Diagonal disorder

In conclusion we propose ASR as an efficient computational scheme to study electronic structure of random alloys which allows us to take into account the coherent scattering from more than one site. Consequently ASR can treat effects such as SRO and essential off-diagonal disorder due to lattice distortion arising out of size mismatch of the constituents. 

For explicitness, let us assume that off-diagonal disorder is caused by chain twists, which randomly diminish the overlap between the n-orbitals of neighboring carbon atoms (see Fig. 3-7). The electron hopping amplitudes that depend both on the interatomic distances and on the relative orientation of the electronic orbitals on neighboring atoms can then be written in the form ... 

The creation of kinks by off-diagonal disorder is illustrated in Figure 3-9. The thick line is the ground state order parameter A = A(2an) = — f2 ,2 +i for... 

Figure 3-9. The numerically obtained order parameter along a polyucciylcne chain of 80 pairs of carbon atoms tor one panicular (but typical) realization of oil-diagonal disorder. The thin line does not account for lattice relaxation, the thick line does (adapted from Ref. 30 )...

Fig. 4. Substitution matrix based on disordered protein families. Below the diagonal are the scores for each amino acid substitution. Above the diagonal are the differences between BLOSUM 62 and the disorder matrix. On the diagonal are the scores/differences. (From Radivojac et al., 2002, PSB 2002 7, 589-600, with permission of World Scientific Publishing Co. Pte Ltd.)...

Effect of diagonal dynamic disorder (DDD). Fluctuations of the polarization and the local vibrations produce the variation of the positions of the electron energy levels eA(Q) and eB(C ) to meet the requirements of the Franck-Condon principle. 

Effect of off-diagonal dynamic disorder (off-DDD). The interaction of the electron with the fluctuations of the polarization and local vibrations near the other center leads to new terms VeP - V P, Vev - Vev and VeAp - VAPd, VA - VAd in the perturbation operators V°d and Vfd [see Eqs. (14)]. A part of these interactions corresponding to the equilibrium values of the polarization P0l and Po/ results in the renormalization of the electron interactions with ions A and B, due to their partial screening by the dielectric medium. However, at arbitrary values of the polarization P, there is another part of these interactions which is due to the fluctuating electric fields. This part of the interaction depends on the nuclear coordinates and may exceed the renormalized interactions of the electron with the donor and the acceptor. The interaction of the electron with these fluctuations plays an important role in processes involving solvated, trapped, and weakly bound electrons. 

Effect of diagonal-off-diagonal dynamic disorder (D-off-DDD). The polarization fluctuations and the local vibrations give rise to variation of the electron densities in the donor and the acceptor, i.e., they lead to a modulation of the electron wave functions A and B. This leads to a modulation of the overlapping of the electron clouds of the donor and the acceptor and hence to a different transmission coefficient from that calculated in the approximation of constant electron density (ACED). This modulation may change the path of transition on the potential energy surfaces. 

Additional effect of diagonal dynamic disorder. The variations of the electron densities near the centers A and B due to polarization fluctuations and local vibrations lead to changes in the interaction of the electron with the medium and, hence, to changes in the shape of the potential energy surfaces Ut and Uf as compared... 

Effects of Diagonal and Off-Diagonal Dynamic Disorder in Reactions Involving Transfer of Weakly Bound Electrons (A Configurational Model)... 

To take into account the additional effect of diagonal dynamic disorder in the improved Condon approximation it was suggested in Ref. 16 that fluctuations of the polarization of the type... 

The first summation incorporates the (non-random) translational invariance, while the second includes random deviations from the lattice on a site-by-site basis. Note that the second summation explicitly indicates that all randomness or disorder is diagonal, not off-diagonal. The corresponding exact GF G = (lu — Hyl satisfies the matrix equation... 

To analyze the negative field dependence of the mobihty in EHO-OPPE within the Gaussian disorder transport formahsm and to determine the diagonal (energetic) disorder parameter a and the off-diagonal (positional) disorder parameter d, the following relation between the charge mobihty p and the disorder parameters was employed [75] ... 

Off-diagonal (positional) disorder parameter. Diagonal (energetic) disorder parameter at 20 °G. 

Fig. 10 Temperature dependence of the diagonal disorder parameter dfor holes (squares) and electrons (circles). Parameters have been obtained by fitting the experimental data (Fig. 9) to Eq. 1 and Eq. 5. Reproduced with permission from [61]...

It was demonstrated by Sundman (1985) and later by Ansara et al. (1988) that an order-disorder transformation could be modelled by setting specific restrictions on the parameters of a two>sublattice phase. One of the first phases to be considered was an A B-ordered compoimd. In such circumstances the sublattice formula A, B)j(A, B) can be applied and the possible relationships between site fiactions and mole fiactions are given in Figure 5.6. The dashed lines denoted xb = 0.25, 0.5 and 0.75 show variations in order of the phase while the composition is maintained constant. When these lines cross the diagonal joining AjA and B3B the phase has disordered completely as Vb Vb As the lines go toward the boundary edge the phase orders and, at the side and comers of the composition square, there is complete ordering of A and B on the sublattices. 

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