Graph Theoretic Molecular Fragmentation for Multidimensional Potential Energy Surfaces Yield an Adaptive and General Transfer Machine Learning Protocol

Abstract

Over a series of publications we have introduced a graph-theoretic description for molecular fragmentation. Here, a system is divided into a set of nodes, or vertices, that are then connected through edges, faces, and higher-order simplexes to represent a collection of spatially overlapping and locally interacting subsystems. Each such subsystem is treated at two levels of electronic structure theory, and the result is used to construct many-body expansions that are then embedded within an ONIOM-scheme. These expansions converge rapidly with many-body order (or graphical rank) of subsystems and have been previously used for ab initio molecular dynamics (AIMD) calculations and for computing multidimensional potential energy surfaces. Specifically, in all these cases we have shown that CCSD and MP2 level AIMD trajectories and potential surfaces may be obtained at density functional theory cost. The approach has been demonstrated for gas-phase studies, for condensed phase electronic structure, and also for basis set extrapolation-based AIMD. Recently, this approach has also been used to derive new quantum-computing algorithms that enormously reduce the quantum circuit depth in a circuit-based computation of correlated electronic structure. In this publication, we introduce (a) a family of neural networks that act in parallel to represent, efficiently, the post-Hartree–Fock electronic structure energy contributions for all simplexes (fragments), and (b) a new k-means-based tessellation strategy to glean training data for high-dimensional molecular spaces and minimize the extent of training needed to construct this family of neural networks. The approach is particularly useful when coupled cluster accuracy is desired and when fragment sizes grow in order to capture nonlocal interactions accurately. The unique multidimensional k-means tessellation/clustering algorithm used to determine our training data for all fragments is shown to be extremely efficient and reduces the needed training to only 10% of data for all fragments to obtain accurate neural networks for each fragment. These fully connected dense neural networks are then used to extrapolate the potential energy surface for all molecular fragments, and these are then combined as per our graph-theoretic procedure to transfer the learning process to a full system energy for the entire AIMD trajectory at less than one-tenth the cost as compared to a regular fragmentation-based AIMD calculation.

Graph-Theoretic Molecular Fragmentation for Potential Surfaces Leads Naturally to a Tensor Network Form and Allows Accurate and Efficient Quantum Nuclear Dynamics

“…In refs , − , , and , we have developed and demonstrated the utility of a graph theory-based molecular fragmentation − approximation that is based on many-body expansions ,,− embedded within an our own n-layered integrated molecular orbital and molecular mechanics ( ONIOM) ,− scheme. We have used the approach to compute conservative Born–Oppenheimer − and extended Lagrangian , ab initio molecular dynamics (AIMD) trajectories at the accuracy of CCSD and MP2 levels of theory but at the computational cost of density functional theory (DFT).…”

“…The implementation of this methodology allows simultaneous use of Gaussian, ORCA, Psi4, Quantum Espresso, and OpenMX within a single electronic structure, dynamics, and potential surface calculation. An efficient approach for quantum computation has also been derived from these methods, and machine-learning generalizations have also been presented in ref . In this section, we utilize the graph-theory-based molecular fragmentation to provide an efficient representation for quantum nuclear dynamical propagation.…”

“…The methodology for obtaining the graphs (and hence fragments that are denoted by the simplexes) is general and allows for treatment of non-uniform systems, based on the needed level of local many-body interactions. Although this graph-based fragmentation approach has been discussed in detail in refs , − , , and , in Appendix A, we present an illustration of how this is done for complex systems. The overarching goal of our study here is to take steps toward performing quantum nuclear dynamics simulations where the potential surface is determined as per eq and its multi-topology generalizations. ,, …”

“…Both levels of theory utilize the Pople style 6-31++g­(d,p) basis set as part of eq . This is not a requirement, and in refs , , , and , other levels of theory and basis have been explored. As discussed in refs and , we use the description of potential energy surfaces using multiple graphs.…”

“…Here, we present two additional discussions for a 3 10 -helix (Figure a) and DNA (Figure b) fragmentation to showcase the generality of the graph-theoretic approach. Our graph-based fragmentation is general, and the electronic structure aspects have been discussed in refs , − , , and . To help clarify, we present two illustrations in Figure , which show how the graph partition is done for more complex systems that may also display quantum nuclear effects.…”