Points In Configuration Space Research Articles

A local Gaussian Processes method for fitting potential surfaces that obviates the need to invert large matrices

In order to compute a vibrational spectrum, one often wishes to start with a set of ab initio Born–Oppenheimer potential values at points, called fitting points, and interpolate or fit to find values of the potential at quadrature or collocation points. It is common to do this once to build a potential energy surface (PES). Once the PES is known, it can be evaluated at any point in configuration space. Gaussian Process (GP) is frequently being used to make a PES. As is the case in other interpolation methods, to use GP one must store and invert a matrix whose size is the number of fitting points. The matrix is sometimes large enough that approximations are introduced to reduce the cost of the calculation. We show that is possible to use many local Gaussian Process fits rather than one global fit. Retaining only local Gaussians and the associated points works well despite the fact that other Gaussians have tails with significant amplitude in the local region. We demonstrate that from the potential values obtained from the local fits it is possible to compute accurate energy levels of formaldehyde. In one calculation, potential values were obtained with N=120,000 fitting points by inverting matrices of size less than m=400. The local idea reduces the cost from N3 to T(m3+N), where T is the number of desired potential points.

Low energy fold paths in multistable origami structures

Origami design concepts are finding pervasive utility in engineering applications due to their ability to map complex shape transformations into a series of folding actions. The interplay between stretching, folding, and facet bending modes in origami structures also generates a complex energy landscape of multistable states to leverage for engineering applications. However, identifying rigid and deformable folding paths in this high-dimensional and non-convex energy landscape remains a challenge. To help address this challenge, we first introduce a global, constraint-based approach to modeling origami that uses a redundant kinematic description of the facets and nodes, and treats the kinematic compatibility between these redundant descriptors as a constraint. This approach allows for complex facet shapes without increasing the dimensionality of the system, as would be necessary in truss-based and other node-based formulations in order to stiffen the facet. Secondly, we adopt the nudged elastic band method, that is widely used in computational chemistry, to identify minimum energy folding paths. This strategy addresses, from a global perspective, the difficulty of piecing together sequences of local folding steps in order to connect two different points in configuration space. We implement this path finding approach on both the kinematic constraint formulation and a truss-based model, and compare their behaviors on a series of folding and multistable origami examples.

Ising-like and Fibonacci anyons from KZ-equations

In this work we present solutions to Knizhnik-Zamolodchikov (KZ) equations corresponding to conformal block wavefunctions of non-Abelian Ising-like and Fibonacci Anyons. We solve these equations around regular singular points in configuration space in terms of hypergeometric functions and derive explicit monodromy representations of the braid group action. This confirms the correct non-Abelian statistics of the solutions. One novelty of our approach is that we explicitly keep track of spin basis states and identify conformal blocks uniquely with such states at relevant points in moduli space.

Intrinsic Dimension of Path Integrals: Data-Mining Quantum Criticality and Emergent Simplicity

Quantum many-body systems are characterized by patterns of correlations that define highly-non trivial manifolds when interpreted as data structures. Physical properties of phases and phase transitions are typically retrieved via simple correlation functions, that are related to observable response functions. Recent experiments have demonstrated capabilities to fully characterize quantum many-body systems via wave-function snapshots, opening new possibilities to analyze quantum phenomena. Here, we introduce a method to data mine the correlation structure of quantum partition functions via their path integral (or equivalently, stochastic series expansion) manifold. We characterize path-integral manifolds generated via state-of-the-art Quantum Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of distances from nearest neighbors (NN): the former is related to dataset complexity, while the latter is able to diagnose connectivity features of points in configuration space. We show how these properties feature universal patterns in the vicinity of quantum criticality, that reveal how data structures {\it simplify} systematically at quantum phase transitions. This is further reflected by the fact that both ID and variance of NN-distances exhibit universal scaling behavior in the vicinity of second-order and Berezinskii-Kosterlitz-Thouless critical points. Finally, we show how non-Abelian symmetries dramatically influence quantum data sets, due to the nature of (non-commuting) conserved charges in the quantum case. Complementary to neural network representations, our approach represents a first, elementary step towards a systematic characterization of path integral manifolds before any dimensional reduction is taken, that is informative about universal behavior and complexity, and can find immediate application to both experiments and Monte Carlo simulations.

