Electronic Structure Calculations: Pseudopotentials and Kohn-Sham Equations
Explore the construction of pseudopotentials and the Kohn-Sham equations in electronic structure calculations, including topics like plane wave basis sets, Bloch theorem, and norm-conserving pseudopotentials. Learn about the development and usage of pseudopotentials in solid state physics.
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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 16: Reading: Chapter 5 in GGGPP Ingredients of electronic structure calculations 1. Plane wave basis sets 2. Construction of pseudopotentials 3. Projector Augmented Wave (PAW) method 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 1
9/30/2015 PHY 752 Fall 2015 -- Lecture 16 2
9/30/2015 PHY 752 Fall 2015 -- Lecture 16 3
9/30/2015 PHY 752 Fall 2015 -- Lecture 16 4
Some practical considerations in electronic structure calculations Bloch theorem + = k k r T = k T k r i i ( ) r ( ) r ( ) e representation = k G e e u k n n n Plane wav ( ) + k G r i ( ) r ( ) G C e k n n In practice, summation is truncated: 2 + 2 k G E cut 2 m 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 5
Kohn-Sham equations (assuming local potential) 2 2 ( ) r ( ) r ( ) r + = V E eff nk nk nk 2 m G = ( ) r ( ) eff V = G r i ( ) G V V e eff eff d r ( ) r G r 3 i G V e ef f Convenient representation provided that G G ( ) for V G max e f f Strong motivation for the development of pseudopotentials 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 6
How can we construct a pseudopotential? Norm-conserving pseudopotentials 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 7
9/30/2015 PHY 752 Fall 2015 -- Lecture 16 8
rV(r) (Bohr * Ry Local pseudopotential for C Self-consistent full potential for C r (Bohr) 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 9
Some details of pseudopotential construction from Troullier and Martins Atomic Kohn-Sham equation for all-electron potential (atomic (Hartree) units AE AE AE Corresponding atomic Kohn-Sham equation for pseudoptential PP PP PP 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 10
Conditions of AE and PP functions: 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 11
Benefit of these conditions Pseudo-wavefunction accurately represents correct functional form in valence region Kohn-Sham equations correctly solved at energy of reference state and its neighborhood 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 12
Troullier-Martins recipe Conditions to determine coefficients cn 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 13
Conditions on coeffients, continued: 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 14
Example of Troullier-Martin pseudopotential for C rV(r) (Bohr * Ry Local pseudopotential for C Self-consistent full potential for C r (Bohr) 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 15
Pseudopotential form in terms of polynomial p(r): Properties of pseudopotential and corresponding pseudo- wavefunctions Correct logarithmic derivatives at energy E and at nearby energies ( ) 1 ( dP r d dr ( ) = ( ) ln ( ) L P r P r nl dr n l c nl ) P r r nl c r c c 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 16
L[P] E (Ry) Construction energy 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 17
Useful relationship in Rydberg uni + ts 2 ( 1) l l d dr + = ( ) r ( ) r ( ) r V P E P i i r eff n i i l n i i l n i i l 2 2 Formally take energy derivative: ( ) r dP d + 2 ( 1) l l d d n E i i l + = ( ) r ( ) r V E P i i r ef f n i i l n i i l 2 2 r n i i l r d ( ) ( ) c 2 2 = ( ) r [ ( )] r ( ) r P L P d r P n l n l n l dE i i i i i i n l 0 r i i c The construction ensures that PS and AE have same log derivatives near E What about other partial waves? (Non-local contributions to pseudopotential) 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 18
PAW representation of Kohn-Sham orbitals Evaluation of the energy of the system 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 19
Example of atomic basis and projector functions 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 20
Clever secret of PAW method; partitioning of planewave and one center contributions 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 21
Construction of atom centered basis and projector functions: ( r H ) = KS( ) a i a i ( ) r 0 s 2s 2s rc s r 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 22
Construction of atom centered basis and projector functions continued (scheme developed by David Vanderbilt for ultra- soft pseudopotentials; for each l channel at at time): 4 1 2 ( ) ( ) i r Construct auxiliary function: + l m r C r r r i m c = Let r = 1 m i r r c ( ) = KS H ( ) r ( ) r i i i = Calculate overlap mat rix: B ij i j ( ) es that 1 B Form projector function: This construction ensur ( ) ( ) r p r i j ji j = a j a i r ( ) P ij 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 23
Example for C p 2s 2s 2s r 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 24
Example for C -- continued p s s s 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 25
Example for C -- continued log derivatives for l=0 E 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 26
Example for C -- continued log derivatives for l=1 E 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 27
Example for C -- continued log derivatives for l=2 E 9/30/2015 PHY 752 Fall 2015 -- Lecture 16 28
