Saddle Point Geometry Research Articles

Surface-Like Diffusion of Fast Ions in Framework Energy Materials for Li- and Na-Ion Batteries.

The rate performance, power density, and energy efficiency of electrochemical devices are often limited by ionic conductivities in electrolyte and electrode materials. Framework Prussian blue analogs and dense niobium oxides have been identified as high-rate electrodes for sodium- and lithium-ion batteries, respectively, yet the origin of the extremely high solid-state Na+/Li+ transport is not fully understood. Of critical importance is the fact that their ultra-low activation energy and anomalous pre-exponent factor cannot be satisfactorily rationalized from conventional theory of solid-state diffusion in the crystal lattice. Here, assisted by density-functional-theory calculations, we argued that the true origin is a unique surface-like diffusion mechanism of the intercalation ions. In a surface-like migration event, a mobile ion moves along the channel wall via a low coordination number and low migration barrier experiencing minimal steric hindrance. It is similar to surface diffusion in the conventional picture and contrasts with lattice diffusion from one interstitial/vacancy site to another one with high coordination number, crowded saddle-point geometry and high migration barrier. We found that the shifting from solid-state lattice diffusion to surface-like diffusion is determined by the size difference between the mobile ion and the diffusion channel, and a lowest migration energy barrier can be reached by mediating the channel size. The analogy to gas diffusion in molecular sieves shall be discussed. Additionally, the effects of defects and crystal water in Prussian blue analogs were also discussed for better understanding their rate performances in experimental scenarios.

Sella, an Open-Source Automation-Friendly Molecular Saddle Point Optimizer.

We present a new algorithm for the optimization of molecular structures to saddle points on the potential energy surface using a redundant internal coordinate system. This algorithm automates the procedure of defining the internal coordinate system, including the handling of linear bending angles, for example, through the addition of dummy atoms. Additionally, the algorithm supports constrained optimization using the null-space sequential quadratic programming formalism. Our algorithm determines the direction of the reaction coordinate through iterative diagonalization of the Hessian matrix and does not require evaluation of the full Hessian matrix. Geometry optimization steps are chosen using the restricted step partitioned rational function optimization method, and displacements are realized using a high-performance geodesic stepping algorithm. This results in a robust and efficient optimization algorithm suitable for use in automated frameworks. We have implemented our algorithm in Sella, an open-source software package designed to optimize atomic systems to saddle point structures. We also introduce a new benchmark test comprising 500 molecular structures that approximate saddle point geometries and show that our saddle point optimization algorithm outperforms the algorithms implemented in several leading electronic structure theory packages.

Enabling Rapid and Accurate Construction of CCSD(T)-Level Potential Energy Surface of Large Molecules Using Molecular Tailoring Approach.

The construction of a potential energy surface (PES) of even a medium-sized molecule employing correlated theory, such as CCSD(T), is arduous due to the high computational cost involved. The present study reports the possibility of efficiently constructing such a PES of molecules containing up to 15 atoms and 550 basis functions by employing the fragment-based molecular tailoring approach (MTA) on off-the-shelf hardware. The MTA energies at the CCSD(T)/aug-cc-pVTZ level for several geometries of three test molecules, viz., acetylacetone, N-methylacetamide, and tropolone, are reported. These energies are in excellent agreement with their full calculation counterparts with a time advantage factor of 3-5. The energy barrier from the ground to transition state is also accurately captured. Further, we demonstrate the accuracy and efficiency of MTA for estimating the energy gradients at the CCSD(T) level. As a further application of our MTA methodology, the energies of acetylacetone at ∼430 geometries are computed at the CCSD(T)/aug-cc-pVTZ level and used for generating a Δ-machine learning (Δ-ML) PES. This leads to the H-transfer barrier of 3.02 kcal/mol, well in agreement with the benchmarked barrier of 3.19 kcal/mol. The fidelity of this Δ-ML PES is examined by geometry optimization and normal mode frequency calculations of global minima and saddle point geometries. We trust that the present work is a major development for the rapid and accurate construction of PES at the CCSD(T) level for molecules containing up to 20 atoms and 600 basis functions using off-the-shelf hardware.

Stability-Order Reversal in FSiY and FYSi (Y = N and P) Molecules after the Insertion of a Noble Gas Atom.