Exploring the effects of quantum decoherence on the excited-state dynamics of molecular systems

Fewest-switches surface hopping (FSSH) is employed in order to investigate the nonadiabatic excited-state dynamics of thiophene and related compounds and hence to establish a connection between the electronic system, the critical points in configuration space and the deactivation dynamics. The potential-energy surfaces of the studied molecules were calculated with complete active space self-consistent field and time-dependent density-functional theory. They are analyzed thoroughly to locate and optimize minimum-energy conical intersections, which are essential to the dynamics of the system. The influence of decoherence on the dynamics is examined by employing different decoherence schemes. We find that irrespective of the employed decoherence algorithm, the population dynamics of thiophene give results which are sound with the expectations grounded on the analysis of the potential-energy surface. A more detailed look at single trajectories as well as on the excited-state lifetimes, however, reveals a substantial dependence on how decoherence is accounted for. In order to connect these findings, we describe how ensemble averaging cures some of the overcoherence problems of uncorrected FSSH. Eventually, we identify carbon–sulfur bond cleavage as a common feature accompanying electronic transitions between different states in the simulations of all thiophene-related compounds studied in this work, which is of interest due to their relevance in organic photovoltaics.

Symmetry of the phononic landscape of twisted kagome lattices across the duality boundary

In this work, we investigate the symmetry of the phonon landscape of twisted kagome lattices across their duality boundary. The study is inspired by recent work by Fruchart et al. [Nature, 577, 2020], who specialized the notion of duality to the mechanistic problem of kagome lattices and linked it to the existence of duality transformations between configurations that are symmetrically located across a critical transition point in configuration space. Our first goal is to elucidate how the existence of matching phonon spectra between dual configurations manifest in terms of observable wavefield characteristics. To this end, we explore the possibility to aggregate dual kagome configurations into bi-domain lattices that are geometrically heterogeneous but retain a mechanically homogeneous response. Our second objective is to extend the analysis to structural lattices of beams, which are representative of realistic cellular metamaterials. We show that, in this case, the symmetry of the phonon landscape across the duality boundary is broken, implying that the conditions for dual behavior do not merely depend on the geometry, but also on the dominant mechanisms of the cell, suggesting a dichotomy between geometric and functional duality.

Minimal Geodesics of the Isosceles Three Body Problem

The isosceles three-body problem with nonnegative energy is studied from a variational point of view based on the Jacobi–Maupertuis metric. The solutions are represented by geodesics in the two-dimensional configuration space. Since the metric is singular at collisions, an approach based on the theory of length spaces is used. This provides an alternative to the more familiar approach based on the principle of least action. The emphasis is on the existence and properties of minimal geodesics, that is, shortest curves connecting two points in configuration space. For any two points, even singular points, a minimal geodesic exists and is nonsingular away from the endpoints. For the zero energy case, it is possible to use knowledge of the behavior of the flow on the collision manifold to see that certain solutions must be minimal geodesics. In particular, the geodesic corresponding to the collinear homothetic solution turns out to be a minimal for certain mass ratios.

A Nonrelativistic Quantum Field Theory with Point Interactions in Three Dimensions

We construct a Hamiltonian for a quantum-mechanical model of nonrelativistic particles in three dimensions interacting via the creation and annihilation of a second type of nonrelativistic particles, which are bosons. The interaction between the two types of particles is a point interaction concentrated on the points in configuration space where the positions of two different particles coincide. We define the operator, and its domain of self-adjointness, in terms of co-dimension-three boundary conditions on the set of collision configurations relating sectors with different numbers of particles.

Chaos and variance in galaxy formation

The evolution of galaxies is governed by equations with chaotic solutions: gravity and compressible hydrodynamics. While this micro-scale chaos and stochasticity has been well studied, it is poorly understood how it couples to macro-scale properties examined in simulations of galaxy formation. In this paper, we show how perturbations introduced by floating-point roundoff, random number generators, and seemingly trivial differences in algorithmic behaviour can produce non-trivial differences in star formation histories, circumgalactic medium (CGM) properties, and the distribution of stellar mass. We examine the importance of stochasticity due to discreteness noise, variations in merger timings and how self-regulation moderates the effects of this stochasticity. We show that chaotic variations in stellar mass can grow until halted by feedback-driven self-regulation or gas exhaustion. We also find that galaxy mergers are critical points from which large (as much as a factor of 2) variations in quantities such as the galaxy stellar mass can grow. These variations can grow and persist for more than a Gyr before regressing towards the mean. These results show that detailed comparisons of simulations require serious consideration of the magnitude of effects compared to run-to-run chaotic variation, and may significantly complicate interpreting the impact of different physical models. Understanding the results of simulations requires us to understand that the process of simulation is not a mapping of an infinitesimal point in configuration space to another, final infinitesimal point. Instead, simulations map a point in a space of possible initial conditions points to a volume of possible final states.

