## Introduction

In 1959, during a famous lecture entitled "There's Plenty of Room at the Bottom",1 Richard Feynman focused on the startling possibilities that would exist at the limit of miniaturization, that being atomically precise devices with dimensions in the nanometer range. "Molecular electronics", also refered to as "nanoelectronics", denotes the goal of shrinking electronic devices, such as diodes and transistors, as well as intergrated circuits that can perform logical operations, down to dimensions in the range of 100 nanometers.2 The forty-year, and growing, hiatus in the development of molecular electronics can be figuratively seen as a period of waiting for the bottom-up and atomically precise construction skills of synthetic chemistry to meet the top-down reductionist aspirations of device physics. The sub-nanometer domain of nineteenth-century classical chemistry has steadily grown, and state-of-the-art supramolecular chemistry can achieve atomic precision in non-repeating molecular assemblies of the size desired for nanotechnology.3 For molecular electronics in particular, a basic understanding of the electron transport properties of molecules themselves must also be developed. The goal of the current research is to investigate the slow (chemically valence) electron dynamics of molecules that are possible prototypes of molecular wires and diode.4 We refrain from basing our analysis on any of the assorted definitions of molecular orbitals. The orbital model was originally devised to explain spectroscopic properties of simple atomic systems in the gaseous phase, a phenomenon far removed from the operation of any kind of electronic device. Instead, our investigation is based on physical properties of model systems. The particular properties chosen for study are those that we expect to yield new, and general insight into the chemical factors governing electron transport at the nanoscale. This expectation arises from recent advances in the understanding of molecular structure and chemical reactivity that were obtained by analogous methods. This unexpected parallel is briefly outlined below.

A tremendous amount of information about the bonding in a molecule, and about its chemical reactivity towards other molecules, can be extracted from the molecule's electron density distribution, p(r), by topological and graphical examinations of the Laplacian of this function, A2p(r).5 This method of analysis is based on an observable property of matter, the electron density, and is currently employed by both computational6 and experimental scientists.7 The term "electron density" is almost always assumed to refer to the probability density of electrons in the real, three-dimensional space of r, the position vector. Strictly speaking, p(r) is the coordinate-space representation of the electronic charge density. It is simply the probability of finding any electron in some elemental volume, dr, multiplied by the total number of electrons in the system. It has the units e/ao3. Equally "real", and also observable, is the momentum-space representation of the electronic charge density, n(p), often called the "electron momentum density" to avoid confusion.8 It is analogously defined as the probability of finding any electron in some elemental unit of momentum space, dp, multiplied by the total number of electrons in the system. It has the units eao3/n3. Although they are simple to conceive, such "single-particle density distributions" are rich in information, since they are shaped by the forces and quantum mechanical symmetry requirements that are many-body in nature. For instance, even at the Hartree-Fock level the motions of all electrons with the same spin must be correlated with one another to ensure obeyance of the anti-symmetry requirement of the Pauli principle.9 Computationally, in practice this so-called "Fermi correlation" is limited by the size of the basis set of atomic orbital-like basis functions. There is of course no such limitation on experimentally determined single-particle density distributions, the limit in this case being how many X-ray reflections, or detector coincidences, the experimentalist is able to obtain.

In keeping with the fundamental complementarity of position and momentum, a corresponding topological analysis of the Laplacian of the electron density in momentum space, A2n(p), has been shown to yield important information about the dynamics of electrons in matter.10 In particular, the directions (with respect to the molecular framework) that slow, or valence, electrons may flow with the least effective resistance, are predicted by the regions in momentum-space near the origin where A2n(p) is most negative. In general, regions in momentum-space where A2n (p) is positive, correspond to directions within the molecule that have nonlaminar electron flow, while regions wherein A2n (p) is negative correspond to directions with laminar electron flow.10 This local (in momentum-space) criterion for exhaustively partitioning the momentum distribution of a molecule, or crystal, into laminar and nonlaminar regions is necessarily nonlocal in coordinate-space, as demanded by the Heisenberg uncertainty principle. Thus although the A2n (p) topology and its three-dimensional isovalue surfaces presented below are relatively easy to visualize (compared to multi-dimensional wavefunctions of many-electron systems), these new entities are difficult to relate to conventional chemical concepts. The usual points of reference for chemists, nuclear positions and bond axes, are all but gone. Only the orientation of the molecule and the direction of the bond axes are directly related to fixed points in momentum-space.11

There has been some effort, led by Schmider, to probe the topological properties of the "fuzzy" density in phase-space for molecular systems. 12 In these studies, the electronic position and momentum are both specified, but the probabilities are uncertain, hence the "fuzziness". In this report we limit our focus to the topological properties of the well-defined A2n (p) distributions. We feel that further investigations of this function will not only foster familiarity among chemists with momentum-space itself, but more importantly such investigations will breed entirely new momentum-based concepts, such as laminar versus nonlaminar electronic flow. We further anticipate that continuing advances in the attainable precision of electron momentum density measurement by electron momentum spectroscopy, gamma-ray Compton scattering, and positron annihilation techniques,13 will reinforce the knowledge gained by the analytical procedure that we discuss below.

Finally in this Introduction, we wish to reiterate a summary conclusion from earlier work,10 but which is further supported by the current results and interim research. The extent to which the correlations among many electrons lead to local electron pairing of either the Lewis type in coordinate-space,14 or of the Cooper type in momentum-space,15 appears to be physically manifested as local concentration of the electronic charge density in the corresponding space. Thus charge density analysis provides a model-independent tool for relating observable information to models of electronic structure that have proven themselves paramount. The range of chemical and material behavior that have been shown to correlate with topological features of electron density distributions is not exhausted. Eberhart has recently reviewed the insight into material failure that is yielded by topological analysis of p(r) inalloys.16 We now consider analysis of the electronic charge density in momentum space of molecules that have properties that are analogous to those of electronic devices.

## Post a comment