Order Saddle Point Research Articles

Automatic Determination of the Non-Covalent Stable Conformations of the NO2-Pyrene Cluster in Full Dimensionality (81D) Using the vdW-TSSCDS Approach.

We present the detailed topographical characterisation (stationary points and minimum energy paths connecting them) of the full dimensional (81D) intermolecular potential energy surface associated with the non-covalent interactions between the NO2 radical and the pyrene (C16H10) molecule. The whole procedure is (quasi) fully automated. We have used our recent algorithm vdW-TSSCDS as implemented on the freely-available AutoMekin software package. To this end, a series of inexpensive classical trajectories using forces from a low-level (semi-empirical) theory are used to sample the configuration space of the system in the search for candidates to first order saddle points. These guess structures are determined by means of a graph-theory based algorithm using the concept of adjacency matrix. Low-level optimizations are followed by re-optimizations at a final high-level of theory (DFT and CCSD(T)-F12 in our case.). The resulting set of stationary points and paths connecting them constitutes the so-called reaction network. In the case of NO2-pyrene, this network exhibits four major basins which can be characterized by their point-group symmetry. A central one, of global C2 symmetry, comprises the global minimum (as well as all other permutationally related conformers) together with the corresponding C2v saddle points connecting them. This central basin is connected to three others of lower C1 symmetry. The latter can be distinguished by the projection of the position of the NO2 nitrogen atom on the pyrene plane in combination with the relative orientation of the oxygen pair pointing either inwards, outwards, upwards or downwards.

COMPASS: Double-ended saddle point search as a constrained optimization problem.

We present an algorithm to find first order saddle points on the potential energy surface (PES). The algorithm is formulated as a constrained optimization problem that involves two sets of atomic coordinates (images), a time-varying distance constraint and a constraint on the energy difference. Both images start in different valleys of the PES and are pulled toward each other by gradually reducing the distance. The search space is restricted to the pairs of configurations that share the same potential energy. By minimizing the energy while the distance shrinks, a minimum of the constrained search space is tracked. In simple cases, the two images are confined to their respective sides of the barrier until they finally converge near the saddle point. If one image accidentally crosses the barrier, the path is split at suitable locations and the algorithm is repeated recursively. The optimization is implemented as a combination of a quasi-Newton optimization and a linear constraint. The method was tested on a set of Lennard-Jones-38 cluster transitions and a set of 121 molecular reactions using density functional theory calculations. The efficiency in terms of energy and force evaluation is better than with competing methods as long as they do not switch to single-ended methods. The construction of a continuous search path with small steps and the ability to focus on arbitrary subsegments of the path provide an additional value in terms of robustness and flexibility.

A reliability-based design and optimization strategy using a novel MPP searching method for maritime engineering structures

PurposeIn order to solve the problems faced by First Order Reliability Method (FORM) and First Order Saddlepoint Approximation (FOSA) in structural reliability optimization, this paper aims to propose a new Reliability-based Design Optimization (RBDO) strategy for offshore engineering structures based on Original Probabilistic Model (OPM) decoupling strategy. The application of this innovative technique to other maritime structures has the potential to substantially improve their design process by optimizing cost and enhancing structural reliability.Design/methodology/approachIn the strategy proposed by this paper, sequential optimization and reliability assessment method and surrogate model are used to improve the efficiency for solving RBDO. The strategy is applied to the analysis of two marine engineering structure cases of ship cargo hold structure and frame ring of underwater skirt pile gripper. The effectiveness of the method is proved by comparing the original design and the optimized results.FindingsIn this paper, the proposed new RBDO strategy is used to optimize the design of the ship cargo hold structure and the frame ring of the underwater skirt pile gripper. According to the results obtained, compared with the original design, the structure of optimization design has better reliability and stability, and reduces the risk of failure. This optimization can also better balance the relationship between performance and cost. Therefore, it is recommended for related RBDO problems in the field of marine engineering.Originality/valueIn view of the limitations of FORM and FOSA that may produce multiple MPPs for a single performance function, the new RBDO strategy proposed in this study provides valuable insights and robust methods for the optimization design of offshore engineering structures. It emphasizes the importance of combining advanced MPP search technology and integrating SORA and surrogate models to achieve more economical and reliable design.

