Spectral functions independency

We note that in other work, some of the frequency-independent factors appearing in the equation may be included in the definition of the spectral function, g(v). In such a case, the spectral function may be scaled relative to our definition and units may differ. 

H2-H2 rototranslational spectra. For the significant A1A2AL induction components, Table 4.11, values of the various spectral functions have been computed at frequencies from 0 to 1800 cm-1 and for temperatures from 40 to 300 K, Fig. 6.3 [282]. As a test of these line shape computations, the zeroth, first and second spectral moments have been computed in two independent ways by integration of the spectral functions with respect to frequency, Eq. 3.4, and also from the quantum sum formulae, Eqs. 6.13, 6.16, and 6.21. Agreement of the numerical results within 0.3% is observed for the 0223, 2023 components, and 1% for the other less important components. This agreement indicates that the line shape computations are as accurate as numerical tests with varying grid widths, etc., have indicated, namely about 1% see Table 6.2 as an example (p. 293). 

From Eq. (C.2) we conclude that the square integrable solution / contains the independent solution spectral function p(E) to be used in the completeness relation and the eigenfunction expansion. The former gives... 

Now we want to consider some general properties of interacting systems. In equilibrium the lesser function is not independent and is simply related to the spectral function by the relation... 

The equation (386) is not closed again and produces new Green functions of higher order. And so on. These sequence of equations can not be closed in the general case and should be truncated at some point. Below we consider some possible approximations. The other important point is, that average populations and lesser Green functions should be calculated self-consistently. In equilibrium (linear response) these functions are easy related to the spectral functions. But at finite voltage they should be calculated independently. 

For valence structure this range is up to about 50 eV. The momentum distribution is observed for each resolved cross-section peak corresponding to an ion eigenvalue —cf. In order to characterise the observation of the target—ion structure we choose a quantity that is as independent as possible of the probe characteristics such as total energy. In conditions where the plane-wave impulse approximation is valid we consider the reaction as a perfect probe for the energy—momentum spectral function... 

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. 

Of course this distinction is somewhat arbitrary, but a way of doing it is the following. Let us suppose that we switch off the interaction, so having a system of independent particles whose eigenstates can be described using one-electron orbitals 4>j ( ) with eigenenergies j. In this case, the matrix elements of the spectral function in the orbital basis set are... 

As long as one can assume that the Hamiltonian is temperature independent, e.g., that the protein structure and, hence, the coupling terms as well as the spectral function J uj) do not change with temperature, one can expect that the classical simulations allow one to determine a suitable quantum mechanical model. For this purpose, one carries out a classical simulation at high temperature, characterizes J uj) corresponding to the simulated AE t) and employs the the resulting J uj) at all temperatures. Since at physiological temperatures (T = 300 K), the majority of frequencies of modes satisfy the property huJa/ksT << 1 one can assume the classical limit to be realized at T =... 

One way to reduce the number of independent variables in the FRET-adjusted spectral equation is to use samples with a fixed donor-to-acceptor ratio. Under these conditions, the values of d and a are no longer independent, but rather the concentration of d is now a function of a and vice-versa. This approach is typical for the situation of FRET-based biosensor constructs. These sensors normally are designed to have a donor fluorophore attached to an acceptor by a domain whose structure is altered either as a result of a biological activity (such as proteolysis or phosphorylation), or by its interaction with a specific ligand with which it has high affinity. In general, FRET based biosensors have a stoichiometry of one... 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

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