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## Spring School on Path Integrals Molecular Dynamics Simulations

#### University of Toulouse, France

#### Organisers

Computer simulations at the atomistic level have been developed for many years with the purpose of improving our understanding of molecular systems met in physics, chemistry, biology, and materials science. Progresses in hardware and algorithms have made simulations becoming a predictive approach intermediate between analytical and experimental methods.

In conventional simulations, the nuclei are treated as classical particles, and the electronic degrees of freedom are either treated implicitly through classical-like descriptions of force fields, or more explicitly through quantum mechanical methods. The classical assumption for the nuclei is sound and usually justified as long as the systems under study consist of heavy particles, or that the temperature is high enough. However, light atoms or molecules at low temperatures are likely to exhibit strong quantum mechanical effects such as tunneling or zero-point energy residual motion. Those effects can alter static structures, and can play an important role in the equilibrium and dynamical properties.

Quantum mechanical effects on nuclear motion can be rigorously included by solving the vibrational Schrödinger equation, for instance using wavepacket propagation methods, but such schemes are not practical for systems with many degrees of freedom. For a few decades, the method of choice for large-scale atomic and molecular compounds has lied in the path-integral (PI) framework [1]. The path integral description of quantum mechanics is based on the isomorphism between the quantum mechanical partition function of the physical system and the classical partition function of a fictitious, higher-dimensional system in which each quantum particle is replaced by a ring polymer consisting of beads (or replicas) connected by harmonic springs [2]. The physical extension of the ring polymers conveys the delocalization of the nuclear wavefunctions.

Historically, the path-integral strategy was first implemented within the Monte Carlo (MC) framework to sample configuration space at thermodynamical equilibrium [3]. Difficulties arising from the additional degrees of freedom and the harmonic coupling, particularly concerning ergodicity and sufficient sampling of the quantum statistical distribution, were then identified and addressed using carefully constructed collective moves. It was then suggested by Parrinello and Rahman [4] that the same sampling of the configuration space could be achieved by performing molecular dynamics (MD) trajectories, assigning momenta to all coordinates of the replicas. The numerous techniques developed for classical MD, especially those related to thermostatting or multiple time step integration, have since been largely exploited in path-integral MD [5].

The advent of the Car-Parrinello method accounting for an explicit treatment of electronic degrees of freedom in density-functional theory enabled during the 1990s the full quantum treatment of electrons and nuclei in a common computational framework [6,7]. Since their introduction, path-integral methods have been applied to a broad variety of compounds exhibiting varying degrees of quantum mechanical delocalization at thermal equilibrium, including helium and van der Waals systems, metals and covalent materials, molecular liquids such as water, and biomolecular systems (see for instance [8-25]).

An important class of path-integral molecular dynamics techniques, also initiated in the 1990s, is concerned with dynamical (time-dependent) properties, involved e.g. in transport processes, rate theories or vibrational spectroscopy. The approximate centroid molecular dynamics [26] and the more recent ring-polymer molecular dynamics [27] schemes have both been shown to be accurate for estimating quantum time correlation functions, and their performances have also been the subject of specific studies [28]. These techniques have largely contributed to reviving and spreading path-integral MD approaches beyond their original communities, as obviously seen from the surge of publications reporting uses of PIMD during the last couple of years.

Although increasingly widespread, several problems make the efficient implementation of path-integral molecular dynamics more complex and challenging than classical MD. In addition to the aforementioned algorithmic aspects, parallelism is an important element that is naturally suitable to the path-integral framework. Recent developments also include a novel estimator for the end-to-end distribution of the Feynman paths for the evaluation of the momentum distribution [29] and schemes for reducing the number of beads on each polymer chain [30,31], among others. These algorithmic improvements and the availability of large-scale computers able to handle such calculations, have made path-integral molecular dynamics a robust and appealing method in numerous fields where a quantum mechanical treatment of the nuclear variables is needed, possibly with explicit descriptions of electronic structure. A tutorial on this subject, which has not been often covered by CECAM so far, seems thus important and timely.

Computer simulations at the atomistic level have been developed for many years with the purpose of improving our understanding of molecular systems met in physics, chemistry, biology, and materials science. Progresses in hardware and algorithms have made simulations becoming a predictive approach intermediate between analytical and experimental methods.

In conventional simulations, the nuclei are treated as classical particles, and the electronic degrees of freedom are either treated implicitly through classical-like descriptions of force fields, or more explicitly through quantum mechanical methods. The classical assumption for the nuclei is sound and usually justified as long as the systems under study consist of heavy particles, or that the temperature is high enough. However, light atoms or molecules at low temperatures are likely to exhibit strong quantum mechanical effects such as tunneling or zero-point energy residual motion. Those effects can alter static structures, and can play an important role in the equilibrium and dynamical properties.

