Mathematical Challenges from Theoretical/Computational Chemistry

A major mathematical landmark in the eighteenth century was Euler's introduction and exploitation of the famous gamma function. One of its basic and striking properties is that it provides a natural "smooth" extension of the factorials n! that are defi ned nominally just for the positive integers to all positive numbers, and indeed even into the complex plane. The pervasive appearance of the Euler gamma function throughout classical mathematical analysis constitutes a powerful paradigm suggesting that analogous extensions from the discrete positive integers to the complex plane in other contexts might generate analogous intellectual benefits.

During roughly the last two decades, simultaneous developments in several distinct areas of physical science appear to point to the necessity (or at least the desirability) of just such an extension. Specifically, this involves generalizing the famili ar notion of Euclidean D-dimensional spaces from positive integer D at least to the positive reals, if not to the complex D-plane. This is not an empty pedantic exercise; at least one serious proposal has been published (Zeilinger and Svozil, 1985) claim ing that accurate spectroscopic measurements of the electron "g-factor" indicate that the space dimension of our world is less than 3 by approximately 5 x 10. Furthermo re, in various theoretical applications that have so far been suggested for the continuous-D concept, D itself or its inverse appears to be a natural expansion parameter for various fundamental quantities of interest. However, most of the work along these lines thus far has been ad hoc, lacking rigorous mathematical underpinning. Naturally this calls into question the validity of claimed results.

Three physical science research areas deserve mention in this context. The first is quantum field theory; dimension D has been treated as a continuously variable "regularizing parameter" whose manipulation avoids embarrassing divergences in perturbatio n expansions (Bollini and Giambiagi, 1972; t'Hooft and Veltman, 1972; Ashmore, 1973). The second is the statistical mechanics of phase transitions (specifically involving critical point phenomena); because of rigorously known results for D = 2 and D = 4, 5, 6, . . . , series expansions in the quantity 4-D have been developed for various quantities of interest to access the physical case D = 3 (Wilson and Fisher, 1972; Gorishny et al., 1984). The third area holds perhaps the greatest promise for chemical progress, namely, the development of atomic and molecular quantum mechanics (with useful computational algorithms) in spaces of arbitrary D (Goodson et al., 1992; Herschbach et al., 1992).

As in the other applications, the notion of atomic and molecular quantum mechanics is unambiguously defined for D a positive integer; in other words, the Schrödinger wave equation and its boundary conditions have an immediate and clear meaning. Th e desire to embed these problems in the arbitrary-D context arises primarily from the observation that solutions to the Schrödinger equation adopt a simple limiting form as D approaches infinity, namely, those for simple harmonic oscillators localize d in multidimensional space (Goodson et al., 1992; Herschbach et al., 1992). Eigenfunction and eigenvalue expansions in 1/D have then been formally generated, with the hope that series summation techniques (e.g., Padé approximants) would permit ex tension to the case of ultimate interest D = 3. This strategic approach to real chemistry in the real world is emboldened by the facts that (a) D = 1 is often an exactly solvable case (or at least amenable to very accurate numerical study), and (b) exact interdimensional identities for D and D + 2 are known (Herrick, 1975). These latter afford convenient fixed points for refining the series summation attempts.

Box 4.4

The presumption that spaces with noninteger dimension were available as analytic tools for atomic and molecular quantum mechanics rests largely on simple observations such as the fact that the D-dimensional (hyper)spherical volume element,

is an obvious analytic function of the variable D. The implicit assumption in the various applications to date, quantum mechanical and otherwise, seems to have been that the same expression can be invested with mathematical legitimacy for noninteger D, in the sense that it is an attribute of a family of precisely defined spaces. This is by no means an obvious proposition, since any quantity such as K(D) above could be augmented by any function of D that vanishes at the positive integers, such as sin (2 D), without affecting the situation for conventional Euclidean geometry.

The published literature reveals some attempts to axiomatize spaces of noninteger dimension (Wilson, 1973; Stillinger, 1977), but it is clear that the subject requires deeper mathematical insight than it has thus far experienced. In particular, it is de sirable to determine the extent to which arbitrary-D spaces are uniquely definable as uniform and isotropic metric spaces and what their relation to conventional vector spaces might be. It has been suggested (Wilson, 1973) that noninteger-D spaces can be viewed as embedded in an infinite-dimensional vector space, but whether this is uniquely possible or even necessary to perform calculations remains open.

It is important to stress the distinction between the general-D spaces that may be obtained by interpolation between the familiar Euclidian spaces for integer D on the one hand and the so-called fractal sets to which a generally noninteger Hausdorff-Be sicovitch dimension can be assigned (Mandelbrot, 1983). The latter are normally viewed as point sets contained in a Euclidean host space; furthermore, they fail to display translational and rotational invariance, and are therefore not uniform and isotrop ic.

References

Ashmore, J.F., 1973, On renormalization and complex space-time dimensions, Commun. Math. Phys. 29:177-187.

Bollini, C.G., and J.J. Giambiagi, 1972, Dimensional renormalization: The number of dimensions as a regularizing parameter, Nuovo Cimento B 12:20.

Goodson, D.Z., M. Lopez-Cabrera, D.R. Herschbach, and J.D. Morgan III, 1992, Large-order dimensional perturbation theory for two-electron atoms, J. Chem. Phys. 97:8481.

Gorishny, S.G., S.A. Larin, and F.V. Tkachov, 1984, Phys. Lett. 101A:120.

Herrick, D.R., 1975, Degeneracies in energy levels of quantum systems of variable dimensionality, J. Math. Phys. 16:281.

Herschbach, D.R., J. Avery, and O. Goscinski, eds., 1992, Dimensional Scaling in Chemical Physics, Kluwer Academic, Dordrecht, Holland.

Mandelbrot, B.B., 1983, The Fractal Geometry of Nature, W.H. Freeman, San Francisco.

Stillinger, F.H., 1977, Axiomatic basis for spaces with noninteger, J. Math. Phys. 18:1224.

t'Hooft, G., and M. Veltman, 1972, Nuclear Phys. B 44:189.

Wilson, K.G., 1973, Phys. Rev. D 7:2911.

Wilson, K.G., and M.E. Fisher, 1972, Phys. Rev. Lett. 28:240.

Zeilinger, A., and K. Svozil, Measuring the dimension of spacetime, Phys. Rev. Lett. 54:2553.

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