On F and E, in DFT

Rigorous mathematical foundations of density functional theory are revisited, with some use of infinitesimal (nonstandard) methods. A thorough treatment is given of basic properties of internal energy and ground-state energy functionals along with several improvements and clarifications of known results.A simple metrizable topology is constructed on the space of densities using a hierarchy of spatial partitions. This topology is very weak, but supplemented by control of internal energy, it is, in a rough sense, essentially as strong as $L^1$. Consequently, the internal energy functional $F$ is lower semicontinuous with respect to it. With separation of positive and negative parts of external potentials, very badly behaved, even infinite, positive parts can be handled. Confining potentials are thereby incorporated directly into the density functional framework.

[1]  A. E. Hurd,et al.  An Introduction to Nonstandard Real Analysis , 2011 .

[2]  Wayne Reitz,et al.  A Review of: “Electronic Structure: Basic Theory and Practical Methods” , 2006 .

[3]  K. Capelle.  A bird's-eye view of density-functional theory , 2002, cond-mat/0211443.

[4]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[5]  S. Albeverio.  Nonstandard Methods in Stochastic Analysis and Mathematical Physics , 1986 .

[6]  G. Folland.  Introduction to partial differential equations , 1976 .

[7]  W. Luxemburg.  What Is Nonstandard Analysis , 1973 .

[8]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[9]  R. Leeuwen,et al.  Density functional approach to the many-body problem : Key concepts and exact functionals , 2003 .

[10]  Peter A. Loeb,et al.  Nonstandard analysis for the working mathematician , 2000 .

[11]  R. Goldblatt.  Lectures on the hyperreals : an introduction to nonstandard analysis , 1998 .

[12]  Helmut Eschrig,et al.  The fundamentals of density functional theory , 1996 .

[13]  R. Dreizler,et al.  Density Functional Theory: An Approach to the Quantum Many-Body Problem , 1991 .

[14]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

[15]  Martin Davis,et al.  Applied Nonstandard Analysis , 1977 .

[16]  K. D. Stroyan,et al.  Introduction to the theory of infinitesimals , 1976 .

[17]  F. Trèves.  Basic Linear Partial Differential Equations , 1975 .