Frequency matrix, mode coupling theory

The photoabsorption spectrum a(co) of a cluster measures the cross-section for electronic excitations induced by an external electromagnetic field oscillating at frequency co. Experimental measurements of a(co) of free clusters in a beam have been reported, most notably for size-selected alkali-metal clusters [4]. Data for size-selected silver aggregates are also available, both for free clusters and for clusters in a frozen argon matrix [94]. The experimental results for the very small species (dimers and trimers) display the variety of excitations that are characteristic of molecular spectra. Beyond these sizes, the spectra are dominated by collective modes, precursors of plasma excitations in the metal. This distinction provides a clear indication of which theoretical method is best suited to analyze the experimental data for the very small systems, standard chemical approaches are required (Cl, coupled clusters), whereas for larger aggregates the many-body perturbation methods developed by the solid-state community provide a computationally more appealing alternative. We briefly sketch two of these approaches, which can be adapted to a DFT framework (1) the random phase approximation (RPA) of Bohm and Pines [95] and the closely related time-dependent density functional theory (TD-DFT) [96], and (2) the GW method of Hedin and Lundqvist [97]. 

In analogy to Marcus theory, we also postulate that the nuclear degrees of freedom of the medium are too inert to change during the electron transfer step, i.e., the Born-Oppenheimer approximation can be applied for the nuclear degrees of freedom. Let us assume that the protein matrix coupled to electron transfer can be represented through N different oscillators where N is of the order of magnitude of the number of atoms in the protein, i.e., about 10 in the case of the photosynthetic reaction center. We denote the frequencies of these oscillators by a = 1,2,..., A and the associated vibrational coordinates by o = 1 2,..., AT. Let us assume further that each mode... 

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