The effective-particle model

The weak-coupling limit takes as its starting point the conventional semiconductor noninteracting band picture, introduced in Chapter 3. The ground state is an occupied valence-band and an empty conduction-band. A bound conduction band electron and valence band hole move through the lattice as an effective-particle. In this section we derive the effective-particle model, discuss its solutions and compare them to essentially exact calculations on the same Hamiltonian (Barford et al. 2002b). We develop this theory for a linear, dimerized chain. 

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.  

The general exciton eigenstate, therefore a linear superposition of 

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

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Fig. 6.2. The effective-particle model of excitons on a linear chain. The total exciton wavefunction, = ipn r) j R), where ipn r) is the relative wavefunction...

Notice that two quantum numbers specify the exciton eigenstates, eqn (6.13) or eqn (6.16) the principle quantum number, n, and the (pseudo) momentum quantum number, K (or fUj). For every n there are a family of excitons with different centre-of-mass momenta, and hence different centre-of-mass kinetic energy. Odd and even values of n correspond to the relative wavefunction, tl>n r), being even or odd under a reversal of the relative coordinate, respectively. We refer to even and odd parity excitons as excitons whose relative wavefunction is even or odd imder a reversal of the relative coordinate. This does not mean that the overall parity of the eigenstate (eqn (6.12)), determined by both the centre-of-mass and relative wavefiictions, is even or odd. The number of nodes in the exciton wavefunction, V rt( ), is n— 1. Figure 6.2 illustrates the wavefunctions and energies of excitons in the effective-particle model. 

In this chapter we have described the effective-particle models for excitons in the weak and strong coupling limits, and compared them to essential exact, numerical (DMRG) calculations. We saw that there is good agreement between the effective-particle models and the computational results in these limits. We used these extreme limits to understand the numerical calculations in the intermediatecoupling regime. We summarize the main points as follows ... 

Formally, the exciton binding energy is defined relative to the energy of a widely separated uncorrelated electron-hole pair. In practice, excitons whose particle-hole separation exceeds the length of the polymer (or more correctly, the conjugation length) can be considered unbound. This marks the breakdown of the effective-particle model. 

As described in Chapter 6, n = 1 corresponds to the Sx and T% families of intrachain excitons, while n = 2 corresponds to the ct and Tct families of intrachain excitons. The lowest energy branch of each family has the smallest pseudo-momentum, namely, j =. The effective-particle model is illustrated in Fig. 6.2. 

The Schodinger equation for the effective-particle model of excitons was introduced in Chapter 6. In this appendix we derive that equation. 

In section 2, we provide a description of the methods employed in the present study the generation of Gaussian-type basis sets, the independent particle model and the treatment of electron correlation effects, and, the computational details. Results are presented and discussed in section 3. Section 4 contains our conclusions. 

Fig. 3.5.1 The minimum of the first derivative of UV/visible absorbance spectra of CdS particles as a function of the particle diameter. The data points have been collected from literature sources where particle sizes were determined by EM or XRD. If specific data about the minimum of the first derivative were not expressly provided, they were estimated from spectra supplied. The estimated error in this technique is less than 5 nm. For clarity, only data from groups with the greatest number of data points have been used here. A curve has been fitted to the data using a theoretical relationship between particle diameter and wavelength using the effective mass model (6). Inset Absorbance spectra of colloidal CdS produced by exposure of a CdAr film, or a Cd2+/HMP solution, to H S. The minima of the first derivative (380 nm in film, 494 nm in solution) correspond to particle sizes of approximately 2.5 nm and 6.0 nm, respectively. (From Ref. 5.)...

Estimate of the HETP intradiffusion coefficients, when based on HETP dependence on concentration, has been shown to be close in value to the quantity calculated with the theoretical formula obtained by using the model of centrosymmetrical diffusion in an ion-exchanger bead [89,90]. The values in this formula of r /D, where r is the effective particle radius, were determined in thin-layer kinetic experiments with the same ion-exchanger fraction. 

As explained earlier, most authors quote nominal mbber contents rather than mbber phase volumes, and there is therefore very little information in the literature on the relationship between Oyc and 0 for mbber-modified plastics. A rare exception occurs in the work of Oxborough and Bowden vdio measured yield stresses in tension and compression for a series of HIPS polymers ccmtaining composite rubber particles. Their results are presented in Fig. 7. Equation (9) underestimates the yield stresses both in tension and compression, and it must be concluded fiiat the effective area model does not provide a satisfactory basis for correlating yield data in this class of material. Either the model itself must be modified in some way, or some allowance must be made for load sharing with the mbber particles, if the effective area apprcrach is to be retained. 

A very important conceptual step within the MO framework was achieved by the introduction of the independent particle model (IPM), which reduces the AT-electron problem effectively to a one-electron problem, though a highly nonlinear one. The variation principle based IPM leads to Hartree-Fock (HF) equations [4, 5] (cf. also [6, 7]) that are solved iteratively by generating a suitable self-consistent field (SCF). The numerical solution of these equations for the one-center atomic problems became a reality in the fifties, primarily owing to the earlier efforts by Hartree and Hartree [8]. The fact that this approximation yields well over 99% of the total energy led to the general belief that SCF wave functions are sufficiently accurate for the computation of interesting properties of most chemical systems. However, once the SCF solutions became available for molecular systems, this hope was shattered. 

In order to define orbitals in a many-electron system, two approaches are possible, which we may refer to as constructive and analytic . The first approach is more common one makes the ad hoc postulate that every electron can be associated with one orbital and the total wave function can be constructed from these orbitals. Then, one is led to an effective one-electron Schrodinger equation from one electron in the field of the other electrons. The underlying model is the independent particle model (IPM). When following the constructive way, one does not know a priori whether the model is a good approximation to the actual physical situation one only knows that it cannot be rigorously correct. The merit of this approach is its relative simplicity from both the mathematical and physical points of view. 

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