Search results for "Saddle point"
showing 10 items of 35 documents
The transition state and cognate concepts
2019
Abstract This review aims firstly to clarify the meanings of key terms and concepts associated with the idea of the transition state, as developed by theoreticians and applied by experimentalist, and secondly to provide an update to the meaning and significance of the transition state in an era when computational simulation, in which complexity is being increasingly incorporated, is commonly employed as a means by which to bridge the realms of theory and experiment. The relationship between the transition state and the potential-energy surface for an elementary reaction is explored, with discussion of the following terms: saddle point, minimum-energy reaction path, reaction coordinate, acti…
High excitations in coupled-cluster series: vibrational energy levels of ammonia
2004
The ammonia molecule containing large amplitude inversion motion is a revealing system in examining high-order correlation effects on potential energy surfaces. Correlation contributions to the equilibrium and saddle point geometries, inversion barrier height and vibrational energy levels, including inversion splittings, have been investigated. A six-dimensional Taylor-type series expansion of the Born–Oppenheimer potential energy surface, which is scaled to different levels of theory, is used to determine vibrational energy levels and inversion splittings variationally. The electronic energies are calculated by coupled-cluster methods, combining explicitly correlated R12 theory (which incl…
Computation of Unstable Binodals Not Requiring Concentration Derivatives of the Gibbs Energy
1998
The equilibrium of three liquid phases in a binary mixture implies the existence of tie lines and binodals that are different from the normal experimentally observable ones. First of all, there are the metastable extensions of the binodal built up by S/S tie lines. These S/S tie lines fulfill the equilibrium condition of the minimum of the Gibbs energy of the entire two-phase system. Both coexisting phases are located within the meta(stable) region. There are two additional types of tie lines: U/U (maximum of the Gibbs energy; both end points within the unstable area) and U/S tie lines (saddle point; one end point within the (meta)stable, the other within the unstable region). All types of…
Stationary Point Processes
2008
Time-dependent alignment of molecules trapped in octahedral crystal fields.
2006
The hindered rotational states of molecules confined in crystal fields of octahedral symmetry, and their time-dependent alignment obtained by pulsed nonresonant laser fields, are studied computationally. The control over the molecular axis direction is discussed based on the evolution of the rotational wave packet generated in the cubic crystal-field potential. The alignment degree obtained in a cooperative case, where the alignment field is applied in a favorable crystal-field direction, or in a competitive direction, where the crystal field has a saddle point, is presented. The investigation is divided into two time regimes where the pulse duration is either ultrashort, leading to nonadia…
On the time function of the Dulac map for families of meromorphic vector fields
2003
Given an analytic family of vector fields in Bbb R2 having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.
Semi-empirical simulations of F-center diffusion in KCl crystals
1997
Abstract The semi-empirical method and 224 atom quantum clusters were used for calculating the activation energy for diffusion of cation and anion vacancies and F-centers in KCl crystals. The relevant activation energies of 1.19 eV, 1.44 eV and 1.64 eV, respectively agree well with the experimental data.
Quasi-conformal mapping theorem and bifurcations
1998
LetH be a germ of holomorphic diffeomorphism at 0 ∈ ℂ. Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze 2πi )=H○S(z) (1). IfH λ is an unfolding of diffeomorphisms depending on λ ∈ (ℂ,0), withH 0=Id, one introduces its ideal $$\mathcal{I}_H$$ . It is the ideal generated by the germs of coefficients (a i (λ), 0) at 0 ∈ ℂ k , whereH λ(z)−z=Σa i (λ)z i . Then one can find a parameter solutionS λ (z) of (1) which has at each pointz 0 belonging to the domain of definition ofS 0, an expansion in seriesS λ(z)=z+Σb i (λ)(z−z 0) i with $$(b_i ,0) \in \mathcal{I}_H$$ , for alli. This result may be applied to the…
Detecting tri‐stability of 3D models with complex attractors via meshfree reconstruction of invariant manifolds of saddle points
2018
In mathematical modeling it is often required the analysis of the vector field topology in order to predict the evolution of the variables involved. When a dynamical system is multi-stable the trajectories approach different stable states, depending on the initialmconditions. The aim of this work is the detection of the invariant manifolds of thesaddle points to analyze the boundaries of the basins of attraction. Once that a sufficient number of separatrix points is found a Moving Least Squares meshfree method is involved to reconstruct the separatrix manifolds. Numerical results are presented to assess the method referring to tri-stable models with complex attractors such as limit cycles o…
Planar systems with critical points: multiple solutions of two-point nonlinear boundary value problems
2005
Abstract Two-point boundary value problems for the second-order ordinary nonlinear differential equations are considered. First, we consider the planar systems equivalent to equation x ″ = f ( x ) , where f ( x ) has multiple zeros and the respective system has centers and saddle points in various combinations. Estimations of the number of solutions are given. Then results are extended to nonautonomous equations which have superlinear behavior at infinity.