lett. 314(1-2), 141–151 (1999)

DOI: 10.1016/s0009-2614(99)01123-9 — BibTeX: fetch

T. Okabe, M. Kawata, Y. Okamoto, M. Mikami, “Replica-exchange monte carlo method for the isobaric–isothermal ensemble”, Chem. Phys. Lett. 335(5-6), 435-439 (2001)

DOI: 10.1016/ s0009-2614(01)00055-0 — BibTeX: fetch

Thermodynamic integrations

Thermodynamic integrations are made easy with i-PI, through the connection of different sockets and the different weights one can assign to different forces. It is possible to do harmonic (Debye model) to anharmonic integration, or integrations between different kinds of potentials. Also, quantum thermodynamic integrations relative to mass are easily done through the manual input of each atom’s masses.

Main contributors: Mariana Rossi, Michele Ceriotti

Implementation:

M. Rossi, P. Gasparotto, M.Ceriotti “Anharmonic and Quantum Fluctuations in Molecular Crystals: A First-Principles Study of the Stability of Paracetamol”, Phys. Rev. Lett. 117, 115702 (2016)

DOI: 10.1103/PhysRevLett.117.115702 — BIBTEX: fetch

Finite-differences Suzuki-Chin PIMD

Suzuki-Chin PIMD gives better convergence w.r.t. the number of imaginary time slices as compared to the standard Trotter scheme. The implementation uses a symplectic and time-reversible finite-difference algorithm to compute high order corrections to traditional PIMD for any empirical or ab initio forcefield.

Main contributors: Venkat Kapil, Michele Ceriotti

Implementation and Theory:

V.Kapil, J.Behler, M.Ceriotti “High order path interals made easy”, J. Chem. Phys. 145, 234103 (2016)

DOI: 10.1063/1.4971438 — BIBTEX: fetch

Theory:

S.Jang, S.Jang, G.A.Voth “Applications of higher order composite factorization schemes in imaginary time path integral simulations”, J. Chem. Phys. 115, 7832 (2001)

DOI: 10.1063/1.1410117 — BIBTEX: fetch

S.A.Chin “Symplectic integrators from composite operator factorizations”, Phys. Lett. A 226, 344 (1997)

DOI: 10.1016/s0375-9601(97)00003-0 — BIBTEX: fetch

M.Suzuki “Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations”, Phys. Lett. A 201, 425 (1995)

DOI: 10.1016/0375-9601(95)00266-6 — BIBTEX: fetch

Reweighting-based high-order PIMD

The Boltzmann weight assciated with the high order correction to standard PIMD is printed out as a property so that the high order estimate of an arbitrary position-dependent observable can be computed as a weighted average.

Main contributors: Michele Ceriotti , Guy A. R. Brian

Implementation:

M.Ceriotti, G.A.R.Brian, O.Riordan, D.E.Manolopolous “The inefficiency of re-weighted sampling and the curse of system size in high-order path integration”, Proc. R. Soc. A 468, 2-17 (2011)

DOI: 10.1098/rspa.2011.0413 — BIBTEX: fetch

Theory:

S.Jang, S.Jang, G.A.Voth “Applications of higher order composite factorization schemes in imaginary time path integral simulations”, J. Chem. Phys. 115, 7832 (2001)

DOI: 10.1063/1.1410117 — BIBTEX: fetch

S.A.Chin “Symplectic integrators from composite operator factorizations”, Phys. Lett. A 226, 344 (1997)

DOI: 10.1016/s0375-9601(97)00003-0 — BIBTEX: fetch

M.Suzuki “Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations”, Phys. Lett. A 201, 425 (1995)

DOI: 10.1016/0375-9601(95)00266-6 — BIBTEX: fetch

Perturbed Path Integrals

Effectively a zeroth-order cumulant expansion of the high-order PI Hamiltonian, perturbed path integrals offer an attractive approach to compute thermochemistry of materials and molecules including quantum nuclei, as a post-processing of a Trotter trajectory.

