Physics 250-06 “Advanced Electronic Structure”

Solution of Electronic Structure for Muffin-Tin Potential

Contents:

1. Augmented Plane Wave Method (APW) of Slater

2. Green Function Method of Korringi, Kohn, and Rostoker (KKR)

3. Tail Cancellation Condition in KKR Method (Andersen)

b. J. M. Ziman, the Calculation of Bloch functions, Solid State Phys. 26, 1 (1971).

a. J. M. Ziman, Principles of the Theory of Solids (Chapter 3)

Solving Schrodinger’s equation for solidsSolving Schrodinger’s equation for solids

2( ( ) ) ( ) 0kj kjV r E r Solution of differential equation is required

Properties of the potential ( )V r

LCAO method is great since it gives as Tight-BindingDescription, however:

Problems with LCAO Method:

Atomic wave functions are tailored to atomic potential(not to self-consistent potential). Atomic wave functions have numerical tails whichare difficult to handle with.

Muffin-tin sphere MTS

Muffin-tin potential

0

( ) ( ),

( ) ( ) ,

MT sph MT

MT sph MT MT

V r V r r S

V r V S V r S

Space is partitioned intonon-overlapping spheres and interstitial region.potential is assumed to be spherically symmetric insidethe spheres, and constant in the interstitials.

0V

Muffin-tin Construction:

Muffin-tin sphere MTS

E

With muffin tin potential solutions are known:

Radial Schroedinger equation inside the sphere

ˆ( , ) ( )ll lmr E i Y r

2( ( ) ) ( , ) 0rl sph lV r E r E

Helmholtz equation outsidethe sphere: spherical waves

2 20

20

( ) ( , ) 0rl lV E r

E V

2( , ) ( ) ( )l l l l lr a j r b h r

( )i k G re Or plane waves

Hence two methods have been invented to solvethe electronic structure problem:

• Augmented plane wave method (APW) of Slater using planewave representation for wave functions in the Interstitial region

• Green function method of Korringi, Kohn and Rostocker(KKR) which uses spherical waves in the interestitialregion.

You can also say it is augmented spherical wave (ASW) methodbut historically this name ASW appeared much later.

APW Method of SlaterAPW Method of Slater

( ) ( )kj kkj r A r

( )( )k i k G rr e

Trying to solve variationally:

2( ( ) ) ( ) 0kj kjV r E r

in the interstitial region.

( ) ( , ) ( )kl Lr r E Y r inside the spheres

Construction of augmented plane wavesConstruction of augmented plane waves

( )i k G re

( )

*4 (| | ) ( ) ( )

i k G r

l L LL

e

j k G r Y r Y k G

S S S S

*

( , )

4 ( , ) ( ) ( )

k G

k Gl l L L

L

r E

r E a Y r Y k G

( ) ( )kj kkj G G

G

r A r

*( , ) 4 ( , ) ( ) ( )k k GG l l L L

L

r E r E a Y r Y k G Resulting APW

is used to construct Hamiltonian and overlapmatrices and lead to generalized eigenvalue problem:

2' | |k k k

G G GG

V E A

' '( ) 0k k kjG G G G G

G

H EO A

Discussed in J. M. Ziman, the Calculation of Bloch functions, Solid State Phys. 26,1. (1971)

Since APWs are not smooth additional surfaceIntergrals should be added in the energy functionalLeading to variational solutions.

Matrix elements of H and O inside the spheres

2' '

2 ' *

2 2

0

2' '

2 ' *

2 2

0

( ) | |

(4 ) ( ') ( )

( , )( ( )) ( , )

( ) | |

(4 ) ( ') ( )

( , )

k kG G G G S

k G k Gl l L L

L

S

l rl l

k kG G G G S

k G k Gl l L L

L

S

l

H k V

a a Y k G Y k G

r E V r r E r dr

O k V

a a Y k G Y k G

r E r dr

=E

=1

Matrix Elements in the interstitial

'

2 ( ') 2 ( )' int int

( ') 2 ( )

( ') 2 ( )

2 2 *'

2

0

( )

| | | ( ) |

| ( ) |

| ( ) |

(| | ( )) (4 ) ( ') ( )

