Reduced Density Matrices: Coulson’s Challenge (2024)

Brian R. Judd

Reduced Density Matrices: Coulson’s Challenge  

A. J.

Coleman

Springer-Verlag

New York

In a 1959 after-dinner speech at a conference on molecular quantum mechanics held at the University of Colorado, the late Charles Coulson reviewed the state of molecular-structure calculations. Among the topics he discussed was the feeling that a conventional many-electron wavefunction tells us more than we need to know. For energy calculations involving one-electron and two-electron interactions, the equivalence of all the electrons in an N-electron system allows a particular pair of electrons to be selected for study, while integration over the coordinates of the remaining N − 2 electrons removes them from the analysis. This procedure amounts to a construction of the pth-order reduced-density matrix (RDM) with p = 2 for the two selected electrons. In terms of these, all calculations of physical quantities can be made. The problem is that, whereas it is straightforward to construct the RDM, given the wavefunction, it is not obvious what conditions the RDM itself should satisfy. These would be needed if the RDM were to replace the wavefunction as the basis for a calculation. The way forward was not obvious to Coulson.

John Coleman was stimulated to take up the challenge set by Coulson, that is, to make the RDM the basic starting point for all molecular calculations. In doing so, he established a very active group at Queen’s University in Kingston, Ontario. The book under review represents an account of the progress that has been made in the past 40 years, principally from Coleman’s perspective. Coleman is joined as co-author by Vyacheslav Yukalov, from the Joint Institute for Nuclear Research in Dubna, Russia, whose collaboration with him on RDM work dates from 1990. The reader’s appetite is whetted by the publisher’s note on the cover of the book, as well as by the authors’ comments in the preface, that there are now algorithms that make it possible to calculate “nearly all the properties of matter which are of interest to chemists and physicists.” To bring the reader to that point, the authors set out the steps taken along the way.

The key problem is to ensure N-representability, that is, to impose fermionic conditions on the RDM appropriate to a wavefunction describing the N-electron system. In discussing this difficulty, the authors adopt a rather formal mathematical style, as well as a specialized vocabulary that reflects the long development of the field. This seems somewhat inappropriate for a text designed (as they say) for first- or second-year graduate students. One wonders how Coulson, whose writing was a model of transparency, would have reacted to it. To get started, students may prefer Density Functional Theory of Atoms and Molecules, by Robert G. Parr and Weitao Yang (Oxford U. Press, 1989).

Despite its use of specialized vocabulary, Reduced Density Matrices has the great advantage of bringing the subject up to the present. Its final section tells an exciting story: Coleman and Yukalov argue that the extreme practitioners of density functional theory, who aim to go beyond RDM theory and eliminate everything but the electron density as an input to their calculations, are whistling in the dark. They see the future presaged in the work of Carmela Valdemoro, Hiroshi Nakatsuji, Koji Yasuda, and David Mazziotti, who base their work on the demand that a recurrence relation between RDMs of orders p, p + 1, and p + 2 be self-consistent, thereby guaranteeing N-representability. Their results for the ground-state energies of light atoms like beryllium and neon and small molecules like H₂O, NH₃, and CH₃F, are competitive with values obtained by traditional multiconfigurational Hartree–Fock methods in which the wavefunction plays the central role.

The numerical comparisons are impressive: Agreements to four or five significant figures are common. However, to learn of these remarkable results, the reader is urged merely to turn to articles in the literature. The book would have gained from a detailed comparison of the two theoretical methods, but there is no doubt that the authors have achieved their goal of describing how Coulson’s challenge can be met.