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Creating analytic representations of multiple potential energy surfaces for modeling electronically nonadiabatic processes is a major challenge being addressed in various ways by the chemical dynamics community. In this work, we introduce a new method that can achieve convenient learning of multiple potential energy surfaces (PESs) and their gradients (negatives of the forces) for a polyatomic system. This new method, called compatibilization by deep neural network (CDNN), is demonstrated to be accurate and, even more importantly, to be automatic. The only required input is a database with geometries and potential energies. The method produces a matrix, called the compatible potential energy matrix (CPEM), that may be interpreted as the electronic Hamiltonian in an implicit nonadiabatic basis, and the analytic adiabatic potential energy surfaces and their gradients are obtained by diagonalization and automatic differentiation. We show that the CPEM, which is neither adiabatic nor necessarily diabatic, can be discovered automatically during the learning procedure by the special design of a CDNN architecture. We believe that the CDNN method will be very useful in practice for learning coupled PESs for polyatomic systems because it is accurate and fully automatic. 

Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.