Main contributors: Igor Poltavski

Theory:

I. Poltavsky and A. Tkatchenko, “Modeling Quantum Nuclei with Perturbed Path Integral Molecular Dynamics”, Chem. Sci. 7, 1368 (2016)

DOI: 10.1039/C5SC03443D — BibTeX: fetch

Open Path Integrals

Open path integrals and momentum distribution estimators for the computation of the particle momentum distribution including quantum fluctuations of nuclei.

Main contributors: Kapil, Cuzzocrea, Ceriotti

Implementation and Theory:

V. Kapil, A. Cuzzocrea, M. Ceriotti, “Anisotropy of the Proton Momentum Distribution in Water”, J. Phys. Chem. B 122, 6048-6054 (2018)

DOI: 10.1021/acs.jpcb.8b03896 — BibTeX: fetch

Theory:

J. A. Morrone, R. Car, “Nuclear Quantum Effects in Water”, Phys. Rev. Lett. 101, 017801 (2008)

DOI: 10.1103/PhysRevLett.101.017801 — BibTeX: fetch

Geometry Optimization

Several standard algorithms for geometry optimization have been implemented to give the convenience of static calculations that are fully compatible with (PI)MD and other advanced sampling techniques.

Main contributors: Benjamin Helfrecht, Sophie Mutzel, Riccardo Petraglia, Yair Litman, Mariana Rossi

Implementation:

M. Rossi, P. Gasparotto, and M. Ceriotti, “Anharmonic and Quantum Fluctuations in Molecular Crystals: A First-Principles Study of the Stability of Paracetamol”, Phys. Rev. Lett. 117, 115702 (2016)

DOI: 10.1103/PhysRevLett.117.115702 — BibTeX: fetch

Theory:

W. H. Press, “Numerical Recipes: The Art of Scientific Computing”, (Cambridge University Press, 2007)

Finite-differences Vibrational Analysis

Harmonic vibrations through finite differences for simple evaluation of the harmonic Hessian.

Main contributors: Kapil, Bienvenue

Implementation:

M. Rossi, P. Gasparotto, M. Ceriotti, “Anharmonic and Quantum Fluctuations in Molecular Crystals: A First-Principles Study of the Stability of Paracetamol”, Phs. Rev. Lett. 117, 115702 (2016)

DOI: 10.1103/PhysRevLett.117.115702 — BibTeX: fetch

Multiple Time Step integrators

A multiple time step integration scheme allows for integration of different components of forces with different time steps. It becomes advantageous when the total force can be decomposed into a slowly varying expensive part and a rapidly varying cheap part. A larger time step can be used to integrate the former, there by reducing the number of expensive computations.

Main contributors: Venkat Kapil

Implementation:

V.Kapil, J.VandeVondele, M.Ceriotti “Accurate molecular dynamics and nuclear quantum effects at low cost by multiple steps in real and imaginary time: using density functional theory to accelerate wavefunction methods”, J. Chem. Phys. 144, 054111 (2016)

DOI: 10.1063/1.4941091 — BibTeX: fetch

Theory:

M.Tuckerman, B.J.Berne “Reversible multiple time scale molecular dynamics”, J. Chem. Phys. 97, 1990 (1992)

DOI: 10.1063/1.463137 — BibTeX: fetch

Ring-polymer instantons

Semiclassical instanton theory is an efficient way of simulating tunneling contributions to reaction rate constants and tunneling splittings, based on a well-defined dominant tunneling pathway. It can be much more efficient than RPMD rate theory, but it is not applicable to condensed phases and includes anharmonicities only along the reaction coordinate.

Main contributors: Yair Litman, Jeremy O. Richardson, Mariana Rossi

Implementation:

Y. Litman, J. O. Richardson, T. Kumagai, M. Rossi, Elucidating the Quantum Dynamics of Intramolecular Double Hydrogen Transfer in Porphycene, arXiv:1810.05681 (2018).