Recent theoretical prediction and experimental identification of fluorinated noble gas cyanides and isocyanides motivate us to explore a unique novel series of neutral noble gas-inserted heavier cyanofluoride isomers, FNgYSi and FNgSiY (Ng = Kr, Xe, and Rn; Y = N and P), theoretically using quantum chemical calculations. The concerned minima and saddle point geometries have been optimized using DFT, MP2, and CCSD(T) methods. The precursor molecule FSiY is more stable than its isomer FYSi, and the stability order is found to be reversed after the insertion of a noble gas (Ng) atom into them which is in contrast to the previously reported FCN/FNC systems where the stability order in the precursors remains intact after the insertion of a Ng atom into them. The predicted FNgYSi molecules are metastable in nature as they are kinetically stable but thermodynamically unstable with respect to the global minima products (FYSi and Ng). All the calculations for the corresponding FNgSiY molecules clearly indicate that the less stable FNgSiY behaves similarly to the FNgYSi in all respects. The energetics, force constant, and spectroscopic data strongly reinforce the possibility of occurrence of these predicted FNgYSi and FNgSiY molecules which might be experimentally realized under suitable cryogenic condition(s).

Islands in multiverse models

We consider multiverse models in two-dimensional linear dilaton-gravity theories as toy models of false vacuum eternal inflation. Coupling conformal matter we calculate the Von Neumann entropy of subregions. When these are sufficiently large we find that an island develops covering most of the rest of the multiverse, leading to a Page-like transition. This resonates with a description of multiverse models in semiclassical quantum cosmology, where a measure for local predictions is given by saddle point geometries which coarse-grain over any structure associated with eternal inflation beyond one’s patch.

The Role of Zero-Point Vibration and Reactant Attraction in Exothermic Bimolecular Reactions with Submerged Potential Barriers: Theoretical Studies of the R + HBr → RH + Br (R = CH3, HO) Systems.

The dynamics of the reactions CH3 + HBr → CH4 + Br and HO + HBr → H2O + Br have been studied using the quasiclassical trajectory method to explore the interplay of the vibrational excitation of the breaking bond and the potential energy surface characterized by a prereaction van der Waals well and a submerged barrier to reaction. The attraction between the reactants is favorable for the reaction, because it brings together the reactants without any energy investment. The reaction can be thought to be controlled by capture. The trajectory calculations indeed provide excitation functions typical to capture: the reaction cross sections diverge when the collision energy is reduced toward zero. Excitation of reactant vibration accelerates both reactions. The barrier on the potential surface is so early that the coupling between the degrees of freedom at the saddle point geometry is negligible. However, the trajectory calculations show that when the breaking bond is stretched at the time of the encounter, an attractive force arises, as if the radical approached a HBr molecule whose bond is partially broken. As a result, the dynamics of the reaction are controlled more by the temporary “dynamical”, vibrationally induced than by the “static” van der Waals attraction even when the reactants are in vibrational ground state. The cross sections are shown to drop to very small values when the amplitude of the breaking bond’s vibration is artificially reduced, which provides an estimate of the reactivity due to the “static” attraction. Without zero-point vibration these reactions would be very slow, which is a manifestation of a unique quantum effect. Reactions where the reactivity is determined by dynamical factors such as the vibrationally enhanced attraction are found to be beyond the range of applicability of Polanyi’s rules.

Gravitational and gravitoscalar thermodynamics

Gravitational thermodynamics and gravitoscalar thermodynamics with S2 × ℝ boundary geometry are investigated through the partition function, assuming that all Euclidean saddle point geometries contribute to the path integral and dominant ones are in the B3 × S1 or S2 × Disc topology sector. In the first part, I concentrate on the purely gravitational case with or without a cosmological constant and show there exists a new type of saddle point geometry, which I call the “bag of gold(BG) instanton,” only for the Λ > 0 case. Because of this existence, thermodynamical stability of the system and the entropy bound are absent for Λ > 0, these being universal properties for Λ ≤ 0. In the second part, I investigate the thermodynamical properties of a gravity-scalar system with a φ2 potential. I show that when Λ ≤ 0 and the boundary value of scalar field Jφ is below some value, then the entropy bound and thermodynamical stability do exist. When either condition on the parameters does not hold, however, thermodynamical stability is (partially) broken. The properties of the system and the relation between BG instantons and the breakdown are discussed in detail.

