# The ground state energy of atoms: Functionals of the one-particle-reduced density matrix and their relation to the full Schrödinger equation

Wednesday, February 18, 2009 - 11:15am - 12:15pm

Lind 409

Heinz Siedentop (Ludwig-Maximilians-Universität München)

To have an explicit formula for the ground state energy (lowest spectral point) E(Z) of the Schrödinger operator of (neutral) atoms of atomic number Z is an elusive goal for Z>1 since it is a matrix differential operator in 3N dimensions with 2z components.

Shortly after the advent of quantum mechanics efforts were made to reduce the dimensions to 3 and the components to one. The first steps were taken by Thomas and Fermi and by Hartree and Fock. A modern version of this idea is due to Hohenberg and Kohn (density functional theory) and Gilbert (density matrix functional theory). The price to pay is to give up the linearity of the problem.

In this talk I will explain the general idea of density matrix functional theory and show how a particular type of density matrix functionals (Müller functional and variants thereof, see the kick-off meeting of IMA's year on mathematics and chemistry) can be used to get information on the asymptotic behavior of E(Z). Among other things, we will show, that the infimum EM(Z) has the same asymptotic expansion

EM(Z) = a Z^(7/3) + 1/4 Z^2 - c Z^(5/3)+ o(Z^(5/3))

as the quantum case.

Shortly after the advent of quantum mechanics efforts were made to reduce the dimensions to 3 and the components to one. The first steps were taken by Thomas and Fermi and by Hartree and Fock. A modern version of this idea is due to Hohenberg and Kohn (density functional theory) and Gilbert (density matrix functional theory). The price to pay is to give up the linearity of the problem.

In this talk I will explain the general idea of density matrix functional theory and show how a particular type of density matrix functionals (Müller functional and variants thereof, see the kick-off meeting of IMA's year on mathematics and chemistry) can be used to get information on the asymptotic behavior of E(Z). Among other things, we will show, that the infimum EM(Z) has the same asymptotic expansion

EM(Z) = a Z^(7/3) + 1/4 Z^2 - c Z^(5/3)+ o(Z^(5/3))

as the quantum case.