Hole-particle symmetry

As Figs 1.1 and 1.2 illustrate for fra ns-polyacetylene, poly (para-phenylene viny-lene), and polydiacetylene, most conjugated polymers possess a two-fold rotation symmetry about an axis of symmetry through their centre and normal to their plane of symmetry. Such polymers are said to possess C2h symmetry (Atkins and Friedman 1997). 

A many body state that is even under inversion (with a positive eigenvalue) is denoted Ag, while a many body state that is odd under inversion (with a negative eigenvalue) is denoted 

As illustrated in Fig. 1.2 poly(po/p-phenylene), possesses planes of symmetry through both the major and minor axes. This is denoted as D2h symmetry. The character table for D2h symmetry is shown in Table 11.3. 

If a Hamiltonian has particle-hole (or charge-conjugation) S3unmetry then it is invariant under the transformation of a particle into a hole under the action of the particle-hole operator, J  

There are two requirements for an interacting model to posses particle-hole symmetry. The first requirement applies to the kinetic energy, and states that the lattice must be composed of two interpenetrating sublattices, with nearest neighbour one-electron hybridization between the two sub-lattices. As shown in 

The PPP method has been extensively used in quantum chemistry to study the electronic structure of planar 7c-electron systems some years ago, when more sophisticated semiempirical all-valence electron models or abinitio methods were not available. Recently, the PPP model received considerable attention in solid state physics (see, e.g. Soos Ramasesha 1984 Ramasesha Soos 1984a, b), where, due to the size of the systems ulider study, it represents perhaps the most advanced general computational method taking into account electron-electron interaction explicitly.  

The PPP Hamiltonian is usually not solved exactly—this would be rather time consuming in most cases—but it is solved at the SCF level under the Hartree-Fock approximation. If desirable, these SCF PPP wave functions can be improved by the configuration interaction (Cl) method. The full Cl solution which corresponds to an exact diagonalization of the PPP Hamiltonian of Eqs. (10.9) or (10.24) has also been calculated in some cases numerically (Mayer 1980). 

A Tc-electron model is necessarily oversimplified because, among other reasons, it lies upon the so-called a — n separation. This means that the n electrons are described by an effective model Hamiltonian as discussed in the previous sections while the effect of the a electrons (a core) is taken into account only by modifying the matrix elements of the effective Hamiltonian for the n electrons. Another drastic simplification is the first-neighbor approximation discussed also in the previous sections. These approximations may considerably limit the power of the 

10 Some Model Hamiltonians in Second Quantized Form 

Needless to say that realistic systems do not follow such symmetries exactly these are to be considered as approximate symmetries in this sense. Real systems and more sophisticated Hamiltonians obey the approximate symmetries only to the extent the approximations inherent in the model are valid. The approximate symmetry is, however, an exact symmetry of the model Hamiltonian. 

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Particularly noteworthy is the particle—hole symmetry. Let us define (one- and two-) hole density matrices [17]... 

Consider the particle-hole symmetry of the one-dimensional Hubbard model. One can obtain w —U) = — w U), so the entanglement is an even function of U, Ej —U) = Ej U). The minimum of the entanglement is 1 as 1/ oo. As U +00, all the sites are singly occupied the only difference is the spin of the electrons on each site, which can be referred to as spin entanglement. As U —oo, all the sites are either doubly occupied or empty, which is referred... 

For the other cases of lower symmetry, the number of neighbors of any given order must be complemented by some extra connectivity information. First, we observe that the minima for n = 2. 5=1 and for n = 3, S = are exactly the same. Indeed, these two cases are related by a particle-hole symmetry applied only to one spin flavor. For all nonequivalent cases, the complete topological information about the wells is contained in the connectivity matrix C(n,S), whose matrix elements... 

For the half-filled case, p is fixed to be U/2. Moreover, by using the particle-hole symmetry, E = 2, the renormalization group equations for U and K take the form... 

In addition to extending 2fa-like theory to admit Fukutome s (1981) eight classes of solutions, Eq. (44) clears up a number of problems such as what Gunnarsson and Jones (1980) call lack of particle-hole symmetry, which is the idea that for fixed orbitals the multiplet splittings of a number of electrons... 

When the discussion is limited to a single open subshell and to perturbations within such a shell, an interesting formulation of the many-electron atomic problem can be achieved that exhibits useful particle-hole symmetry. The summations in the perturbation term of the hamiltonian then run only over electron states nlmu) of the open subshell (nQ. This means that the electron repulsion integrals can be expressed as... 

The introduction of the pair operators allows a simple way to demonstrate that corresponding particle and hole configurations have the same terms and similar energy expressions. This particle-hole symmetry has been known since the early work of Heisenberg. A imitary transformation U is employed, which is expressed in terms of the pair operator tt = (0000) and its adjoint... 