Homogeneous transitions during inflation: A description in quantum cosmology

The usual description of inflationary fluctuations uses the framework of quantum field theory (QFT) in curved spacetime, in which quantum fluctuations are superimposed on a classical background spacetime. Even for large fluctuations, such as those envisioned during a regime of eternal inflation, this framework is frequently used. In the present work we go one step beyond this description by quantising both the scalar field and the scale factor of the universe. Employing the Lorentzian path integral formulation of semi-classical gravity we restrict to a simplified minisuperspace setting by considering homogeneous transitions. This approach allows us to determine the dominant geometry and inflaton evolution contributing to such amplitudes. We find that for precisely specified initial scale factor and inflaton values (and uncertain momenta), two distinct saddle point geometries contribute to the amplitude, leading to interference effects. However, when the momenta of both scale factor and inflaton are specified with sufficient certainty, only a single saddle point is relevant and QFT in curved spacetime is applicable. In particular we find that for inflaton transitions up the potential, meaningful results are only obtained when the initial uncertainty in the inflaton value is large enough, allowing the dominant evolution to be a complexified slow-roll solution \emph{down} from a comparatively unlikely position higher up in the potential.

On a class of Hamiltonian Choquard-type elliptic systems

This paper is concerned with the existence of solutions for a class of Hamiltonian Choquard-type elliptic systems in the plane with exponential critical growth. Typical features of this class of problems are a lack of compactness because of the unboundedness of the domain and that the critical growth and the energy functional associated with this class of systems are strongly indefinite, that is, the domain has a saddle-point geometry where both positive and negative subspaces of the quadratic form are infinite-dimensional. Moreover, the presence of the nonlocal term in the nonlinearity provides a series of difficulties in obtaining certain estimates.

No-Boundary Proposal as a Path Integral with Robin Boundary Conditions.

Realizing the no-boundary proposal of Hartle and Hawking as a consistent gravitational path integral has been a long-standing puzzle. In particular, it was demonstrated by Feldbrugge, Lehners, and Turok that the sum over all universes starting from a zero size results in an unstable saddle point geometry. Here we show that, in the context of gravity with a positive cosmological constant, path integrals with a specific family of Robin boundary conditions overcome this problem. These path integrals are manifestly convergent and are approximated by stable Hartle-Hawking saddle point geometries. The price to pay is that the off-shell geometries do not start at a zero size. The Robin boundary conditions may be interpreted as an initial state with Euclidean momentum, with the quantum uncertainty shared between the initial size and momentum.

English

We consider the implicit discretization of Nagumo equation on finite lattices and show that its variational formulation corresponds in various parameter settings to convex, mountain-pass or saddle-point geometries. Consequently, we are able to derive conditions under which the implicit discretization yields multiple solutions. Interestingly, for certain parameters we show nonuniqueness for arbitrarily small discretization steps. Finally, we provide a simple example showing that the nonuniqueness can lead to complex dynamics in which the number of bounded solutions grows exponentially in time iterations, which in turn implies infinite number of global trajectories.

Automated Transition State Theory Calculations for High-Throughput Kinetics.

A scarcity of known chemical kinetic parameters leads to the use of many reaction rate estimates, which are not always sufficiently accurate, in the construction of detailed kinetic models. To reduce the reliance on these estimates and improve the accuracy of predictive kinetic models, we have developed a high-throughput, fully automated, reaction rate calculation method, AutoTST. The algorithm integrates automated saddle-point geometry search methods and a canonical transition state theory kinetics calculator. The automatically calculated reaction rates compare favorably to existing estimated rates. Comparison against high level theoretical calculations show the new automated method performs better than rate estimates when the estimate is made by a poor analogy. The method will improve by accounting for internal rotor contributions and by improving methods to determine molecular symmetry.

Hamiltonian elliptic systems in $$\mathbb {R}^2$$ R 2 with subcritical and critical exponential growth

Our aim in this paper was to establish by a variational method the existence of nontrivial solutions for strongly coupled Hamiltonian systems of the form $$\begin{aligned} \left\{ \begin{array}{llll} -\Delta u +V(x) u&{} = g(x,v), &{}\quad v > 0 &{} \quad \text {in}\; \mathbb {R}^2, \\ -\Delta v +V(x) v&{} = f(x,u), &{}\quad u > 0 &{} \quad \text {in}\; \mathbb {R}^2, \end{array} \right. \end{aligned}$$when the potential \(V\) is neither bounded away from zero, nor bounded from above. The nonlinear terms \(g(x,s)\) and \(f(x,s)\) are superlinear at infinity and have exponential subcritical or critical growth of the Trudinger–Moser type. Typical features of this class of problems are the lack of compactness because of the unboundedness of the domain and the critical growth. Moreover, the Lagrangian functional associated with this class of systems is strongly indefinite, that is, it has a saddle-point geometry where both positive and negative subspaces of the quadratic form are infinite-dimensional. To overcome these difficulties, we use an inequality of Trudinger–Moser type combined with Galerkin methods and a linking theorem.