Partition Function for a Volume of Space.

We consider the quantum gravity partition function that counts the dimension of the Hilbert space of a spatial region with topology of a ball and fixed proper volume, and evaluate it in the leading order saddle point approximation. The result is the exponential of the Bekenstein-Hawking entropy associated with the area of the saddle ball boundary, and is reliable within effective field theory provided the mild curvature singularity at the ball boundary is regulated by higher curvature terms. This generalizes the classic Gibbons-Hawking computation of the de Sitter entropy for the case of positive cosmological constant and unconstrained volume, and hence exhibits the holographic nature of nonperturbative quantum gravity in generic finite volumes of space.

A quantum-classical correspondence in the dynamics around higher order saddle points: a Bohmian perspective

A quantum-classical correspondence in the dynamics around higher order saddle points: a Bohmian perspective

Isomerization of Silylnitrene to Silanimine: A Density Functional Study

Density functional theory is used to study the isomerization of closed- shell singlet and triplet silylnitrene to corresponding silanimine. Our calculation predicted that singlet silylnitrene is a first order saddle point and silylnitrene has a triplet ground state. The energy barrier of 79.0 Kcal mol-1 is predicted for the 1,2- hydrogen migration (i.e. isomerization) of triplet H3SiN → H2SiNH. It is also predicted that intermediate silylnitrene is not present during the direct photolysis of silyl azide for Curtius type rearrangement.

Density Functional Study of the Reaction Pathways of the Decomposition of Azidoborane: A Curtius-type Rearrangement

Density functional study is used for the decomposition of azidoborane and the possibility of borylnitrene being an intermediate in the formation of iminoborane both in the singlet and triplet state. Our calculations has shown that in the one step concerted manner, both singlet and triplet azidoborane can lead to the formation of iminoborane. A two step path (nitrogen loss followed by isomerization of borylnitrene) is a first order saddle point on the energy surface. However, such a two step process is possible for the triplet azidoborane, since triplet borylnitrene corresponds to a energy minimum on the potential energy surface.

A theoretical mechanistic study of [K ⊂ [2.2.2]]+ enantiomerization

AbstractDensity functional calculations at three different levels of theory (B3LYP/LANL2DZp, B3LYP‐GD3/LANL2DZp, and ωB97XD/def2tzvp) were applied to different conformers of the Lehn‐type macrobicyclic polyether cryptate complex [K ⊂ [2.2.2]]+ to construct the enantiomerization pathway and to compute the energy barrier for the change in symmetry. Changes in the conformation of the investigated complex are discussed in terms of energy and geometric criteria, where changes in the structures follow a trend of activation energies prohibiting the achiral pathways. The investigated D3‐symmetric cryptate [K ⊂ [2.2.2]]+ shows enantiomerization through an all chiral five step pathway with two local minima (C2 and C2′) and three C1 transition states ending in the D3′‐symmetric cryptate [K ⊂ [2.2.2]]+, the mirror image of the starting structure. The potential achiral transition states are higher order saddle points with a minimum twice as high in energy as an all‐chiral path. This study unambiguously accounts for Lehn's NMR data showing fluxionality of the studied complex. The obtained results were further rationalized by means of energy decomposition analysis (EDA) and natural bond orbital (NBO) calculations performed for the metal cation and the host in its different conformations.

Hysteresis and Fréedericksz thresholds for twisted states in chiral nematic liquid crystals: Minimum-energy path approach

We study minimum-energy pathways (MEPs) between the branches of metastable helical structures in chiral nematic liquid crystals (CNLCs) subjected to the electric field applied across the cell. By performing stability analysis we have found that, for the branches with non-vanishing half-turn number, the threshold (critical) voltage of the Fréedericksz transition is an increasing function of the free twisting wave number. The curves for the threshold voltage depend on the elastic anisotropy and determine the zero-field critical free twisting number where the director out-of-plane fluctuations destabilize the CNLC helix. For each MEP passing through a first order saddle point we have computed the energy barrier as the energy difference between the saddle-point and the initial structures at different values of the applied field. In our calculations, where the initial approximation for a MEP at the next step was determined by the MEP obtained at the previous step, the electric field dependence of the energy barrier is found to exhibit the hysteresis. This is the hysteresis of electrically driven transition of the saddle-point configuration between the planar and the tilted structures involving out-of-plane director deformations. It turned out that, by contrast to the second-order Fréedericksz transition, this transition is first order and we have studied how it depends on the zenithal anchoring energy strength.

