Solid State Physics Lecture on Density Functional Theory
Explore the concepts of density functional theory, Thomas-Fermi theory, and practical considerations in solid state physics. The lecture covers topics such as electronic energy expressions, jellium extensions, and explicit density functional models based on the extended jellium model.
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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 13: Reading: Chapter 9 in MPM Density functional theory 1. Thomas-Fermi theory 2. Some practical considerations of density functional theory 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 1
Note: Take-home exam scheduled for the week of March 2nd. 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 2
Summary of results for expressing the electronic energy in terms of the density: [ ] = + 3 ( ) ( ) v r r [ ] [ ] n E E vn F d r n v = + + + [ ] n [ ] [ ] n [ ] n [ ] n E T n E E E v ee ex ext 3 ( ) ( ) r v r r [ ] n E d n ex t General forms 2 r r ( ) ( ') n r e n d r d r = 3 3 E e e r 2 ' 2 V 3 5 V ( ) 2 /3 = 2 5/3 [ ] 3 T n n Special for jellium 2 m 2 3 e n ( ) 1/ 3 2 [ ] = n 3 E n ex 4 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 3
Jellium extension containing V Box of volume electrons N i i 2 V 33 5 ( ) 2/3 = + 2 5/3 with energy E V in i i 2 m Local potential at site i Slowly varying potential and electron density: i j l k 2 V 33 5 ( ) 2/3 = + 2 5/3 ( ) r ( ) r ( ) r E V n 2 m 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 4
Explicit density functional based on extended jellium model = + + + [ ] n [ ] [ ] [ ] n [ ] n E T n E n E E v ee ex ex t 2 3 5 ( ) 2/3 = 2 3 5/3 r [ ] 3 ( ) T n d r n 2 m 2 r r ( ) ( ') n r e n = 3 3 E d r d r e e r 2 ' 2 3 4 d e ( ) 1/3 2 3 4 /3 r [ ] = 3 ( ) r n E n d ex 3 ( ) ( v r r [ ] n ) E r n e xt 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 5
Thomas-Fermi theory Thomas, Proc. Cambridge Phil. Soc. 23, 542 (1927) Fermi, Z. Physik 48, 73 (1928) Minimize [ ] subject to the constraint n E v = 3 ( ) r [ ] d r n N n [ ] n n [ ] n n E N = v 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 6
Fermi-Thomas-Dirac equation: ( 3 2 m 2 r ( ') n r ) 2/3 + 2 2/3 2 3 ( ) r n e d r r ' 2 e ( ) 1/3 + = 2 1/3 ) r ( ) r 3 ( n v r Relationship of electron density ( ) to local potential ( ) in Fermi-Thomas V n i j l r k ( ) 2/3 2 2 ( ) r 3 n + = ( ) r picture: 0 V 2 m 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 7
Unfortunately, Thomas-Fermi theory predicts that atoms never bind into molecules. Modern extensions Orbital Free DFT 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 8
Kohn-Sham formulation of density functional theory 2 ( ) ( ) Let i i = r r n n r r Resulting equations for orbitals ( ): + + [ ] ( ) ee n i 2 + = i i 2 ( ) r ( ) r r ( ) r ( ) r ( ) V V v ee ex i 2 m r ( ') E n = = 2 3 r V e d r ee r ' 2 [ ] n n E e ( ) 1/ 3 = = 2 1 3 / ( ) r ( ) r 3 V n ex ex [ ] n n E = = ( ) r ( ) r V v ext ext 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 9
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 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 10
Kohn-Sham formulation of density functional theory Results of self-consistent calculations Variationally determined -- Ground state energ Electron density y [ ] ( ) n r E vn Some remaining issues Theory for Eexc[n] still underdevelopment This formalism does not access excited states Strongly correlated electron systems are not well approximated 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 11
Examples of Eexc -- Local Density Approximation (LDA) 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 12
Examples of Eexc -- Generalized Gradient Approximation (GGA) 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 13
Some details of the Generalized Gradient Approximation 2 2 2 n x n y n z + + Note that n n / n x = ( ) / n x n 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 14
Comparison of LDA and GGA binding energy curves 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 15
LDA vs GGA performance wrt normal modes of vibration A: B. N. Mavrin et al, J. Exp. Theor. Phys. 96,53 (2003); B: F. Harbach and F. Fischer, Phys. Status Solidi B 66, 237 (1974) room temp. C: Ref. B at liquid nitrogen temp.; D: L. Popovi et al, J. Raman Spectrosc. 34,77 (2003). 2/13/2015 PHY 752 Spring 2015 -- Lecture 13 16