Coincidence Structures and Hard-Core Few-Body Interactions

The symmetry and topology of the coincidence structure, i.e. the locus of points in configuration space corresponding to particles in the same position, plays a critical role in extracting universal properties for few-body models with hard-core interactions. The coincidence structure is a scale-invariant union of manifolds possessing rich symmetry. When there are zero-range hard-core two-body interactions, the coincidence structure forms a nodal surface for finite-energy wave functions in configuration space. More generally, it acts like a defect that changes the topology of configuration space in a way that depends on the dimension of the underlying space, the total number of particles, and the number of particles in the hard-core interaction. We show that for the specific case of three-body hard-core interactions in one-dimension, the configuration space is no longer simply-connected, providing a topological explanation for several models that exhibit anyonic behavior.

Importance of eigenvector sign consistency in computations of expectation values via mixed quantum-classical surface-hopping dynamics

For simulations of systems undergoing nonadiabatic dynamics between multiple quantum states, one must often compute nonadiabatic coupling vectors, which involve the overlap of one adiabatic state with the gradient (with respect to the nuclear positions) of another. Because these states are usually computed numerically using a matrix diagonalization routine, the signs of the eigenvectors and the resulting nonadiabatic coupling vectors can change erratically over the course of a trajectory. To ensure a smooth evolution of the eigenvectors, one usually tracks the sign of the overlap of the current eigenvector with that of the previous time step and corrects the sign if needed. However, when computing time-dependent expectation values based on an ensemble of trajectories (which are typically initialized at different points in configuration space), one now has to ensure eigenvector sign continuity across the entire ensemble. In this work, we present a simple procedure for accomplishing this in mixed quantum-classical surface-hopping dynamics. The utility of this procedure is demonstrated by calculating both adiabatic state populations in a reduced model of a condensed phase proton-coupled electron transfer (PCET) reaction and subsystem state populations in an excitonic model of an alpha-helical chain undergoing vibrational energy transport (VET), using mixed quantum-classical Liouville surface-hopping dynamics. Our results show that when eigenvector sign continuity across the ensemble is neglected, as is commonly done, one can obtain large deviations from the correct values. However, after implementing this procedure, the results agree better with the exact results in the case of the PCET model and behave physically in the case of the VET model. These results illustrate the importance of a careful treatment of the signs in mixed quantum-classical dynamics calculations of expectation values in the adiabatic representation.

Direct grid-based quantum dynamics on propagated diabatic potential energy surfaces

We present a method for performing non-adiabatic, grid-based nuclear quantum dynamics calculations using diabatic potential energy surfaces (PESs) generated “on-the-fly”. Gaussian process regression is used to interpolate PESs by using electronic structure energies, calculated at points in configuration space determined by the nuclear dynamics, and diabatising the results using the propagation diabatisation method reported recently (Richings and Worth, 2015). Our new method is successfully demonstrated using a grid-based approach to model the non-adiabatic dynamics of the butatriene cation. Overall, our scheme offers a route towards accurate quantum dynamics on diabatic PESs learnt on-the-fly.

Prediction of conformationally dependent atomic multipole moments in carbohydrates.

The conformational flexibility of carbohydrates is challenging within the field of computational chemistry. This flexibility causes the electron density to change, which leads to fluctuating atomic multipole moments. Quantum Chemical Topology (QCT) allows for the partitioning of an “atom in a molecule,” thus localizing electron density to finite atomic domains, which permits the unambiguous evaluation of atomic multipole moments. By selecting an ensemble of physically realistic conformers of a chemical system, one evaluates the various multipole moments at defined points in configuration space. The subsequent implementation of the machine learning method kriging delivers the evaluation of an analytical function, which smoothly interpolates between these points. This allows for the prediction of atomic multipole moments at new points in conformational space, not trained for but within prediction range.In this work, we demonstrate that the carbohydrates erythrose and threose are amenable to the above methodology. We investigate how kriging models respond when the training ensemble incorporating multiple energy minima and their environment in conformational space. Additionally, we evaluate the gains in predictive capacity of our models as the size of the training ensemble increases. We believe this approach to be entirely novel within the field of carbohydrates. For a modest training set size of 600, more than 90% of the external test configurations have an error in the total (predicted) electrostatic energy (relative to ab initio) of maximum 1 kJ mol−1 for open chains and just over 90% an error of maximum 4 kJ mol−1 for rings. © 2015 Wiley Periodicals, Inc.

What is a wavefunction?

