Approximation, Gaussian

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... 

Part B. Repeat the entire process of Part A using a Gaussian approximation to the wave function for the ground state of the hydrogen atom... 

Eirst decide what the integral equation corresponding to Eq. (6-29) is for the approximate wave function (6-30), then integrate it for various values of y. Report both Y at the minimum energy and Cmin for the Gaussian approximation function. This is a least upper bound to the energy of the system. Your report should include a... 

Figure 8-2 The li Orbital Shape and a Gaussian Approximation (dotted line).

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

In the Gaussian approximation the water-water structure factor iS vro (A ) for (1) is given by... 

FIG. 12 The behavior of the internal energy U (per site), heat capacity Cy (per site), the average Euler characteristic (x) and its variance (x") — (x) close to the transition line and at the transition to the lamellar phase for/o = 0. The changes are small at the transition and the transition is very weakly first-order. The weakness of the transition is related to the proliferation of the wormhole passages, which make the lamellar phase locally very similar to the microemulsion phase (Fig. 13). Note also that the values of the energy and heat capacity are not very much different from their values (i.e., 0.5 per site) in the Gaussian approximation of the model [47]. (After Ref. 49.)... 

The field t is not coupled to any other fields and plays no role in the formation of ordered structures [15] within the Gaussian approximation. 

Having determined the effect of the diffusive interfaces on the structure parameters, we now turn to the calculation of H and K in microemulsions. In the case of oil-water symmetry three-point correlation functions vanish and = 0. In order to calculate K from (77) and (83) we need the exphcit expressions for the four-point correlation functions. In the Gaussian approximation... 

McWeeny, R., Nature 166, 21, Gaussian approximation to wave functions. ... 

The traditional analysis has then used a Gaussian to approximate both the Airy disk and sinc distribution as in Fig. 6 (Primot et al., 1990 Welsh and Gardner, 1989). The value of a in Eq. 41 is determined by matching the Gaussian approximation and the actual intensity at both the peak and the exp[—1] points (Welsh and Gardner, 1989). When this is done we find... 

An interesting result occurs when instead of using the Gaussian approximation the Airy disk is used directly in the calculations. In this case the variance goes to infinity,... 

Figure 5. Simulation of photon arrival for 100 photons for (a) the Airy disk and (b) the Gaussian approximation.

Determine the copolymer composition for a styrene-acrylonitrile copolymer made at the azeotrope (62 mol% styrene). Assume = 1000. One approach is to use the Gaussian approximation to the binomial distribution. Another is to synthesize 100,000 or so molecules using a random number generator and to sort them by composition. 

Theoretical treatment of the viscosity-concentration relationship for polyelectrolyte solutions would involve both the cumbersome statistics of highly elongated chains beyond the range of usefulness of the Gaussian approximation and the even more difficult problem of their electrostatic interactions when highly charged. There appears to be little hope for a satisfactory solution of this problem from theory. Fuoss has shown, however, that experimental data may be handled satisfactorily through the use of the empirical relation ... 

The same parameter for the unperturbed state. Corresponding parameters for the distribution of chain segments about the center of gravity of the molecule, in the Gaussian approximation. 

On the other hand concentration detector (DRI) data represent at least 10% of the maximum on the low molecular weight side of the chromatogram (Results), and the Gaussian approximation is probably applicable. 

Gaussian approximation, heat flow, 60 Gaussian distribution, transition state trajectory, white noise, 206-207 Gaussian-Markov process, linear... 

Gaussian approximation, 61 Monte Carlo simulations, 67-81 dynamics, 75-81 Metropolis algorithms, 70-71 nonequilibrium molecular dynamics, 71-74 structure profiles, 74-75 system details, 67-70... 

According to Benoit and Hadziioannou [148] the partial structure factors of a f functional star polymer with diblock arms of A- B-type in the Gaussian approximation are given by... 

Figure 1 Distributions for the end-to-end distance of a PDMS chain having n = 20 skeletal bonds of length / = 1.64 A. The Fixman-Alben distribution (dotted curve) and that from a Monte Carlo simulation (solid curve) are compared with the Gaussian approximation (dashed curve).

We first require some assumptions. Let the cubes into which the elastomer is decomposed be large relative to the dimensions of a single chain, so that edge and corner effects are negligible. Furthermore, the chains will be represented in the Gaussian approximation, so that for any stretch of a chain containing n segments, the end-to-end separation is described by the probability distribution... 

---