Static expansion method

When time resolution is required, then it is time-dependent probabilities rather than rates (= probability per unit of time) that have to be computed. For the case of the static field, the SSEA has not been applied yet. However, for the one-electron hydrogen atom, the problem of the time-resolved evolution of its ground and its first excited states was tackled a few years ago, first by Geltman [25], who used an expansion method (the TDSE turns into a set of coupled equations), with a box-normalized discretized continuum, and then, more extensively, by Durand and Paidarova [26], who approached the problem in terms of Eq. (6c) and the complex coordinate rotation method. In addition to their numerical results, the publications [25, 26] contain interpretations and critical remarks on aspects of the dynamics. 

An alternative approach may be made without using a differential equation, but treating the problem completely statically. Such an approach was first made by Yamamoto and Teramoto as early as 1952 3). These authors used the Ursell-Mayer expansion method and evaluated the first order coefficient which agreed with Grimley s result. This method has been pursued by Yamakawa and Kurata and also by Fixman. 

The foregoing designs were discussed as ring expansion joints by Kopp and Sayre, Expansion Joints for Heat Exchangers (ASME Misc. Pap., vol. 6, no. 211). All are statically indeterminate but are subjected to analysis by introducing various simplifying assumptions. Some joints in current industrial use are of lighter wall constniction than is indicated by the method of this paper. 

Pyroelectricity of several kinds of alternating LB films consisting of phenylpyrazine derivatives and stearic acid was measured by the static method at various temperatures. Effects of thermal expansion and molecular packing density of the film on pyroelectricity were also examined. The following conclusions were derived. 

As we shall have occasion to note in dealing with solutions, the composition of the surface phase is very different from that of the bulk liquid. When a liquid interface is newly formed the system is unstable until the surface phase has acquired its correct excess or deficit of solute by diffusion from or into the bulk of the solution. This process of diffusion is by no means instantaneous and, as has been observed in discussing the drop weight method, several minutes may elapse before equilibrium is established. In the ripple method the surfece is not renewed instantaneously but may be regarded as undergoing a series of expansions and contractions, thus we should anticipate that the value of the surface tension of a solution determined by this method would lie between those determined by the static and an ideal dynamic method respectively. 

The PPD and shell models are nearly equivalent in this sense, because they model the electrostatic potential via static point charges and polarizable dipoles (of either zero or very small extent). Accuracy can be improved by extending the expansion to include polarizable quadrupoles or higher order terms.The added computational expense and difficulty in parameterizing these higher order methods has prevented them from being used widely. The accuracy of the ESP for dipole-based methods can also be improved by adding off-atom dipolar sites. 

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit w— 0. /3 values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B ) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth phenomenological convention (X) is converted to the T convention by multiplication by a factor of 4. 

Solution of the Kohn-Sham equations as outlined above are done within the static limit, i.e. use of the Born-Oppenheimer approximation, which implies that the motions of the nuclei and electrons are solved separately. It should however in many cases be of interest to include the dynamics of, for example, the reaction of molecules with clusters or surfaces. A combined ab initio method for solving both the geometric and electronic problem simultaneously is the Car-Parrinello method, which is a DFT dynamics method [52]. This method uses a plane wave expansion for the density, and the inner ions are replaced by pseudo-potentials [53]. Today this method has been extensively used for studies of dynamic problems in solids, clusters, fullerenes etc [54-61]. We have recently in a co-operation project with Andreoni at IBM used this technique for studying the existence of different isomers of transition metal clusters [62,63]. 

The basic idea of the dynamic pressurization (DP) experiment is similar to the dynamic gas expansion (DGE) method. Both use a sudden pressure change around a gel specimen to initiate gas flow into or out of the sample, thus avoiding the delicate leakage problems typically encountered in static gas flow setups. While DGE monitors the gas pressure outside the gel as a function of time and deduces the permeability from its equilibration behavior (in principle a dynamic pycnometry experiment). DP utilizes the dynamics of the elastic deformation of the gel to deduce both elastic modulus and permeability. The deformation, or strain, is a consequence of the pressure difference between the interior and the exterior of the specimen. For example, after a sudden increase in pressure, the gas in the gel pores is initially only slightly compressed along with the elastic compression of the gel. After a characteristic time, the pressure equilibrates and the gel ideally springs back to its original dimensions. 

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