Closed orbits uniqueness

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

Assume the hypotheses of Dulac s criterion, except now suppose that R is topologically equivalent to an annulus, i.e., it has exactly one hole in it. Using Green s theorem, show that there exists at most one closed orbit in R. (This result can be useful sometimes as a way of proving that a closed orbit is unique.)... 

The argument above proves the existence of a closed orbit, and almost proves its uniqueness. But we haven t excluded the possibility that Ply) = y on some in-... 

By specifying an orbital, we come pretty close to uniquely describing each electron in an atom We can say that a particular electron is in a particular principal level, in a particular sublevel, and in a particular orbital. Any given orbital can only hold two electrons. In order to complete this unique description, we only need to differentiate between the two electrons in the orbital. The quantum-mechanical model states that these two electrons have opposite spin direction. 

Then, in the extended phase space the direct product of the phase space and the parameter space) near the origin there exists a uniquely defined -smooth invariant surface of the form p = 0(a ), V (0) = 0, such that each its intersection with the plane p = constant consists of a set of closed orbits of the system (11.5.17), lying in a neighborhood of the origin at the given p. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

Notice that dz2 is unique among the d-orbitals in that lz does not couple it to any other orbital. Thus if the major metal contribution to the SOMO is dzi, g2 will be close to the free electron value. Accordingly, when one g-matrix... 

Hydrogen bonds may be considered special types of 3c/4e interactions, closely related to other forms of hypervalency in main-group (Section 3.5) and d-block (Section 4.6) compounds. However, the fundamental nB— oah interaction of B - HA hydrogen bonding displays unusual characteristics compared with other three-center MO phenomena, due mainly to the unique properties of the H atom, whose valence shell contains only the isotropic Is orbital for construction of ctah and ctah NBOs. 

Shape selectivity and orbital confinement effects are direct results of the physical dimensions of the available space in microscopic vessels and are independent of the chemical composition of nano-vessels. However, the chemical composition in many cases cannot be ignored because in contrast to traditional solution chemistry where reactions occur primarily in a dynamic solvent cage, the majority of reactions in nano-vessels occur in close proximity to a rigid surface of the container (vessel) and can be influenced by the chemical and physical properties of the vessel walls. Consequently, we begin this review with a brief examination of both the shape (structure) and chemical compositions of a unique set of nano-vessels, the zeolites, and then we will move on to examine how the outcome of photochemical reactions can be influenced and controlled in these nanospace environments. 

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

The second statement of Theorem 3.1 follows from the following fact the closure of a G -orbit is a union of orbits of smaller dimensions. Hence any orbit contains a closed G -orbit in its closure. (Moreover, it is unique by Theorem 3.3.)... 

An extensive review with many examples125 shows that the reactivity-selectivity principle cannot be used to predict the selectivity of a reaction except in unique systems where one reaction is close to or diffusion controlled. The relative importance of the Hammond effect and the frontier-orbital effects determines the reactivity-selectivity relationship that will be found in a particular system. The review also concludes that the Hammond-Leffler a-value cannot be used as an indicator of transition-state structure. 

Conclusions We have established that the light Br and Rb isotopes presented here have very large quadrupole deformations of s 0.4 and moments of inertia close to the rigid body values. The odd proton in the 431 3/2+ Nilsson orbit polarizes and stabilizes the y-soft, shape coexistent Se and Kr cores into definite prolate triaxial shapes. This effect sets in at rather low spin and seems to be intimately connected with the suppression of pairing correlations near the N = Z = 38 gap developing at 82 = 0.4. We thus face a cumulative suppression of both proton and neutron pairing correlations in the same oscillator shell, a fairly unique feature in the periodic table. 

This is transfer covariant if all quadratically integrable functions are represented in the same orbital basis. Requiring fps to be orthogonal to all radial factor riPa(r)) enforces a unique representation, but introduces Lagrange multipliers in the close-coupling equations. An alternative is to require... 

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