New Perspectives in Liquid State Theories for Complex Molecular Systems
Institute Henri Poincaré, Paris, France
Integral equations (IE), classical density functional theory (DFT), and field theoretical methods (FTM) in liquid state theory are tantalizing tools to study equilibrium properties of simple and complex liquids [1-4]. Yet, despite numerous important advances, they are still trailing behind computer simulations when it comes to practical applications ranging from simple solutions to more complex nanoscale and biological systems. Besides frequently encountered numerical problems, this is due in part to the intrinsic limitations which were discovered in the 90’s: when its comes to associated liquids such as water, none of the existing IE, DFT or FTM are able to reproduce accurately the structure of the liquid in par with computer simulations, irrespective of the water model . The reason for this is well known for IE in a diagrammatic representation: they are missing an important set of diagrams, known as bridge diagrams, that play an important role in describing the exact properties of any system, even for the canonical hard spheres model . IE have been tested on various complex liquids, ranging from simple polar molecules to water, and the quality of the results quickly deteriorates when explicit highly directional interactions are present, such as hydrogen bonding interactions, for example. None of the existing approximations for the missing bridge function are able to close this gap with computer simulations . The same kind of limitations are reached when trying to define excess free-energy functionals for water: it appears hard to keep track of the subtle balance between tetrahedral order at short range and dipolar order at longer range [6,7].
On the other hand, it turns out that computer simulations also face severe problems when handling complex systems, the most obvious being computational issues related to the exploration of configurational space for systems with slow relaxation times. In contrast, theoretical liquid state methods are orders of magnitude faster to run. The recent development of 3D-RISM (reference interaction site model) [4,8-9], or other liquid-state 3D approaches based on DFT or FTM [7,11-12], has shown that sensible results and very useful insight can be gained for complex three-dimensional problems, such as protein hydration or host-guest molecular recognition. This is equally true for chemical engineering applications such as ionic liquids at charged interfaces (supercapacitors) or molecular adsorption in porous materials [13-14]. All those works pave the way to the broad applicability of liquid state theory to problems of biological or industrial relevance.
However, these new approaches still suffer from the same shortcomings described above, and despite the impressive advances in application domains that were previously reserved to computer simulations, they must be improved for a quantitative agreement.
We happen to think that liquid state theories approaches and direct computer simulations do not need to be competing alternatives, but could rather be used to perfect each other’s predictive capabilities. In particular, availability of an optimal, highly accurate and computationally manageable liquid state theory serves several purposes: on one hand, reducing the number of degrees of freedom of a large solvated system by accurately modeling the solvent effect (in the form of the excess chemical potential of a given solute geometry) would facilitate drastically larger simulation time and length scales to be studied. On the other hand, inverting simulation data to obtain effective coarse-grained interaction potentials for mesoscale simulations benefits from less severely approximated liquid state theories that form the conceptual basis of the inversion process.
The timely aim of this workshop is thus to review and discuss the latest developments in IE, DFT, and FTM approaches, to assess the future theoretical as well as numerical challenges, and to push their applications towards complex molecular systems, for example room temperature ionic liquids, molecular liquid mixtures, molecular liquids in confinement, or aqueous solutions in complex environments.
All approaches, IE, DFT, as well as FTM have been developed to handle many simple model liquids, such as hard spheres or Lennard-Jones particles embedding charges and dipoles. In IE theory for example, bridge functions are known for those systems by direct inversion of the simulation correlation functions. None of the theoretical guesses are in par with the accuracy of such direct approaches. A priori, the most reliable theoretical approach to the evaluation of the bridge function would be a term-by-term evaluation of the diagrams in the series density expansion [15-16]. An important caveat of this approach is the lack of information on the convergence of such series. It has been shown recently  that the exact evaluation of the bridge function for even the simplest system, hard rods in one dimension, faces a diverging series expansion, that can fortunately be re-summed in that precise case. Such re-summation seems impossible for more realistic models and, despite the past effort, it is about time we all simply accept that the bridge function cannot be evaluated by analytical methods. Some recent developments of 3D-RISM are based on a mixed PY/HNC closure relation that has an unclear theoretical foundation . Although, it should be recognized, again, that 3D-RISM is able to tackle complex systems, almost beyond expectation [4,8-9], the status of the method is not completely clarified and it is certainly perfectible. A possible route to improve integral equations is to couple them with simulations, with the aim of producing reliable and thermodynamically consistent  molecular bridge functions that would serve as template to investigate systems not accessible by simulations alone. Direct inversion of the correlation functions obtained from the simulation results has been recently carried out successfully for single component molecular liquids [7,18-19]. This procedure might be more difficult for liquid mixtures because of the strong local order, which is not described by current theories, and which needs very large system sizes to be dealt by simulations, one area of discussion during the workshop.
Concerning DFT approaches, The advent of a quasi-exact functional for inhomogeneous hard sphere mixtures, the fundamental measure theory originally proposed by Rosenfeld in 1989, and improved over the last 20 years (see Ref.  for a review), has promoted recently a great deal of applications to atomic-like and polymeric fluids in bulk or confined conditions or at interfaces. Classical ”atomic" DFT can be considered nowadays as a method of choice for many chemical engineering problems . Much less applications exist for molecular fluids, and molecular density functional theory (MDFT) has been limited to generic dipolar solvents or recently to dipolar solvent/ions mixtures . Recently a 3D-MDFT method has been proposed to describe the solvation of complex molecular objects in polar solvents [7,23] and water [7,24]. Such an approach relies on the proper description of the so-called excess free energy. Although this theory is able to describe satisfactorily polar, aprotic liquids such as acetonitrile, it is not appropriate for complex liquids such as water. The problems that are met are reminiscent of those occurring in MOZ integral equation approaches: how to account properly in the functional for the subtle balance between dipolar order at long range and tetrahedral order at short range?
At last, various field theoretical approaches to molecular solvation have been proposed recently. Chandler and collaborators  have developed a Gaussian field theory of hydration making it possible to compute the solvation properties of hydrophobic solutes of arbitrary shape and size. Electrostatic interactions are not incorporated in the model yet and the way to do it, and account properly for H-bonding, is not clear. On the other hand, Orland and collaborators  have developed a field theoretical approach to dipolar solvent-ions mixtures, that leads to a generalization the Poisson-Boltzmann equation accounting for particle size and dielectric saturation. Although this is still a crude model of aqueous solutions with no account whatsoever of hydrogen-bonding (but, nevertheless, better than continuum theories), it is able to provide a qualitative description of hydration of protein surfaces and protein channels, in agreement with experimental structural results. Such approaches define the scope of what liquid state theories should be able to contribute in the field of structural biology, and the degree of refinement of the models that are eventually needed.
In all approaches, it seems that the heart of the problem is to simultaneously describe the strong localized molecular order in terms of correlations, and keeping correctly the long-range electrostatic correlations. This is one of the key theoretical issues to be addressed in the workshop.
Finally, it should be mentioned that another important field where liquid state theories have made recent advances is the bottom-up derivation of coarse-grained effective potentials to be injected in mesoscale simulations of complex molecular assemblies. This can be done by inversion of atomistic simulations using integral equation closures or by direct computation of effective pair potentials in solution. Such procedures were applied up to now to simple models of polymers[25-26] or colloids . The use of liquid state theories to compute the effective interaction between solvated complex molecular entities, such as proteins or nano-objects, remains a challenging problem.
Daniel BORGIS (CNRS) - Organiser
Aurélien PERERA (Sorbonne Université) - Organiser
Stefan Kast (Technical University Dortmund) - Organiser