# Research

Our interests lie in the development of theoretical and simulation approaches and their application to explain effects observed in the classical and quantum dynamics of chemical systems. Particular themes which occur frequently in our research are hydrogen bonding, the interplay between structure and dynamics, systems with multiple time and length-scales and quantum mechanical effects.

Please contact us for full details of upcoming projects and openings. Ongoing interests include:

## Dynamics and structure in hydrogen bonded systems

Hydrogen bonded systems play vital roles in areas ranging from chemistry to biology to materials science and geology.

Our recent and current interests in this area include:

- Quantum effects in water.
- Hydrogen and proton transfer in water.
- Isotope fractionation.
- Supercooled water, heterogenous nucleation of ice and the effects of cryogenic species.

### Isotope Fractionation

When two phases of water are at equilibrium, the ratio of hydrogen isotopes in each is slightly altered because of their different phase affinities. This isotopic fractionation process has a number of fortuitous consequences, which are utilized in hydrology and geology. For instance, by comparing the ratio of H to D, one can estimate the origins of a water sample, the temperature at which it was formed, and the altitude at which precipitation occurred. Fractionation therefore provides both an excellent test of the ability of theory to accurately predict the magnitude of quantum effects in hydrogen bonded systems and also allows insights into fractionation processes in the world's climate. Our previous work in this area has demonstrated the importance of anharmonicty in correctly describing H/D fractionation between liquid water and its vapor. Our current work is focused on ab initio prediction of fractionation ratios for systems found in the earth's climate.

- Above: A depiction of the equilibrium which determines H/D fractionation between liquid water and its vapor. The free energy change associated with the process would be zero in the absence of quantum effects and so is entirely a consequence of quantum mechanics.

### Quantum Effects in Hydrogen Bonded Systems

We have recently demonstrated that water has two different and opposite quantum effects. The former of these is due to the long appreciated disruption of the hydrogen bond network which leads to destructing of the liquid and faster dynamics. However, a second effect exists in which the quantum kinetic energy in the OH covalent bond allows it to stretch and form shorter and stronger hydrogen bonds, which partially cancels the disruptive effect. We are currently extending this work to investigate quantum effects on hydrogen bonding in materials and biological systems.

- Above: Schematic representation of the two competing quantum effects present in water. The degree of cancellation is sensitive to properties such as the OH bond anharmonicity, temperature and pressure.

##### Representative publications

Quantum fluctuations and isotope effects in ab initio descriptions of water. L. Wang, M. Ceriotti, T. E. Markland.,J. Chem. Phys., 141, 104502 (2014)

Unraveling quantum mechanical effects in water using isotopic fractionation. T. E. Markland and B. J. Berne, Proc. Natl. Acad. Sci., 109, 7988-7991 (2012)

Competing quantum effects in the dynamics of a flexible water model. S. Habershon, T. E. Markland and D. E. Manolopoulos, J. Chem. Phys. 131, 024501 (2009)

Interface limited growth of heterogeneously nucleated ice in supercooled water. R. A. Nistor, T. E. Markland, B. J. Berne, J. Phys. Chem. B, 118 (3), 752-760 (2014)

Isotope effects in water as investigated by neutron diffraction and path integral molecular dynamics. A. Zeidler, P. S. Salmon, H. E. Fischer, J. C. Neuefeind, J. M. Simonson and T. E. Markland, J. Phys. Condens. Mat. 24, 284126 (2012)

Oxygen as a Site Specific Probe of the Structure of Water and Oxide Materials. A. Zeidler, P. S. Salmon, H. E. Fischer, J. C. Neuefeind, J. M. Simonson, H. Lemmel, H. Rauch, and T. E. Markland, Phys. Rev. Lett., 107, 145501 (2011)

## Quantum dynamics

Quantum dynamics plays an important role in many fundamental processes such as hydrogen and electron transfer, vibrational relaxation and electronic energy transfer processes in photosynthetic systems. However, due to the large computational cost associated with solving the many-body quantum problem for systems of more than a few degrees of freedom approximations are necessary. Given the wide range of problems of interest in this area one must pick the correct quantum dynamics approach for a given application or often create new ones to obtain the correct balance between speed and accuracy. In particular we are interested in applying and developing our reduced density matrix hybrid formalism and the ring polymer molecular dynamics approach.

