The One-Dimension Limit

David Z. Goodson and Mario Ldpez-Cabrera Department of Chemistry Harvard University 12 Oxford Street Cambridge, MA 02138, USA 

With appropriate dimensional scalings, theD — 1 limit of Schrodinger equations for coulombic systems provides important information about the dimension dependence of energy eigenvalues. The energy typically has a second-order pole at D = 1, the residue of which can often be exactly determined. We demonstrate this with some simple examples and then review a systematic procedure for characterizing a class of dimensional singularities found in coulombic problems. 

Dimensional continuation of the Schrodinger equation has yielded valuable insights into the properties of atoms and molecules. If we consider the energy eigenvalues to be a function E D) of the 

Herschbach et al. (eds.). Dimensional Scaling in Chemical Physics, 115-130. 1993 Kluwer Academic Publishers. 

We begin with the hydrogen atom. The first step is to define what we mean by a one-dimensional Schrodinger equation. Perhaps the most straightforward approach would be simply to choose 

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Eqs. (5) and (7) can be thought of as changes in the distance and energy units, respectively. In these new, dimension-scaled, units the radial expectation value, (r) = has a much tamer dimension dependence, and the new groimd-state energy, E = — is completely independent of D. Thus, the one-dimension limit of the scaled Schrodinger equation, Eq. (6), is not such a bad model for the physical, three-dimensional, problem. 

The second and third terms on the right-hand side are finite at 2 = 1, so they become insignificant compared to the first term in the one-dimension limit. Note that 8 r) = 2S(x), since the former is normalized on the domain 0 < r < oo while the latter is normalized on a domain that is twice as large, —oo < x < oo. Thus,... 

Hence, our prescription for constructing the one-dimension limit of a Schiddinger equation is to replace the coulomb potential 1/r with (r). 

Thus, the one-dimension limit for the polarizability does not give the full solution however, it differs only by a quadratic polynomial in D. 

The one-dimension limit of Eq. (2) as written is also not well-defined. The problem now is that the expectation value (V), and hence the energy, becomes infinite at particle coalescences [18], as we discussed in Chapter 4.1. V) can be rendered finite through... 

One kind of a multicritical point is a point over a critical line where more than two different states coalesce. The common multicritical points in statistical mechanics theory of phase transition are tricritical points (the point that separates a first order and a continuous line) or bicritical points (two continuous lines merge in a first order line) (see, for example, Ref. 166). These multicritical points were observed in quantum few-body systems only in the large dimension limit approximation for small molecules [10,32]. For three-dimensional systems, this kind of multicritical points was not reported yet. 

But what are we to make of the new dimension-scaled potential energy function, V r) = D — l)/2r, when D = Herrick and Still-inger [10] have observed that (U—l)/2r = S(x) in the one-dimensional limit, where x is the cartesian coordinate, —oo < x < oo, and S(x) is the Dirac delta function. This surprising result is a consequence of the dimension dependence of the Jacobian volume element, r dr, which leads to the identity [14]... 

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. 

Nanoscience involves materials where some critical properly is attributable to a structure with at least one dimension limited to the nanometer size scale, 1 -100 nanometers . Below that size die disciplines of Chemistry and Atomic/Molecular Physics have already provided detailed scientific understanding. Above that size scale, in the last SO years Condensed Matter Physics and Materials Science have provided detailed scientific understanding of microstructures. So die nanoscale is the last size fixintier for materials science. 

One of the more recent advances in XPS is the development of photoelectron microscopy [ ]. By either focusing the incident x-ray beam, or by using electrostatic lenses to image a small spot on the sample, spatially-resolved XPS has become feasible. The limits to the spatial resolution are currently of the order of 1 pm, but are expected to improve. This teclmique has many teclmological applications. For example, the chemical makeup of micromechanical and microelectronic devices can be monitored on the scale of the device dimensions. 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

The siipercell plane wave DFT approach is periodic in tliree dimensions, which has some disadvantages (i) thick vacuum layers are required so the slab does not interact with its images, (ii) for a tractably sized unit cell, only high adsorbate coverages are modelled readily and (iii) one is limited in accuracy by the fonn of the... 

In this problem, the integral over all spaee dx is in only one dimension, x. The limits of integration are the dimensions of the box, 0 and 1 in whatever unit was ehosen. 

Electrodriven techniques are useful as components in multidimensional separation systems due to their unique mechanisms of separation, high efficiency and speed. The work carried out by Jorgenson and co-workers has demonstrated the high efficiencies and peak capacities that are possible with comprehensive multidimensional electrodriven separations. The speed and efficiency of CZE makes it possibly the best technique to use for the final dimension in a liquid phase multidimensional separation. It can be envisaged that multidimensional electrodriven techniques will eventually be applied to the analysis of complex mixtures of all types. The peak capacities that can result from these techniques make them extraordinarily powerful tools. When the limitations of one-dimensional separations are finally realized, and the simplicity of multidimensional methods is enhanced, the use of multidimensional electrodriven separations may become more widespread. 

It is easy to see that this expression has two minima within the Brillouin zone. One minimum is at fc = 0 and gives the correct continuum limit. The other, however, is at k = 7t/a and carries an infinite momentum as the lattice spacing a 0. In other words, discretizing the fermion field leads to the unphysical problem of species doubling. (In fact, since there is a doubling for each space-time dimension, this scheme actually results in 2 = 16 times the expected number of fermions.)... 

Blind holes in molded plastics are created by a core supported by only one side of the mold. The length of the core and depth of the hole are limited by the ability of the core to withstand the bending forces produced by the flowing plastic without excessive deflection. For this reason, the depth of a blind hole should not exceed three times its diameter or minimum cross-sectional dimension. For small blind holes with a minimum dimension below 1/4 in., the L/D ratio should be kept to two. With through holes the cores can be longer, since the opposite side of the mold cavity supports them (3). 

In the second section a classification of the different kinds of polymorphism in polymers is made on the basis of idealized structural models and upon consideration of limiting models of the order-disorder phenomena which may occur at the molecular level. The determination of structural models and degree of order can be made appropriately through diffraction experiments. Polymorphism in polymers is, here, discussed only with reference to cases and models, for which long-range positional order is preserved at least in one dimension. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

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