Density Functional Theory in Solid State Physics
Explore the intricacies of density functional theory in solid-state physics, focusing on approximations to the many-electron problem, exchange energy and potential for jellium, and formal proofs of key theorems. Discover how this theory describes electron interactions and independent electron treatments. Dive into the relationships between electron density, external potentials, ground state energies, and unique functionals. Gain insights into the foundational principles behind density functional theory and its applications in understanding complex systems.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
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 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 1
Note: Take-home exam scheduled for the week of March 2nd. 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 2
2/11/2015 PHY 752 Spring 2015 -- Lecture 12 3
Density functional theory Describes the relationship between the many electron problem and independent electron treatments. Proof of theorm Estimates of F[n]. 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 4
Density functional theory -- continued 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 5
Hohenberg and Kohn: formal proof of basic theorem The system consists of electrons interacting via their mutual Coulomb repulsion in the presence of an "external" single particle potential ( ). H T V = + + Kinetic energy potential N r v Coulomb interaction U External Consider a many Fermion wavefunction The (many electron) density can be calculated from ( ) ( i r N d . = r r r ) n i 3 3 * r r = .... ( , ..... ) r r ( ) ( , ..... ) r r d r r r 1 1 2 1 2 N N i N i 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 6
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 . = Ground state energies: E H = an Note that ' d H + ' ' ' H ' E ' H = + ' ' ' E = ' ' H H V V = ' H V V ( ) + 3 ( ) r '( ) r ( ) r = E d r n v v ( ) + 3 r ( ) r ( ) r ' ( r n ) ' E E d v v 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 7
We can also show: = = + Note that ' ' E H H ' ' ' ' ' ' H H V + V = ' H V V ( ( ) ) + 3 r ( ) r r = ' '( ) n '( ) E d r v v + 3 r ( ) r r '( ) r ' '( r n ( ( ( ) ( v ) E + E d r v v ) ) 3 '( ) r ' ( ) r n ( ) E E d v v + 3 r r '( ) r ' '( ) r n E E d v v ( ) r '( ) if r r '( ) r ) n n v 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 8
The theorem implies that the ground state energy E can be considered as a functional of the density n(r) + [ ] = 3 ( ) ( r v r r [ ] n ) E F d n v 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. 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 9
Determination of F[n] for jellium V Assume we have particles in a volume with / : Kinetic energy contribution: 2 2 m N = V n N 2 2 2 5 F V V 2 2 k k = = 3 4 T d k ( ) ( ) 3 3 2 2 5 m ( ) 1/3 = 2 Recall that 3 k n F 2 V 33 5 ( ) 2/3 = 2 5/3 T n 2 m 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 10
Determination of F[n] for jellium -- continued The Coulomb (Hartree) contribution : 2 r r ( ) ( ' n r ) e n d r d r = 3 3 E ee r 2 ' 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 11
Determination of F[n] for jellium continued Exchange contribution within Hartree-Fock approximation Previously we have shown: r * ( ') r r ( ') r 2 e i k k n n = 3 * 3 ( ) r ( ) r E d r d r j j j i i k k ex n n 2 | | ' i j i i i j j j , i j For jellium: It can be shown that ( 2 ) 4/3 1 V 2 2 V 2 3 e n 2 4 F V 2 e k k r i = ( ) r e = = j E k n ( ) ( n ) ex 3 3 j j j 2 2 V 3 e ( ) 1 / 3 2 = 3 n 4 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 12
Summary of results for jellium: [ ] = + 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 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 13
Variational equations [ ] v n n E = 0 Constrain t on density: = = 3 r [ ] ( ) r n N n d N 2 = ( ) r ( ) r L e t n i i ( ): r Resulting equations for orbital + s i 2 + + = i i 2 ( ) r ( ) r ( ) r ( ) r r ( ) V V v ee ex i 2 m 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 14
Summary of Kohn-Sham equations: 2 = ( ) r ( ) r Let n i i r Resulting equations for orbitals ( ): + + i 2 + = i i 2 ( ) r ( ) r r ( ) r ( ) r ( ) V V v ee ex i 2 m r ( ') n r = 2 3 ( ) r V e d r e e r ' 2 e ( ) 1 / 3 2 ( ) = r 3 V n For jellium; exchange only e x 2/11/2015 PHY 752 Spring 2015 -- Lecture 12 15
