Energy Saddle Research Articles

Energetics of phase transitions in homogeneous antiferromagnets

Spatially uniform static phases in an antiferromagnet (AFM) at 0 K accessible by varying an external magnetic field along the anisotropy axis are investigated. Using the macrospin model, the energy contributions are comprised of the external field, effective anisotropy, and spatially homogeneous AFM exchange. The critical energy configurations are fully cataloged, and local energy concavity is used to identify stable states. Relative energy levels are taken into account to classify phase transitions. Phase diagrams for energetic stability and phase transitions are provided in terms of the strength of the applied field and the ratio between anisotropy and the AFM exchange. Two nonstandard critical energy states are identified as energy saddle points so are not stable but function as energy barriers between multiple stable states. The results determine thermal switching rates and suggest interesting AFM textures and solitons.

Topological van Hove singularities at phase transitions in Weyl metals

We show that in three-dimensional (3D) topological metals, a subset of the van Hove singularities of the density of states sits exactly at the transitions between topological and trivial gapless phases. We may refer to these as topological van Hove singularities. By investigating two minimal models, we show that they originate from energy saddle points located between Weyl points with opposite chiralities, and we illustrate their topological nature through their magnetotransport properties in the ballistic regime. We exemplify the relation between van Hove singularities and topological phase transitions in Weyl systems by analyzing the 3D Hofstadter model, which offers a simple and interesting playground to consider different kinds of Weyl metals and to understand the features of their density of states. In this model, as a function of the magnetic flux, the occurrence of topological van Hove singularities can be explicitly checked.

Bifurcation of Dividing Surfaces Constructed from a Pitchfork Bifurcation of Periodic Orbits in a Symmetric Potential Energy Surface with a Post-Transition-State Bifurcation

In this work, we analyze the bifurcation of dividing surfaces that occurs as a result of a pitchfork bifurcation of periodic orbits in a two degrees of freedom Hamiltonian System. The potential energy surface of the system that we consider has four critical points: two minima, a high energy saddle and a lower energy saddle separating two wells (minima). In this paper, we study the structure, the range, and the minimum and maximum extent of the periodic orbit dividing surfaces of the family of periodic orbits of the lower saddle as a function of the total energy.

An algorithm for computing phase space structures in chemical reaction dynamics using Voronoi tessellation

An efficient sampling algorithm is proposed for computing reactive islands, which consist of intersections of reactive trajectories and a Poincaré surface of section (PSOS), and are characterized by the number of times the reactive trajectories intersect the PSOS before/after reactions take place. The boundaries of reactive islands are series of intersections of cylindrical stable and unstable invariant manifolds emanated from a normally hyperbolic invariant manifold located, typically, nearby a rank-one potential energy saddle, which forms “reactivity boundaries”. The reactive islands structures enable us to predict the fate of trajectories. The proposed algorithm estimates regions of reactive islands on the PSOS as a Voronoi diagram constructed from the set of labeled intersections of computed molecular dynamics trajectories. The algorithm iteratively refines the boundaries of the estimated reactive islands efficiently, by computing trajectories of which initial condition is sampled from the estimated boundaries of reactive islands in the Voronoi diagram of the previous iteration step. The efficiency of the proposed algorithm is scrutinized, compared with random sampling algorithm and using a two-degrees of freedom Hamiltonian system of a double well potential, and further possible extensions of the proposed algorithm are addressed.

The dynamical significance of valley-ridge inflection points

In this paper we demonstrate that valley-ridge inflection (VRI) points of a potential energy surface (PES) have a dynamical influence on the fate of trajectories of the underlying Hamiltonian system. These points have attracted the attention of chemists in the past decades when studying selectivity problems in organic chemical reactions whose energy landscape exhibits a post-transition-state bifurcation in the region between two sequential saddles without an intervening energy minimum. To address the dynamical significance of valley-ridge inflection points, we construct a symmetric potential energy function that allows us to move the location of the VRI point while keeping the locations and energies of the critical points fixed. In this setup, we carry out a parametric study of the dynamical behavior of ensembles of trajectories in terms of the energy of the chemical system and the position of the VRI point. Our analysis reveals that the location of the VRI point controls the fraction of trajectories that recross the high energy saddle region of the PES without entering either of the potential wells that are separated by the low energy saddle.