Quantum mechanical effects on nuclear motion can be rigorously included by solving the vibrational Schrödinger equation, for instance using wavepacket propagation methods, but such schemes are not practical for systems with many degrees of freedom. For a few decades, the method of choice for large-scale atomic and molecular compounds has lied in the path-integral (PI) framework [1]. The path integral description of quantum mechanics is based on the isomorphism between the quantum mechanical partition function of the physical system and the classical partition function of a fictitious, higher-dimensional system in which each quantum particle is replaced by a ring polymer consisting of beads (or replicas) connected by harmonic springs [2]. The physical extension of the ring polymers conveys the delocalization of the nuclear wavefunctions.Historically, the path-integral strategy was first implemented within the Monte Carlo (MC) framework to sample configuration space at thermodynamical equilibrium [3]. Difficulties arising from the additional degrees of freedom and the harmonic coupling, particularly concerning ergodicity and sufficient sampling of the quantum statistical distribution, were then identified and addressed using carefully constructed collective moves. It was then suggested by Parrinello and Rahman [4] that the same sampling of the configuration space could be achieved by performing molecular dynamics (MD) trajectories, assigning momenta to all coordinates of the replicas. The numerous techniques developed for classical MD, especially those related to thermostatting or multiple time step integration, have since been largely exploited in path-integral MD [5].

The advent of the Car-Parrinello method accounting for an explicit treatment of electronic degrees of freedom in density-functional theory enabled during the 1990s the full quantum treatment of electrons and nuclei in a common computational framework [6,7]. Since their introduction, path-integral methods have been applied to a broad variety of compounds exhibiting varying degrees of quantum mechanical delocalization at thermal equilibrium, including helium and van der Waals systems, metals and covalent materials, molecular liquids such as water, and biomolecular systems (see for instance [8-25]).

An important class of path-integral molecular dynamics techniques, also initiated in the 1990s, is concerned with dynamical (time-dependent) properties, involved e.g. in transport processes, rate theories or vibrational spectroscopy. The approximate centroid molecular dynamics [26] and the more recent ring-polymer molecular dynamics [27] schemes have both been shown to be accurate for estimating quantum time correlation functions, and their performances have also been the subject of specific studies [28]. These techniques have largely contributed to reviving and spreading path-integral MD approaches beyond their original communities, as obviously seen from the surge of publications reporting uses of PIMD during the last couple of years.Although increasingly widespread, several problems make the efficient implementation of path-integral molecular dynamics more complex and challenging than classical MD. In addition to the aforementioned algorithmic aspects, parallelism is an important element that is naturally suitable to the path-integral framework. Recent developments also include a novel estimator for the end-to-end distribution of the Feynman paths for the evaluation of the momentum distribution [29] and schemes for reducing the number of beads on each polymer chain [30,31], among others. These algorithmic improvements and the availability of large-scale computers able to handle such calculations, have made path-integral molecular dynamics a robust and appealing method in numerous fields where a quantum mechanical treatment of the nuclear variables is needed, possibly with explicit descriptions of electronic structure. A tutorial on this subject, which has not been often covered by CECAM so far, seems thus important and timely.

The purpose of this tutorial is to bring together PhD students, young and experienced researchers wishing to learn the basics and recent developments in path-integral molecular dynamics methods and start setting up their own projects. The tutorial will be organized as a series of topical lectures given by experts in the field and several practicals and exercises on various aspects covered by the lectures, thus providing some broad hands-on-experience supervised by expert users. More specialized lectures will also be given in order to cover specific topics related to path integrals in molecular simulation, or to alternative methods which we feel as important background knowledge on the subject of quantum nuclear effects.

The tutorial will involve general introductions about conventional (classical) simulation methods, emphasizing practical tricks involved in molecular dynamics and, to a lesser extent, Monte Carlo. Of particular importance will be the discussions about thermostats and multiple time step integrators. Introduction to Feynman's theory of path integrals in quantum statistics will also be provided, as well as a more practically oriented illustration of the circumstances under which nuclear effects need to be quantized in many-body systems. A key session will be devoted to conventional path-integral MD at equilibrium, detailing the method, its most relevant algorithmic ingredients and tricks of computational interest. Straightforward applications on simple atomic systems with pairwise forces will be given to illustrate these algorithmic aspects. The more involved situation involving explicit descriptions of the electronic structure will be discussed in a separate lecture, covering first the case of classical nuclei, and encompassing the path-integral extension. Ab initio MD and PIMD will be illustrated in a specific hands-on session. The tutorial will then focus on two more recent versions of the path-integral MD technique aimed at extracting dynamical (time-dependent) information, namely centroid MD and ring-polymer MD. Dedicated applications to transport and spectroscopic properties will subsequently support this lecture. Finally, other aspects connected to PIMD will be given at the end of the tutorial. Consistently with the main theme of this school, these shorter lectures will highlight related topics such as path-integral Monte Carlo, other computational methods for the quantum vibrational problem, or important recent developments in PIMD.

In addition to the main lectures and practical sessions, more specialized talks on various applications of the PIMD framework will be given by experts in order to highlight the current state-of-the-art of the field. In order to contribute to the overall scientific discussion, and for a deeper involvement into the school, the participants will also be given the opportunity to present their own work during dedicated poster sessions.

The deadline for application is April, 15th 2012

## References

**France**

Magali Benoit (CNRS) - Organiser

Florent Calvo (CNRS - University of Grenoble) - Organiser

Nathalie TARRAT (CEMES-CNRS) - Organiser

**United States**

Mark E. Tuckerman (New York University) - Organiser