Total photoionization cross-sections of excited electronic states by the algebraic diagrammatic construction-Stieltjes-Lanczos method

Here, we extend the L2 ab initio method for molecular photoionization cross-sections introduced in Gokhberg et al. [J. Chem. Phys. 130, 064104 (2009)] and benchmarked in Ruberti et al. [J. Chem. Phys. 139, 144107 (2013)] to the calculation of total photoionization cross-sections of molecules in electronically excited states. The method is based on the ab initio description of molecular electronic states within the many-electron Green's function approach, known as algebraic diagrammatic construction (ADC), and on the application of Stieltjes-Chebyshev moment theory to Lanczos pseudospectra of the ADC electronic Hamiltonian. The intermediate state representation of the dipole operator in the ADC basis is used to compute the transition moments between the excited states of the molecule. We compare the results obtained using different levels of the many-body theory, i.e., ADC(1), ADC(2), and ADC(2)x for the first two excited states of CO, N2, and H2O both at the ground state and the excited state equilibrium or saddle point geometries. We find that the single excitation ADC(1) method is not adequate even at the qualitative level and that the inclusion of double electronic excitations for description of excited state photoionization is essential. Moreover, we show that the use of the extended ADC(2)x method leads to a substantial systematic difference from the strictly second-order ADC(2). Our calculations demonstrate that a theoretical modelling of photoionization of excited states requires an intrinsically double excitation theory with respect to the ground state and cannot be achieved by the standard single excitation methods with the ground state as a reference.

Accurate ab initio potential energy surfaces for the 3A′′ and 3A′ electronic states of the O(3P)+HBr system

In this work, we report the construction of potential energy surfaces for the (3)A('') and (3)A(') states of the system O((3)P) + HBr. These surfaces are based on extensive ab initio calculations employing the MRCI+Q/CBS+SO level of theory. The complete basis set energies were estimated from extrapolation of MRCI+Q/aug-cc-VnZ(-PP) (n = Q, 5) results and corrections due to spin-orbit effects obtained at the CASSCF/aug-cc-pVTZ(-PP) level of theory. These energies, calculated over a region of the configuration space relevant to the study of the reaction O((3)P) + HBr → OH + Br, were used to generate functions based on the many-body expansion. The three-body potentials were interpolated using the reproducing kernel Hilbert space method. The resulting surface for the (3)A('') electronic state contains van der Waals minima on the entrance and exit channels and a transition state 6.55 kcal/mol higher than the reactants. This barrier height was then scaled to reproduce the value of 5.01 kcal/mol, which was estimated from coupled cluster benchmark calculations performed to include high-order and core-valence correlation, as well as scalar relativistic effects. The (3)A(') surface was also scaled, based on the fact that in the collinear saddle point geometry these two electronic states are degenerate. The vibrationally adiabatic barrier heights are 3.44 kcal/mol for the (3)A('') and 4.16 kcal/mol for the (3)A(') state.

High‐temperature rate constants for H/D + C2H6 and C3H8

AbstractThe reactions of H/D with C2H6 and C3H8 have been studied with both shock‐tube experiments and ab initio transition‐state theory calculations. Rate constants for the reactions of D with C2H6 and C3H8 have been measured in reflected shock wave experiments over the temperature range, 1128–1299 K, at pressures 0.3–1 atm. D atoms are detected using atomic resonance absorption spectrometry. The measured D‐atom profiles are sensitive to only the thermal dissociation of the D‐atom source molecule, C2D5I, and to the title reactions. Since the dissociation has been previously studied in this laboratory, modeling the temporal evolution of the D‐atom profiles allows determinations of the rate constants for the title reactions. Over the T range, 1128–1228 K, rate constants from the present experiments for D + C2H6 can be represented by the Arrhenius expression equation image The experimental total rate constants for D + C3H8 over the T range, 1128–1299 K, can be represented by the Arrhenius expression equation image The title reactions have been studied at the CCSD(T)/aug‐cc‐pv∞z level of theory using saddle point geometries at B3LYP/6‐311++G(d,p) and MP2/6‐311++G(d,p) levels of theory. The reaction endothermicities are in good agreement with current Active Thermochemical Tables values. Theoretical rate constants were estimated using transition state theory. The theoretical rate constants are in good agreement with the present experiments and lower temperature literature data. Over the T range of the present experiments, the theoretically predicted isotope effects are close to unity. © 2012 Wiley Periodicals, Inc. Int J Chem Kinet 44: 194–205, 2012

A quantum reaction dynamics study of the translational, vibrational, and rotational motion effects on the HD + ${\rm H}_3^+$H3+ reaction

