From Atoms to Crystals: a Mathematical Journey

We present an overview of some works on the models of computational quantum chemistry. We examine issues such as the existence of ground states (both for the electronic structure and the configuration of nuclei), the foundations of the models of the crystalline phase, and the macroscopic limits. We emphasize the connections between the physical modelling, the numerical concerns and the mathematical analysis of the problems. 1. Outline This article presents an expository review of some collective work by a group of researchers (X. Blanc, E. Cancès, I. Catto, M. Esteban, E. Séré) to which we belong on the mathematical aspects of the models used for the simulation of matter at the microscopic scale. The relation of these models to models for the continuum description of matter at the macroscopic scale will also be examined. The mathematical material reported on here covers almost two decades, starting in the late 1980s with our contribution to a general endeavour (mainly initiated by E.H. Lieb, B. Simon, W. Thirring) to put the models of quantum chemistry on a sound mathematical basis (see Sections 3 to 5). It continues in the 1990s with our work on the thermodynamic limit of molecular models in order to establish and/or justify models for the crystal phase (see Section 6). The most recent step (in fact hardly addressed here, only outlined in Section 7, and postponed until future publications) is the definition of the energy of general infinite microscopic sets of particles and the passage to the macroscale with the tentative definition of the density of mechanical energy on the basis of models at the atomistic scale. The program described above is ambitious. However, we are in a position where on the one hand some building blocks of the whole program are indeed known from the mathematical standpoint for most of the models used in practice, and on the other hand where we are able to accomplish all the steps for some models, rather academic in nature but nevertheless illustrative and of some physical relevance. Therefore, we believe that this is the proper time for such a survey article. Enough convincing arguments are available to prove that it is indeed possible to analyze models of microscopic matter and to determine the macroscopic limit, on a mathematical ground, provided one restricts one’s ambition. A large body Received by the editors November 20, 2004. 2000 Mathematics Subject Classification. Primary 35Bxx, 35Jxx, 35Pxx, 49Kxx, 81Q05, 81Q10, 82Bxx. This article is an extended version by the two authors of notes based upon a series of lectures given by PLL at Collège de France during the fall semester of the academic year 2003/04. c ©2005 American Mathematical Society Reverts to public domain 28 years from publication