2/18/2015PHY 752 Spring 2015 -- Lecture 151 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 15: Reading: Chapter 10 in MPM Numerical.

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Presentation transcript:

2/18/2015PHY 752 Spring Lecture 151 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 15: Reading: Chapter 10 in MPM Numerical Realizations of Density functional theory 1.Electronic structure of atoms 2.Integration of the radial equations 3.Frozen core approximation 4.Extension of formalism to multi-center analysis

2/18/2015PHY 752 Spring Lecture 152 Note: Take-home exam scheduled for the week of March 2 nd.

2/18/2015PHY 752 Spring Lecture 153

2/18/2015PHY 752 Spring Lecture 154

2/18/2015PHY 752 Spring Lecture 155 Kohn-Sham equations for spherical atom

2/18/2015PHY 752 Spring Lecture 156 Self-consistent solution Numerical methods for solving the Kohn-Sham equations

2/18/2015PHY 752 Spring Lecture 157 Example for carbon

2/18/2015PHY 752 Spring Lecture 158 Results for carbon r (Bohr) P nl (r) 1s 2s 2p

2/18/2015PHY 752 Spring Lecture 159 n core (r) n val (r) r (Bohr) n r 2 /4 

2/18/2015PHY 752 Spring Lecture 1510 Example for Cu 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 r (Bohr) n core (r) n vale (r) n r 2 /4 

2/18/2015PHY 752 Spring Lecture 1511 r (Bohr) 4s 4p 3d 3s 3p Results for Copper P nl (r)

2/18/2015PHY 752 Spring Lecture 1512 Frozen core approximation Example for Cu 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 r (Bohr) n core (r) n vale (r) n r 2 /4 

2/18/2015PHY 752 Spring Lecture 15b13 Systematic study of frozen core approximation in DFT

2/18/2015PHY 752 Spring Lecture 1514 Variational relations for DFT in frozencore approximation (Kohn-Sham formulation)

2/18/2015PHY 752 Spring Lecture 1515 Practical solution to Kohn-Sham equations for single particle orbitals:

2/18/2015PHY 752 Spring Lecture 1516 Practical solution of Kohn-Sham equations in solids

2/18/2015PHY 752 Spring Lecture 1517 Muffin tin potential construction Augmented Plane Wave (APW) approximation

2/18/2015PHY 752 Spring Lecture 1518 Muffin tin potential construction

2/18/2015PHY 752 Spring Lecture 1519 Muffin tin model continued: Problems with APW and KKR Green’s function schemes 1.Difficult numerically to find Kohn-Sham energies  i 2.Potential form unrealistic especially for covalent materials  Linearized equations – O. K. Andersen

2/18/2015PHY 752 Spring Lecture 1520

2/18/2015PHY 752 Spring Lecture 1521 Modern software based on LAPW method --

2/18/2015PHY 752 Spring Lecture