Saddle points, localization

The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

A technical difference from other Gaussian wavepacket based methods is that the local hamionic approximation has not been used to evaluate any integrals, but instead Maiti nez et al. use what they term a saddle-point approximation. This uses the localization of the functions to evaluate the integrals by... 

Fig. 5. The left hand side figure shows a contour plot of the potential energy landscape due to V4 with equipotential lines of the energies E = 1.5, 2, 3 (solid lines) and E = 7,8,12 (dashed lines). There are minima at the four points ( 1, 1) (named A to D), a local maximum at (0, 0), and saddle-points in between the minima. The right hand figure illustrates a solution of the corresponding Hamiltonian system with total energy E = 4.5 (positions qi and qs versus time t).

A study of Table 1.1 reveals interesting features as to the mobility of the adsorbed atoms. Thus, for an argon atom on the (100) face, the easiest path from one preferred site S to the next is over the saddle point P, so that the energy barrier which must be surmounted is (1251 — 855) or 396 X 10 J/molecule. Since the mean thermal energy kT at 78 K is only 108 J/molecule, the argon molecule will have severely limited mobility at this temperature and will spend nearly all of its time in the close vicinity of site S its adsorption will be localized. On the other hand, for helium on the... 

The PES found by Smedarchina et al. [1989] has two cis-form local minima, separated by four saddle-points from the main trans-form minima. The step-wise transfer (trans-cis, cis-trans) - because of endoergicity of the first stage - displays Arrhenius behavior even at T < T. . The concerted transfer of two hydrogen atoms was supposed to become prevalent at sufficiently low temperatures. However, because of too high a barrier for the concerted trans-trans transition, this... 

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

The present work aims to derive fully microscopic expressions for the nucleation rate J and to apply the results to realistic estimates of nucleation rates in alloys. We suppose that the state with a critical embryo obeys the local stationarity conditions (9) dFjdci — p, but is unstable, i.e. corresponds to the saddle point cf of the function ft c, = F c, — lN in the ci-space. At small 8a = c — cf we have... 

There are numerous algorithms of different kinds and quality in routine use for the fast and reliable localization of minima and saddle points on potential energy surfaces (see 47) and refs, therein). Theoretical data about structure and properties of transition states are most interesting due to a lack of experimental facts about activated complexes, whereas there is an abundance of information about educts and products of a reaction. 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

We have outlined how the conceptual tools provided by geometric TST can be generalized to deterministically or stochastically driven systems. The center-piece of the construction is the TS trajectory, which plays the role of the saddle point in the autonomous setting. It carries invariant manifolds and a TST dividing surface, which thus become time-dependent themselves. Nevertheless, their functions remain the same as in autonomous TST there is a TST dividing surface that is locally free of recrossings and thus satisfies the fundamental requirement of TST. In addition, invariant manifolds separate reactive from nonreactive trajectories, and their knowledge enables one to predict the fate of a trajectory a priori. 

Finally, the hexagonal interstitial (H) site, which lies in the (111) direction halfway between two T sites, was found to be a local minimum along the (111) direction but only a saddle point when considered in three dimensions (Van de Walle et al., 1989). 

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