Nondegenerate systems

Trans-polyacetylene has the unusual property of exhibiting no extrinsic dimerization, and thus has no extrinsic band gap. The dimerization arises entirely from TT-electrons coupling to the lattice. Consequently, the A and B phases are degenerate. Most polymers, however, have an extrinsic semiconducting band gap as a result of their stereochemistry independent of the of 7r-electrons. Examples of polymers that are extrinsically semiconducting include, cis-polyacetylene (because of the structure caused by the a orbitals), polydiacteylene (because of the 

The detailed effects of the stereochemistry vary from polymer to polymer -the details of particular polymers will be described in their relevant chapters. However, to understand the qualitative consequences of extrinsic dimerization we can use the linear chain to model its effects. In this model the 7r-electrons are coupled to the extrinsic dimerization via the bond integral (Bishop et al. 1981 Brazovoskii and Kirova 1981), 

Assuming a uniform staggered distortion in the ground state, = (—1) A, and minimizing the ground state energy, we obtain the self-consistent equation. 

The extrinsic dimerization has two effects. First, it causes an increased intrinsic dimerization, as shown in Fig 4.10. Second, it lifts the degeneracy of the A and B phases, as shown in the plot of the ground state energy in Fig. 4.1. This causes a linear confinement of the soliton-antisoliton pair, because the energy to create a B phase relative to the A phase increases linearly with the soliton-antisoliton separation. This new property of soliton-antisoliton confinenment is illustrated by the localized Wannier orbitals associated with the soliton, and antisoliton, These are obtained from the molecular orbitals associated with the mid-gap electronic states, V n (described in Section 4.5) by inverting eqn (4.33). Thus, 

These become confined in the presence of extrinsic dimerization. 

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In this context equations (50) and (53) can be considered forming a completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [lej. 

The occupation numbers will be 0,1, or 2 for a nondegenerate system. Since n, is an excited-state density of a noninteracting system whose potential is wh at least... 

While the T — 0 limit has been taken formally, it should be noted that the electronic fluctuation correlations at room temperature are essentially equivalent to those at T = 0 for normal, nondegenerate systems. (Thermal photons at room temperature are generally unable to excite electronic transitions )... 

For nondegenerate systems, single reference coupled cluster (CC) methods have proven to be quite successful in their description of the electron correlation problem (1-3). This is primarily due to their (size) extensivity and ability to efficiently incorporate higher-order correlation effects—CC models including triple excitation effects are generally within 1-2 kcal mol 1 of full Cl (3). 

Consider now the case of a very large distortion parameter compared to the other ones and to the thermal energy available for the system. D is taken to be positive so that the ground-term of each isolated ion is the orbital singlet A1 well-separated from the orbital doublet E. We are then dealing with a nondegenerate system with half-filled S-orbitals. 

This term (Equation 19.31), linear in F, is zero for a nondegenerate system with no permanent electric dipole moment, whose Hamiltonian is unaffected by the parity operation [94]. In centrosymmetric nondegenerate polymers with no permanent dipole moment, the linear Stark effect ensues from disorder [95]. In a semiclassical approach, the shift in energy caused by a permanent dipole moment my can be expressed as ... 

Note that in (2.58-2.60) a nondegenerate system has been assumed. 

High activation barriers for such reactions give rise to a particularly interesting situation in the case of nondegenerate systems, namely, a thermodynamically unstable tautomer can be isolated under conditions that would exclude all but the intramolecular interconversion channels. Such conditions occur in gas-phase reactions under deep vacuum, they are particularly effective in interstellar space. 

Within the RSPT computational scheme for nondegenerate systems [7] the eigenvalue problem 77 ° [/) = [/) for the unperturbed Hamiltonian is pre-... 

The properties we have included are the electric mnltipole polarizabilities - a , dipole-dipole, Aa py, dipole-quadrupole and C p ys, quadrupole-quadrupole - as well as the mixed polarizabilities - G p, electric dipole-magnetic dipole, Dy p, magnetic dipole-electric quadrupole and the magnetizability (magnetic dipole-magnetic dipole polarizability) i, p. The frequency-dependent response properties labeled with a prime vanish for w = 0, and the properties specified in the second rows of the equations ( Eqs. 11.75, O 11.77,0 11.79) are zero for reference states described by real wave functions (in practice, closed-shell nondegenerate systems). 

As a result of the resonance tunneling between the equivalent wells, the zeroth-order energy is split into many energy levels. Since for the interaction of nondegenerate systems all of these levels, except for one, correspond to states violating the Pauli exclusion principle, the wave function approximating the physical state can be obtained by antisymmetrization of any of the equivalent resonance structures, in particular by antisymmetrization of 0Q. Thus, the antisymmetrized function Azeroth-order wave function than 00 itself. 

The expressions in Equations 4-18 and 4-19 are valid only for nondegenerate systems (Ej,° ). Also note that it is possible to have a... 

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