### Hybrid master-equation semiclassical approaches to quantum dynamics

We have recently introduced the reduced density matrix hybrid (RDM-H) approach to quantum dynamics. In the RDM-H approach the quantum degrees of freedom are treated by evolving a reduced density matrix which is then coupled to a classically evolved reservoir to treat the remaining modes. This leads to an accurate and efficient approach which avoids the use of wave functions and scales well with the number of system states.

RDM-H is thus widely applicable to subsystems embedded in a surrounding thermal bath, which forms the basis for the investigation of condensed phase energy and electron transfer as well as spin and charge transport in nanoscale devices. We have previously applied this approach to investigate electronic energy transfer processes such as those occurring in photosynthetic complexes.

- Above: Schematic diagram of excitation dynamics in a model of the FMO complex. Q1 and Q2 correspond to collective bath coordinates. The solid line arrows correspond to an un-shifted bath initial condition and subsequent excitation dynamics, whereas the dotted line arrow corresponds to an initial bath condition shifted to the minimum of site 1 in the complex, which leads to trapping of the excitation.

##### Representative publications

Efficient and accurate surface hopping for long time nonadiabatic quantum dynamics. A. Kelly and T. E. Markland, J. Chem. Phys. 139, 014104 (2013)

Reduced density matrix hybrid approach: Application to electronic energy transfer. T. C. Berkelbach, T. E. Markland and D. R. Reichman, J. Chem. Phys., 136, 084104 (2012)

Reduced density matrix hybrid approach: An efficient and accurate method for adiabatic and non-adiabatic quantum dynamics. T. C. Berkelbach, D. R. Reichman and T. E. Markland, J. Chem. Phys. 136, 034113 (2012)

### Ring Polymer Molecular Dynamics

The Ring Polymer Molecular Dynamics approach allows the inclusion of quantum fluctuations such as zero-point energy and tunneling in the dynamics of liquids and glasses. The formalism exploits the mapping of a quantum mechanical particle onto a classical "ring polymer" and provides an accurate and physically insightful way to approximate the effect of quantum fluctuations on reaction rates, diffusion coefficients and spectra.

We have previously used RPMD to elucidate isotope effects observed for hydrogen atom transfer in water and ice, quantum effects in the dynamics of water and the effect of quantum fluctuations on systems near the liquid-glass transition. We are also interested in developing methods to make RPMD more efficient such as our ring polymer contraction approach (see section below) and to improve its accuracy.

- Above: Snapshots from an RPMD simulation of a supercooled liquid. In the left image the polymer is extended as it tunnels between cavities in the liquid and in the right it is localized in one well.

##### Representative publications

Ring Polymer Molecular Dynamics: Quantum Effects in Chemical Dynamics from Classical Trajectories in an Extended Phase Space. S. Habershon, D. E. Manolopoulos, T. E. Markland and T. F. Miller, Annu. Rev. Phys. Chem., 64, 387-413 (2013)

Theory and simulations of quantum glass forming liquids. T. E. Markland, J. A. Morrone, K. Miyazaki, B. J. Berne, D. R. Reichman and E. Rabani, J. Chem. Phys., 136, 074511 (2012)

Quantum fluctuations can promote or inhibit glass formation. T. E. Markland, J. A. Morrone, B. J. Berne, K. Miyazaki, E. Rabani and D. R. Reichman, Nature Phys., 7, 134-137 (2011)

Competing quantum effects in the dynamics of a flexible water model. S. Habershon, T. E. Markland and D. E. Manolopoulos, J. Chem. Phys. 131, 024501 (2009)

Quantum diffusion of hydrogen and muonium atoms in liquid water and hexagonal ice. T. E. Markland, S. Habershon and D. E. Manolopoulos, J. Chem. Phys. 128, 194506 (2008)

## Path integral quantum mechanics methods

Imaginary time path integral simulations provide an elegant method by which quantum fluctuations, such as zero-point energy and tunneling, can be included in complex systems. This formalism exploits the exact isomorphism between a set of quantum mechanical particles and that of a classical set of "ring polymers". As such path integral simulations can be used to calculate exactly static equilibrium properties of quantum systems and also provides the basis to many successful approaches to treat quantum dynamics in the condensed phase.

To improve the efficiency of these methods we have introduced the ring polymer contraction approach which reduces the cost of performing path integral simulations be a factor of ~30-40 for typical systems. This allows one to perform a simulation including the effects of zero-point energy and tunneling for a computational cost barely more than a classical simulation of the same system. We have also developed efficient path integral Langevin equation schemes to thermostat path integral simulations.