Uncertainty-Based Design and Optimization Using First Order Saddle Point Approximation Method for Multidisciplinary Engineering Systems

AbstractThe widely used First Order Reliability Method (FORM) is efficient for uncertainty quantification and safety assessment in Uncertainty-based Multidisciplinary Design Optimization (UBMDO). H...

Novel probabilistic model for searching most probable point in structural reliability analysis

The Most Probable Point (MPP) is typically searched by transforming the basic random variables from original to standard normal space in the traditional first order reliability method (FORM). However, the FORM formulation may provide inaccurate MPP for those reliability problems with several local MPPs. The FORM based probabilistic model is a computational method and it may provide inaccuracy and unstable results for complex problems. In this paper, a probabilistic model is proposed to evaluate the MPP using cumulative distribution function (CDF) of basic random variables. The MPP is determined by solving an optimization problem in the original space which maximizes a multiplier function in terms of the CDF of the random variables. The performance of the proposed probabilistic model, usual FORM and first order saddle point approximation method (FOSAM) using SQP optimization solver is demonstrated by several numerical and engineering problems with normal and non-normal variables. The results of the numerical study illustrate that the proposed model provides an efficient approach to obtain the MPP which is simpler and more accurate than the usual FORM and FOSAM; particularly for reliability problems with non-normal random variables.

Multistate ring polymer instantons and nonadiabatic reaction rates.

We present two multistate ring polymer instanton (RPI) formulations, both obtained from an exact path integral representation of the quantum canonical partition function for multistate systems. The two RPIs differ in their treatment of the electronic degrees of freedom; while the Mean-Field (MF)-RPI averages over the electronic state contributions, the Mapping Variable (MV)-RPI employs explicit continuous Cartesian variables to represent the electronic states. We compute both RPIs for a series of model two-state systems coupled to a single nuclear mode with electronic coupling values chosen to describe dynamics in both adiabatic and nonadiabatic regimes. We show that the MF-RPIs for symmetric systems are in good agreement with the previous literature, and we show that our numerical techniques are robust for systems with non-zero driving force. The nuclear MF-RPI and the nuclear MV-RPI are similar, but the MV-RPI uniquely reports on the changes in the electronic state populations along the instanton path. In both cases, we analytically demonstrate the existence of a zero-mode, and we numerically find that these solutions are true instantons with a single unstable mode as expected for a first order saddle point. Finally, we use the MF-RPI to accurately calculate rate constants for adiabatic and nonadiabatic model systems with the coupling strength varying over three orders of magnitude.

Multiple minimum-energy paths and scenarios of unwinding transitions in chiral nematic liquid crystals.

We apply the minimum-energy paths (MEPs) approach to study the helix unwinding transition in chiral nematic liquid crystals. A mechanism of the transition is determined by a MEP passing through a first order saddle point on the free energy surface. The energy difference between the saddle point and the initial state gives the energy barrier of the transition. Two starting approximations for the paths are used to find the MEPs representing different transition scenarios: (a) the director slippage approximation with in-plane helical structures and (b) the anchoring breaking approximation that involves the structures with profound out-of-plane director deviations. It is shown that, at sufficiently low voltages, the unwinding transition is solely governed by the director slippage mechanism with the planar saddle-point structures. When the applied voltage exceeds its critical value below the threshold of the Fréedericksz transition, the additional scenario through the anchoring breaking transitions is found to come into play. For these transitions, the saddle-point structure is characterized by out-of-plane deformations localized near the bounding surface. The energy barriers for different paths of transitions are computed as a function of the voltage and the anchoring energy strengths.

Accelerated Saddle Point Refinement through Full Exploitation of Partial Hessian Diagonalization.

