Particle-hole excitations

The matrices A, B, Q are of infinite dimension since there are an infinite number (2N+1, N — oo) of k-values and thus an infinite number of k-states in each band. Moreover, there is an equation for each triplet formed by a k-value and two band indices. This triplet represents a particle-hole excitation that is vertical in order to preserve the momentum. As is the case in many polymeric techniques, the infinite sum over k is transformed into an integration in the first Brillouin zone ... 

M. Rosina and M. V. Mihailovic, The determination of the particle—hole excited states by using the variational approach to the ground state two-body density matrix, in International Conference on Properties of Nuclear States, Montreal 1969, Les Presses de I Universite de Montreal, 1969. 

Detailed descriptions of the model is given elsewhere [HEY82], [HEY83]. The basic idea, however, is to calculate two sets of states where one corresponds to the vibrational states and the other corresponds to the rotation-like intruder states which arise from the particle-hole excitations, and then to mix the two resulting configurations. The hamiltonians for both configurations are esssentially the same ... 

In many nuclei near single closed shells low lying intruder states have been observed. These intruder states are known to be due to particle-hole excitations across the closed shell. The additional particle-hole degree of freedom causes dramatic changes in the properties of the nuclear states. It is for example a characteristic feature of the intruder states that they act as band heads for rotational bands. 

At this point it is worth rephrasing some of the issues of the above discussions. The UPS spectra are a measure of the single-particle excitation spectrum of the molecule, in so far as removal of an electron is concerned, while UAS data are a measure of the particle-hole excitation spectrum. In other terms, UPS measures the molecular-ion states while UAS measures excited states of the neutral molecule. For a molecule in isolation, in a one-electron picture the valence electron molecular cation states are comprised of the set of one-electron molecular orbitals (mo s) containing one half-filled (usually non-degenerate) molecular orbital and the totality of other fully occupied orbitals, distorted from their situation in the neutral molecule due to the removal of an electron from the molecule in a photoelectron... 

Both P and Q are sums of excitation operators (with weighting coefficients p and 9 )- Thus, P and Q applied to 0> create a polarization of 0> and we call P 6 a polarization propagator. In the special case where P and Q are both single particle-hole excitations, i.e. only one term in Eqs (5) and (6), we talk about the particle-hole propagator. It is important to note that only the residues of the polarization propagator and not of the particle-hole propagator determine transition moments (Oddershede, 1982). We must have the complete summations in Eqs (5) and (6) in order to represent the one-electron operator that induces the transition in question. 

This means that we have a chance to find a logarithmic (infrared) divergence and thus an infrared problem if dimension d and dispersion S coincide. As the normal dispersions are S = 1 for quantummechanical systems (massless particle-hole excitations in Fermi systems, massless Bose systems) and 6 = 2 for classical systems with short range forces, we conclude ... 

It plays a special role since the single-particle densities involved mix particle (r, r ) and hole (s, s ) indices. This term thus originates from the possibility of particle-hole excitations to annihilate. Note also that this term vanishes for a target in a singlet state if the particle-hole pair forms a triplet excitation, e.g., if the spins are opposite [r = (r, t)i and s = (s,),)]. 

In a simple and very commonly used approximation to the PP, the reference state 0> is chosen to be a single-configuration (but not necessarily single determinant) HF wavefunction. The operator manifold T then is taken as the set of particle-hole excitation and deexcitation operators used for optimizing the reference state ... 

Two major approximations must be made to obtain the response functions a choice of a reference function and a choice of an operator manifold. For the linear response function, the choices of a Hartree-Fock reference state and simple particle-hole excitation operators (in the second quantization sense) lead to an approximation known as the random phase approximation (RPA) and is equivalent to the TDFiF method discussed earlier. 

Diagram (a) in this figure is the third-order TDA diagram, while diagrams (b), (c) and (d) are examples of third-order diagrams involving other particle-hole excitations. 

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.

Fig. 3.5. The energy spectrum of the valence and conduction molecular-orbital states for a dimerized, linear chain. A particle-hole excitation and its degenerate counterpart, connected by the paxticle-hole transformation, are shown. 2A is the charge gap, shown as a function of inverse chain length in Fig. 3.6.

The particle-hole excitations, defined in Section 3.5, are eigenstates of the noninteracting Hamiltonian, but they are not eigenstates of the particle-hole operator, J, introduced in Section 2.9.2. To see this, consider the operation of J on the singlet excitation, A ) ... 

Thus, a particle-hole excitation from the HOMO to the LUMO must have overall odd symmetry. This is the state. The first Ag excitation (the 2Ag state) will be HOMO—1 to LUMO (or, equivalently HOMO to LUMO+1). Such an excitation will lie higher in energy than the 1B state.These transitions are shown in Fig. 3.8. 

For this particle-hole excitation the A -body problem has thus been mapped onto the two-body problem, described by,... 

The study of excitons in conjugated polymers has often been inspired by the treatment of excitons in bulk three-dimensional semiconductors (as described in Knox (1963)). A particle-hole excitation from the valence band to the conduction band in a semiconductor leaves a positively charged hole in the valence band and a negatively charged electron in the conduction band. The Coulomb attraction between these particles results in bound states, or excitons. In three-dimensional semiconductors the excitons are usually weakly bound, with large particle-hole separations, and are well described by a hydrogenic model. Excitons in this limit are known as Mott- Wannier excitons. 

An opposite, strong-coupling limit has also been used to describe excitons in conjugated polymers (Gallagher and Mazumdar 1997 Gebhard et al. 1997 Essler et al. 2001 Harford 2002). As described in the previous chapter, in this limit a correlation gap separates the electron removal spectral weight (the lower Hubbard band) from the electron addition spectral weight (the upper Hubbard band). Now the bound particle-hole excitations are Mott-Huhhard excitons. That is, a particle excited from the lower Hubbard band to the upper Hubbard band... 

Since excitons are bound particle-hole excitations, a convenient basis for their description are the particle-hole basis states introduced in Chapter 3. In A -space these basis states are ke,kh), defined by... 

This particle-hole excitation is illustrated in Fig. 3.4. Now, for translationally invariant Hamiltonians iF is a good quantum number. However, unlike the noninteracting Hamiltonian, the interacting Hamiltonian mixes states with different k. ... 

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.

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