Machine Learning Coupled Potential Energy Surfaces

Abstract

Recent advancements in machine learning (ML) for potential energy surfaces (PESs) have yielded promising results, with much of the focus on isolated ground-state surfaces. Extending machine learning to coupled excited state surfaces introduces significant challenges, however, particularly in regions of conical intersections. At these points, the adiabatic potentials are non-differentiable, complicating the application of standard ML techniques. In this work, we build on a previously proposed approach that overcomes this issue by learning the nuclear coordinate dependent characteristic polynomial coefficients of the potential matrix instead, which enables the construction of smooth, accurate machine learning models even at seams of conical intersections. The proposed approach is first validated at the ab initio multi-reference configuration interaction (MRCI) level of theory for various molecules. We examine the ability of the proposed model to accurately reproduce energies near a minimum energy conical intersection (MECI) and analyze its performance in capturing seams of conical intersection against analytical MRCI solutions. The results demonstrate that, through this approach, quantitatively accurate machine learning models of seams of conical intersection may be constructed. We further demonstrate the application of this framework to the combined density functional theory and multi-reference configuration interaction (DFT/MRCI) method. The selected configuration interaction nature of the method enables the calculation of accurate excitation energies at a low computational cost. However, the potential energy surfaces produced resemble smooth underlying surfaces contaminated with noise, rendering them locally non smooth. To mitigate this, we treat the local discontinuities in the potential surfaces as noise by explicitly optimizing a whitenoise kernel within a Gaussian process regression framework. We apply this method to optimize DFT/MRCI minimum energy conical intersection geometries and compare the results to ab initio MRCI solutions. While treating the locally discontinuous surface as noise limits the ability to achieve arbitrarily small energy gap at nominal intersections, the structures and branching spaces obtained are in strong agreement with ab initio data. This approach thus proves to be a viable method for generating smooth, accurate representations of DFT/MRCI(2) PESs.