Electron spectral densities

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

Table 2.4 shows a comparison of the experimental and PPP-MO calculated electronic spectral data for azobenzene and the three isomeric monoamino derivatives. It is noteworthy that the ortho isomer is observed to be most bathochromic, while the para isomer is least bathoch-romic. From a consideration of the principles of the application of the valence-bond approach to colour described in the previous section, it might have been expected that the ortho and para isomers would be most bathochromic with the meta isomer least bathochromic. In contrast, the data contained in Table 2.4 demonstrate that the PPP-MO method is capable of correctly accounting for the relative bathochromicities of the amino isomers. It is clear, at least in this case, that the valence-bond method is inferior to the molecular orbital approach. An explanation for the failure of the valence-bond method to predict the order of bathochromicities of the o-, m- and p-aminoazobenzenes emerges from a consideration of the changes in 7r-electron charge densities on excitation calculated by the PPP-MO method, as illustrated in Figure 2.14. 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

The symbol Re(K ((o)) denotes the real part of the complex spectral density, corresponding to the autocorrelation of the dipolar interactions, while Re(i (co)) is its counterpart for the scalar interaction. The symbol Re(K (a>)) denotes the spectral density describing the cross-correlation of the two parts of the hyperfine interaction. The cross-correlation vanishes at the MSB level of the theory, but in the more complicated case of the lattice containing the electron spin, the cross term may be non-zero. A general expression for the dipolar spectral density is ... 

Fig. 3. Variation of the completely reduced dipole-dipole spectral density (see text) for the model of a low-symmetry complex for S = 3/2. Reprinted from J. Magn. Reson., vol. 59,Westlund, RO. Wennerstrom, H. Nordenskiold, L. Kowalewski, J. Benetis, N., Nuclear Spin-Lattice and Spin-Spin Relaxation in Paramagnetic Systems in the Slow-Motion Regime for Electron Spin. III. Dipole-Dipole and Scalar Spin-Spin Interaction for S = 3/2 and 5/2 , pp. 91-109, Copyright 1984, with permission from Elsevier.

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

An analytical theory of the outer-sphere PRE for slowly rotating systems with an arbitrary electron spin quantum number S, appropriate at the limit of low field, has been proposed by Kruk et al. (144). The theory deals with the case of axial as well as rhombic static ZFS. In analogy to the inner sphere case (95), the PRE for the low field limit could be expressed in terms of the electron spin spectral densities s ... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

A more general theory for outer-sphere paramagnetic relaxation enhancement, valid for an arbitrary relation between the Zeeman coupling and the axial static ZFS, has been developed by Kruk and co-workers (96 in the same paper which dealt with the inner-sphere case. The static ZFS was included, along with the Zeeman interaction in the unperturbed Hamiltonian. The general expression for the nuclear spin-lattice relaxation rate of the outer-sphere nuclei was written in terms of electron spin spectral densities, as ... 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

Figure 6.6 Two-state quantum system driven on resonance by an intense ultrashort (broadband) laser pulse. The power spectral density (PSD) is plotted on the left-hand side. The ground state 11) is assumed to have s-symmetry as indicated by the spherically symmetric spatial electron distribution on the right-hand side. The excited state 12) is ap-state allowing for electric dipole transitions. Both states are coupled by the dipole matrix element. The dipole coupling between the shaped laser field and the system is described by the Rabi frequency Qji (6 = f 2i mod(6Iti-...

There is now fairly good evidence from EPR/ENDOR on the S2 state MLS that the observed signal and the 55Mn hyperfine structure results from a single Miu cluster with electron spin density distributed over all 4 Mn ions.462 465 In this work 55Mn ENDOR experiments, which deliver more precise hfc and nqc values463 and spectral simulations,462-466 have been of great importance. Independent support for the model came from simulations of the NH3-modified MLS that showed a spin redistribution in the Mm cluster rather than spin transfer to other atoms.462-463... 

Exercise. A cathode emits electrons at independent random times. Derive for the spectral density of the current fluctuations... 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

S. Mukamel In order to represent situations in which nuclear and electronic dynamics take place on the same time scale, one needs to incorporate nuclear degrees of freedom into the description. A frequency-dependent Redfield superoperator can capture some effects, but in general is very limited and may even yield negative probabilities. A method for decomposing a given spectral density into a few collective coordinates and identifying these coordinates was presented in Ref. 1. 

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