Identification and refinement of first order saddle point (FOSP) structures on the potential energy surface (PES) of chemical systems is a computational bottleneck in the characterization of reaction pathways. Leading FOSP refinement strategies for modestly sized molecular systems require calculation of the full Hessian matrix, which is not feasible for larger systems such as those encountered in heterogeneous catalysis. For these systems, the standard approach to FOSP refinement involves iterative diagonalization of the Hessian, but this comes at the cost of longer refinement trajectories due to the lack of accurate curvature information. We present a method for incorporating information obtained by an iterative diagonalization algorithm into the construction of an approximate Hessian matrix that accelerates FOSP refinement. We measure the performance of our method with two established FOSP refinement benchmarks and find a 50% reduction on average in the number of gradient evaluations required to converge to a FOSP for one benchmark and a 25% reduction on average for the second benchmark.

Jahn-Teller effect versus spin-orbit coupling: The structure of the free molybdenum pentafluoride molecule

The complete active space self-consistent field (CASSCF) and multiconfigurational quasi-degenerate second-order perturbation (MCQDPT2) calculations have been used to study the ground and low-lying excited electronic states of MoF5. Scalar-relativistic effects and spin-orbit coupling (SOC) have been taken into account employing the third-order Douglas-Kroll-Hess (DKH) Hamiltonian and full Breit-Pauli operator, respectively. The optimization and the calculation of the Hessian matrix have been performed by non-routine procedure employing the symmetry coordinates for all considered (D3h, C2v and C4v) geometrical configurations. The trigonal bipyramidal D3h structure of molybdenum pentafluoride MoF5 possesses two-fold degenerate ground first excited electronic orbital state and undergoes Jahn-Teller distortion to the C2v configurations. Accounting of SOC quenches the JT effect. As a result the D3h structure of MoF5 is no longer the conical intersection of two adiabatic potential energy surfaces (APES) but two separated APES corresponding to ground (12E1/2) and nearest excited (12E3/2) spin-orbit states are appeared instead. The D3h trigonal bipyramidal structure in the ground spin-orbit state corresponds to the 2-nd order saddle point on APES and lies above the minima (C2v structure, 12E1/2 state) by 244 cm−1. The warping barrier (another C2v structure) in the trough of APES and the Berry pseudorotation barrier (C4v structure, ground spin-orbit state 2E3/2) are 19 and 310 cm−1, respectively. Therefore MoF5 molecule is characterized in the gas phase by a large-amplitude motion of fluorine atoms in the equatorial plane and non-rigid intramolecular rearrangement by Berry mechanism. Despite such non-rigid nuclei dynamic, the calculated harmonic frequencies of MoF5 molecule do not contradict to available experimental IR spectrum. The repeat refinement of MoF5 gas electron diffraction data has been carried out using static and pseudoconformer models based on the SO-MCQDPT2 results. Corresponding to only extreme points on APES the configurations have been included into the pseudoconformer model. The both models have been fitted to the experimental data by the same quality.

Inference of the kinetic Ising model with heterogeneous missing data.

We consider the problem of inferring a causality structure from multiple binary time series by using the kinetic Ising model in datasets where a fraction of observations is missing. Inspired by recent work on mean field methods for the inference of the model with hidden spins, we develop a pseudo-expectation-maximization algorithm that is able to work even in conditions of severe data sparsity. The methodology relies on the Martin-Siggia-Rose path integral method with second-order saddle-point solution to make it possible to approximate the log-likelihood in polynomial time, giving as output an estimate of the couplings matrix and of the missing observations. We also propose a recursive version of the algorithm, where at every iteration some missing values are substituted by their maximum-likelihood estimate, showing that the method can be used together with sparsification schemes such as lasso regularization or decimation. We test the performance of the algorithm on synthetic data and find interesting properties regarding the dependency on heterogeneity of the observation frequency of spins and when some of the hypotheses that are necessary to the saddle-point approximation are violated, such as the small couplings limit and the assumption of statistical independence between couplings.

