PHY 752 Fall 2015 -- Lecture 16 19/30/2015

PHY 752 Solid State Physics11-11:50 AM MWF Olin 103

Plan for Lecture 16:Reading: Chapter 5 in GGGPP

Ingredients of electronic structure calculations1. Plane wave basis sets2. Construction of pseudopotentials3. Projector Augmented Wave (PAW) method

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Some practical considerations in electronic structure calculations

( ) ( ) ( )Plane wav

Bloch

e representation

( )

theorem

( )

i in n n

in n

e e u

C e

k T k rk k k

k G rk k

G

r T r r

r G

22

In practice, summation is truncated:

2 cutEm

k G

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Kohn-Sham equations (assuming “local” potential)

2 2

2

( )

nk nk nk

ieff

eff

eff V e

V Em

V

G r

G

r r r

r G

3

max

( )

Convenient representation provided that

( ) for

ieff

e

f

f

ef

f

V d r eV

V G

G rG

G G

r

òStrong motivation for the development of pseudopotentials

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How can we construct a pseudopotential?

Norm-conserving pseudopotentials

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PHY 752 Fall 2015 -- Lecture 16 99/30/2015r (Bohr)

rV(r

) (B

ohr *

Ry

Self-consistent full potential for C

Local pseudopotential for C

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Some details of pseudopotential construction from Troullier and Martins

Atomic Kohn-Sham equation for all-electron potential (atomic (Hartree) units

AEAEAE

Corresponding atomic Kohn-Sham equation for pseudoptential

PP PPPP

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Conditions of AE and PP functions:

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Benefit of these conditionsPseudo-wavefunction accurately represents correct functional form in valence regionKohn-Sham equations correctly solved at energy of reference state and its neighborhood

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Troullier-Martins recipe

Conditions to determine coefficients cn

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Conditions on coeffients, continued:

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rV(r

) (B

ohr *

Ry

Self-consistent full potential for C

Local pseudopotential for C

Example of Troullier-Martin pseudopotential for C

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Properties of pseudopotential and corresponding pseudo-wavefunctions

Correct logarithmic derivatives at energy E and at nearby energies

(1( ln ((

)) ))

c c

cr

nln

c rl nl

nl

dP rdL P r P rdr P r dr

Pseudopotential form in terms of polynomial p(r):

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Construction energyE (Ry)

L[P

]

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2

2 2

2

2 2

( 1) ( ) ( ) ( )

Formally take energy

Useful relationship in Rydberg uni

derivative:

( )( 1) ( ) ( )

ts

i i i i i i

i i

i i i i

i i

ieff l l l

lief

in n n

nin n

nf l l

l

l ld V r P r E P rdr r

dP rl ld V r E P rd Er dr

2 2

0

( ) ( )] ( )[c

i i i i i i

i ic

r

n l n l n ln l r

r r ddP L P PdE

r r The construction ensures that

PS and AE have same log derivatives near E

What about other partial waves? (Non-local contributions to pseudopotential)

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PAW representation of Kohn-Sham orbitals

Evaluation of the energy of the system

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Example of atomic basis and projector functions

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Clever secret of PAW method; partitioning of planewave and one center contributions

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Construction of atom centered basis and projector functions: KS( ) ( ) 0a a

i i r rH

rc

2s2ss

s

r

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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):

41 2

1( )(

Let )

i ml

cmmi

i c

rr C r r

rr

r

r

KS

1

( ) ( )

Calculate overlap mat

Construct auxiliary function:

Form projector function:

rix:

( ) ( )

i i i

ij i j

i j jij

r r

B

rp r

B

H

This construction ensur

(

es that

)

a aj i ijP r

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Example for C

2s2s 2sp

r

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Example for C -- continued

s

s

sp

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Example for C -- continued log derivatives for l=0

E

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Example for C -- continued log derivatives for l=1

E

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Example for C -- continued log derivatives for l=2

E