Due to the particle-hole symmetry, the range of the parameters N and / can be limited, so that... 

Huckel theory for the even alternant hydrocarbons leads to the Coulson-Rushbrooke theorem and some other characteristic results shown by McLach-lan to be valid also in the Pariser-Parr-Pople model. These are the well-known pairing relations between electronic states of alternant hydrocarbon cat-and anions. This particle-hole symmetry is analogous to the situation discussed in Chapter 4 for electrons and holes in atomic subshells. 

Most linear conjugated molecules and polymers possess spatial symmetries, while cyclic polymers possess axial symmetry. Conjugated systems also possess an approximate particle-hole symmetry. These symmetries characterize the electronic... 

Fig. 2.8, this requirement is satisfied for a one-dimensional chain with nearest neighbour hybridization. As a consequence of particle-hole symmetry the kinetic energy for a uniform cyclic chain satisfies = —efc-, r/a shown in Fig. 3.1. Similarly, for a linear uniform chain the kinetic energy satisfies tj = as...

The second requirement for a model to posses particle-hole symmetry is that the electron-electron interactions must be balanced - on average - by electron-nuclear interactions. For a chain with translational symmetry every site is equivalent with the same potential energy. For a linear chain, with open boundary conditions, however, the sites are not equivalent. The electrons on sites in the middle of a chain experience a larger potential energy from the nuclei than electrons on sites towards the ends of the chain. This potential energy, Vj = is shown in Fig. 2.9. Correspondingly, the electrons on sites in the middle of the chain experience a larger electron-electron repulsion than electrons towards the end of the chain. When this repulsion is equal and opposite to the electron-nuclei attraction, there is particle-hole symmetry, and every site is essentially equivalent. 

Systems which posses particle-hole symmetry satisfy a number of properties. First, the expectation value of the occupancy of each site is unity, or. 

Fig. 2.9. The on-site pseudopotential, Vj, of a linear chain of 100 sites, using the Ohno potential, eqn (2.55). On average, this is balanced by the electron-electron repulsion in systems with particle-hole symmetry.

Table 2.2 States and symmetry character table for linear polymers described by TT-electron Hamiltonians with inversion and particle-hole symmetries...

We show in Chapter 8 that the dipole operator connects states of the same spin with opposite spatial and particle-hole symmetries. 

Fig. 3.1. The tight-binding band structure of a cyclic chain, eqn (3.11). As a consequence of particle-hole symmetry, Cfc = —eh+njai while ejt = e fc is a consequence of time reversal invariance.

Particle-hole symmetry and particle-hole parity... 

There is an important connection between particle-hole symmetry and the relative parity of the particle-hole pair. Consider a basis state created by the removal of an electron from a valence band Wannier orbital on the repeat unit at i — r/2 and the creation of an electron on a conduction band Wannier orbital at R + r/2. This is illustrated in Fig. 6.1. This particle-hole pair has a centre-of-mass coordinate, R, and a relative coordinate, r ... 

These geometric properties of the chain are also associated with mid-gap states (Pople and Walmsley 1962). To see this, consider the energy spectrum of an even-site chain. There are N/2 states in each of the valence and conduction bands. As a result of particle-hole symmetry, every valence band state with energy e" = e maps into a conduction band state with energy = —e. Thus, the energy spectrum is symmetric about e = 0, as shown in Fig. 3.4. Now, for an odd-site chain there are N — l)/2 states in each of the valence and conduction bands, and one localized gap state. As a consequence of particle-hole symmetry the localized state lies at e = 0. This mid-gap state is occupied by one electron, and is associated with the soliton, as shown in Fig. 4.3. 

As shown in Section 3.6.1, as a consequence of particle-hole symmetry, the amplitudes satisfy... 

V n( ) is the relative wavefunction that describes the internal structure of the exciton. Owing to particle-hole symmetry it satisfies... 

There is an important observation to be made about this effective-particle model. This is that since the exchange interaction, X, is local (i.e. it is only nonzero when r = 0), we immediately see that this term vanishes for odd parity excitons (namely, (r) = —i/ ni—r)), as tl>n 0) = 0. Now, since the parity of the exciton is determined by the particle-hole symmetry, and odd singlet and... 

Table 6.1 The classification of the many body singlet exciton states with particle-hole symmetry in terms of their Mott- Wannier exciton quantum numbers (The corresponding triplet states with the same spatial symmetry but opposite particle-hole symmetry have the same quantum numbers)...

Fig. 6.7. A schematic energy level diagram of the j = 1 members of the Mott-Waimier exciton families. The symmetry assignments refer to centro-symmetric pol5miers with particle-hole symmetry.

Particle-hole s5Tnmetry No particle-hole symmetry n 3... 

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