Expansion method

Altschuler M.D and Herman G. Fully three-dimensional image reconstruction using series expansion methods., A Review of Information Processing in Medical Imaging, Oak Ride National Lab., Oak Ride, TN, 1977, p.124-142. 

The above expansion of the full N-eleetron wavefunetion is termed a "eonfiguration-interaetion" (Cl) expansion. It is, in prineiple, a mathematieally rigorous approaeh to expressing F beeause the set of aU determinants that ean be formed from a eomplete set of spin-orbitals ean be shown to be eomplete. In praetiee, one is limited to the number of orbitals that ean be used and in the number of CSFs that ean be ineluded in the Cl expansion. Nevertheless, the Cl expansion method forms the basis of the most eommonly used teehniques in quantum ehemistry. 

IV. RECENT DEVELOPMENTS CONCLUDING REMARKS A. Survey of the Expansion Methods... 

TABLE X. Different Expansion Methods and Extended Hartree-Fock Schemes g = l+ar12... 

It is essential for the success of the expansion methods, that they are based on sets which are complete in a mathematical sense. This... 

Bohm-Pines" plasma model seems at first sight to be very different from the expansion methods treated here, but in Section III.E(2b) it was shown that it is rather closely connected with the method using correlation factor. Similar to Wigner s formula, it probably gives reasonable values for the correlation energy for the... 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

Kubo R. Generalized cumulant expansion method. J. Phys. Soc. Japan. 

Fortunately the Fourier expansion method helps us for small and intermediate h but large n. We get in the limit n - oo... 

Where the vessel wall will be at a significantly higher temperature than the skirt, discontinuity stresses will be set up due to differences in thermal expansion. Methods for calculating the thermal stresses in skirt supports are given by Weil and Murphy (1960) and Bergman (1963). 

Rabinowitz, J. R., K. Namboodiri, and H. Weinstein. 1986. A Finite Expansion Method for the Calculation and Interpretation of Molecular Electrostatic Potentials. Int. J. Quant. Chem. 29,1697. 

Church-Rosser property. That is, if and Eg are expressions derived from an expression E by alternative expansion methods, then there is an expression Eg which can be derived from both and Eg (of course, Eg might be either or Eg ). In particular, as long as the inside-out restriction is maintained the order of expansion of functional terms cannot affect the answer. So we shall arbitrarily select whatever expansion method seems most convenient at the moment usually we shall expand from left to right, always expanding the leftmost defined function letter whose inner terms are all terminal. 

The isentropic expansion method assumes that the gas expands isentropically from its initial to final state. The following equation represents this case ... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

Here we have introduced an auxiliary expansion parameter 5 to be set equal to unity after calculations, similar to 5 expansion method (de Souza,2002).The first order term AVq b) in this equation will... 

Friedman, H. L., Ionic Solution Theory Based on Cluster Expansion Methods, Interscience Publishers, New York, 1962. 

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

In Section II, we summarize the ideas and the results of Bogolubov,3 Choh and Uhlenbeck,6 and Cohen.8 Bogolubov and Choh and Uhlenbeck solved the hierarchy equations and derived two- and three-body generalized Boltzmann operators Cohen used a cluster expansion method and obtained two-, three-, and four-body explicit results which he was able to extend to arbitrary concentrations. 

The Variation Principle is the main point of departure all questions of symmetry, approximation etc. are judged from the point of view of their likely effect on the variational form of the Schrodinger equation. We attempt to take the minimal basis AO expansion method as far as possible while remaining within a family of well-defined conceptual models of the electronic structure which is theoretically and numerically underpinned by the variation principle. 

Everything we have said in this section applies to the integro-differential Eq. (22). We therefore now examine the consequences of the transition to an orbital basis expansion method. 

The simple orbital basis expansion method which is used in the implementation of most models of molecular electronic structure consists of expanding each R as a linear combination of determinants of a set of (usually) atom-centred functions of one or two standard forms. In particular most qualitative and semi-quantitative theories restrict the terms in this expansion to consist of the (approximate) occupied atomic orbitals of the constituent atoms of the molecule. There are two types of symmetry constraint implicit in this technique. 

We see therefore that, however desirable it is to abandon symmetry constraints from the point of view of the variation method, we shall be involved in radical departures from the conventional AO expansion method - particularly in its minimal basis form. Indeed most of the changes required to the usual AO basis method are already implicit in any decision to lower the symmetry of the AO basis from its local spherically symmetric form to that of the molecular point group. We shall see later that these considerations are too pessimistic. 

It must be stressed that the use of GHOs in no way depends on this latter Gaussian expansion procedure since all molecular integrals reduce to standard STO forms. It has merely proved expedient to make use of the expansion method in calculations reported earlier. This ab initio technique provides the raw data with which to establish the patterns of behaviour of the GHOs. We can now address ourselves to one of our stated aims the development of approximation methods for a quantitative theory of valence. 

Secondly, we might mention that there are two possible attitudes to a theory of valence based on the AO expansion method. The chemist uses the electron-pair model essentially as one of his axioms or, at least, as a good working hypothesis. This model is extremely familiar and useful any theory of such a model of valence should be capable of providing at least some foundation for and extension of his qualitative concepts. To the physicist, however, the simplest molecules are quite complex many-particle systems and he would perhaps find it surprising if we are able to obtain any useful results from our coarse-grained minimal basis model in view of the complexity of the interactions involved. We must try to balance these views in any evalution we make. 

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