Spectral density comparisons

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

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. 

Figure 8. Comparison between the adiabatic spectral density and the standard one (with or without the exchange approximation), (a) and (c) display the spectral density Isf from Eq. (81), using dashed lines, (b) and (d) display the spectral density /sf ex from Eq. (88), within the exchange approximation, using dashed lines. Comparison is made with the adiabatic spectral density If (thin plain lines) obtained from (96). Spectra (a) and (b) are computed with Oo = 1.2, whereas spectra (c) and (d) are computed with a0 = 0.6. Common parameters A = 120cm-1, (n0 = 3000cm-1, C05 = 1440 cm-1, co00 = 150 cm-1, y0 = y5 = 60 cm-1, and T = 300 K.

Equations (1-3) are widely used for protein dynamics analysis from relaxation measurements. The primary goals here are (A) to measure the spectral densities J(co) and, most important, (B) to translate them into an adequate picture of protein dynamics. The latter goal requires adequate theoretical models of motion that could be obtained from comparison with molecular dynamics simulations (see for example Ref. [23]). However, accurate analysis of experimental data is an essential prerequisite for such a comparison. 

Fig. 12. Comparison of longitudinal and transverse cross-relaxation rates as a function of the measurement frequency (vq) for a pair of protons and a normalized spectral density of the form 2tc/(1 -I- (see Eqs. (58) and (59)).

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

One can also compare the spectral densities for the three models as was done by Bendler and Yaris for their model relative to the VJGM model (22). The comparison of the spectral densities produced by the three models using the parameters given in Table I... 

These relations between spectral densities and experiments furnish only the formal framework for a comparison of theory and experiment. The most difficult step still remains How can one evaluate the relevant correlation functions and spectral densities from a theoretical microscopic... 

Comparison with measurement. Measurements of the absorption of rare gas mixtures exist for some time. This fact has stimulated a good deal of theoretical research. A number of ab initio computations of the induced dipole moment of He-Ar are known, including an advanced treatment which accounts for configuration interaction to a high degree see Chapter 4 for details. Figure 5.5 shows the spectral density profile computed... 

Fig. 4.5. Plot of the spectral density functions of Eqs. (4.46) and (4.47) as a function of magnetic field (expressed as proton Larmor frequency log scale), rp = 2 x 10-9 s. The Solomon profiles obtained for xc = 2 x 10 9 s are also reported (dotted lines) for comparison purposes.

Comparison of the NHDH and CEL Spectral Densities with the Reference Ones [71]... 

Figure 6. Comparisons of the spectral densities according to the quantum and different semiclassical models. co° = 3000cm-1, El = 100cm-1, a° = 1, T = 300K, y° = 0.200 for all situations there is superposed the same SD computed by aid of eq. 126 with y = 0, grayed, y = 0.80 n.

Figure 24. Effects of temperature and isotopic substitution on the spectral densities of crystalline adipic acid in the absence of Fermi resonance. Comparison between theoiy (Eq. (279)) (thick Line) and experiment [101] (grayed spectra). Left column calculations using the breaking of the IR selection rule (r)° = 0). Right column same calculations but without the breaking of the IR selection rule (r 0 = 0).

More detailed information can be obtained from noise data analyzed in the frequency domain. Both -> Fourier transformation (FFT) and the Maximum Entropy Method (MEM) have been used to obtain the power spectral density (PSD) of the current and potential noise data [iv]. An advantage of the MEM is that it gives smooth curves, rather than the noisy spectra obtained with the Fourier transform. Taking the square root of the ratio of the PSD of the potential noise to that of the current noise generates the noise impedance spectrum, ZN(f), equivalent to the impedance spectrum obtained by conventional - electrochemical impedance spectroscopy (EIS) for the same frequency bandwidth. The noise impedance can be interpreted using methods common to EIS. A critical comparison of the FFT and MEM methods has been published [iv]. 

The interpretation of relaxation data is most often performed either with reduced spectral density or the Lipari-Szabo approach. The first is easy to implement as the values of spectral density at discrete frequencies are derived from a linear combinations of relaxation rates, but it does not provide any insight into a physical model of the motion. The second approach provides parameters that are related to the model of the internal motion, but the data analysis requires non-linear optimisation and a selection of a suitable model. A graphical way to relate the two approaches is described by Andrec et al Comparison of calculated parametric curves correlating 7h and Jn values for different Lipari-Szabo models of the internal motion with the experimental values provides a range of parameter values compatible with the data and allows to select a suitable model. The method is particularly useful at the initial stage of the data analysis. 

Figure 26. Comparison for the absorption spectrum and three-pulse-echo peak shift determined from experiment for the B850 band of LH2 with that calculated using the model described in the text. Parameters are the same as in Fig. 23 right panel), plus a spectral density (see Ref. 40).

Figure 17. Comparison between exact spectral densities (-----) and asymptotic...

Figure 1 Comparison between the experimental and calculated reduced spectral densities. The two spectra are normalized with respect to the intensities of their lowest frequency peak.

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

Figure 3.2 refers to displacement measurements and shows a comparison among (i) the theoretical spectral density function Sy corresponding to the continuous process y = x (solid line). According to Equation (3.5), this is given by ... 

Comparison of Spectral Density Approach and Time-domain Approach... 

Chapter 3 and Chapter 4 presented the Bayesian spectral density approach and Bayesian time-domain approach. The comparison can be summarized as follows ... 

Even though the spectral density approach requires computation of the inverse and determinant of a number of matrices, the size of these matrices is only No y. No. They are significantly smaller than the NgNp x NgNp matrix Eyj j (= E22) required in the time-domain approach. Comparison of the computational efficiency between the two methods depends on the number of the elements in the frequency index set and the number of data points in a fundamental period. The ratio of the computations required by the Bayesian spectral density approach and the Bayesian time-domain approach can be approximated by ... 

It can be seen that in comparison to Equation [1.56] the spectral density for the output sliver will have a periodic component and a phase shift ... 

Fig. 1. Comparison of measured and calculated ratios of spectral densities for a S%- U-enriched uranyl fluoride solution experiment.

Figure 1. The bath spectral density function J oj) [Eq. (2.29)] plotted in terms of J (ju)/ r]uJc) (solid) as the function of (jojujc. Included for comparison are also the corresponding curve in dash from the Ohmic Johm(c< ) =. ...

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