Correlation function calculation

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

The paper is organized as follows. In See.2 we consider the frustrated spin chain at F-AF transition point and describe the exact singlet ground-state wave function as well as details of the spin correlation function calculations. We discuss the phase diagram of this model and its magnetic properties in the AF phase. In Sec.3 the special spin ladder will be considered. A two-dimensional frustrated spin model with the exact ground state is considered in Sec.4. Sec. 5 is devoted to the construction of the electronic models with the SB type of wave function. The results of this paper are summarized in Sec.6. 

Comparisons of the correlation functions calculated quantum mechanically and semiclassically, like those presented in Fig. 6.2, show that the correction due to the dipole moment gradient included in (6.34) sometimes improves the accuracy especially for short propagation times. This correction affects not only the amplitude of the correlation function oscillation, but also its frequency and distortions due to the presence of high harmonics in the spectrum. An analysis of the spectrum of the correlation function indicates that including this correction in the formula enables additional quantum effects to be taken into account. 

Formulae (366)—(368) show that higiher-order correlations modify considerably the shape of the correlation functions, calculated on the basis of dipole interaction alone. This enables us to achieve better agreement with the experimental data for nitrobenzme and benzene."" ... 

Spectra of other dynamical variables like CoM velocities and zeolite window diameters can also be obtained by Fourier transformation of the appropriate time correlation functions. Calculation of spectra for different spatial components of a time dependent quantity provides useful information about the anisotropy of the corresponding motion. 

This method was used to determine the structure of a number of PE-contain-ing diblocks. An example of a correlation function calculated for a PE-poly(ethyl ethylene) diblock, together with its interpretation in terms of amorphous and crystalline layer thicknesses is shown in Fig. 2. The PE crystal thickness in these... 

The scattered light is detected by a photomultipler tube (PMT). The area that is seen by the PMT encloses the spot of the laser beam on the film. The PMT signal is fed either to a correlator or to a spectrum analyzer. The correlator consists of a fast analogue-to-digital converter and a minicomputer, programmed to calculate the time autocorrelation function of the signal. The correlation function, calculated in this way, is proportional to the time autocorrelation function of the fluctuations in the scattered intensity. 

Most macro molecules in solution are neither as stiff as the rigid rod nor as flexible as the Gaussian coil. For particular systems of interest a dynamical model should be made and the corresponding spectrum (or time correlation function) calculated. Measurement of the spectrum and a fit to the theoretical form then allows extraction of the model dynamic constants. These dynamic constant may then be related to equilibrium structural properties of the molecule (end-to-end distances, backbone curvature, etc.). 

Figure 5.24 The correlation functions calculated for lamellar structures in which the thickness of the lamellae varies according to a Gaussian distribution function. The solid curve is based on the model that gave the solid intensity curve in Figure 5.22, and the broken curve here matches the broken curve in Figure 5.22.

Figure 7. Plot of the real-time position correlation function calculated by the extended Lagrangian CMD method of Section III.C.2 for the three-dimensional potential given in Eq. (3.86) at a temperature of /3 = 5. Also shown is the classical MD result.

The subscripts V , C , A , E , and S denote water , cation , anion , salt solution , and solute , respectively. It is assumed that the solute molecules are present at infinite dilution. The calculation process is then split into two steps where bulk salt solution (step 1) and salt solution near a solute molecule (step 2) are treated, respectively. The site-site intermolecular total correlation functions calculated in step 1 are used as input variables for step 2. The calculation in step 1 is performed using the basic equations and the hybrid algorithm described in 1.1. We consider step 2 hereafter. It is assumed that the solute molecule, a water molecule, a cation, and an anion have m, 3, 1, and 1 interaction sites, respectively. The SSOZ equation is expressed as... 

Among the most useful quality tests are those for self-consistency, which determine whether the pair correlation functions calculated using a particular approximation method give identical results when used to calculate a certain thermodynamic function by different routes. For example, in an ionic solution calculation we can apply the virial-compressibility consistency test as follows From the pair correlation functions, we calculate the osmotic coefficient 4> ... 

Fig. 4 The a,/ -two-point pair correlation functions calculated with DQG and DQGT constraints are given for the (a) ladder 2 x 10 and (b) linear 1 x 10 lattices at half filling (n) = 1...

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