Particle hole transformation

One can see from this result that a complete particle-hole transformation occurs when... 

The generators tt and of the transformation commute with the total orbital and spin angular momenta such that the transformation only connects states with the same quantum numbers (LMlSMs). The various operators appearing in the hamiltonian in Eq. (4.150) under a complete particle-hole transformation become... 

This means that apart from a term involving the number operator, the hamil-tonian is invariant under the complete particle-hole transformation and the relative energies of the terms of a configuration (nl) are the same as those of a configuration... 

Consider the complete particle-hole transformation by the unitary operator U = exp (iS), where... 

It is easy to demonstrate that the kinetic energy term is invariant under the transformation, eqn (2.58). Also, because under the particle-hole transformation... 

For the 7r-electron models this means that the average number of 7r-electrons per site is unity. This result is proved in Appendix B. A second property is that singlet particle-hole excitations that are negative under a particle-hole transformation have an even particle-hole spatial parity, while singlet particle-hole... 

Fig. 3.4. The valence and conduction bands of a dimerized, cyclic chain. The particle-hole excitation at k, and its degenerate counterpart at —k, connected by the particle-hole transformation, are shown.

Note that under the particle-hole transformation k and [Pg.35]

Fig. 6.1. The real-space particle-hole excitation, R- -r/2, R—r/2), labelled 1, from the valence band Wannier orbital at R — r/2 to the conduction band valence orbital at R+r/2. Its degenerate counterpart, R—r/2, R+r/2), connected by the particle-hole transformation, is labelled 2. R = (ve + rh)/2 is the centre-of-mass coordinate and r = (re — Vh) is the relative coordinate. A Mott-Wannier exciton is a bound particle-hole pair in this representation.

This implies that the molecular orbitals are related by a particle-hole transformation. In particular, numbering the sites as indicated in Fig. 11.4, it is readily shown that a molecular orbital l) is transformed to its complement ) by the particle-hole transformation as follows,... 

Eig. 11.4. The molecular orbitals of benzene and their electronic occupation in the ground state. The shading indicates the atomic orbital amplitudes on each site. The site labelling defines the particle-hole transformation rule, eqn (11.7). 

To prove this, note that under a particle-hole transformation,... 

The particle-hole transformation of Eq. (10.25) appears to be rather formal but we shall see its advantages below. Let us first study the symmetry of the second quantized Hamiltonian of alternant hydrocarbons under the particle-hole transformation. 

Let us apply now the particle-hole transformation to this Hamiltonian. The transformation consists of substituting the particle operators aj, aj by the corre-... 

Application of the particle-hole transformation on the Hamiltonian of Eq. (10.35) yields ... 

It follows that, if Q = 0, the expectation values of H and H are the same the energy is invariant to the particle-hole transformation. Moreover, many-electron states of the corresponding positive and negative ions of the molecule are in a one-to-one correspondance, their energy difference is determined by 2 Q. The corresponding electron affinites and ionization potentials have the same absolute value. These are the most important consequences of the particle-hole symmetry of the PPP Hamiltonian for alternant hydrocarbons. Unfortunately, this symmetry cannot be generalized to more sophisticated Hamiltonians which will be discussed in the forthcoming sections. 

Another special case of the general quasiparticle transformation is when the coefficients Aj = 0. This corresponds to interchanging the creation and annihilation operators. The canonical conditions of Eq. (16.3) then require matrix B to be unitary. With B being the unit matrix, this is the particle-hole transformation we have considered in Sect. 10.2. 

---