Accurate densities of states for disordered systems from free probability: Live Free or Diagonalize

We investigate how free probability allows us to approximate the density of states in tight binding models of disordered electronic systems. Extending our previous studies of the Anderson model in neighbor interactions [J. Chen et al., Phys. Rev. Lett. 109, 036403 (2012)], we find that free probability continues to provide accurate approximations for systems with constant interactions on two- and three-dimensional lattices or with next-nearest-neighbor interactions, with the results being visually indistinguishable from the numerically exact solution. For systems with disordered interactions, we observe a small but visible degradation of the approximation. To explain this behavior of the free approximation, we develop and apply an asymptotic error analysis scheme to show that the approximation is accurate to the eighth moment in the density of states for systems with constant interactions, but is only accurate to sixth order for systems with disordered interactions. The error analysis also allows us to calculate asymptotic corrections to the density of states, allowing for systematically improvable approximations as well as insight into the sources of error without requiring a direct comparison to an exact solution

Person-specific theory of mind in medial pFC

Although research on theory of mind has strongly implicated the dorsomedial pFC (incuding medial BA 8 and BA 9), the unique contributions of medial pFC (MPFC; corresponding to medial BA 10) to mentalizing remain uncertain. The extant literature has considered the possibility that these regions may be specialized for self-related cognition or for reasoning about close others, but evidence for both accounts has been inconclusive. We propose a novel theoretical framework: MPFC selectively implements "person-specific theories of mind" (ToMp) representing the unique, idiosyncratic traits or attributes of well-known individuals. To test this hypothesis, we used fMRI to assess MPFC responses in Democratic and Republican participants as they evaluated more or less subjectively well-known political figures. Consistent with the ToMp account, MPFC showed greater activity to subjectively well-known targets, irrespective of participants' reported feelings of closeness or similarity. MPFC also demonstrated greater activity on trials in which targets (whether politicians or oneself) were judged to be relatively idiosyncratic, making a generic theory of mind inapplicable. These results suggest that MPFC may supplement the generic theory of mind process, with which dorsomedial pFC has been associated, by contributing mentalizing capacities tuned to individuated representations of specific well-known others

Disconfirmation modulates the neural correlates of the false consensus effect: A parametric modulation approach

The false consensus effect (FCE) - the tendency to (erroneously) project our attitudes and opinions onto others - is an enduring bias in social reasoning with important societal implications. In this fMRI investigation, we examine the neural correlates of within-subject variation in consensus bias on a variety of social and political issues. Bias demonstrated a strong association with activity in brain regions implicated in self-related cognition, mentalizing, and valuation. Importantly, however, recruitment of these regions predicted consensus bias only in the presence of social disconfirmation, in the form of feedback discrepant with participants' own attitudes. These results suggest that the psychological and neural mechanisms underlying the tendency to project attitudes onto others are crucially moderated by motivational factors, including the desire to affirm the normativity of one's own position. This research complements social psychological theorizing about the factors contributing to the FCE, and further emphasizes the role of motivated cognition in social reasoning

Transferability in Machine Learning for Electronic Structure via the Molecular Orbital Basis

We present a machine learning (ML) method for predicting electronic structure correlation energies using Hartree-Fock input.The total correlation energy is expressed in terms of individual and pair contributions from occupied molecular orbitals, and Gaussian process regression is used to predict these contributions from a feature set that is based on molecular orbital properties, such as Fock, Coulomb, and exchange matrix elements. With the aim of maximizing transferability across chemical systems and compactness of the feature set, we avoid the usual specification of ML features in terms of atom- or geometry-specific information, such atom/element-types, bond-types, or local molecular structure. ML predictions of MP2 and CCSD energies are presented for a range of systems, demonstrating that the method maintains accuracy while providing transferability both within and across chemical families; this includes predictions for molecules with atom-types and elements that are not included in the training set. The method holds promise both in its current form and as a proof-of-principle for the use of ML in the design of generalized density-matrix functionals.Comment: 8 pages, 5 figure

Even-handed subsystem selection in projection-based embedding

Projection-based embedding offers a simple framework for embedding correlated wavefunction methods in density functional theory. Partitioning between the correlated wavefunction and density functional subsystems is performed in the space of localized molecular orbitals. However, during a large geometry change—such as a chemical reaction—the nature of these localized molecular orbitals, as well as their partitioning into the two subsystems, can change dramatically. This can lead to unphysical cusps and even discontinuities in the potential energy surface. In this work, we present an even-handed framework for localized orbital partitioning that ensures consistent subsystems across a set of molecular geometries. We illustrate this problem and the even-handed solution with a simple example of an S_N2 reaction. Applications to a nitrogen umbrella flip in a cobalt-based CO_2 reduction catalyst and to the binding of CO to Cu clusters are presented. In both cases, we find that even-handed partitioning enables chemically accurate embedding with modestly sized embedded regions for systems in which previous partitioning strategies are problematic

Performance of Bootstrap Embedding for long-range interactions and 2D systems

Fragment embedding approaches offer the possibility of accurate description of strongly correlated systems with low-scaling computational expense. In particular, wave function embedding approaches have demonstrated the ability to subdivide systems across highly entangled regions, promising wide applicability for a number of challenging systems. In this paper, we focus on the wave function embedding method Bootstrap Embedding, extending it to the Pariser–Parr–Pople and 2D Hubbard models in order to evaluate the behaviour of the method in systems that are less amenable to local fragment embedding. We find that Bootstrap Embedding remains accurate for these systems, and we investigate how fragment size, shape, and choice of matching conditions affect the results. We also evaluate the properties of Bootstrap Embedding that lead to the method's favourable convergence properties. Keywords: Embedding; correlation; Bootstrap; DMETNational Science Foundation (U.S.) (Grant CHE-1464804

A universal density matrix functional from molecular orbital-based machine learning: Transferability across organic molecules

We address the degree to which machine learning (ML) can be used to accurately and transferably predict post-Hartree-Fock correlation energies. Refined strategies for feature design and selection are presented, and the molecular-orbital-based machine learning (MOB-ML) method is applied to several test systems. Strikingly, for the second-order Møller-Plessett perturbation theory, coupled cluster with singles and doubles (CCSD), and CCSD with perturbative triples levels of theory, it is shown that the thermally accessible (350 K) potential energy surface for a single water molecule can be described to within 1 mhartree using a model that is trained from only a single reference calculation at a randomized geometry. To explore the breadth of chemical diversity that can be described, MOB-ML is also applied to a new dataset of thermalized (350 K) geometries of 7211 organic models with up to seven heavy atoms. In comparison with the previously reported Δ-ML method, MOB-ML is shown to reach chemical accuracy with threefold fewer training geometries. Finally, a transferability test in which models trained for seven-heavy-atom systems are used to predict energies for thirteen-heavy-atom systems reveals that MOB-ML reaches chemical accuracy with 36-fold fewer training calculations than Δ-ML (140 vs 5000 training calculations)