Spectral function representation

The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

Representation of Spectral Function in Terms of Fourier Amplitudes... 

In accord with the planar libration approximation, we first come from representation of the spectral function for motion of a dipole in a plane, where integration over l is lacking by definition, so only integration over energy h is employed. We shall find in this way the function (203). As a next step we carry out integration over l, so that a rather simple expression (171) for the spectral function L(z) will be obtained. 

The representation (6.69) of e s indicates that this quantity includes contributions from all the resonance frequencies (C0 =P ) of the frequency dependent permittivity (6.68). Moreover it also is determined by some dissipative properties of the material reflected in the spectral function (,(p)) and also in the optical permittivity =n p,), [6.6, 6.29]. 

If such a truncated basis set i> is used in the actual calculations, the resulting spectral function p (e) has a Lehmann representation similar to (50) but with exact (JV — l)-electron eigenstates and eigenvalues replaced by the approximate eigenstates n,(iV - 1)> and eigenvalues E N — 1) obtained by diagonalizing H in the truneated spaee This representation shows that independently of the quality... 

The spectral function actually selected diagonal matrix elements Ann ( ) in a suitable one-electron basis representation - may exhibit well-defined structures reflecting the existence of highly probable one-electron excitations. Due to the Coulomb interaction, we cannot assign each excitation to an independent particle (electron or hole) added to the system with the excitation energy. Nonetheless, some of these structures can be explained approximately in terms of a particle-like behaviour, so having a quasiparticle (QP) peak. Where a second peak is required we may have what is called a satellite. 

One assumes here that the molecular Hamiltonian H is the same for all electronic states j, still the notation Hj with index j is useful to identify the electronic shell in which the wavepacket evolves. One can then also apply directly the general theory to nuclear degrees of freedom with the nuclear Hamiltonian depending on the electronic state j. A corresponding time-dependent representation for the RXS cross section (3.93) can be obtained by a Fourier transform of the spectral function,... 

Calculations of the lowest-lying attractive and repulsive states of the electron-pair bond (H2) illustrate the attributes of the formalism and the convergence achieved. In this case ritotai = 2), the spin functions factor out, there are no unphysical irreducible representations to contend with, and the development deals only with spatial functions which are symmetric (singlet) or antisymmetric (triplet) under electron transposition (6). The spectral-product representation spans these spatially symmetric and antisymmetric... 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

For a correlated N-electron system with a non-degenerate ground state > the one-particle Green s function has the spectral representation (20,21) ... 

Figure 1.15 Time domain representation of a hard rectangular pulse and its frequency domain excitation function. The excitation profile of a hard pulse displays almost the same amplitude over the entire spectral range.

Figure 1.16 Time domain representation and frequency excitation function of a soft pulse. The soft pulse selectively excites a narrow region of a spectral range and leads to a strong offset-dependent amplitude of the excitation function.

We may recall and emphasize that the autocorrelation function obtained in the three representations I, II, and III must be equivalent, from the general properties of canonical transformation which must leave invariant the physical results. Thus, because of this equivalence, the spectral density obtained by Fourier transform of (43) and (45) will lead to the same Franck-Condon progression (51). 

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