Energy barriers for Dy and H penetrating graphene on 6H-SiC(0001) and freestanding bilayer graphene from first-principles calculations

Currently, intercalation of foreign guest atoms into two-dimensional (2D) layered van der Waals materials is an active research area motivated in part by the development of next-generation energy-storage technologies and optoelectronic devices. One such extensively studied 2D material is the graphene-on-SiC system. To realize and control the desired intercalated structures, it is fundamentally important to understand the kinetic process of intercalation. For the intercalation of a guest atom into graphene layers on SiC substrate, a critical kinetic parameter is the energy barrier of a guest atom penetrating the perfect graphene top layer into the gallery under it. However, accurate theoretical calculations for such penetration barriers are unavailable in literature. From our first-principles density functional theory calculations, we obtain the global energy barriers of 3.47 and 1.80 eV for single Dy and H atoms penetrating the graphene top layer on a graphene buffer layer supported by a Si-terminated 6H-SiC(0001) substrate, respectively. For comparison as well as for examining the lateral strain effects, we also obtain the global barriers of 5.05 and 1.50 eV for single Dy and H atoms penetrating freestanding bilayer graphene with a tensile strain of about 8.8% to match our model for supported graphene, as well as the global barriers of 7.21 and 4.18 eV for penetrating unstrained freestanding bilayer graphene, respectively. From corresponding minimum energy paths with multiple energy minima and saddle points, we can also obtain various local energy barriers and the global backward barrier from the graphene gallery back to the top surface.

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

We consider the first exit point distribution from a bounded domain \(\Omega \) of the stochastic process \((X_t)_{t\ge 0}\) solution to the overdamped Langevin dynamics $$\begin{aligned} d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t \end{aligned}$$starting from deterministic initial conditions in \(\Omega \), under rather general assumptions on f (for instance, f may have several critical points in \(\Omega \)). This work is a continuation of the previous paper [14] where the exit point distribution from \(\Omega \) is studied when \(X_0\) is initially distributed according to the quasi-stationary distribution of \((X_t)_{t\ge 0}\) in \(\Omega \). The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincare maps for revealing the phase space structure of two degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincare maps are compared for their ability to determine the phase space structures that govern this dynamical process.

Two-dimensional overdamped simulations of droplet solitons activated by spin transfer torque

We present overdamped micromagnetic simulations of the time evolution of magnetic droplet solitons which are formed in a thin film with perpendicular magnetic anisotropy by injecting a spin polarized current through a circular nanocontact. The overdamped dynamics help explore the effective energy landscape of these structures and permit identification of stationary states that are either energy extrema or saddle configurations. Our micromagnetic simulations start with configurations that are numerical solutions of a one-dimensional model where the magnetization depends only on the radial distance to the center of the nanocontact. We verify that these solutions correspond either to energy minima or saddle states, and use them to estimate thermal activation barriers for various current strengths. From the two-dimensional micromagnetic simulations, we identify a new persistent state which does not appear in the one-dimensional model.

Tuning the branching ratio in a symmetric potential energy surface with a post-transition state bifurcation using external time dependence

Chemical selectivity, as quantified by a branching ratio, is a phenomenon relevant for many organic chemical reactions. It may be exhibited on a potential energy surface (PES) that features a valley-ridge inflection point (VRI) in the region between two sequential index-1 saddles, with one saddle having higher energy than the other. Reaction occurs when a trajectory crosses the region of the higher energy saddle (the “entrance channel”) and approaches the lower energy saddle. On both sides of the lower energy saddle, there are two wells and the question we address is that given an initial ensemble of trajectories, what is the relative fraction of trajectories that enter each well. For a symmetric PES this fraction is 1:1. We consider a symmetric PES subject to a time-periodic forcing characterized by an amplitude, frequency, and phase. In this letter we analyze how the branching ratio depends on these three parameters.

The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1

We consider the first exit point distribution from a bounded domain Ω of the stochastic process (Xt)t≥0 solution to the overdamped Langevin dynamicsdXt=−∇f(Xt)dt+hdBt starting from the quasi-stationary distribution in Ω. In the small temperature regime (h→0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.