V. Kapil et al. i-PI 2.0: A Universal Force Engine for AdvancedMolecular Simulations, Comp. Phys. Comm. (2018)

Theory:

W. H. Miller, Semiclassical limit of quantum mechanical transition state theory for nonseparable systems, J. Chem. Phys. 62(5) 1899–1906 (1975)

DOI: 10.1063/1.430676 — BibTeX: fetch

J. O. Richardson, Ring-polymer instanton theory, Int. Rev. Phys. Chem. 37, 171 (2018)

DOI: 10.1080/0144235X.2018.1472353 — BibTeX: fetch

Thermostatted RPMD

By introducing an internal mode thermostat to RPMD it is possible to reduce the well-known artifacts in the simulation of dynamical properties by path integral methods.

Main contributors: Mariana Rossi, Michele Ceriotti

Implementation:

M.Rossi, M.Ceriotti, D.E.Manolopoulos, “How to remove the spurious resonances from ring polymer molecular dynamics”, J. Chem. Phys. 140, 234116 (2014)

DOI: 10.1063/1.4883861 — BibTeX: fetch

Theory:

I.R.Craig, D.E.Manolopoulos “Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics”, J. Chem. Phys. 121, 3368 (2004)

DOI: 10.1063/1.1777575 — BibTeX: fetch

Fast-Forward Langevin Thermostat

This is a modified form of Langevin dynamics in which sluggish high-friction behaviour is corrected for by flipping a particle’s momentum when the action of the thermostat causes it to change direction.

Main contributors: Mahdi Hijazi, David Wilkins, Michele Ceriotti

Implementation and Theory:

M. Hijazi, D. M. Wilkins, “Fast-forward Langevin dynamics with momentum flips”, J. Chem. Phys. 148, 184109 (2018)

DOI: 10.1063/1.5029833 — BibTeX: fetch

Ring-Polymer Contraction

A ring-polymer contraction makes it possible to compute different components of the forces on different number of imaginary time slices. In order to reap maximum benefits, the implementation is fully compatible with the multiple time step integrators.

Main contributors: Michele Ceriotti, Venkat Kapil

Implementation:

V.Kapil, J.VandeVondele, M.Ceriotti “Accurate molecular dynamics and nuclear quantum effects at low cost by multiple steps in real and imaginary time: using density functional theory to accelerate wavefunction methods”, J. Chem. Phys. 144, 054111 (2016) DOI: 10.1063/1.4941091 — BibTeX: fetch

Theory:

T.Markland, D.E.Manolopoulos “An efficient ring polymer contraction scheme for imaginary time path integral simulations”, J. Chem. Phys. 129, 024105 (2008)

DOI: 10.1063/1.2953308 — BibTeX: fetch

Direct Estimators for Isotope Fractionation

A direct estimator to evaluate the isotope fractionation ratios using a single operation (and a single keyword in the input file), without the need for a thermodynamic integration with respect to the mass of the isotope.

Main contributors: Bingqing Cheng, Michele Ceriotti

Implementation and Theory:

B.Cheng, M.Ceriotti, “Direct path integral estimators for isotope fractionation ratios.” The Journal of chemical physics 141, 244112 (2015)

DOI: 10.1063/1.4904293 — BibTeX: fetch

Free-energy Perturbation Estimators for Isotope Fractionation

Computing isotope fractionation using the thermodynamic integration method requires evaluating the quantum kinetic energy of several systems containing atoms that have different fictitious masses between the physical masses of two isotopes, meaning that a number of PIMD simulations have to be performed. With the help of re-weighting, one has the option of running just one set of simulation with a certain fictitious mass, and obtain the quantum kinetic energy for systems with other masses.

Main contributors: Michele Ceriotti, Thomas Markland

Theory and implementation:

Michele Ceriotti, Thomas E. Markland, “Efficient methods and practical guidelines for simulating isotope effects.” The Journal of chemical physics 138(1), 014112 (2013).

DOI: 10.1063/1.4772676 — BibTeX: fetch

Quantum Alchemical Transformation

An algorithm that performs Monte Carlo moves to change a chemical species into its isotopes.