(| ' | )( ( )) (| |

c

G G

k k i k G r i k G rG G

i k G r i k G r

i k G r i k G rS

G G L LL

S

l rl l

H k

V e V S e

e V S e

e V S e

k G V S Y k G Y k G

j k G r V S j k G

2)r r dr

Integrals between Bessel functions

2 2 2 2 2 2

0 0

2 2

0

2 2

2 2

2 2

( ' )( ) ( ) ( ' )( ) ( )

( ' ) ( )

( ' ) | | ( ) ( ' ) | ( )

( ) | | ( ' ) ' ( ' ) | ( )

( ' ) ( ' ) | ( )

S S

l rl l l rl l

S

l l

l rl l l l

l rl l l l

l l

j r j r r dr j r j r r dr

j r j r r dr

Consider

j r j r j r j r

j r j r j r j r

j r j r

2 2

22 20

( ' ) | | ( ) ( ) | | ( ' )

[ ' ( ' ) ( ) ( ' ) ' ( )] [ ( ' ) ( )]

[ ( ') ( )]( ' ) ( )

( ' )

l rl l l rl l

l l l l l l

Sl l

l l

j r j r j r j r

S j S j S j S j S W j S j S

W j jj r j r r dr

Major difficulty of APW approach: implicit energy dependence

2'

2'

| |

( ) | | ( ) 0

k k kG G G

G

k k kG G G

G

V E A

E V E E A

' 'det[ ( ) ( )] 0k kG G G GH E EO E

Therefore, to find the roots, the determinant should be evaluated as a function of E on some energy grid and seeat which E it goes to zero:

In practice, this determinant is very strongly oscillating which can lead to missing roots!

Each APW by construction is continuous but not smooth.

Alternative view on APWs: kink cancellationAlternative view on APWs: kink cancellation

*( , ) 4 ( , ) ( ) ( )k Gk G l l L L

L

r E r E a Y r Y k G

( ) *4 (| | ) ( ) ( )i k G rl L L

L

e j k G r Y r Y k G for r>S

for r<S

Request that linear combination of APWs becomes smooth:

( ) ( )kj kkj G G

G

r A r This occurs for selected set of energies only!

Discussed in J. M. Ziman, the Calculation of Bloch functions, Solid State Phys. 26,1. (1971)

Green Function (KKR) Method

2

2

( ( ) ) ( ) 0

( ) ( ) ( ) ( )

k

k k

V r E r

E r V r r

First rewrite Schroedinger equation to intergral form

20

| '|

0

( ) ( , ', ) ( ')

1( , ', )

4 | ' |

i E r r

E G r r E r r

eG r r E

r r

0( ) ( , ', ) ( ') ( ') 'k kr G r r E V r r dr

Introduce free electron Green function (of Helmgoltz equation)

We obtain:

Green Function (KKR) Method

Second, using Bloch property of wave functions rewrite the integral over crystal to the integral over a single cell

0

0

0

| '|

| ' |

0

( ) ( , ', ) ( ') ( ') '

( , ' , ) ( ' ) ( ' ) '

( , ' , ) ( ') ( ') '

1( ') ( ') '

4 | ' |

1{ } ( ') ( ') '

4 | ' |

cell

cell

cell

ce

k kV

kR

ikRk

R

i E r r

k

i E r r RikR

kR

r G r r E V r r dr

G r r R E V r R r R dr

e G r r R E V r r dr

eV r r dr

r r

ee V r r dr

r r R

ll

Use expansion theorems:

| '|

| ' |

' ' ''

''' ' '' ''

''

( ) ( ' ) ( ) ( ')| ' |

( ) ( ) ( ) ( ') ( ')| ' |

( ) ( ) ( )

i r r

l l L LL

i E r r R

l L LL l LLL

LLL LL l L

L

ej r h r Y r Y r

r r

ej r Y r F R j r Y r

r r R

where

S R C h R Y R

Summation over lattice is trivial

| ' |

' ' '0 '

''' ' '' ''

0 ''

