Close-coupling method time-dependent

In this section we outline the coupled cluster-molecular mechanics response method, the CC/MM response method. Ref. [51] considers CC response functions for molecular systems in vacuum and for further details we refer to this article. The identification of response functions is closely connected to time-dependent perturbation theory [51,65,66,67,68,69,70], Our starting point is the quasienergy and we identify the response functions as simple derivatives of the quasienergy. For a molecular system in vacuum where Hqm is the vacuum Hamiltonian for the unperturbed molecule and V" is a time-dependent perturbation we have the following time-dependent Hamiltonian, H,... 

If the coupling is zero, the bound states will live forever. However, immediately after we have switched on the coupling they start to decay as a consequence of transitions to the continuum states until they are completely depopulated. Our goal is to derive explicit expressions for the depletion of the bound states l iz) and the filling of the continuum states 2(E,0)). The method we use is time-dependent perturbation theory in the same spirit as outlined in Section 2.1, with one important extension. In Section 2.1 we explicitly assumed that the perturbation is sufficiently weak and also sufficiently short to ensure that the population of the initial state remains practically unity for all times (first-order perturbation theory). In this section we want to describe the decay process until the initial state is completely depleted and therefore we must necessarily go beyond the first-order treatment. The subsequent derivation closely follows the detailed presentation of Cohen-Tannoudji, Diu, and Laloe (1977 ch.XIII). 

The set of coupled equations (15.11) represents an example of time-dependent close-coupling as described in Section 4.2.3. It is formally equivalent to (4.25), for example, and can be solved by exactly the same numerical recipes. The dependence on the two stretching coordinates R and r is treated by discretizing the two nuclear wavepackets on a two-dimensional grid and the Fourier-expansion method is employed to evaluate the second-order derivatives in R and r. If we additionally include the rotational degree of freedom, we may expand each wavepacket in terms of... 

Both close-coupling approaches (h q)erspherical or with absorbing potentials) and iterative/time-dependent absorbing-potential arrangement-decoupling approaches are readily extended to three-dimensional atom-molecule and molecule-molecule scattering. The wavefunction representation becomes more complicated and includes rotational matrices, but the essence and application of the method remains analogous [58. 65. 75. 76]. 

Different theoretical methods have been used to calculate the complex energies, Eq. (8.1), for compound-state resonances. They can be divided into time-independent and time-dependent methods. A standard quantum mechanical time-independent method is a close-coupling calculation (Stechel et al., 1978) which considers resonant state formation as a result of a collision such as A + BC —> ABC AB + C. Determined... 

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]. 

---