Insights into the deviation from piecewise linearity in transition metal complexes from supervised machine learning models.

Virtual high-throughput screening (VHTS) and machine learning (ML) with density functional theory (DFT) suffer from inaccuracies from the underlying density functional approximation (DFA). Many of these inaccuracies can be traced to the lack of derivative discontinuity that leads to a curvature in the energy with electron addition or removal. Over a dataset of nearly one thousand transition metal complexes typical of VHTS applications, we computed and analyzed the average curvature (i.e., deviation from piecewise linearity) for 23 density functional approximations spanning multiple rungs of "Jacob's ladder". While we observe the expected dependence of the curvatures on Hartree-Fock exchange, we note limited correlation of curvature values between different rungs of "Jacob's ladder". We train ML models (i.e., artificial neural networks or ANNs) to predict the curvature and the associated frontier orbital energies for each of these 23 functionals and then interpret differences in curvature among the different DFAs through analysis of the ML models. Notably, we observe spin to play a much more important role in determining the curvature of range-separated and double hybrids in comparison to semi-local functionals, explaining why curvature values are weakly correlated between these and other families of functionals. Over a space of 187.2k hypothetical compounds, we use our ANNs to pinpoint DFAs for which representative transition metal complexes have near-zero curvature with low uncertainty, demonstrating an approach to accelerate screening of complexes with targeted optical gaps.

Ligand Additivity and Divergent Trends in Two Types of Delocalization Errors from Approximate Density Functional Theory.

The predictive accuracy of density functional theory (DFT) is hampered by delocalization errors, especially for correlated systems such as transition-metal complexes. Two complementary strategies have been developed to reduce delocalization error: eliminating the global curvature with change in charge, and applying a linear response Hubbard U as a measure of local curvature at a metal center at fixed charge in a DFT+U framework. We investigate the relationship between the two delocalization error measures as the ligand field strength is varied with the number of strong-field ligands in a series of heteroleptic complexes or by geometrically constraining the metal-ligand bond length in homoleptic octahedral complexes. We show that across these sets of complexes an inverse relationship generally exists between global and local curvatures. We find that effects of ligand substitution on both measures of delocalization are typically additive, but the quantities seldom coincide.

Large-scale comparison of 3d and 4d transition metal complexes illuminates the reduced effect of exchange on second-row spin-state energetics.

Density functional theory (DFT) is widely used in transition-metal chemistry, yet essential properties such as spin-state energetics in transition-metal complexes (TMCs) are well known to be sensitive to the choice of the exchange-correlation functional. Increasing the amount of exchange in a functional typically shifts the preferred ground state in first-row TMCs from low-spin to high-spin by penalizing delocalization error, but the effect on properties of second-row complexes is less well known. We compare the exchange sensitivity of adiabatic spin-splitting energies in pairs of mononuclear 3d and 4d mid-row octahedral transition-metal complexes. We analyze hundreds of complexes assembled from four metals in two oxidation states with ten small monodentate ligands that span a wide range of field strengths expected to favor a variety of ground states. We observe consistently lower but proportional sensitivity to exchange fraction among 4d TMCs with respect to their isovalent 3d TMC counterparts, leading to the largest difference in sensitivities for the strongest field ligands. The combined effect of reduced exchange sensitivities and the greater low-spin bias of most 4d TMCs means that while over one-third of 3d TMCs change ground states over a modest variation (ca. 0.0-0.3) in exchange fraction, almost no 4d TMCs do. Differences in delocalization, as judged through changes in the metal-ligand bond lengths between spin states, do not explain the distinct behavior of 4d TMCs. Instead, evaluation of potential energy curves in 3d and 4d TMCs reveals that higher exchange sensitivities in 3d TMCs are likely due to the opposing effect of exchange on the low-spin and high-spin states, whereas the effect on both spin states is more comparable in 4d TMCs.

Stochastic Optimally Tuned Range-Separated Hybrid Density Functional Theory.

We develop a stochastic formulation of the optimally tuned range-separated hybrid density functional theory that enables significant reduction of the computational effort and scaling of the nonlocal exchange operator at the price of introducing a controllable statistical error. Our method is based on stochastic representations of the Coulomb convolution integral and of the generalized Kohn-Sham density matrix. The computational cost of the approach is similar to that of usual Kohn-Sham density functional theory, yet it provides a much more accurate description of the quasiparticle energies for the frontier orbitals. This is illustrated for a series of silicon nanocrystals up to sizes exceeding 3000 electrons. Comparison with the stochastic GW many-body perturbation technique indicates excellent agreement for the fundamental band gap energies, good agreement for the band edge quasiparticle excitations, and very low statistical errors in the total energy for large systems. The present approach has a major advantage over one-shot GW by providing a self-consistent Hamiltonian that is central for additional postprocessing, for example, in the stochastic Bethe-Salpeter approach.

Metropolis Evaluation of the Hartree-Fock Exchange Energy.

We examine the possibility of using a Metropolis algorithm for computing the exchange energy in a large molecular system. Following ideas set forth in a recent publication (Baer, Neuhauser, and Rabani, Phys. Rev. Lett. 111, 106402 (2013)) we focus on obtaining the exchange energy per particle (ExPE, as opposed to the total exchange energy) to a predefined statistical error and on determining the numerical scaling of the calculation achieving this. For this we assume that the occupied molecular orbitals (MOs) are known and given in terms of a standard Gaussian atomic basis set. The Metropolis random walk produces a sequence of pairs of three-dimensional points (x,x'), which are distributed in proportion to ρ(x,x')(2), where ρ(x,x') is the density matrix. The exchange energy per particle is then simply the average of the Coulomb repulsion energy υC(|x-x'|) over these pairs. To reduce the statistical error we separate the exchange energy into a short-range term that can be calculated deterministically in a linear scaling fashion and a long-range term that is treated by the Metropolis method. We demonstrate the method on water clusters and silicon nanocrystals showing the magnitude of the ExPE standard deviation is independent of system size. In the water clusters a longer random walk was necessary to obtain full ergodicity as Metropolis walkers tended to get stuck for a while in localized regions. We developed a diagnostic tool that can alert a user when such a situation occurs. The calculation effort scales linearly with system size if one uses an atom screening procedure that can be made numerically exact. In systems where the MOs can be localized efficiently the ExPE can even be computed with "sublinear scaling" as the MOs themselves can be screened.