## Strong electron correlation

Most traditional quantum chemistry methods fail for systems that are not well-described by a single electron configuration, that is, a single Slater determinant. When a single Slater determinant is not even a good starting point for further approximations, it becomes necessary to treat multiple electron configurations either explicitly or implicitly. At the explicit level, we consider efficient factorizations of the wavefunction, chiefly matrix product states or their generalization, tensor network states. While these factorization methods allow us to consider much larger systems than brute-force complete-active-space approaches, they are still limited to relatively small systems. For large systems, we are considering an alternative factorization, where the wavefunction is written as a product of two-electron states, or geminals. Antisymmetric products of geminals can be constructed with mean-field cost, but still capture most of the effects of strong electron correlation.

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## Density functional theory and its generalizations

Instead of using the electronic wavefunction as the fundamental descriptor for molecular quantum mechanics, one can use other properties. We have explored many approaches of this sort, concentrating on theories based on electron distribution functions and, especially, the electron density. Within the purview of density functional theory, we have sought to develop new approaches based on nonlocal exchange-correlation functionals and quantitatively accurate exchange-correlation potentials. More recently, our primary focus has been on extending density functional theory to excited states and deriving constraints that can be imposed on approximate density functionals to improve their accuracy for systems with degenerate, and nearly-degenerate, ground states.

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## Reaction-Path Finding

In order to predict the products and mechanisms of chemical reactions, we must be able to locate the molecular structures of the reactant, product, reactive intermediate(s), and the transition-states connecting them. More generally, we wish to find the chemical reaction pathway on the potential energy surface. To do this, we have developed new methods for optimizing transition states on potential energy surfaces. When good guesses for transition states are not available but a plausible reaction mechanism can be found, methods like the quadratic string method can be used to locate the minimum energy pathway on the potential energy surface. When one cannot even specify a plausible mechanism, approaches based on the fast-marching method can be used to predict the products and mechanism of a chemical reaction using only the initial reagents. Our work on all these approaches is united by our philosophy of specifying the “reduced” coordinates that are of greatest chemical relevance, and then using these coordinates to improve the robustness and efficiency of the resulting methods.

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## Conceptual Tools

Chemists tend to think of molecules as a set of atoms connected by bonds, with chemical properties determined by the atoms’ properties and bonds’ multiplicities. However, in quantum mechanics molecules are merely assemblages of atomic nuclei and electrons: there are no atoms and there are no bonds. Many other critical concepts— (electronegativity, chemical hardness, atomic charge, oxidation state, -are similarly mysterious from the view of molecular quantum mechanics. We try to elucidate how chemical concepts emerge from quantum mechanics, usually (but not always) within the mathematical framework provided by density-functional theory. Sometimes we work deductively: we start with the exact mathematical framework of conceptual density functional theory and then derive appropriate reactivity indicators for a chemical phenomena. At other times, we work axiomatically: first we list the characteristics a chemist wishes a concept to possess, and then we try to find a mathematical reification of that concept. Using these approaches, we have developed new approaches for predicting regioselectivity and nucleofugality, new perspectives on the hard/soft acid/base principle, new perspectives on the quantum theory of atoms in molecules, and new population analysis methods.

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## Machine Learning

A chemist can guess the properties of an unknown molecule based on its similarity to other, more familiar, molecules. Machine-learning mimics this process: given a set of training data, machine-learning predicts the properties of unknown molecules by assessing their similarity to molecules in the training set. Alternatively, a machine- learning algorithm can be used to uncover underlying trends—and even discover new chemical concepts—from a set of chemical training data. We are using machine-learning methods to make predictions for properties that are difficult (biological binding free energies) or even impossible (carcinogenicity) to compute. We are also using machine- learning methods to uncover chemical reactivity rules.

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## Molecular Mechanics Force Fields

Molecular mechanics (MM) force fields are useful when conformational sampling and entropic effects are important, or when molecules are too large for quantum chemical methods to be routinely applicable. Molecular mechanics force fields model the molecular potential energy surface as combinations of chemically-intuitive terms like bond-stretches, angle-bends, torsions, and through-space atom-atom interactions. The explicit parameterization of the potential energy surface provided by molecular- mechanics (MM) methods is many orders of magnitude faster than quantum-mechanics (QM) methods. MM models are typically constructed using chemical intuition and parameterized by (over)fitting to empirical and/or computational data. When the models are inadequate or the fits are poor, MM methods fail catastrophically. We have developed new methods for automatically parameterizing force fields using QM data, including methods for updating MM force fields on-the-fly during a molecular dynamics simulation. We recently derived the ACKS2 MM model, which directly parameterizes molecular polarization and electron-transfer using Kohn-Sham DFT, and which avoids the “polarization catastrophe” that afflicted previous electronegativity equalization methods.

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