Jahn-Teller effect enhanced by spin-orbit coupling. From theory to Experiment:Example of niobium tetrachloride NbCl4

The ground and lower-lying excited electronic states of NbCl4 molecule were systematically studied by the complete active space self-consistent field (CASSCF) and multiconfigurational quasi-degenerate second-order perturbation (MCQDPT2) methods. Scalar-relativistic effects and spin-orbit coupling (SOC) have been taken into account employing the third-order Douglas-Kroll-Hess (DKH) Hamiltonian and full Breit-Pauli operator, respectively. The high-symmetrical Td structure with spin-orbit ground 2G3/2 state undergoes Jahn-Teller distortion along all non-totally symmetric vibrational coordinates. As a result the compressed (2E1/2 state) and elongated (2E3/2 state) D2d structures correspond to minimum and first order saddle point on the lower sheet of adiabatic potential energy surface (APES). The height of the warping barrier in the trough of APES and the Jahn-Teller stabilization energy are 179 and 438 cm−1, respectively. For two-mode 2E⊗e Jahn-Teller problem the vibronic Schrödinger equation has been solved in the diabatic basis. It was shown, the topology of the lower sheet determined mainly the energies and wave functions ground and few low-lying vibronic levels. The original methodology of the simulation of gas-phase electron diffraction (GED) data was developed. This methodology is based on the analytic description of some APES numerically obtained. The computed radial distribution function and thermal average parameters were compared with the experimental GED data. A good agreement between experimental and theoretical results supports the reliability of the theoretical data.

Photoionization and dissociation study of 2-methyl-2-propen-1-ol: Experimental and theoretical insights

The photoionization and dissociation of 2-methyl-2-propen-1-ol (MPO) have been investigated by using molecular beam experimental apparatus with tunable vacuum ultraviolet synchrotron radiation in the photon energy region of 8.0 $-$ 15.5 eV. The photoionization efficiency (PIE) curves for molecule ion and fragment ions: C $_4$ H $_8$ O $^+$ , C $_4$ H $_7$ O $^+$ , C $_3$ H $_5$ O $^+$ , C $_4$ H $_7$ $^+$ , C $_4$ H $_6$ $^+$ , C $_4$ H $_5$ $^+$ , C $_2$ H $_4$ O $^+$ , C $_2$ H $_3$ O $^+$ , C $_3$ H $_6$ $^+$ , C $_3$ H $_5$ $^+$ , C $_3$ H $_3$ $^+$ , CH $_3$ O $^+$ , CHO $^+$ have been measured, and the ionization energy (IE) and the appearance energies (AEs) of the fragment ions have been obtained. The stable species and the first order saddle points have been calculated on the CCSD(T)/cc-pvTZ//B3LYP/6-31+G(d, p) level. With combination of theoretical and experimental results, the dissociative photoionization pathways of 2-methyl-2-propen-1-ol are proposed. Hydrogen migrations within the molecule are the dominant processes in most of the fragmentation pathways of MPO.

The SMFA program for quantum chemistry calculations on large molecules

SMFA is a general program package for performing quantum chemistry calculations on large molecules, using an energy‐based fragmentation approach. The program can calculate electronic energies, energy gradients and second derivatives; perform geometry optimization; find first order saddle points (transition states); perform energy optimized scans along a user‐defined path; and evaluate various molecular properties. The program can use any of the following quantum chemistry packages: GAMESS(US), GAUSSIAN, NWChem and Q‐Chem. In addition, SMFA provides a number of utility programs that, inter alia, calculate vibrational frequencies and infrared spectra with isotopic substitutions, the electrostatic potential on the solvent‐accessible‐surface, and isodesmic and higher order near‐iso‐energetic reaction schemes. Calculations of the electronic energy and related properties can be carried out using a scheme that provides a computation time that is linearly dependent on the size of the molecule or, if the user has enough processing units available, in a walltime that is independent of the size of the molecule.This article is categorized under: Software > Quantum Chemistry Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Computational Biochemistry and Biophysics

Duplication, Collapse, and Escape of Magnetic Skyrmions Revealed Using a Systematic Saddle Point Search Method

Various transitions that a magnetic Skyrmion can undergo are found in calculations using a method for climbing up the energy surface and converging onto first order saddle points. In addition to collapse and escape through a boundary, the method identifies a transition where the Skyrmion divides and forms two Skyrmions. The activation energy for this duplication process can be similar to that of collapse and escape. A tilting of the external magnetic field for a certain time interval is found to induce the duplication process in a dynamical simulation. Such a process could turn out to be an important avenue for the creation of Skyrmions in future magnetic devices.