A model for a driven Frenkel–Kontorova chain

We study a Frenkel–Kontorova (FK) model of a finite chain with free-end boundary conditions. The model has two competing potentials. Newton trajectories are an ideal tool to understand the circumstances under a driving of an FK chain by external forces. To reach the insights we calculate some stationary structures for a chain with 23 particles. We search the lowest energy saddle points for a complete minimum energy path of the chain for a movement over the full period of the on-site potential, a sliding. If an additional tilting is set, then one is interested in barrier breakdown points (BBPs) on the potential energy surface for a critical tilting force named the static frictional force. In symmetric cases, such BBPs are often valley-ridge inflection points of the potential energy surface. We explain the theory and demonstrate it with an example. We propose a model for a DC drive, as well as an AC drive, of the chain using special directional vectors of the external force.

Half-Projected σ Self-Consistent Field For Electronic Excited States.

Fully self-consistent mean-field solutions of electronic excited states have been much less accessible compared to ground state solutions (e.g., Hartree-Fock). The main reason for this is that most excited states are energy saddle points, and hence energy-based optimization methods such as Δ-SCF often collapse to the ground state. Recently, our research group has developed a new method, σ-SCF [ J. Chem. Phys. 2017 , 147 , 214104 ], that successfully solves the "variational collapse" problem of energy-based methods. Despite the success, σ-SCF solutions are often spin-contaminated for open-shell states due to the single-determinant nature; unphysical behaviors such as disappearing solutions and discontinuous first-order energy derivatives are also observed along with the spontaneous breaking of spin or spatial symmetries. In this work, we tackle these problems by partially restoring the broken spin-symmetry of a σ-SCF solution through an approximate spin-projection scheme called half-projection. Orbitals of the projected wave function are optimized in a variation-after-projection (VAP) manner. The resulting theory, which we term half-projected (HP) σ-SCF, brings substantial improvement to the description of singlet and triplet excitations of the original σ-SCF method. Numerical simulations on small molecules suggest that HP σ-SCF delivers high-quality excited-state solutions that exist in a wide range of geometries with smooth potential energy surfaces. We also show that local excitations in HP σ-SCF can be size-intensive.

Phase Space Structure and Transport in a Caldera Potential Energy Surface

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

Reversal time of jump-noise magnetization dynamics in nanomagnets via Monte Carlo simulations

The jump-noise is a nonhomogeneous Poisson process which models thermal effects in magnetization dynamics, with special applications in low temperature escape rate phenomena. In this work, we develop improved numerical methods for Monte Carlo simulation of the jump-noise dynamics and validate the method by comparing the stationary distribution obtained empirically against the Boltzmann distribution. In accordance with the Néel-Brown theory, the jump-noise dynamics display an exponential relaxation toward equilibrium with a characteristic reversal time, which we extract for nanomagnets with uniaxial and cubic anisotropy. We relate the jump-noise dynamics to the equivalent Landau-Lifshitz dynamics up to second order correction for a general energy landscape and obtain the analogous Néel-Brown theory's solution of the reversal time. We find that the reversal time of jump-noise dynamics is characterized by Néel-Brown theory's solution at the energy saddle point for small noise. For large noise, the magnetization reversal due to jump-noise dynamics phenomenologically represents macroscopic tunneling of magnetization.

Understanding the viscosity of supercooled liquids and the glass transition through molecular simulations

We discuss a potential energy landscape (PEL) approach to calculate the characteristically sluggish viscous behaviour of supercooled liquids. This phenomenon is central to the theoretical understanding of the liquid-to-glass transition, a long-standing challenge in non-equilibrium statistical mechanics of amorphous states of matter. Experimentally, the shear viscosity of supercooled glass-forming liquids shows strong variations with the quench temperature. In particular, two types of behaviour are observed, Arrhenius and highly super-Arrhenius. Conceptually, the molecular description of the structural and dynamical processes underpinning the viscosity behaviour has become a topic of interest because of general implications for the kinetics of slow relaxation in the glassy state. We regard the supercooled liquid as an assembly of interacting particles undergoing thermal fluctuations such that the system evolves in space and time by crossing a series of potential energy barriers. To find these barriers and their corresponding atomic configurations, we apply an energy landscape sampling algorithm that maps out the evolution trajectories in the form of alternating sequences of local energy minima and saddle points. The energy sequences and the atomic coordinates constitute a body of atomic-scale data which we then process using two distinct methods. One is based on an effective activation barrier extracted from the transition-state pathway data, and the other is based on the linear response theory of statistical mechanics. The former is heuristic by being more physically transparent, while the latter is theoretically more rigorous. Through these two complementary calculations, an understanding of the temperature variations of shear viscosities of supercooled liquids, as well as the nature of fragile and strong behaviour of the glass transition, emerges. Our calculation provides a molecular-level account of the viscosities of supercooled liquids in a unified and consistent manner without invoking ad hoc assumptions. Relative to the nature of the glass transition, the usefulness of the PEL perspective is demonstrated, along with the concept of crossover between strong and fragile behaviour. In terms of advancing atomistic simulation capability, we believe the time-scale limitations of traditional molecular dynamics may be significantly extended through the use of metadynamics algorithms for sampling transition-state pathways.

