Although the observable universe strictly obeys the laws of quantum mechanics, in many instances, a classical description that either ignores quantum effects entirely or accounts for them at a very crude level is sufficient to describe a wide variety of phenomena. However, when this approximation breaks down, as is often the case for processes involving light nuclei, a full quantum treatment becomes indispensable. This Special Topic in The Journal of Chemical Physics showcases recent advances in our understanding of nuclear quantum effects in condensed phases as well as novel algorithmic developments and applications that have enhanced the capability to study these effects.

The forty new research articles collected in this Special Topic report algorithms and simulations for which nuclear quantum effects are important. Quantum nuclear effects, particularly the anomalies presented by the heat capacities of molecular systems at low temperatures, played a key role in the discovery of quantum mechanics more than a century ago. At that time, diatomic molecules and nearly classical molecules could be understood with textbook methods. While some properties, e.g., phonon spectra in atomic and certain molecular crystals, can, under appropriate conditions, be computed within a harmonic approximation, when higher accuracy is needed or when nuclear quantum effects are expected to constitute the dominant contribution, proper theoretical treatments are needed. Nuclear quantum effects can be crucial in modeling a variety of phenomena, especially those involving light elements such as hydrogen, helium, and, in certain problems, even lithium. Examples include proton transfer reactions, transport of protons and hydroxide ions via structural diffusion in bulk and confined environments, hydrogen diffusion in nanoconfined materials, H/D isotopic substitution, separation, and fractionation, and numerous other phenomena of importance in materials science, chemistry, and biology. In a very recent study, it was shown that a full quantum treatment is necessary in the calculation of isotope fractionation of dissolved lithium.1 

“Imaginary time” path integrals, as introduced by Feynman in 19532 to treat superfluid helium, were exploited beginning in the 1980s by several groups.3–5 In this method, a classical particle becomes a special type of ring polymer. For distinguishable particles and systems well approximated by Boltzmann statistics, this “isomorphism” is an exact mapping of the equilibrium quantum statistical mechanical problem onto a pseudo-classical framework, thereby allowing classical simulation techniques to be exploited in the calculation of quantum static properties without further approximation. Quantum exchange effects, when important, are obtained by particular crosslinkings of the ring polymers6 derived from fundamental quantum rules pertaining to systems of identical particles. However, the computational resources needed to implement these approaches can be formidable without advanced algorithms, a problem that remains a considerable challenge in the field. Some of the articles in this Special Topic address problems for which this approach, despite the considerable difficulties, are critical.7,8

The experimental and computational treatment of nuclear quantum effects in condensed phase systems is entering an exciting era. New and efficient computer simulation techniques, particularly those based on instantons, the aforementioned path-integral, and more advanced path-integral approaches, combined with and/or complementing experimental methods such as neutron Compton scattering, helium spin echo, high-resolution scanning tunneling microscopy, and vibrational spectroscopy, are now shedding new light on the quantum nature of nuclei in condensed phases. With the availability of ever-more powerful computational resources, modeling techniques, once limited to simple systems, are applicable to realistic systems. Nevertheless, connecting the theory to some of these experiments requires addressing another important, outstanding challenge, namely, that of computing quantum dynamical properties. The calculation of exact quantum time correlation functions for complex systems requires taming the wild fluctuations associated with the real-time propagators that enter the expressions for these functions. Semiclassical and purely imaginary time methods deliver approximate time correlation functions that are accurate when quantum or anharmonic effects are relatively weak or, as occurs in many condensed-phase environments, rapid decoherence is expected. Although these methods have become quite popular, going beyond these approximations has proven to be a considerable challenge. Several of the articles in this Special Topic are concerned with this question.9–13 

Beyond what has already been mentioned, many of the articles in this Special Topic describe new methodology, for example, methods to couple electronic computation with nuclear evolution,14,15 for calculating transition rates,16 to perform coarse graining,17 and, as noted, to estimate quantum dynamical effects. Other articles are applications of these algorithms to challenging systems. A particular focus is the application to water in various forms and conditions: these studies remain a “grand challenge” for simulations because the interactions between water molecules are complex, its structural evolution is slow, and the quantum nature of the protons must often be accounted for accurately.18–21 Additional applications include systems containing graphene,22 hydrogen,23–26 helium,7 high temperature plasmas,27 and various other problems.

Overall, the articles in this Special Topic highlight the current state-of-the-art with an emphasis on how computational approaches are impacting and complementing both experiments and other areas of theory. They also point to problems for future investigation and exploration.

In his 1965 book The Character of Physical Law, Feynman wrote, “… I think I can safely say that nobody understands quantum mechanics.” Whether or not you agree with this view, one thing is clear: There is a fascination with quantum phenomena that drives many researchers to pursue questions of how to treat them and to discover new problems in which they may lurk, hitherto undetected. In this light, we hope you enjoy this Special Topic on nuclear quantum effects.

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