Spectral density coupling

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

More pertinent to the present topic is the indirect dissipation mechanism, when the reaction coordinate is coupled to one or several active modes which characterize the reaction complex and, in turn, are damped because of coupling to a continuous bath. The total effect of the active oscillators and bath may be represented by the effective spectral density For instance, in the... 

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

Dealing with the restrictive situation of equal dampings, where yQ = y8 = y, the spectral density (107) may be written as the limit of /sf (oj) ex [Eq. (88)] when neglecting the anharmonic coupling—that is, when aG = 0 ... 

Then, neglecting the anharmonic coupling ac, the spectral density 7sf [Eq. (81)] reduces to... 

Figure 10. Pure Fermi coupling within or beyond the exchange approximation. Left column spectra were obtained from expression (110) of the spectral density /sf (m, V. = 0) ex (a) Resonant case A — 0. A — 60 cm-1 (b) nonresonant case A — 120cm 1 with A — 60 cm-1 (dotted line),...

Owing to the above remarks, we shall study in the following the features of the lineshapes of weak H bonds in which there is a dephasing of the fast mode and simultaneously a damping of the slow mode to which the fast one is anharmo-nically coupled. The quantum spectral density is then [96]... 

Figure 15 gives the superposition of RR (full line) and RY (dotted plot) spectral densities at 300 K. For the RR spectral density, the anharmonic coupling parameter and the direct damping parameter were taken as unity (a0 = 1, y0 = ffioo), in order to get a broadened lineshape involving reasonable half-width (a = 1 was used systematically, for instance, in Ref. 72). For the RY spectral density, the corresponding parameters were chosen aD = 1.29, y00 = 0.85angular frequency shift (the RY model fails to obtain the low-frequency shift predicted by the RR model) and a suitable adjustment in the intensities that are irrelevant in the RR and RY models. 

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]... 

Ky represent the frequency and the linear coupling of the yth vibrational mode, respectively. Here, we consider specifically an Ohmic bath described by the spectral density... 

The quantity bk has a physical meaning as the spectral density of photons emitted by the dressed excited particle, and its explicit form is rather complicated [11]. However, for weak coupling case it reduces to a simple form ... 

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 ... 

In the presence of both order-disorder and displacive, as in the KDP family, the two dynamic concepts have somehow to be merged. It could well be that the damping constant Zs becomes somewhat critical too (at least in the over-damped regime of the soft mode), because of the bihnear coupling of r/ and p. It would, however, lead too far to discuss this here in more detail. The corresponding theory of NMR spin-lattice relaxation for the phase transitions in the KDP family has been worked out by Blinc et al. [19]. Calculation of the spectral density is here based on a collective coordinate representation of the hydrogen bond fluctuations connected with a soft lattice mode. Excellent and comprehensive reviews of the theoretical concepts, as well as of the experimental verifications can be found in [20,21]. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

For analytic purposes, it is usefiil to define a spectral density of the bath modes coupled to the reaction coordinate in a given frequency range ... 

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-...

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