Main contributors: Bingqing Cheng, Michele Ceriotti

Implementation:

Cheng, Bingqing, J”{o}rg Behler, Michele Ceriotti, “Nuclear Quantum Effects in Water at the Triple Point: Using Theory as a Link Between Experiments.” J. Phys. Chem. Lett. 7(12), 2210-2215 (2016)

DOI: 10.1021/acs.jpclett.6b00729 — BibTeX: fetch

Theory:

Michael R. Shirts, David L. Mobley, John D. Chodera, “Alchemical Free Energy Calculations: Ready for Prime Time?”, Ann. Rep. Comp. Chem. 41-59 (2007)

DOI: 10.1016/S1574-1400(07)03004-6 — BibTeX: fetch

Jian Liu, Richard S Andino, Christina M Miller, Xin Chen, David M Wilkins, Michele Ceriotti, David E Manolopoulos, “A surface-specific isotope effect in mixtures of light and heavy water”, J. Phys. Chem. C 117(6), 2944-2951 (2013)

DOI: 10.1021/jp311986m — BibTeX: fetch

Langevin Sampling for Noisy or Dissipative Forces

A modified Langevin thermostat that allows for constant-temperature dynamics with noisy or dissipative forces by applying additional damping or noise for compensation. The implementation contains a method to adjust the amount of compensation automatically.

Main contributors: Jan Kessler, Thomas D. Kühne

Theory:

T. D. Kühne, M. Krack, F. R. Mohamed, M. Parrinello, “Efficient and Accurate Car-Parrinello-like Approach to Born-Oppenheimer Molecular Dynamics”, Phys. Rev. Lett. 98, 066401 (2007)

DOI: 10.1103/PhysRevLett.98.066401 — BibTeX: fetch

F. R. Krajewski, M. Parrinello, “Linear scaling electronic structure calculations and accurate statistical mechanics sampling with noisy forces”, Phys. Rev. B 73, 041105 (2006)

DOI: 10.1103/PhysRevB.73.041105 — BibTeX: fetch

Y. Luo, A. Zen, S. Sorella, “Ab initio molecular dynamics with noisy forces: Validating the quantum Monte Carlo approach with benchmark calculations of molecular vibrational properties”, J. Chem. Phys. 141, 194112 (2014)

DOI: 10.1063/1.4901430 — BibTeX: fetch

Path Integral GLEs

Generalized Langevin Equations can be combined with a PIMD framework to accelerate convergence of quantum observables while retaining systematic approach to the quantum limit. Parameters formatted for i-PI input can be obtained from the GLE4MD website.

Main contributors: Michele Ceriotti, Joshua More

Implementation:

M. Ceriotti, J. More, D. Manolopoulos, “i-PI: A Python interface for ab initio path integral molecular dynamics simulations”, Comp. Phys. Comm. 185(3), 1019 (2014)

DOI: 10.1016/j.cpc.2013.10.027 — BibTeX: fetch

Theory:

PIGLET — M. Ceriotti and D. E. Manolopoulos, “Efficient First-Principles Calculation of the Quantum Kinetic Energy and Momentum Distribution of Nuclei”, Phys. Rev. Lett. 109, 100604 (2012)

DOI: 10.1103/PhysRevLett.109.100604 — BibTeX: fetch

PI+GLE — M. Ceriotti, D. E. Manolopoulos, and M. Parrinello, “Accelerating the Convergence of Path Integral Dynamics with a Generalized Langevin Equation”, J. Chem. Phys. 134, 84104 (2011)

DOI: 10.1063/1.3556661 — BibTeX: fetch

Generalized Langevin Equation Thermostats

The Generalized Langevin Equation provides a very flexible framework to manipulate the dynamics of a classical system, improving sampling efficiency and obtaining quasi-equilibrium ensembles that mimic quantum fluctuations. Parameters for the different modes of operation can be obtained from the GLE4MD website.