Much of the the discussion of the metaphysics of quantum mechanics focusses on the status of wavefunctions. This paper is about how to think about wavefunctions, when we bear in mind that quantum mechanics—that is, the nonrelativistic quantum theory of systems of a fixed, finite number of degrees of freedom—is not a fundamental theory, but arises, in a certain approximation, valid in a limited regime, from a relativistic quantum field theory. We will explicitly show how the wavefunctions of quantum mechanics, and the configuration spaces on which they are defined, are constructed from a relativistic quantum field theory. Two lessons will be drawn from this. The first is that configuration spaces are not fundamental, but rather are derivative of structures defined on ordinary spacetime. The second is that wavefunctions are not as much like classical fields as might first appear, in that, on the most natural way of constructing wavefunctions from quantum field-theoretic quantities, the value assigned to a point in configuration space is not a local fact about that point, but rather, depends on the global state.

Frozen in Beta

Frozen in Beta

Calculations of the structure of basin volumes for mechanically stable packings

Experimental and computational model systems composed of frictionless particles in a fixed geometry have a finite number of distinct mechanically stable (MS) packings. The frequency of occurrence for each MS packing is highly variable and depends strongly on preparation protocol. Despite intense work, it is extremely difficult to predict a priori the MS packing probabilities. We describe a novel computational method for calculating the volume and other geometrical properties of the "basin of attraction" for each MS packing. The basin of attraction for an MS packing contains all initial conditions in configuration space that map to that MS packing using a given preparation protocol. We find that the basin is a highly complex structure. For a compressive-quench-from-zero-density protocol, we show the existence of a small core volume of the basin around each MS packing for which all points map to that MS packing. However, in contrast to previous studies for supercooled liquids, glasses, and over-compressed jammed systems, we find that the MS packing probabilities are very weakly correlated with this core volume. Instead, MS packing probabilities obtained from compression protocols that use initially dilute configurations and do not allow particle overlaps (i.e., those relevant to granular media) are determined by complex geometric features of the basin of attraction that are distant from the MS packing. In particular, we find that the shape of the average basin profile function S(l), which gives the probability for a point on a hyperspherical shell a distance l from a given MS packing to map back to that packing, can be described by a Γ distribution with a peak that increases as the system size increases and as the quench rate decreases. We find a simple model which predicts S(l) for the extreme cases of very slow and fast quench rates.

Experimental demonstration of nonuniform frequency distributions of granular packings

We developed an experimental method to generate mechanically stable (MS) packings of frictionless disks and performed coordinated experiments and simulations to characterize MS packings in small systems. For a given system geometry, MS packings occur as discrete, well-separated points in configuration space with probabilities that vary by many orders of magnitude and are robust with respect to the packing preparation. Over a continuous range of system geometries, MS packings occur as distinct geometrical families and only a small fraction of families are sampled via quasistatic dynamics. These results suggest that the most frequent MS packings may dominate the structural and mechanical properties of dense granular media.

Pointwise and generalized virial theorems

Given an eigenstate of a time-independent Hamiltonian, an equation is derived which generates the corresponding energy by means of a pointwise application of only the kinetic operator. Satisfaction of the equation at all points in configuration space, for arbitrary values of a scale factor, constitutes a necessary eigenstate condition. Explicit knowledge of the potential operator, other than knowledge of its homogeneity, is not required. In addition, integration of the pointwise equation yields a generalization of the familiar virial theorem. 2 references.

Particle segregation by Stokes number for small neutrally buoyant spheres in a fluid

It is a commonly observed phenomenon that spherical particles with inertia in an incompressible fluid do not behave as ideal tracers. Due to the inertia of the particle, the planar dynamics are described in a four-dimensional phase space and thus can differ considerably from the ideal tracer dynamics. Using finite-time Lyapunov exponents, we compute the sensitivity of the final position of a particle with respect to its initial velocity, relative to the fluid, and thus partition the relative velocity subspace at each point in configuration space. The computations are done at every point in the relative velocity subspace, thus giving a sensitivity field. The Stokes number, being a measure of the independence of the particle from the underlying fluid flow, acts as a parameter in determining the variation in these partitions. We demonstrate how this partition framework can be used to segregate particles by Stokes number in a fluid. The fluid model used for demonstration is a two-dimensional cellular flow.

From transition paths to transition states and rate coefficients.

Transition states are defined as points in configuration space with the highest probability that trajectories passing through them are reactive (i.e., form transition paths between reactants and products). In the high-friction (diffusive) limit of Langevin dynamics, the resulting ensemble of transition states is shown to coincide with the separatrix formed by points of equal commitment (or splitting) probabilities for reaching the product and reactant regions. Transition states according to the new criterion can be identified directly from equilibrium trajectories, or indirectly by calculating probability densities in the equilibrium and transition-path ensembles using umbrella and transition-path sampling, respectively. An algorithm is proposed to calculate rate coefficients from the transition-path and equilibrium ensembles by estimating the frequency of transitions between reactants and products.