Spectral density computation

Here C is the gas phase (uncoupled) flux autocorrelation function, Zbath is the bath partition function, J(co) is the bath spectral density (computed as described above from a classical molecular dynamics computation), Bi and B2 are combinations of trigonometric functions of the frequency a> and the inverse barrier frequency, and Anally ... 

Application of this methodology to this model of horse liver alcohol dehydrogenase yields the results shown in Fig. 9.4. In fact we do see strong numerical evidence for the presence of a promoting vibration - intense peaks in the spectral density for the reaction coordinate are greatly reduced at a point between the reactant and product wells. This is defined as a point of minimal coupling. As we have described, the restraint on the hydride does not impact the spectral density computation. This computation measures the forces on the reaction coordinate, not those... 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

Fig. 8. Spectral densities for BPTI as computed by cosine Fourier transforms of the velocity autocorrelation function by Verlet (7 = 0) and LN (7 = 5 and 20 ps ). Data are from [88].

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. 

The eigenvalues o>(r and the four sets of scalar products may be computed by full diagonalization of the Hamiltonian Hgi. Then, the spectral density of a medium-strength H bond involving a Fermi resonance is... 

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.

We kept the same structure in Figs. 8(a) and 8(b), but the spectra were computed for a greater value a0 = 1.2. As may be seen, some differences between the adiabatic and nonadiabatic spectral densities appear in all cases, whether applying the exchange approximation or not. Within the exchange approximation, Fig. 8(b), these discrepancies may be safely attributed to the... 

Looking at (110), it appears that the spectral density /Sf(oo, aG = 0) ex is composed of two sub-bands, as is also shown by the four sample spectra of Fig. 10 which were computed for various parameters A and A. The frequency and intensity of these two sub-bands do not depend on the temperature, and are similar to those which may be obtained within the simpler undamped treatment. Solving the eigenvalue problem (111), we obtain... 

These behaviors may be observed in Fig. 10(b), which displays three spectra computed with an increasing parameter A and A = —120 cm-1. We shall see in the following that beyond the exchange approximation the spectral density behaves in an opposite way. 

Note that the spectral densities (a) to (e) were computed from the same expression (103) of If, by zeroing some of the physical parameters. The undamped spectra (f) to (j) were computed from the following expression ... 

Figure 13. Hydrogen bond involving a Fermi resonance relative influence of the damping parameters. Spectral densities 7sf(co) computed from Eq. (81). Common parameters a0 = 1, A = 160cm-1, co0 = 3000cm-1, co00 = 150cm-1, 2t05 = 2790cm-1, and T = 300K.

These conclusions must be considered keeping in mind that the general theoretical spectral density used for the computations, in the absence of the fast mode damping, reduces [8] to the Boulil et al. spectral density and, in the absence of the slow mode damping, reduces to that obtained by Rosch and Ratner one must also rember that these two last spectral densities, in the absence of both dampings [8], reduce to the Franck-Condon progression involving Dirac delta peaks that are the result of the fundamental work of Marechal and Witkowski. Besides, the adiabatic approximation at the basis of the Marechal... 

The expansion method described above enables one to compute the spectral density of fluctuations in successive orders of O 1, provided the Master Equation is known.14 In the linear case, however, it was sufficient to know the macroscopic equation and the equilibrium distribution, as... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

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... 

Starting from the assumption that the only known information concerning the line shape is a finite number of spectral moments, a quantity called information is computed from the probability for finding a given spectral component at a given frequency this quantity is then minimized. Alternatively, one may maximize the number of configurations of the various spectral components. This process yields an expression for the spectral density as function of frequency which contains a small number of parameters which are then related to the known spectral moments. If classical (i.e., Boltzmann) statistics are employed, information theory predicts a line shape of the form... 

Interest in spectral hole burning comes from its potential application as high density computer memory. An external electric field can be used to shift the spectral holes so as to obtain an even larger number of detectable holes than with the excitation wavelength alone. 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

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.

All of these direct NOE penalty functions have several features in common. First, they are all, in principle, more correct than refinement against calculated distances. Unfortunately, they vary from moderately to extremely costly in terms of CPU time. Each implementation also raises the question of the nature of the spectral density function used to model the molecule s motions. In each case, except that of Baleja et al.,66 isotropic motion was assumed, although all the methods could be readily adapted to use more elaborate spectral density functions. As experience with these methods accumulates, it should become clear when the extra computational time is worth investing and what the effects of different motional models are. 

Much effort has been put into the explanation of the spectral line shapes but it seems that the definite theory has yet to be established. In the meantime one can extract useful information from the first few moments of the spectral density, by applying the elegant theory developed by Van Kranendonk and Poll and Van Kranendonk This theory relates the first moment to the derivative of the dimer dipole moment with respect to the intermolecular distance. The zeroth moment yields information about the square of the dipole moment. As this review is not the place to go extensively into the Van Kranendonk theory, we only note that, once the intermolecular potential surface and the interaction dipole field are known, — for instance by ab initio calculations — it is relatively easy to compute the moments of the spectral density. Since these are directly observable, the experiment of pressure induced absorption may serve as a check on the correctness of ab initio calculations, not only of the interaction energy, but also of the interaction dipole. 

Here we apply the LAND-map approach to compute of the time dependent average population difference, A t) = az t)), between the spin states of a spin-boson model. Here az = [ 1)(1 — 2)(2 ]. Within the limits of linear response theory, this model describes the dissipative dynamics of a two level system coupled to an environment [59,63-65]. The environment is represented by an infinite set of harmonic oscillators, linearly coupled to the quantum subsystem. The characteristics of the system-bath coupling are completely described by the spectral density J(w). In the following, we shall restrict ourselves to the case of an Ohmic spectral density... 

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