Machine learning dynamic correlation in chemical kinetics.

Lattice models are a useful tool to simulate the kinetics of surface reactions. Since it is expensive to propagate the probabilities of the entire lattice configurations, it is practical to consider the occupation probabilities of a typical site or a cluster of sites instead. This amounts to a moment closure approximation of the chemical master equation. Unfortunately, simple closures, such as the mean-field and the pair approximation (PA), exhibit weaknesses in systems with significant long-range correlation. In this paper, we show that machine learning (ML) can be used to construct accurate moment closures in chemical kinetics using the lattice Lotka-Volterra model as a model system. We trained feedforward neural networks on kinetic Monte Carlo (KMC) results at select values of rate constants and initial conditions. Given the same level of input as PA, the ML moment closure (MLMC) gave accurate predictions of the instantaneous three-site occupation probabilities. Solving the kinetic equations in conjunction with MLMC gave drastic improvements in the simulated dynamics and descriptions of the dynamical regimes throughout the parameter space. In this way, MLMC is a promising tool to interpolate KMC simulations or construct pretrained closures that would enable researchers to extract useful insight at a fraction of the computational cost.

A pyridinic Fe-N4 macrocycle models the active sites in Fe/N-doped carbon electrocatalysts.

Iron- and nitrogen-doped carbon (Fe-N-C) materials are leading candidates to replace platinum catalysts for the oxygen reduction reaction (ORR) in fuel cells; however, their active site structures remain poorly understood. A leading postulate is that the iron-containing active sites exist primarily in a pyridinic Fe-N4 ligation environment, yet, molecular model catalysts generally feature pyrrolic coordination. Herein, we report a molecular pyridinic hexaazacyclophane macrocycle, (phen2N2)Fe, and compare its spectroscopic, electrochemical, and catalytic properties for ORR to a typical Fe-N-C material and prototypical pyrrolic iron macrocycles. N 1s XPS and XAS signatures for (phen2N2)Fe are remarkably similar to those of Fe-N-C. Electrochemical studies reveal that (phen2N2)Fe has a relatively high Fe(III/II) potential with a correlated ORR onset potential within 150 mV of Fe-N-C. Unlike the pyrrolic macrocycles, (phen2N2)Fe displays excellent selectivity for four-electron ORR, comparable to Fe-N-C materials. The aggregate spectroscopic and electrochemical data demonstrate that (phen2N2)Fe is a more effective model of Fe-N-C active sites relative to the pyrrolic iron macrocycles, thereby establishing a new molecular platform that can aid understanding of this important class of catalytic materials.

Bootstrap Embedding for Molecules.

Fragment embedding is one way to circumvent the high computational scaling of accurate electron correlation methods. The challenge of applying fragment embedding to molecular systems primarily lies in the strong entanglement and correlation that prevent accurate fragmentation across chemical bonds. Recently, Schmidt decomposition has been shown effective for embedding fragments that are strongly coupled to a bath in several model systems. In this work, we extend a recently developed quantum embedding scheme, bootstrap embedding (BE), to molecular systems. The resulting method utilizes the matching conditions naturally arising from using overlapping fragments to optimize the embedding. Numerical simulation suggests that the accuracy of the embedding improves rapidly with fragment size for small molecules, whereas larger fragments that include orbitals from different atoms may be needed for larger molecules. BE scales linearly with system size (apart from an integral transform) and hence can potentially be useful for large-scale calculations.

Incremental embedding: A density matrix embedding scheme for molecules.

