As electrons in matter obey the laws of quantum mechanics, one should in principle solve the Schrödinger equation when performing molecular simulations. On the nanoscale (tracking thousands of molecules during multiple nanoseconds) this is completely unfeasible. Even for approximate electronic structure methods, such as Density Functional Theory (DFT), the computational burden is too high to study phenomena such as diffusion of gas molecules in nanoporous materials (a phenomenon typical for the nanoscale). Classical force fields can resolve this issue, because atomic forces are calculated with very simple analytical expressions, such as the Lennard-Jones potential to model van der Waals interactions between atoms. The major disadvantage is the loss of accuracy compared to electronic structure methods. Indeed, determining appropriate analytic expression and corresponding parameters is a daunting task. This has lead to a situation where force fields are often deemed unreliable, as one cannot anticipate to what extent a force field is able to correctly predict certain properties for the molecular systems at hand.

In recent years, quantifying uncertainty in simulation predictions has attracted the attention of several scientific communities. The Technology Roadmap for Computational Chemistry (http://energy.gov/eere/amo/downloads/itp-chemicals-technology-roadmap-co...) ranked as most critical the “lack of methods to estimate the intrinsic accuracy of calculations, leading to a critical barrier to the more widespread use of these methods for solving practical engineering problems.” Despite the fact that the necessity for uncertainty management in constructing force-field models has been expressed some years ago, the topic has received very little attention so far.

Goal

The goal of this thesis is to investigate how we can predict the uncertainty on simulation results obtained with force fields derived from ab initio reference data. At least the following two test cases will be considered: (i) a simple Lennard-Jones model for Argon to gain initial experience and (ii) a more realistic application to a hydrocarbon force field. Plenty of experimental data for basic hydrocarbons is available and they are practically relevant in production of fuel and chemical feedstocks.

The methodological workflow is as follows. The first step is to construct a database of ab initio reference data using electronic structure methods. Next, force-field parameters will be fitted to this database, where several analytical expression for the force field can be considered. The resulting force field will however never be able to precisely reproduce the reference data, and the biggest challenge is to construct a valid statistical model for the remaining errors between force-field and ab initio results. Once such a statistical model is available, errors on force field simulations can be deduced using standard error propagation techniques. Finally, it will be checked whether the error between simulation predictions and experimental values are within the estimated error.

Fig 1. Workflow of error modelling in ab initio force field development

Only a few examples of force-field error modeling can be found the literature, which typically make use of Bayesian statistics [1]. In this thesis, a new type of maximum-likelihood estimate of the error model will be developed, which may (but must not) be combined with Bayesian inference. The error model can also be validated by using cross-validation techniques on the database of reference calculations.

Mobility
This research topic will be conducted in the framework of a strong international network and if possible the student will be actively involved in work discussions with collaborative partners.

Motivation Appl. Phys.
This thesis requires physical insight because a thorough understanding of interactions at the atomic level is necessary to construct force fields and perform molecular simulations. The methods that will be developed are however more broadly applicable to solve general practical engineering problems where uncertainty of the model parameters must be quantified.