Dynamically expanding context

ACD = Available Chemicals Directory CAS = Chemical Abstracts Service CSD = Cambridge Structural Database DEC = dynamically expanding context FA = factor analysis LVQ = learning vector quantization MDS = multidimensional scaling NNT = nearest-neighbor table PCA = principal component analysis QSAR = quantitative structure-activity relationships SOM = self-organized map. 

Equations (4.329) for a solid assembly and (4.332) for a magnetic suspension are solved by expanding W with respect to the appropriate sets of functions. Convenient as such are the spherical harmonics defined by Eq. (4.318). In this context, the internal spherical harmonics used for solving Eq. (4.329) are written Xf (e, n). In the case of a magnetic fluid on this basis, a set of external harmonics is added, which are built on the angles of e with h as the polar axis. Application of a field couples [see the kinetic equation (4.332)] the internal and external degrees of freedom of the particle so that the dynamic variables become inseparable. With regard to this fact, the solution of equation (4.332) is constructed in the functional space that is a direct product of the internal and external harmonics ... 

Time-independent approaches to quantum dynamics can be wxriational where the wavefunction for all coordinates is expanded in some basis set and the parameters optimized. The best knowm variational implementation is perhaps the S-matrix version of Kohn s variational prineiple which was introduced by Miller and Jansen op de Haar in 1987[1]. Another time-independent approach is the so called hyperspherical coordinate method. The name is unfortunate as hyperspher-ical coordinates may also be used in other contexts, for instance in time-dependent wavepacket calculations [2]. 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

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