( ) ( ) ( ) ( ') ( ')| ' |

( ) ( ) ( )

i E r r RikR

l L LL l LR LL

ikR LLL LL l L

R L

ee j r Y r F k j r Y r

r r R

F k e C h R Y R

are called structure constants

Consider now the solutions in the form of linear combinationsof radial Schroedinger’s equation with a set of unknown coefficients:

( ) ( , ) ( )kk L l L

L

r A r E Y r

0

0

( ) ( , ', ) ( ') ( ') '

[ ( , ) ( ) ( , ', ) ( ') ( ', ) ( ') '] 0

k kV

kl L l L LV

L

r G r r E V r r dr

r E Y r G r r E V r r E Y r dr A

Using expansion theorems and many other trickswe finally obtain the conditions of consistency:

' ' '

' ( )[ ( , ) ] ( , ) 0

' ( )k kl l l

L L L L L L L LL Ll l l

h h D ES E k E A M k E A

j j D E

Consequences of KKR equations:

'

'

( , ) 0

det ( , ) 0

kL L L

L

L L

M k E A

M k E

Spectrum is obtained from highly non-linear eigenvalue problem

' ' '

' ( )( , ) ( , )

' ( )l l l

L L L L L Ll l l

h h D ES k E E M k E

j j D E

Information about the crystal structure and potentialis split:

Structure constants Potential Parameters

Muffin-tin sphere MTS

E

Reformulation of KKR method as tail-cancellationcondition (Andersen, 1972)

Radial Schroedinger equation inside the sphere

ˆ( , ) ( )ll lmr E i Y r

2( ( ) ) ( , ) 0rl sph lV r E r E

Helmholtz equation outsidethe sphere: spherical waves

2 20

20

( ) ( , ) 0rl lV E r

E V

MTSMTS

Solution of Helmholtz equation outsidethe sphere

2( , ) ( ) ( )l l l l lr a j r b h r

where coefficients provide smoothmatching with

,l la b( , )l r E

{ , }

{ , }

{ , } ' '

l l l

l l l

a W j

b W h

W f g f g g f

Linear combinations of local orbitals should be considered.

( , ) ( , )k ikRL L

R

r E e r R E

ˆ( , ) ( , ) ( ),

ˆ( , ) { ( ) ( )} ( ),

lL l L MT

lL l l l l L MT

r E r E i Y r r S

r E a j r b h r i Y r r S

However, it looks bad since Bessel does not fall off sufficiently fast! Consider instead:

ˆ( , ) { ( , ) ( )} ( ),

ˆ( , ) ( ) ( ),

lL l l l L MT

lL l l L MT

r E r E a j r i Y r r S

r E b h r i Y r r S

Our construction is thus

Using expansion theorem, the Bloch sum is trivial:

0

' ''

' ' ''

( , ) ( , )

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( )

( , ) ( , ){ ( ) }

k ikRL L

R

ikRL l L l L

R

kL l L L L L l

L

kL L L L l L L l

L

r E e r R E

r E a j r e b h r R

r E a j r j r S b

r E j r S b a

''' ' ''

0 ''

( ) ( , )k ikR LL L LL L

R L

S e C h R

where structure constants are:

ˆ( , ) ( )} ( )

ˆ( , ) ( )} ( )

ˆ( , ) ( ) ( )

lL l L

lL l L

lL l L

r E r i Y r

j r E j r i Y r

h r E h r i Y r

Convenient notations which aquire spherical harmonics inside spherical functions:

A single L-partial wave

' ' ''

( , ) ( , ) ( , ){ ( ) }k kL L L L L l L L l

L

r E r E j r S b a is not a solution:

2( ( ) ) ( , ) 0kMT LV r E r E

( , ) ( , ) ( )k k kL L L L k

L L

A r E A r E r However, a linear combination can be a solution

Tail cancellation is needed

' '{ ( ) ( ) ( )} 0k kL L l L L l L

L

S E b E a E A which occurs at selected , kj

kj LE A

( , )kL r E

is a good basis, basis of MUFFIN-TIN ORBITALS (MTOs),

which solves Schroedinger equation for MT potential exactly!

For general (or full) potential it can be used with variational principle

2' | | 0k k kj

L MT NMT kj L LL

V V E A

( ) ( , )kj kkj L L kj

L

r A r E