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2/18/2015 PHY 752 Spring 2015 -- Lecture 15 1

PHY 752 Solid State Physics11-11:50 AM MWF Olin 107

Plan for Lecture 15:Reading: Chapter 10 in MPM

Numerical Realizations of Density functional theory

1. Electronic structure of atoms

2. Integration of the radial equations

3. Frozen core approximation

4. Extension of formalism to multi-center analysis

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Note: Take-home exam scheduled for the week of March 2nd.

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Kohn-Sham equations for spherical atom

2

2 2

( 1)( ) ( ) ( ) ( ) ( )

Equations in Rydberg units

i i i i i i

ie

ie exc n lnl ln

l ldV r V r v r P r P r

dr r

2

2 1 3

0

/1/3

[ ]( ) ( ) ( )

[ ] 2( ) ( ) (

12 ' '

)

[ ] 2( )

3

( )

eeee

excexc c

extex

r

t

r

E nr dr n r dr n r

n

E nr n r r

n

E n Zr v r

V r rr

V V

Vn r

p

p

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Self-consistent solution

Iteration 0

2

( )

( ) ( )

i

ii

n

r

r r

1 12

2 ( ) ( ) ( ) ( ) ( )2 i iee ex iV V vm

r r r r r

\alpha

21 1

1 1

( ) ( )

( ) ( ) (1 ) ( )

temp

temp

ii

n

nn x x n

r r

r r r

1

Numerical methods for solving the Kohn-Sham equations

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Example for carbon

2

22 2

22 2

2

1 2 2

1 2 2

=4 2 2 2

4 =

( ) ( )

( ) ( )

2

( )

( ) (2 ) ( )2

i i i i

s s p

s s

n l

p

l ni

n r w r

r r r

P r P rr

P r

p

p

ˆ(

For

)

spherically symmetric a

( ) ( )

( )

:

( )

tom

i i i i i i i

i i

i i

n l m n l l m

n l

n l

r Y

P rr

r

r r

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Results for carbon

r (Bohr)

Pnl(r

)

1s

2s

2p

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ncore(r)

nval(r)

r (Bohr)

nr2

/4p

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Example for Cu

1s22s22p63s23p63d104s1

r (Bohr)

ncore(r)

nvale(r)

nr2

/4p

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r (Bohr)

4s

4p3d

3s

3p

Results for Copper

Pnl(r

)

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Frozen core approximation

Example for Cu

1s22s22p63s23p63d104s1

r (Bohr)

ncore(r)

nvale(r)

nr2

/4p

core vale

vale

( ) ( )

Variationally optimize energy wrt

)

(

(

)

r n r

n r

n r n

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Systematic study of frozen core approximation in DFT

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.21.2222

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core vale

3 core vale

23 3 core-core core-vale vale-vale

core vale

( ) ( ) ( )

(

[ ] [ ] [ ] [ ]

[ ]

[ ]

) ( ')

2 '

= [ ]

v

ee ee e

ext ee exc

ext

exc ex

e

c

ee

E T E n E n E n

E n d

E n E

n

T T T

r v n n

e n nE d r

n

d r E E E

r r r

r r

r r

Variational relations for DFT in frozencore approximation (Kohn-Sham formulation)

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2

22

( ) ( )

Equations for orbitals ( ) :

( ) ( ) ( )2

For ii

i

i i iVm

n

r r

r

r r r

Practical solution to Kohn-Sham equations for single particle orbitals:

2

Numerical problem: near each nuclear center --

( )a

a

Z eV

r

r R

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Practical solution of Kohn-Sham equations in solids

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Muffin tin potential construction

http://journals.aps.org/pr/abstract/10.1103/PhysRev.51.846

Augmented Plane Wave (APW) approximation

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Muffin tin potential construction

0( )V Vr

(|( |)) a aV V r Rr

http://www.jara.org/de/research/jara-hpc/forschung/details/simlab-ai/performance-modeling-for-linear-algebra-in-fleur/

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Muffin tin model continued:

0

for ( )

otherwise

(| |

) |

|a a a aVV

V

r Rr

R r R

Problems with APW and KKR Green’s function schemes1. Difficult numerically to find Kohn-Sham energies ei

2. Potential form unrealistic especially for covalent materials

Linearized equations – O. K. Andersen

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.12.3060

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Modern software based on LAPW method --

http://www.wien2k.at/

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http://elk.sourceforge.net/

http://exciting-code.org/