Time-dependent, quantum reaction dynamics wavepacket approach is employed to investigate the impacts of the translational, vibrational, and rotational motion on the HD+H(3)(+) → H(2)D(+) + H(2) reaction using the Xie-Braams-Bowman potential energy surface [Z. Xie, B. J. Braams, and J. M. Bowman, J. Chem. Phys. 122, 224307 (2005)]. We treat this five atom reaction with a seven-degree-of-freedom model by fixing one Jacobi and one torsion angle related to H(3) (+) at the lowest saddle point geometry of the potential energy surface. The initial state selected reaction probabilities show that the rotational excitations of H(+)-H(2) greatly enhance the reactivity with the reaction probabilities increased double at high rotational states compared to the ground state. However, the vibrational excitations of H(3) (+) hinder the reactivity. The ground state reaction probability shows no reaction threshold for this exoergic reaction, and as the translational energy increases, the reaction probability decreases. Furthermore, reactive resonances and zero point energy play very important roles on the reaction dynamics. The obtained integral cross section has the character of an exoergic reaction without a threshold: it decreases with the translational energy increasing. The calculated thermal rate constants using this seven-degree-of-freedom model are in agreement with a later experiment measurement.

The hidden horizon and black hole unitarity

We motivate through a detailed analysis of the Hawking radiation in a Schwarzschild background a scheme in accordance with quantum unitarity. In this scheme the semi-classical approximation of the unitary quantum - horizonless - black hole S-matrix leads to the conventional description of the Hawking radiation from a classical black hole endowed with an event horizon. Unitarity is borne out by the detailed exclusive S-matrix amplitudes. There, the fixing of generic out-states, in addition to the in-state, yields in asymptotic Minkowski space-time saddle-point contributions which are dominated by Planckian metric fluctuations when approaching the Schwarzschild radius. We argue that these prevent the corresponding macroscopic "exclusive backgrounds" to develop an event horizon. However, if no out-state is selected, a distinct saddle-point geometry can be defined, in which Planckian fluctuations are tamed. Such "inclusive background" presents an event horizon and constitutes a coarse-grained average over the aforementioned exclusive ones. The classical event horizon appears as a coarse-grained structure, sustaining the thermodynamic significance of the Bekenstein-Hawking entropy. This is reminiscent of the tentative fuzzball description of extremal black holes: the role of microstates is played here by a complete set of out-states. Although the computations of unitary amplitudes would require a detailed theory of quantum gravity, the proposed scheme itself, which appeals to the metric description of gravity only in the vicinity of stationary points, does not.

Seven-degree-of-freedom, quantum scattering dynamics study of the H2D++H2 reaction

A quantum scattering dynamics, time-dependent wavepacket propagation method is applied to study the reaction of H(2)D(+)+H(2)-->H(3)(+)+HD on the Xie-Braams-Bowman potential energy surface. The reduced-dimensional, seven-degree-of-freedom approach is employed in this calculation by fixing one Jacobi and one torsion angle related to H(2)D(+) at the lowest saddle point geometry of D(2d) on the potential energy surface. Initial state selected reaction probabilities are presented for various initial rovibrational states. The ground state reaction probability shows no threshold for this reaction, in other words, this reaction can occur without an activation barrier. The vibrational excitation shows that the stretching motion of H(+)-HD only has a small effect on the reaction probability; the vibrational excitation of HD in H(2)D(+) hinders the reactivity. By contrast, rotational excitation of H(+)-HD greatly enhances the reactivity with the reaction probability increased double or triple at high rotational states compared to the ground state. Reactive resonances, seen in all the initial state selected reaction probabilities, are also found in the integral cross section for the ground state of H(2)D(+) and H(2). The thermal rate coefficient is also calculated and is found to be in semiquantitative agreement with experiment; however, quantum scattering approaches including more degrees of freedom, especially including all the angles, are necessary to study this reaction in the future.

Influence of the Core Conformation on the NH-Tautomerism in Porphyrins: A Study of meso-(1,3-Dithian-2-yl)porphyrins

In order to investigate the mechanism of the NH-tautomerism in porphyrins, three meso-dithianyl-substituted porphyrins of different substitution pattern were studied theoretically. The corresponding trans-, cis- and saddle-point geometries were optimized with DFT methods, and the macrocyclic conformations obtained were analyzed using normal-structure-decomposition (NSD) analysis. Special attention was given to the influence of the participating out-of-plane and in-plane conformations on the NH-tautomerism, and the interplay of substituents, core conformations and energies of the transition-state structures was critically evaluated. The calculated energy barriers of the preferred pathways are compared with experimental activation enthalpies determined by variable-temperature (VT) NMR spectroscopy.