Orbital connections invariance

One may also wish to impose an additional requirement on the connection, namely that it is translationally and rotationally invariant. This may seem to be a trivial requirement. However, a connection is conveniently defined in terms of atomic Cartesian displacements rather than in terms of a set of nonredundant internal coordinates. This implies that each molecular geometry may be described in an infinite number of translationally and rotationally equivalent ways. The corresponding connections may be different and therefore not translationally and rotationally invariant. In other words, the orbital basis is not necessarily uniquely determined by the internal coordinates when the connections are defined in terms of Cartesian coordinates. Conversely, a rotationally invariant connection picks up the same basis set regardless of how the rotation is carried out and so the basis is uniquely defined by the internal coordinates. [For a discussion of translationally and rotationally invariant connections, see Carlacci and Mclver (1986).]... 

According to Theorem C.6, the limit set can be deformed to a compact invariant set A, without rest points, of a planar vector field. By the Poin-care-Bendixson theorem, A must contain at least one periodic orbit and possibly entire orbits which have as their alpha and omega limits sets distinct periodic orbits belonging to A. Using the fact that A is chain-recurrent, Hirsch [Hil] shows that these latter orbits cannot exist. Since A is connected it must consist entirely of periodic orbits that is, it must be an annulus foliated by closed orbits. Monotonicity is used to show... 

GAO is an orbital-based graph-theoretical representation of molecules from which the common graph invariants, such as connectivity, Zagreb, and Wiener indices, can be calculated, and thus it represents a source of orbital-based molecular descriptors, which can be generally called GAO descriptors. Note that orbital interaction graph of linked atoms is another representation of molecules, which accounts for atom orbitals. 

When it is impossible to use real functions, the complex description is easily introduced since the reduced matrices and the 6-1 and 9-1 symbols are invariant in the two representations. It is further important that the reduced ligand-field parameters are the same, even though they have been defined on the basis of the real orbitals. In this connection it may be mentioned that Racah s formulae for the 3-j and 6-j symbols (4, 75), which are convenient for computer work, make it possible to generate ligand-field matrices by a rather simple algorithm. 

It has the property that/pp = —IPp when the orbital p is doubly occupied and/pp = —EAp when the orbital is empty. The value will be somewhere between these two extremes for active orbitals. Thus, we have for orbitals with occupation number one /pp = — j(IPp + EAp). This formulation is somewhat unbalanced and will favor systems with open shells, leading, for example, to somewhat low binding energies [52]. The problem is that one would like to separate the energy connected with excitation out from an orbital from that of excitation into the orbital. This cannot be done within a one-electron formulation of the zeroth order Hamiltonian. K. Dyall has suggested to use a two-electron operator for the active part [53], but this leads to a too complicated formalism and also breaks important orbital invariance properties (the result is, for... 

It is possible, therefore, to transform orbits which we have hitherto permitted, and which have been confirmed empirically, into orbits in which the electron approaches indefinitely close to the nucleus. At present no explanation of this difficulty can be given. There is a possibility that the J s need not be strictly invariant for the adiabatic changes considered in this connection, since states are continually traversed where (non-identical) commensurabilities exist between the frequencies ( accidental degeneration, see 15, p. 89, and 16, p. 97). 

Sources of atomic data for determining the Hu are discussed at length below. There are two contributions to the Hij(i j). The neighbor atom potential introduces terms connecting members of a set of p, d, or f orbitals on the same atom if the neighbor atom is not in the direction of a coordinate axis from the reference atom. These Hij terms are necessary to preserve rotational invariance. For atomic orbitals on different atoms,... 

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

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