Periodic orbit expansion

In integrable systems, the periodic orbits are not isolated but form continuous families, which are associated with so-called resonant tori. In action-angle variables, the Hamiltonian depends only on the action variables, similar to the Dunham expansion, ... 

Figure 4. Vibrogram of C2HD calculated with = 2000 cm-1 from all the vibrational energy levels predicted by the Dunham expansion corresponding to the Hamiltonian (3.12) obtained by Herman and co-workers by fitting to high-resolution spectra [112], The periods of the bulk periodic orbits of Table I obtained numerically for the classical Hamiltonian (3.12) are superimposed as circles. On the right-hand side, the main labels (n4,ns) of the periodic orbits are given.

The theory of bifurcations shows that the different types of bifurcations can be described in terms of normal forms, which represent local expansions of the dynamics around the bifurcating periodic orbit [19, 32, 49]. The purpose of the above mapping is to describe the successive bifurcations of the symmetric-stretch periodic orbit, starting from low energies above the saddle point. Appropriate truncation of the Taylor series of the potential v(q) around <7 = 0, which corresponds to the location of the symmetric-stretch orbit, provides us with the normal forms of the bifurcations [144], The bifurcations relevant for the dissociation dynamics under discussion can be described by truncating at the sixth order in q,... 

In a more recent work, Joens [158] has assigned the structures of the Hartley band using a Dunham expansion, that is equilibrium point quantization. The lifetime predicted by his analysis is extremely short, equal to 3.2 fs, while the symmetric stretching period is of 30 fs. Recall, however, that the interpretations in terms of equilibrium point expansions and in terms of periodic orbits are strictly complementary only for regular regimes. 

This result is significant since it applies not only to the shift map but to many other chaotic systems as well. It means that the periodic orbits of a chaotic system can be used as a skeleton for the expansion of physically interesting quantities. In fact, Gutzwiller (1971, 1990) used a periodic... 

For classically chaotic quantum systems the forward application of the trace formula is difiicult because of the following reasons, (i) In a chaotic system the number of periodic orbits proliferates exponentially (ii) the orbits have to be computed numerically and (iii) there are convergence problems with (4.1.72) that have to be circumvented with appropriate resummation prescriptions such as, e.g., cycle expansions (Cvitanovic and Eckhardt (1989), Artuso et al. (1990a,b)). Nevertheless, sometimes valuable information on the structure of atomic states can be obtained by retaining only the shortest orbits in the expansion (4.1.72). This was... 

The most recent advance in the theory of the helium atom was the discovery of its classically chaotic nature. In connection with modern semiclassical techniques, such as Gutzwiller s periodic orbit theory and cycle expansion techniques, it was possible to obtain substantial new insight into the structure of doubly excited states of two-electron atoms and ions. This new direction in the application of chaos in atomic physics was initiated by Ezra et al. (1991), Kim and Ezra (1991), Richter (1991), and Bliimel and Reinhardt (1992). The discussion of the manifestations of chaos in the helium atom is the focus of this chapter. 

When treating periodic systems, the orbital expansion given so far is incomplete in principle. Although, the charge density is necessarly periodic, there can be for the wavefunction itself a phase factor from one periodic image to the other. This is the essence of the Bloch theorem stipulating that orbitals can be written as... 

The elimination of secular terms from the power series expansion of the solution is achieved by the method of Lindstedt. The underlying idea is to pick a fixed frequency p, and to look for a quasi-periodic solution with basic frequencies /i and v. This is actually the same thing as looking for a quasi-periodic orbit on an invariant 2-dimensional torus. The process of solution is the following. Write the Duffing s equation as... 

Thus far we have dealt purely with definitions. We have shown that a periodic orbit is an extremum of a classical adiabatic potential energy surface and that given the periodic orbit and its immediate vicinity of energy dependent periodic orbits one can compute the coefficients of a Taylor expansion of the surface around the extremum. One must now turn to dynamics. In fact, it is not too difficult to show that if translational motion is much slower than vibrational then the Hamiltonian governing the translational motion may be written as... 

A pitch is made for a renewed, rigorous and systematic implementation of the GW method of Hedin and Lundquist for extended, periodic systems. Building on previous accurate Hartree-Fock calculations with Slater orbital basis set expansions, in which extensive use was made of Fourier transform methods, it is advocated to use a mixed Slater-orbital/plane-wave basis. Earher studies showed the amehoration of approximate linear dependence problems, while such a basis set also holds various physical and anal3ftical advantages. The basic formahsm and its realization with Fourier transform expressions is explained. Modem needs of materials by precise design, assisted by the enormous advances in computational capabilities, should make such a program viable, attractive and necessary. 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Expansion of Octet. The octet rule strictly applies to those elements that have only four orbitals available. In those cases, a maximum of four bonds can be formed through overlap. There are, however, a number of molecules where five or six covalent bonds are formed to a central atom. Such behavior is exhibited by elements in the third period and subsequent periods of the periodic table, but never by elements in the second row of the periodic table. A few molecules of this type are listed below. 

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