Main contributors: Michele Ceriotti

Implementation:

M. Ceriotti, G. Bussi, M. Parrinello, “M. Colored-Noise Thermostats à la Carte”, J. Chem. Theory Comput. 6, 1170–1180 (2010)

DOI: 10.1021/ct900563s — BibTeX: fetch

Theory:

Optimal Sampling Efficiency — M. Ceriotti, G. Bussi, and M. Parrinello, “Langevin Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations”, Phys. Rev. Lett. 102, 20601 (2009)

DOI: 10.1103/PhysRevLett.102.020601 — BibTeX: fetch

Quantum Thermostat — M. Ceriotti, G. Bussi, and M. Parrinello, “Nuclear Quantum Effects in Solids Using a Colored-Noise Thermostat”, Phys. Rev. Lett. 103, 30603 (2009)

DOI: 10.1103/PhysRevLett.103.030603 — BibTeX: fetch

Delta Thermostat — M. Ceriotti and M. Parrinello, “The δ-Thermostat: Selective Normal-Modes Excitation by Colored-Noise Langevin Dynamics”, Procedia Comput. Sci. 1, 1607 (2010)

DOI: 10.1016/j.procs.2010.04.180 — BibTeX: fetch

MTS Thermostat — J. A. Morrone, T. E. Markland, M. Ceriotti, and B. J. Berne, “Efficient Multiple Time Scale Molecular Dynamics: Using Colored Noise Thermostats to Stabilize Resonances”, J. Chem. Phys. 134, 14103 (2011)

DOI: 10.1063/1.3518369 — BibTeX: fetch

“Hot-spot” — R. Dettori, M. Ceriotti, J. Hunger, C. Melis, L. Colombo, and D. Donadio, “Simulating Energy Relaxation in Pump-Probe Vibrational Spectroscopy of Hydrogen-Bonded Liquids”, J. Chem. Theory Comput. (2017)

DOI: 10.1021/acs.jctc.6b01108 — BibTeX: fetch

Path-Integral Langevin Equation Thermostats

Simple yet efficient Langevin thermostat for PIMD, with normal-modes thermostats optimally coupled to the ideal ring polymer frequencies

Main contributors: Michele Ceriotti

Implementation and Theory:

M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, “Efficient stochastic thermostatting of path integral molecular dynamics” J. Chem. Phys. 133, 124104 (2010).

DOI: 10.1063/1.3489925 — BibTeX: fetch

Path Integrals at Constant Pressure

The constant-pressure implementation allows for arbitrary thermostats to be applied to the cell degrees of freedom, and work in both constant-shape and variable-cell mode.

Main contributors: Michele Ceriotti, Joshua More, Mariana Rossi

Implementation:

M. Ceriotti, J. More, D. Manolopoulos, “i-PI: A Python interface for ab initio path integral molecular dynamics simulations”, Comp. Phys. Comm. 185(3), 1019 (2014)

DOI: 10.1016/j.cpc.2013.10.027 — BibTeX: fetch

Theory:

G. J. Martyna, A. Hughes, M. Tuckerman, “Molecular dynamics algorithms for path integrals at constant pressure”, J. Chem. Phys. 110(7), 3275 (1999)

DOI: 10.1063/1.478193 — BibTeX: fetch

G. Bussi, T. Zykova-Timan, M. Parrinello, “Isothermal-isobaric molecular dynamics using stochastic velocity rescaling”, J. Chem. Phys. 130(7), 074101 (2009)

DOI: 10.1063/1.3073889 — BibTeX: fetch

P. Raiteri, J. D. Gale, G. Bussi, “Reactive force field simulation of proton diffusion in BaZrO3 using an empirical valence bond approach”, J. Phys. Cond. Matt. 23(33), 334213 (2011)

DOI: 10.1088/0953-8984/23/33/334213 — BibTeX: fetch

Path Integral Molecular Dynamics

The basic PIMD implementation in i-PI relies on a normal-modes integrator, and allows setting non-physical masses, so that both RPMD and CMD can be easily realized.

Main contributors: Michele Ceriotti, Joshua More

Implementation:

M. Ceriotti, J. More, D. Manolopoulos, “i-PI: A Python interface for ab initio path integral molecular dynamics simulations”, Comp. Phys. Comm. 185(3), 1019 (2014) DOI: 10.1016/j.cpc.2013.10.027 — BibTeX: fetch

Theory:

R. Feynman, A. Hibbs, “Quantum Mechanics and Path Integrals”, McGraw-Hill (1964)

M. Tuckerman, “Statistical Mechanics and Molecular Simulations”, Oxford Univ. Press (2008)