Positive orbit

In systems in which the donor and acceptor centers are in direct contact with each other or connected by a conducting bridge (conjugated bonds), electron transfer rates are very fast (kET = 10"13 -10 12 s 1). The transition occurs markedly slower when the donor-acceptor mutual orientation is not favorable for positive orbital overlap and, therefore, the electron coupling V is small. 

Proof of Theorem C.3. If there were an attracting periodic orbit, then one could find a point x in its domain of attraction such that x periodic orbit. As / is a limit point of the positive orbit through x, there exists T > 0 such that Xrest point by Theorem C.2(b), contradicting our assumption that it converges to a nontrivial periodic orbit. ... 

Since Interest attaches here on photolonlzatlon as well as absorption, it is useful to note that molecular Rydberg orbitals are closely related to their continuum Coulombic counterparts, associated with positive orbital energies (16), both sets joining together at the ionization threshold. 

The negative Langevin susceptibility Xju hardly varies with concentration x for both components in Ag Pti-. It more or less compensates for the positive orbital contribution Xvv e corresponding atom. A variation in xSia with concentration should primarily arise from the change of the lattice parameter with concentration (Banhart et al. 1986). It seems that this effect is not very pronounced for Ag Pti-jr. 

As discussed in Section 3.1.1, starting from an orbit generating wavefunction W(ri, , r,v) in position space, we may compute the optimal one-particle density Polt(r) = pii) jj(r) by optimizing the energy functional S[p x) j 1] subject to a normalization condition on the density. In other words, the optimal wavefunction , rV) within the position orbit is obtained by means of a local-scaling transformation of the orbit-generating wavefunction. 

In Eq. (99), the momentum transformation vector function s(p) has the same role in momentum orbit V as f f) in position orbit of. 

Excimers (excited dimers) are formed by pairs of molecules or atoms that do not signihcantly interact in the ground state, but are weakly bonded in the excited state. The bonding in the excimer takes place between an excited molecule and a ground state molecule of the same species. Its origin is in the change of orbital symmetry that accompanies excitation and leads to cooperative (positive) orbital overlap and hence to bonding between the two systems. Examples in resist systems can be found in aromatic and heteroaromatic molecules used in photoactive compounds. Excimers were hrst observed by Eorster and Kasper in 1954 when they observed two kinds of fluorescence in fairly concentrated solutions (10 M) of pyrene. ... 

The Hartree-Fock equations which arise from the application of the variation principle to any of these constrained models guarantees a minimum in the energy of the associated single-determinant wavefimction. What they do not guarantee is that this wavefunction is physically meaningful. This point wiU be taken up in detail later for the moment simply recall the single-determinant solution of the closed-shell model for (H20) obtained with the testbench which had positive orbital energies ... 

Determine the general and special positions (orbits) along with the symmetry elements for the plane group p4gm. 

The shapes of the filled 0 g, Cg, and t g MOs result from the positive orbital overlap formed by taking a linear combination of the appropriate AO on Fe + with one of the abovementioned SALCs having the same symmetry. The shapes of the filled bonding MOs for [Fe(H20)6] are shown here. The filled t2g nonbonding MOs have the same shape as the d-orbitals from which they are derived and are not shown in the diagram. However, because their lobes are pointed between the coordinate axes, it is readily apparent that the d, and dy atomic orbitals do not have the correct symmetry to overlap with any of the SALCs on the tr-donor ligands. 

Koopmans theorem provides the theoretical justification for interpreting Hartree-Fock orbital energies as ionization potentials and electron affinities. For the series of molecules we are using, the lowest virtual orbital always has a positive orbital energy, and thus Hartree-Fock theory predicts that none of these molecules will bind an electron to form a negative ion. Hartree-Fock almost always provides a very poor description of the electron affinity, and we will not consider the energies of virtual orbitals further. 

The first term is the Fermi contact interaction and is only operable for s electrons. The second term is due to the orbital current The third term represents the dipole field due to the electron spin. These two latter terms are generally smaller than the contact term and vanish for s-state ions. For Fe in Fe " (S — 5/2, L — 0), the contact interaction gives about —60 T. For Fe in Fe " (5 = 2, L — 2), the field is somewhat smaller because of smaller spin and also appreciable positive orbital contribution. At room temperature the hyperfine magnetic field at Fe in metallic iron is —33 T and this is the reference value to determined the hyperfine magnetic field in magnetic materials using Fe Mbssbauer spectroscopy. Nuclear levels of Fe under magnetic field and the expected Mbssbauer spectrum are shown in Fig. 1.6c. 

Cz (considered as rotation about the x axis) orbitals 2, 3, 5,6 changed in position orbitals 1 and 4 changed in sign character —2. 

A further essential feature of the equations of motion discussed so far is the fact that the functions Fi xi,jc ) in (1.1,11) are generally non-/mear and that the transition probability w (y <— ) in (1.16) is a function of x and This non-hnear structure can be considered as a consequence of the self-consistency principle , i.e. there exists a cyclic coupling or a feedback between causes and effects , where both may be macro variables. For instance, the units of a system may generate a collective field this field reacts on the units in a static or dynamic way creating and stabilizing a certain space-time order of positions, orbits, excitations, etc. of the units. The ordered structures and the collective field then mutually sustain themselves in a self-consistent way. 

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