Electronic Structure Analysis with LCAO Methods
Explore the intricacies of electronic structure analysis using Linear Combination of Atomic Orbital (LCAO) methods, including the Slater-Koster analysis and Wannier representation. Dive into the concepts of Bloch waves, lattice translations, and angular variations to uncover the fundamentals of LCAO basis functions with Bloch symmetry. Understand how to apply these methods to analyze bond types in diatomic molecules, enhancing your comprehension of solid-state physics.
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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 9: Reading: Chapter 8 in MPM; Electronic Structure 1. Linear combination of atomic orbital (LCAO) method 2. Slater and Koster analysis 3. Wannier representation 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 1
2/04/2015 PHY 752 Spring 2015 -- Lecture 9 2
2/04/2015 PHY 752 Spring 2015 -- Lecture 9 3
Linear combinations of atomic orbitals (LCAO) methods for analyzing electronic structure Ra Bloch wave: k ( ) r ( ) r = k r i e u k n n periodic function = + a a R Let T lattice translation basis vector Bloch wave identity: + k r T LCAO basis functions with Bloch symmetry: k T ( ) ( ) r ( ) = k T ( ) i e a + k i T = a nlm a nlm a r r ( ) e T k n n 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 4
LCAO methods -- continued LCAO basis functions with Bloch symmetry: = T ( ) = = k T r a n m k l i a nlm a ( ) r ( ) r r r Example for Ce ( ) e T 100 nlm k=0 k= /2a k= /a 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 5
LCAO methods -- continued angular variation http://winter.group.shef.ac.uk/orbitron/ l=0 l=1 l=2 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 6
LCAO methods -- continued angular variation While, for atoms the z axis is an arbitrary direction, for diatomic molecules and for describing bonds, it is convenient to take the z axis as the bond direction. Atom Bond symbol l=0 m=0 s l=1 m=0 p m= 1 p l=2 m=0 d m= 1 d m= 2 d symbol l=0 =0 l=1 =0 =1 l=2 =0 =1 =2 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 7
LCAO methods -- continued bond types pp ss pp dd 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 8
2/04/2015 PHY 752 Spring 2015 -- Lecture 9 9
LCAO methods -- continued Slater-Koster analysis LCAO basis funct ions with Bloch symmetry: ( ) ( ) a T + k i T = a nlm k a nlm a r r ( ) e T Approximate ( ) k r Bloch wavefunction k : = k a nlm a nlm r ( ) X a nlm In this basis, we can estimate the electron energy by variationally computing the expectation value of the Hamiltonian: k k H k k = E k Terms in this expansion have the form: T ( ) ( ) ( ) a b + k i T a n l m b a nl a r e H r T ' ' ' m 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 10
LCAO methods Slater-Koster analysis -- continued ( ) ( ) ( ) a b T + k i T a n l m b a nlm a r e H r T ' ' ' Notation in Slater-Koster paper ( k ( ' ' ' n l m r ) + ) = ( + + ) a b T l m n a b a n a : H r T lm 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 11
LCAO methods Slater-Koster analysis -- continued Simple cubic lattice on site NN ( ) = + + + ( / ) s s (000) 2 (100) cos cos cos E E , , s s NNN s s ( ) + cos cos + cos cos + +4 (110) cos cos ... E , s s ( ) + = + + a b T x y z ap aq ra p q r = = = l m n + + + + + + 2 2 2 2 2 2 2 2 2 p q r p q r p q r 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 12
LCAO methods summary ions with Bloch symmetry: LCAO basis funct ( ) ( ) a T + k i T = a nlm k a nlm a r r ( ) e T Approximate ( ) k r Bloch wavefunction k : = k a nlm a nlm r ( ) X a nlm In this basis, we can estimate the electron energy by variationally computing the expectation value of the Hamiltonian: k k H k k = E k Terms in this expansion have the form: T ( ) ( ) ( ) a b + k i T a n l m b a nlm a r e H r T ' ' ' ( ) ( ) ( ) a b T + k i T a n l m b a nlm a r and also e r T ' ' ' 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 13
LCAO methods summary LCAO basis functions with Bloch symmetry: k T ( ) ( ) a + k i T = a nlm a nlm a r r ( ) e T Is there a best choice for atom-centered functions? Introduction to the Wannier representation 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 14
Wannier representation of electronic states Note: This formulation is based on the relationship between the Bloch and Wannier representations and does not necessarily imply an independent computational method. Bloch wave: = k k r r Orthonorma ty: li Bloch wave identity: + k r T ( ) ( ) ( ) ( ) r = k r k T i i e e u k n n n n ( ) r ( ) r ( ) = 3 k k ' k ' ' k n n nn T Wannier function in lattice cell , associated with band is given by: V d k e n = k T 3 i r T ( ) r W ( ) ( ) k n n 3 2 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 15
Wannier representation of electronic states -- continued T Wannier function in lattice cell , associated with band is given by: V d k e n = k T 3 i r T ( ) r W ( ) ( ) k n n 3 2 Note that r : = T r T ( ) ( ') W W TT ' ' n n nn Comment: Wannier functions are not unique since the the Bloch function may be multiplied by a k-dependent phase, which may generate a different function Wn(r-T). 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 16
Example from RMP 84, 1419 (2012) by Mazari, Mostofi, Yates, Souza, and Vanderbilt 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 17
