2/11/2015PHY 752 Spring 2015 -- Lecture 121 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 12: Reading: Chapter 9 in MPM Approximations.

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2/11/2015PHY 752 Spring Lecture 121 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 12: Reading: Chapter 9 in MPM Approximations to the many electron problem -- continued 1.Density functional theory 2.Exchange energy and potential for jellium

2/11/2015PHY 752 Spring Lecture 122 Note: Take-home exam scheduled for the week of March 2 nd.

2/11/2015PHY 752 Spring Lecture 123

2/11/2015PHY 752 Spring Lecture 124 Density functional theory Describes the relationship between the many electron problem and independent electron treatments. Proof of theorm Estimates of F[n].

2/11/2015PHY 752 Spring Lecture 125 Density functional theory -- continued

2/11/2015PHY 752 Spring Lecture 126 Hohenberg and Kohn: formal proof of basic theorem Kinetic energy External potential Coulomb interaction

2/11/2015PHY 752 Spring Lecture 127 Theorem: The density n(r) of the ground state of the system is a unique functional of the external potential v(r). Proof: Consider two Hamiltonians H and H’ differing only by external potentials v and v’.

2/11/2015PHY 752 Spring Lecture 128 We can also show:

2/11/2015PHY 752 Spring Lecture 129 The theorem implies that the ground state energy E can be considered as a functional of the density n(r) Thus, the determination of the ground state energy E is transformed into a minimization of the functional with respect to the density n(r), transforming a many particle minimization into a single particle minimization. In practice, the functional form of F[n] is not known.

2/11/2015PHY 752 Spring Lecture 1210 Determination of F[n] for jellium

2/11/2015PHY 752 Spring Lecture 1211 Determination of F[n] for jellium -- continued

2/11/2015PHY 752 Spring Lecture 1212 Determination of F[n] for jellium – continued Exchange contribution within Hartree-Fock approximation

2/11/2015PHY 752 Spring Lecture 1213 Summary of results for jellium: General forms Special for jellium

2/11/2015PHY 752 Spring Lecture 1214 Variational equations

2/11/2015PHY 752 Spring Lecture 1215 Summary of Kohn-Sham equations: For jellium; exchange only