- Above: A depiction of the isomorphism between a quantum mechanical particle and a classical ring polymer.

##### Representative publications

Efficient methods and practical guidelines for simulating isotope effects. M. Ceriotti and T. E. Markland, J. Chem. Phys. 138, 014112 (2013)

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

A fast path integral method for polarizable force fields. G. S. Fanourgakis, T. E. Markland and D. E. Manolopoulos, J. Chem. Phys. 131, 094102 (2009)

A refined ring polymer contraction scheme for systems with electrostatic interactions. T. E. Markland and D. E. Manolopoulos, Chem. Phys. Lett. 464, 256-261 (2008)

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

## Electric field control of catalytic reactivity

Recent experiments by the Kanan group (Stanford, Chemistry) have shown that static electric fields can be used to change the ratio of reaction products produced at a catalytic surface by up to 2 orders of magnitude. In collaboration with the Kanan group we are investigating the mechanisms by which this process occurs. Describing this requires modeling the interplay between the external field and the ion and solvent structure. This must then be coupled to the reaction dynamics occuring at the catalytic interface.

- Above: A simulation cell showing the catalytic surfaces, ions and solvent.

##### Representative publications

Coming Soon

## Methods for efficient classical simulation

The combination of increasing computing power and efficient algorithms have pushed simulation capabilities to a stage where studying processes on the microsecond or even millisecond time-scale for systems containing millions of atoms is possible. However, even these simulation time-scales are frequently insufficient to obtain a full sampling of the space or to observe rare events necessitating the need for further development of efficient simulation approaches.

Our focus in this area is concentrated on developing multiple time-scale molecular dynamics approaches and targeted thermostat schemes. In particular we have been interested in developing new uses of thermostats based on the Generalized Langevin Equation (GLE). GLE thermostats have many appealing features such as allowing one to develop approaches which selectively couple to certain modes or efficiently sample a wide range of modes. For example we have recently used the former to introduce the GLE-RESPA scheme which allows much larger time-step to be used in molecular simulations and the latter to develop an efficient thermostatting scheme for path integral simulations.

- Above: Using the GLE-RESPA approach the Ramachandran plot of alanine dipeptide can be accurately obtained using time-steps of in excess of 12fs. Using standard multiple time-step techniques only 2fs is possible.

##### Representative publications

Multiple Time Step Integrators in Ab Initio Molecular Dynamics. N. Luehr, T. E. Markland, T. J. Martinez, J. Chem. Phys. 140, 084116 (2014)

Efficient multiple time scale molecular dynamics: Using colored noise thermostats to stabilize resonances. J. A. Morrone, T. E. Markland, M. Ceriotti and B. J. Berne, J. Chem. Phys. 134, 014103 (2011)

## Glassy systems

Glasses are dynamically arrested states of matter that do not exhibit the long-range periodic structure of crystals. While simple structural indicators suggest the structure of glass forming systems to be barely different from the liquid that produced them the dynamics can be as slow as observed in the crystalline state. Hence, investigating the growing static and dynamic length-scales which instill liquids with this rigidity provides an intruiging challenge.

Our interests in glasses incorporate three primary areas:

- Investigating how quantum mechanics affects the liquid-glass transition.
- Understanding the interplay between the structure of glass forming liquids and their dynamics using techniques such as point-to-set correlations.
- Developing methods for efficiently exploring the rough free energy landscapes characteristic of glassy systems.
- Above: A snapshot from a simulation used to obtain the point-to-set correlation of a Lennard Jones glass forming system. The point-to-set correlation can be used to define a static length-scale which is strongly correlated with the dynamics of the system.

##### Representative publications

Quantum fluctuations can promote or inhibit glass formation. T. E. Markland, J. A. Morrone, B. J. Berne, K. Miyazaki, E. Rabani and D. R. Reichman, Nature Phys., 7, 134-137 (2011)

Theory and simulations of quantum glass forming liquids. T. E. Markland, J. A. Morrone, K. Miyazaki, B. J. Berne, D. R. Reichman and E. Rabani, J. Chem. Phys., 136, 074511 (2012)

Growing point-to-set length scale correlates with growing relaxation times in model supercooled liquids. G. M. Hocky, T. E. Markland and D. R. Reichman, Phys. Rev. Lett. 108 225506 (2012)