Large-dimension models

For electronic structure calculations there are mainly three exactly solvable models the Thomas-Fermi statistical model, the noninteracting electron model, and the large-dimension model. With the poineering work of Herschbach, it was recognized that the large dimension limit, where the dimensionality of space is treated as a free parameter, is simple, captures the main physics of the system, and is analytically solvable. In the final section we will summarize the main ideas of the large-dimension model for electronic structure problems. 

For electronic structure calculations of atoms and molecules there are three exactly solvable models the Thomas-Fermi statistical model (the limit A —> oo for fixed N/Z, where N is the number of electrons and Z is the nuclear charge) the noninteracting electron model,the limit of infinite nuclear charge (Z —> oo, for fixed N) and the large-dimension model D - oo for fixed N and Z). ... 

There are many exactly solvable potentials in quantum mechanics and there was no intention to cover all of them in this article. We therefore decided to give a brief description of supersymmetry in quantum mechanics, the analytic solution of the Hooke s law model for two-electron atoms, and the large-dimension models for electronic structure problems. 

Eq. 25 is given in Fig. 14 as a contour plot of AF against m and JZ, and the lamella grows into infinitely large dimensions in all directions. In contrast, the exactly solved model of Fig. 11 and Eqs. 9-23 show that finite lamellar... 

The constructed system of equations is a closed one. It is solved with the preset initial conditions 6j (r — 0), 0 jg(, t — 0), 6i (2, t = 0). The system of equations makes it possible to describe arbitrary distributions of particles on a surface and their evolution in time. The only shortcoming is the large dimension. The minimal fragment of a lattice on which a process with cyclic boundary conditions should be described is 4 x 4. It is, therefore, natural to raise the question of approximating the description of particle distribution to lower the dimension of the system of equations. In this connection, it is reasonable to consider simpler point-like models. 

Haskell E., Nykamp D.Q., Tranchina D. Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics cutting the dimension down to size. Network Comput Neural Syst, 2001 12,141-174. 

If the equalities are used in the above equations, these equations become identical to the equations derived previously by Fisher. Although the equalities are correct for the two-dimensional Ising model, use of the equalities in (54) again fail to match the expected values (j8 =, v = y = I, and A = f) for the limiting van der Waals behavior for large dimensions. 

Such models would be structurally compatible with the other constituents of petroleum and are able to represent the thermal chemistry and other process operations. It must also be recognized that although such models can have the large dimensions that have been proposed for asphaltenes, they can also be molecular chameleons insofar as they can vary in dimensions depending on the angle of rotation around an axis (Figure 9) or the freedom of rotation around one, or more, of the bonds. 

Figure 1. Evolution of the radial electron density for argon from each of the three limits described in Table 1. The limits (top row) are those given by the Thomas-Fermi approximation, the non-interacting electron model, and the large-dimension limit. The abscissae are Hn-ear in y/r, where r is given in units of (IS/JV) for the first column, (18/Z) Co for the second, and (D/Zy a<, for the third. All curves except the Thomas-Fermi limit were obtained by adding Slater distributions whose exponents were determined from the subhamiltonian minima described in Sec. 4.

The large-dimension limit offers an approach that explicitly includes electron correlation [2,3]. It seems to give an excellent qualitative description of correlation effects that are difficult to understand in terms of independent-electron models [3,4]. This is because all the coulombic forces are retained in their exact form. The dimensionality of space is treated as an arbitrary and continuous parameter, D. For any atom or molecule, an exact solution can be obtained in the limit D oo. The problem then becomes how to exploit this limiting result in constructing an approximate solution for the actual physical system with D = 3. Our aim is twofold to find an efficient way to calculate accurate expectation values, and to develop reliable qualitative interpretations of the D oo results that elucidate the correlation effects at i = 3. 

The large-dimension limit has recently resolved at least some of the difficulties of the molecular model. The molecule-like structure falls out quite naturally from the rigid bent triatomic Lewis configuration obtained in the limit D — oo [5], and the Langmuir vibrations at finite D can be analyzed in terms of normal modes, which provide a set of approximate quantum numbers [6,7]. These results are obtained directly from the Schrodinger equation, in contrast to the phenomenological basis of some of the earlier studies. When coupled with an analysis of the rotations of the Lewis structure, this approach provides an excellent alternative classification scheme for the doubly-excited spectrum [8]. Furthermore, an analysis [7] of the normal modes offers a simple explanation of the connection between the explicitly molecular approaches of Herrick and of Briggs on the one hand, and the hyperspherical approach, which is rather different in its formulation and basic philosophy. 

In Section 2 we set forth the model for excited states of two-electron atoms that is provided by the large-dimension limit and then very briefly describe the method that we used to calculate the expansion coefficients. Further deteuls can be found in Re s. [6] emd [7] and in a forthcoming publication [12]. In Section 3 we present our numerical results and discuss implications of this work. 

In aU cases, when a QM description is introduced, the problem of effectively including environmental effects remains, due to the extremely large dimension of the system to be treated and the complex network of interactions to be considered. In these cases, as anticipated in the preceding section, a vahd strategy is to complement a QM description of the MS with a classical description of the ENV by introducing the so-called hybrid QM/classical models. 

Structural models reach easily very large dimensions, many times counted by millions of degrees of freedom. In many circumstances, it is required to reduce the size of the matrices used for the calculation, especially when dealing with dynamic problems. 

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