The idea of using fragment embedding to circumvent the high computational scaling of accurate electronic structure methods while retaining high accuracy has been a long-standing goal for quantum chemists. Traditional fragment embedding methods mainly focus on systems composed of weakly correlated parts and are insufficient when division across chemical bonds is unavoidable. Recently, density matrix embedding theory and other methods based on the Schmidt decomposition have emerged as a fresh approach to this problem. Despite their success on model systems, these methods can prove difficult for realistic systems because they rely on either a rigid, non-overlapping partition of the system or a specification of some special sites (i.e., "edge" and "center" sites), neither of which is well-defined in general for real molecules. In this work, we present a new Schmidt decomposition-based embedding scheme called incremental embedding that allows the combination of arbitrary overlapping fragments without the knowledge of edge sites. This method forms a convergent hierarchy in the sense that higher accuracy can be obtained by using fragments involving more sites. The computational scaling for the first few levels is lower than that of most correlated wave function methods. We present results for several small molecules in atom-centered Gaussian basis sets and demonstrate that incremental embedding converges quickly with fragment size and recovers most static correlation in small basis sets even when truncated at the second lowest level.

σ-SCF: A direct energy-targeting method to mean-field excited states.

The mean-field solutions of electronic excited states are much less accessible than ground state (e.g., Hartree-Fock) solutions. Energy-based optimization methods for excited states, like Δ-SCF (self-consistent field), tend to fall into the lowest solution consistent with a given symmetry-a problem known as "variational collapse." In this work, we combine the ideas of direct energy-targeting and variance-based optimization in order to describe excited states at the mean-field level. The resulting method, σ-SCF, has several advantages. First, it allows one to target any desired excited state by specifying a single parameter: a guess of the energy of that state. It can therefore, in principle, find all excited states. Second, it avoids variational collapse by using a variance-based, unconstrained local minimization. As a consequence, all states-ground or excited-are treated on an equal footing. Third, it provides an alternate approach to locate Δ-SCF solutions that are otherwise hardly accessible by the usual non-aufbau configuration initial guess. We present results for this new method for small atoms (He, Be) and molecules (H2, HF). We find that σ-SCF is very effective at locating excited states, including individual, high energy excitations within a dense manifold of excited states. Like all single determinant methods, σ-SCF shows prominent spin-symmetry breaking for open shell states and our results suggest that this method could be further improved with spin projection.

Sticking to (first) principles: quantum molecular dynamics and Bayesian probabilistic methods to simulate aquatic pollutant absorption spectra.

This work explores the relationship between theoretically predicted excitation energies and experimental molar absorption spectra as they pertain to environmental aquatic photochemistry. An overview of pertinent Quantum Chemical descriptions of sunlight-driven electronic transitions in organic pollutants is presented. Second, a combined molecular dynamics (MD), time-dependent density functional theory (TD-DFT) analysis of the ultraviolet to visible (UV-Vis) absorption spectra of six model organic compounds is presented alongside accurate experimental data. The functional relationship between the experimentally observed molar absorption spectrum and the discrete quantum transitions is examined. A rigorous comparison of the accuracy of the theoretical transition energies (ΔES0→Sn) and oscillator strength (fS0→Sn) is afforded by the probabilistic convolution and deconvolution procedure described. This method of deconvolution of experimental spectra using a Gaussian Mixture Model combined with Bayesian Information Criteria (BIC) to determine the mean (µ) and standard deviation (σ) as well as the number of observed singlet to singlet transition energy state distributions. This procedure allows a direct comparison of the one-electron (quantum) transitions that are the result of quantum chemical calculations and the ensemble of non-adiabatic quantum states that produce the macroscopic effect of a molar absorption spectrum. Poor agreement between the vertical excitation energies produced from TD-DFT calculations with five different functionals (CAM-B3LYP, PBE0, M06-2X, BP86, and LC-BLYP) suggest a failure of the theory to capture the low energy, environmentally important, electronic transitions in our model organic pollutants. However, the method of explicit-solvation of the organic solute using the quantum Effective Fragment Potential (EFP) in a density functional molecular dynamics trajectory simulation shows promise as a robust model of the hydrated organic pollutant. Furthermore, the described protocol can be extended using higher-level equilibration and vertical excitation methods to increase the numerical accuracy and describe multi-reference electronic transitions. Finally, a measure of the accuracy of theoretically derived absorption spectra is discussed as a tool to further develop our capacity to produce accurate a priori simulations of sunlight-driven photochemistry in natural waters.