Variationally fitting the total electron-electron interaction

Density fitting is used throughout quantum chemistry to simplify the electron-electron interaction energy (EE). A fundamental property of quantum chemistry, and DFT in particular, is that a variational principle connects the EE to a potential. Density fitting generally does not preserve this connection. Herein we describe the construction of a robust EE that is variationally connected to fitted potentials in all electronic structure methods. For DFT, this results in fitting equations that are satisfied at an energy saddle point in multidimensional fitting space.

Nonstatistical dynamics on the caldera.

We explore both classical and quantum dynamics of a model potential exhibiting a caldera: that is, a shallow potential well with two pairs of symmetry related index one saddles associated with entrance/exit channels. Classical trajectory simulations at several different energies confirm the existence of the "dynamical matching" phenomenon originally proposed by Carpenter, where the momentum direction associated with an incoming trajectory initiated at a high energy saddle point determines to a considerable extent the outcome of the reaction (passage through the diametrically opposing exit channel). By studying a "stretched" version of the caldera model, we have uncovered a generalized dynamical matching: bundles of trajectories can reflect off a hard potential wall so as to end up exiting predominantly through the transition state opposite the reflection point. We also investigate the effects of dissipation on the classical dynamics. In addition to classical trajectory studies, we examine the dynamics of quantum wave packets on the caldera potential (stretched and unstretched). These computations reveal a quantum mechanical analogue of the "dynamical matching" phenomenon, where the initial expectation value of the momentum direction for the wave packet determines the exit channel through which most of the probability density passes to product.

Biased gradient squared descent saddle point finding method.

The harmonic approximation to transition state theory simplifies the problem of calculating a chemical reaction rate to identifying relevant low energy saddle points in a chemical system. Here, we present a saddle point finding method which does not require knowledge of specific product states. In the method, the potential energy landscape is transformed into the square of the gradient, which converts all critical points of the original potential energy surface into global minima. A biasing term is added to the gradient squared landscape to stabilize the low energy saddle points near a minimum of interest, and destabilize other critical points. We demonstrate that this method is competitive with the dimer min-mode following method in terms of the number of force evaluations required to find a set of low-energy saddle points around a reactant minimum.

Spin polarization and potential energy function of FeH2

Among the three methods (B3LYP, BP86 and B3LYP*) in density functional theory (DFT), the best tools for predicting the ground state of metal hydride, the B3LYP method for predicting the harmonic frequencies and geometric parameters of the ground state of FeH2 gives result in good accordance with the experimental data; so it is employed to optimize the structure of molecules FeH and FeH2 in possible geometries and multiplicities based on 6-311++g(d,p) level in searching of the structure with the lowest energy. Results show that their electronic states in the ground states are FeH(4Δ) and FeH2(5A1), supposing that the two molecules have three and four unpaired electrons respectively, with spin polarization effect, and they are paramagnetic substances, and the stable structure of molecule FeH2 is of C2v symmetry. The Murrell-Sorbie potential energy function-the sufficient analytical potential function form for biatomic molecules-with 4 parameters in molecule FeH is derived via the least square method. Their spectra data and force constants are deduced according to the results. The analytical potential energy function of FeH2 is also obtained from the many-body expansion theory, which gives the analytical potential function of triatom molecules of the single-value potential surface consisting of three parts with single body terms, two body terms, and three body terms. The deduced analytical functions for FeH2 in this paper predict successfully a global minimum stable structure of quintet FeH2 with a 4.68 eV depth potential trap, and other higher energy stable and saddle structures. This potential function predicts the balanced ground structure and the second derivative force constants of this molecule. According to the potential function of FeH2(C2v), when it is formed from H and FeH, a potential trap with its depth being 4.68 eV is excited and the complex molecule of H–Fe–H is easily formed. The reaction of Fe+H2 → HFeH is exothermic with ΔH=-0.08305 eV.