Post-Hartree-Fock methods, such as coupled-cluster singles and doubles (CCSD), provide accurate electronic energies for atoms and molecules, but scale as N6, where N is the number of spin orbitals. Previous work on Data-Driven Quantum Chemistry (DDQC) methods, developed within our group, has built a framework for the accelerated recovery of the CCSD wave function and energy. In this work, we are exploring the underlying data structure related to the electronic correlation using graph neural networks (GNNs). Graphs provide a data structure for finding relationships between pair-energies (nodes), which are defined as the sum of all two-electron excitation energies summed over the virtual orbitals. According to Nesbet’s theorem, the sum of all pair-energies yields the total correlation energy of an atom or molecule. Since pair-energies are an inherently Euclidean object, a metric space can be used to form spatial networks (geometric graphs). Graph data are generated using NetworkX and Deep Graph Library using quantum chemical information by the quantum chemistry package Psi4, through the Psi4Numpy python interface. Our features are derived from Møller-Plesset second-order perturbation theory, using Boys localization, to predict the total CCSD correlation energy. The computational power of the required neural network operations is severely limited on traditional hardware since the number of nodes scales as O2 (where O is the number of occupied orbitals), and in the worst-case fully connected scenario, the number of edges scale as O4. To bypass this bottleneck, we utilize graphical processing units (GPUs) for the training of all GNN models since they offer a highly parallelized, acceleration platform with efficient tensor algebra operations. GPUs allow us to explore a vast amount of chemical space and large graphs without sacrificing accuracy. Within this GNN-DDQC scheme, we have explored the underlining topology of the electron correlation space which could provide potential insight for the efficient development of a DDQC ecosystem.

Exploring the topology of electronic correlation with graph neural networks

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