The definition, properties, and applications of the single-particle (electron) density $\ensuremath{\rho}(\mathrm{r})$ are discussed in this review. Since the discovery of Hohenberg-Kohn theorem, which gave a theoretical justification for considering $\ensuremath{\rho}(\mathrm{r})$, rather than the wave function, for studying both nondegenerate and degenerate ground states of many-electron systems, $\ensuremath{\rho}(\mathrm{r})$ has been acquiring increasing attention. The quantum subspace concept of Bader et al. has further highlighted $\ensuremath{\rho}(\mathrm{r})$ since a rigorous decomposition of the three-dimensional (3D) space of a molecule into quantum subspaces or virial fragments is possible, the boundaries of such subspaces being defined solely in terms of $\ensuremath{\rho}(\mathrm{r})$. Further, $\ensuremath{\rho}(\mathrm{r})$ is a very useful tool for studying various chemical phenomena. The successes and drawbacks of earlier models, such as Thomas-Fermi-Dirac, incorporating $\ensuremath{\rho}(\mathrm{r})$ are examined. The applications of $\ensuremath{\rho}(\mathrm{r})$ to a host of properties---such as chemical binding, molecular geometry, chemical reactivity, transferability, and correlation energy---are reviewed. There has been a recent trend in attempting to bypass the Schr\"odinger equation and directly consider single-particle densities and reduced density matrices, since most information of physical and chemical interest are encoded in these quantities. This approach, although beset with problems such as $N$-representability, and although unsuccessful at present, is likely to yield fresh concepts as well as shed new light on earlier ideas. Since charge density in 3D space is a fundamental quantum-mechanical observable, directly obtainable from experiment, and since its use in conjunction with density-functional theory and quantum fluid dynamics would provide broadly similar approaches in nuclear physics, atomic-molecular physics, and solid-state physics, it is not unduly optimistic to say that $\ensuremath{\rho}(\mathrm{r})$ may be the unifying link between the microscopic world and our perception of it.

The role of single-particle density in chemistry

A model of ethylene and acetylene adsorption on the (111) surfaces of platinum and nickel

0.023 cm-1 to the previously described Dnss curves. When appended by this spin-orbit contritution, the Harrison-Liedtke37 curve predicts D = 0.745 cm-l a t the computed equilibrium angle of 133", while that of Langhoff and Davidsonl4 predicts D = 0.807 cm-1 a t this same angle. While both estimates are in reasonable agreement with the experimental number,23 0.76 f 0.02 cm-1, it is somewhat disconcerting that the more complete theoretical study of Langhoff and Davidson is outside of the experimental range. Indeed, if we ask what HCH angle the theoretical curves would predict, given the experimental D, we see that the Harrison-Liedtke curve is consistent with a range of angles from 145 to 135" while the Langhoff-Davidson calculation suggests an angle less than 125". However, given the complexity of both the experimental interpretation and theoretical calculations, mutual agreement, even a t the rt5% level, must be regarded as a satisfying accomplishment.

Applications of qualitative molecular orbital theory

Ab initio Gaussian type calculations are reported for the ground and excited states of the hypothetical molecule formaldimine (H2CNH). The C=N group is compared with the C=O group.

Theoretical investigation of the carbon nitrogen double bond

While diabatic approaches are ubiquitous for the understanding of electron-transfer reactions and have been mooted as being of general relevance, alternate applications have not been able to unify the same wide range of observed spectroscopic and kinetic properties. The cause of this is identified as the fundamentally different orbital configurations involved: charge-transfer phenomena involve typically either 1 or 3 electrons in two orbitals whereas most reactions are typically closed shell. As a result, two vibrationally coupled electronic states depict charge-transfer scenarios whereas three coupled states arise for closed-shell reactions of non-degenerate molecules and seven states for the reactions implicated in the aromaticity of benzene. Previous diabatic treatments of closed-shell processes have considered only two arbitrarily chosen states as being critical, mapping these states to those for electron transfer. We show that such effective two-state diabatic models are feasible but involve renormalized electronic coupling and vibrational coupling parameters, with this renormalization being property dependent. With this caveat, diabatic models are shown to provide excellent descriptions of the spectroscopy and kinetics of the ammonia inversion reaction, proton transfer in N2H7+, and aromaticity in benzene. This allows for the development of a single simple theory that can semi-quantitatively describe all of these chemical phenomena, as well as of course electron-transfer reactions. It forms a basis for understanding many technologically relevant aspects of chemical reactions, condensed-matter physics, chemical quantum entanglement, nanotechnology, and natural or artificial solar energy capture and conversion.

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R.G. Pearson

Papers

The structures of molecules in excited electronic states