Low-energy Saddle Points Research Articles

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.

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.

Statistical Analysis of the Effect of Deuteration on Quantum Delocalization in CH5.

Statistical approaches are applied to an investigation of how deuterated forms of CH5+ sample the potential energy surface for this ion. In its ground state, CH5+ has been shown to have amplitude in all of the 120 equivalent minima in the potential as well as in the region of each of the 180 low-energy saddle points that connect these minima. Although deuteration quenches this delocalization, the multidimensional nature of the isomerization pathways makes it difficult to visualize and quantify the delocalization of the ground state wave function. In this work, the localization of the ground state wave function is explored through an analysis of projections of the ground state probability amplitude onto various atom-atom distances. Specifically, analysis of two-dimensional projections of the ground state probability amplitude on to the distances between pairs of hydrogen atoms reveals that the ground state probability amplitude is localized near structures in which the hydrogen atoms in the ion can be divided into those that make up a CH3+ part and those that make up in a H2 part. Analysis of fits of projections of the probability amplitude onto distances between pairs of hydrogen atoms, between pairs of deuterium atoms, and between hydrogen and deuterium atoms to sums of Gaussian functions allows us to quantify the increased localization of hydrogen and deuterium atoms in the CH3+ and H2 parts of the ion. In addition, the amplitude of the wave function in the regions of the potential that correspond to the transition state for exchange of hydrogen or deuterium atoms between these two parts of the ion is found to decrease in the partially deuterated forms of CH5+ and is smallest for CH2D3+ and CH3D2+.

Friedel oscillation near a van Hove singularity in two-dimensional Dirac materials

We consider Friedel oscillation in the two-dimensional Dirac materials when the Fermi level is near the van Hove singularity. Twisted graphene bilayer and the surface state of topological crystalline insulator are the representative materials which show low-energy saddle points that are feasible to probe by gating. We approximate the Fermi surface near saddle point with a hyperbola and calculate the static Lindhard response function. Employing a theorem of Lighthill, the induced charge density due to an impurity is obtained and the algebraic decay of is determined by the singularity of the static response function. Although a hyperbolic Fermi surface is rather different from a circular one, the static Lindhard response function in the present case shows a singularity similar with the response function associated with circular Fermi surface, which leads to the at large distance R. The dependences of charge density on the Fermi energy are different. Consequently, it is possible to observe in twisted graphene bilayer the evolution that near Dirac point changes to above the saddle point. Measurements using scanning tunnelling microscopy around the impurity sites could verify the prediction.

Shift-enriched optical properties in bilayer graphene

The relative shift between graphene layers (d) significantly changes the electronic and optical properties of bilayer graphenes (BLGs). It induces band anticrossings, low-energy saddle points, the shift of subbands, and the change of state degeneracies of middle-energy saddle points. The number, frequencies, and spacings of peaks in single-particle excitation spectra are all affected by the relative shift. Moreover d induces anisotropy in the optical excitation, except for AA- and AB-stacked BLGs. The anisotropy and the d-dependent excitation spectra could be used to identify the interlayer shift.

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.

Optimizing transition states via kernel-based machine learning

We present a method for optimizing transition state theory dividing surfaces with support vector machines. The resulting dividing surfaces require no a priori information or intuition about reaction mechanisms. To generate optimal dividing surfaces, we apply a cycle of machine-learning and refinement of the surface by molecular dynamics sampling. We demonstrate that the machine-learned surfaces contain the relevant low-energy saddle points. The mechanisms of reactions may be extracted from the machine-learned surfaces in order to identify unexpected chemically relevant processes. Furthermore, we show that the machine-learned surfaces significantly increase the transmission coefficient for an adatom exchange involving many coupled degrees of freedom on a (100) surface when compared to a distance-based dividing surface.

A stroll in the energy landscape

Abstract We review recent results on the potential energy landscape of model liquids. The role of saddle points in the potential energy surface (PES) in connecting dynamics to statics is investigated, confirming that a change between minima-dominated and saddle-point-dominated regions of the PES explored in equilibrium happens around the mode coupling temperature. The structure of the low-energy saddle points in the basins is found to be simple and hierarchically organized; the presence of saddle points near to the local minimum energies indicates that, at non-cryogenic temperatures, entropic bottlenecks limit the dynamics.

A nonplanar cyclic minimum-energy structure of singlet C9

The second order Moller-Plesset (MP2) perturbation treatment with the 6-31G* basis set was applied to cyclic singlet structures of C9. It has been found that the minimum is a Cs nonplanar structure located only about 26 kJ mol-1 above the related linear structure. The finding was further confirmed at the MP2/DZP and MP2/6-311G* levels (although with the MP2 frozen-core option only). Two other low-energy saddle points of C2v symmetry are reported and some results for the regular ring. The Cs minimum should easily undergo an inversion. The analytically computed MP2/6-31G* frequencies of the Cs structure are probably influenced by an artefact of the method. Relation to the available observed data is discussed.

Fluxionality and low-lying transition structures of the water trimer

The minimum energy structure of the cyclic water trimer, its stationary points, and rearrangement processes at energies <1 kcal/mol above the global minimum are examined by ab initio molecular orbital theory. Structures corresponding to stationary points are fully optimized at the Hartree–Fock and second-order Mo/ller–Plesset levels, using the 6-311++G(d,p) basis; each stationary point is characterized by harmonic vibrational analyses. The lowest energy conformation has two free O–H bonds on one and the third O–H bond on the other side of an approximately equilateral hydrogen-bonded O...O...O (O3) triangle. The lowest energy rearrangement pathway corresponds to the flipping of one of the two free O–H bonds which are on the same side of the plane across this plane via a transition structure with this O–H bond almost within the O3 plane. Six distinguishable, but isometric transition structures of this type connect six isometric minimum energy structures along a cyclic vibrational-tunneling path; neighboring minima correspond to enantiomers. The potential energy along this path has C6 symmetry and a very low barrier V6=0.1±0.1 kcal/mol. This implies nearly free pseudorotational interconversion of the six equilibrium structures. The corresponding anharmonic level structure was modeled using an internal rotation Hamiltonian. Two further low-energy saddle points on the surface are of second and third order; they correspond to crown-type and planar geometries with C3 and C3h symmetries, respectively. Interconversion tunneling vibrations via these stationary points are also important for the water trimer dynamics. A unified and symmetry-adapted description of the intermolecular potential energy surface is given in terms of the three flipping coordinates of the O–H bonds. Implications of these results for the interpretation of spectroscopic data are discussed.