Understanding the Kohn-Sham Equations in Solid-State Physics

Delve into the intricacies of the Kohn-Sham equations in solid-state physics, exploring topics such as electronic structure of atoms, numerical realizations of density functional theory, self-consistent solutions, and more. Dive deep into the theoretical foundations and practical applications of these equations for spherical atoms like carbon.

• Solid-State Physics
• Kohn-Sham Equations
• Electronic Structure
• Density Functional Theory
• Numerical Methods

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1. 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 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 1

2. Note: Take-home exam scheduled for the week of March 2nd. 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 2

3. 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 3

4. 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 4

5. Kohn-Sham equations for spherical atom Equations in Rydberg units + 2 ( 1) l l d dr + + + = ( ) r ( ) r ( ) v r ( ) r ( ) r V V P P i i r e e exc n i i l n i i l n i i l 2 2 r [ ] n n 1 r E dr n r dr n r = = + 2 ( ) r 2 ' ( ) ' ( ) V r r ee ee 0 r [ ] n n n n 2 E ( ) 1/3 = = + 2 1 3 / ( ) r 3 ( ) n r ( ) V V r exc exc c [ ] 2 E Z r = = = ( ) r ( ) v r V ext ex t 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 5

6. Numerical methods for solving the Kohn-Sham equations Self-consistent solution Iteration = ( ) i = 0 r 2 ( ) r ( ) r n i i 2 + + 1 1 + + + = i 2 ( ) r ( ) r ( ) r ( ) r ( ) r V V v ee ex 2 m i i 2 + + = 1 1 ( ) r ( ) r n i temp i n + + = + 1 1 \alpha ( ) r + ( ) r ( ) r (1 ) n x x n temp 1 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 6

7. For spherically symmetric a ( ) i i i n l m n l r Y P r r r tom : = r r ( ) ( ) l m i i i i ( ) n l = ( ) i i n l i i Example for carbon ( ) n r = 2 ( ) r w n i i l n i i l i ) ( 2 2 2 + + =4 2 ( ) r 2 ( ) r 2 ( ) r 1 2 2 s s p ) ( 4 r 2 2 2 + + = 2 ( ) 2 ( ) 2 ( ) r P r P r P 1 2 2 s s p 2 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 7

8. Results for carbon 1s 2p Pnl(r) 2s r (Bohr) 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 8

9. ncore(r) n r2/4 nval(r) r (Bohr) 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 9

10. Example for Cu 1s22s22p63s23p63d104s1 ncore(r) n r2/4 nvale(r) r (Bohr) 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 10

11. Results for Copper 3s 3d 4p Pnl(r) 4s 3p r (Bohr) 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 11

12. Frozen core approximation = + ( ) ( ) r ( ) r n r core n n vale Variationally optimize energy wrt ( ) Example for Cu n r vale 1s22s22p63s23p63d104s1 ncore(r) n r2/4 nvale(r) PHY 752 Spring 2015 -- Lecture 15 r (Bohr) 2/18/2015 12

13. Systematic study of frozen core approximation in DFT http://journals.aps.org/prb/abstract/10.1103/PhysRevB.21.2222 2/18/2015 PHY 752 Spring 2015 -- Lecture 15b 13

14. Variational relations for DFT in frozencore approximation (Kohn-Sham formulation) = + + + [ ] n = [ ] n [ ] n [ ] n E T E E E v ext ee exc + core vale T T T ( ) + 3 core vale ( ) r v r ( ) r ( ) r [ ] n E d n n ext 2 r r r ( ) ( ') n r e n = = + + 3 3 core-core ee E core-vale e e E vale-vale ee E E d r d r ee 2 ' + core vale [ ] n = [ ] E E n exc ex c 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 14

15. Practical solution to Kohn-Sham equations for single particle orbitals: 2 = ( ) r ( ) r For n i i r Equations for orbitals ( ): + i 2 = i i 2 ( ) r ( ) r ( ) r V i 2 m Numerical problem: near each nuclear center -- 2 a Z e r ( ) r V a R 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 15

16. Practical solution of Kohn-Sham equations in solids 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 16

17. Muffin tin potential construction http://journals.aps.org/pr/abstract/10.1103/PhysRev.51.846 Augmented Plane Wave (APW) approximation 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 17

18. Muffin tin potential construction = a a r r R ( ) (| |) V V = ( ) r V V 0 http://www.jara.org/de/research/jara-hpc/forschung/details/simlab-ai/performance-modeling-for-linear-algebra-in-fleur/ 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 18

19. Muffin tin model continued: a a a a R r R r R (| | ) for otherwise | | V = ( ) r V V 0 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 http://journals.aps.org/prb/abstract/10.1103/PhysRevB.12.3060 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 19

20. 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 20

21. Modern software based on LAPW method -- http://www.wien2k.at/ 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 21

22. http://elk.sourceforge.net/ http://exciting-code.org/ 2/18/2015 PHY 752 Spring 2015 -- Lecture 15 22