Aditya Kulkarni

• Water is one of the most important substances and many molecular models of water exist. Widely used molecular models of water are built from a single Lennard-Jones site on which three point charges are positioned, one negative and two positive ones. Models from that class, denoted LJ3PC here, are computationally efficient, but it is well-known that they cannot represent all relevant properties of water simultaneously with good accuracy. Therefore, multi-criteria optimisation (MCO) was used in the present work to parametrise molecular models of water from the LJ3PC class. The set of the best possible compromises between different competing objectives, the Pareto set (PS), as well as its picture in the parameter space (PSPS) were determined. Calculating PS for optimisation problems involving molecular simulations is computationally expensive. Therefore, a novel and highly efficient method was used, which is based on the fact that numerical results from molecular simulations can be interpreted as dimension less numbers. Hence, they carry information on an entire class of models in physical units. Despite the importance of the LJ3PC water model class, its inherent limitations in simultaneously describing different properties of water have never been studied systematically. This was achieved here by calculating PS for this model class. In the first MCO study carried out in this work, the objectives were the quality of the description of the vapour pressure, liquid density and enthalpy of vaporisation. The results were compared to models from the literature. Significant improvements were observed. The new optimisation method for the development of molecular models was shown to be efficient and robust. Having demonstrated the applicability of the new MCO concept, in the second MCO study of the present work, the new optimisation method was applied in a hierarchical scheme to optimize LJ3PC water models taking into account five objectives: the representation of the vapour pressure, saturated liquid density, self-diffusion coefficient, shear viscosity, and relative permittivity. Of the six parameters of the LJ3PC models, five were varied; only the H–O–H bond angle, which is usually chosen based on physical arguments, was kept constant. Our hierarchical approach yields a Pareto set that contains attractive new water models. Furthermore, the results give an idea of what can be achieved by molecular modelling of water with models from the LJ3PC class. In both MCO studies of the present work, we have found characteristic topological features of the PS and PSPS, which depend on the dimensionality of the parameter space M and the objective space N. E.g., M=2 and N=3 yields triangles with needle-like extensions. The reasons for these topological features were unknown so far. It has been shown that they are to be expected if all objective functions of the MCO satisfy two conditions: (a) they can be approximated by quadratic functions and (b) one of the eigenvalues of the Hessian matrix evaluated at the function’s minimum is small compared to the other eigenvalues. Objective functions which meet conditions (a) and (b) have a valley-like topology, for which the valley lies in the direction of the eigenvector corresponding to the lowest eigenvalue. The PSPS can be estimated by starting at the minimum of an objective function, following the valley, and combining these lines for all objective functions. The PS is obtained by evaluating the objective functions. We believe that the conditions (a) and (b) are met in many practical problems and discuss an example from molecular modelling. The improved understanding of the features of these MCO problems opens a route for designing methods for swiftly finding estimates of their PS and PSPS.