Potential energy surfaces: from force fields to neural networks

Almost a century ago, Paul A. M. Dirac remarked that the Schrödinger equation (SE) contains all that is necessary to describe chemical phenomena. Unfortunately, solving the SE, even approximately, remains a time-intensive task and is possible only for systems containing a few atoms. For this reason, potential energy surfaces (PESs) are used to circumvent the solution of the SE altogether: They estimate the energy of a chemical system by evaluating an analytical function. For example, so-called force fields (FFs) model chemical bonds as "springs", i.e. with harmonic potentials. While this is computationally efficient, it limits the accuracy of FFs. A promising alternative are machine learning (ML) methods, such as kernel ridge regression (KRR) and artificial neural networks (NNs), which allow the construction of a PES without assuming a functional form.
In the first part of this thesis, FFs are described in more detail and the minimal distributed charge model (MDCM) is introduced as a way to increase their accuracy. Applications to several challenging molecules are used to demonstrate the utility of this method. In the second part, KRR is reviewed and ways to improve its computational efficiency for PES construction are discussed. Further, the reproducing kernel Hilbert space (RKHS) toolkit is introduced, which largely automates the construction of efficient and accurate PESs for small systems. In the last part, a brief overview of NNs and their historic development is given. The use of NNs to construct PESs is explored and two alternatives are described in more detail. Both variants are applied to various benchmark datasets in order to demonstrate their versatility and accuracy.