Physics 250-06 “Advanced Electronic Structure”

LMTO family: ASA, Tight-Binding and Full Potential Methods

Contents:

1. ASA-LMTO equations

2. Tight-Binding LMTO Method

3. Full Potential LMTO Method

4. Advanced Topics: exact LMTOs, NMTOs.

Linear Muffin-Tin OrbitalsLinear Muffin-Tin Orbitals

Consider envelope function as

Inside every sphere perform smooth augmentation

( ) ( , )k ikRL L

R

r e h r R

' ''

' ''

( ) ( , ) ( , ) ( )

( ) ( ) ( ) ( )

k kL L L L L

L

k h j kL L L L L

L

r h r j r S

r r r S

which gives LMTO construction.

SSSS

LMTO definition ( dependence is highlighted):

' ''

int

( ) ( ) ( ) ( ),

( ) ( , ),

k h j kL L L L L MT

L

k ikRL L

R

r r r S r

r e h r R r

Variational EquationsVariational Equations

which should be used as a basis in expanding

Variational principle gives us matrix eigenvalue problem.2

' ' | | 0k k kjL kj L L

L

V E A

( ) ( )kj kkj L L

L

r A r

Integrals between radial wave functions:

LMTO HamiltonianLMTO Hamiltonian2

' ' ' '

2' ' '' ' '' ' ' ' '' ' '' '

'' ''

' '

' ' ' ' ' '

' '' '' ' ''''

| |

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

|

| ( ') | ( )

( ') |

k k kL L L L

h j k h j kL L L L L L L L

L L

h hL L l l

j h k h j kl l LL l l L L

k jL L l l

L

H V

r r S V r r S

S S

S

'' ( )j kL LS

one-center integrals

two-center integrals

three center integrals

' ' '

2' '

2 2

0

2

0

| ( , ) ( , ) | ( , ) ( , )

( , ) 1

( , ) ( , ) 0

l l l l l l l l l l l l l l

l l l l l

S

l l

S

l l l l

a r E b r E a r E b r E

a a b b

r E r dr

r E r E r dr

How linearization works:How linearization works:

( ) ( , ) ( , { })kj k kj kkj L L L L l

L L

r A r E A r D E

KKR-ASA equations produce energy dependent orbitals

Energy dependence can be factorized to linear order

( , ) ( ) ( , )k kL Lr D D r D const

As a result, a prefactor (D) can be combined with the variational coefficients A! Therefore, energy dependence of the orbitals cancels out with the normalization.

' ' ''

'' ' '

'

( , ) ( , ) ( , ){ ( ) ( )}

( ) 1( , ) ( ) ( ){ ( ) 2(2 1) }

( )

k kL L L L L L L l

L

l k lL L L L L L

L l

r E r E j r S P E

r D E lr E Y r S l

S D E l

Energy dependent MTO in KKR-ASA method

Keeping in mind the use of variational principle (not tail cancellation condition) let us augment the tails

( ) ( ) ( , 0)

( , 0) ( , 0) 1( , ') ( , ) ( , )

( , ') ( , ) ( , )

lL L

jL LL L Lj

L L L

rY r j r

Sj r j r

r E r D r lS E S D S l

'' '

' '

( , ') 1( , ) ( , ) { ( ) 2(2 1) }

( , ')k kLL L L L L L

L L

r l D lr D r D S l

S l D l

Energy dependent MTO in KKR-ASA method becomes

Perform linearization:

( , ) ( , ) ( ) ( , ) ( , )L L L Lr D r E D r E r D

is the function having given logarithmic derivative D:

[ '( , ) ( ) '( , )]'( , ) / ( , )

[ ( , ) ( ) ( , )]

S S E D S ED S S D S D

S S E D S E

We obtain

( , )( )

( , )

S E D DD

S E D D

( ) ( )( , ) ( , 1)

( ) ( 1)k kl lL L

l l

l Dr D r l

l l

Then one can show that the energy dependence hidden in D(E)factorizes

''

' '

( , ')( , 1) ( , 1) ( )

( , ')k kLL L L L

L L

r lr l r l S

S l

' ''

( ) ( ) ( ) ( )k h j kL L L L L

L

r r r S

where LMTOs are given by (kappa=0 approximation)

Or for any fixed kappa:

Accuracy of LMTOs:

LMTO is accurate to first order with respect to (E-E) withinMT spheres.

LMTO is accurate to zero order (2 is fixed) in the interstitials.

Atomic sphere approximation: Blow up MT-spheresuntil total volume occupied by spheres is equal to cell volume.Take matrix elements only over the spheres.

Multiple-kappa LMTOs: Choose several Hankel tails, typically decaying as 0,-1,-2.5 Ry. Frequently, double kappa(tripple kappa) basis is sufficient.

Tight-Binding LMTOTight-Binding LMTO

Tight-Binding LMTO representation (Andersen, Jepsen 1984)

LMTO decays in real space as Hankel function which depends on 2=E-V0 and can be slow.

Can we construct a faster decaying envelope?

Advantage would be an access to the real space hoppings,perform calculations with disorder, etc:

' ' ' '

( ) ( )

( )

k ikRL L

R

k ikRL L L L

R

r e r R

H e H R

Tight-Binding LMTOTight-Binding LMTO

Any linear combination of Hankel functions can be the envelope which is accurate for MT-potential

where A matrix is completely arbitrary. Can we choose A-matrixso that screened Hankel function is localized?

Electrostatic analogy in case 2=0

Outside the cluster, the potential may indeed be screened out.The trick is to find appropriate screening charges (multipoles)

( )' '

'

( , ) ( ) ( , )L LL LRL

h r A R h r R

1/ lLZ r

' 1' /

lLM r

( ) ~ 0scrV r

Screening LMTO orbitals:Screening LMTO orbitals:

( )' ' '

' '

( ) ( ')L LRL R LR L

h r R A h r R

' ''

( ) ( ) ( )L L L LL

h r R j r S R Unscreened (bare) envelopes (Hankel functions)

Screening is introduced by matrix A

Consider it in the form

( )' ' ' ' ' ' 'LRL R L L RR l L R LRA S

where alpha and S_alpha coefficents are to be determined.

( ) ( )' ' ' ' ' ' '

' ' ' '

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

' ' '' '

( ) ( ') ( ) ( ')

( ) ( '') ( ')

L LRL R L L L l LRL RR L R L

L L l LRL R L l LRL RL R R L

h r R A h r R h r R h r R S

h r R h r R S h r R S

We obtain when r is within a sphere centered at R’’

'' '' ''''

( ) ( '')L L L R LRL

h r R j r R S

R

R’

R’’

r

r-R

r-R’

r-R’’

' '' '' '' ' '''

( ') ( '')L L L R L RL

h r R j r R S

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

'' ' '' ' '' '

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

' '' ' '' '

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

( '') ( '') ( )

L L L R LR L l LRL R L L R L R l LRL RL L L R R L

L l LRL R L L R L R LRL R l LRL RL L R R L

h r R j r R S h r R S j r R S S

h r R S j r R S S

= ' 'LRL RA

Demand now that

( ) ( )' ' '' ' ''

'

( ) ( )' ' ''

'

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

( '')

L L l L LRL RL

L LRL RL

h r R h r R j r R S

j r R S

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

' '' '

( )L R L R LRL R l LRL R L R LRR R L

S S S

we obtain one-center like expansion for screened Hankel functions

( )'( ) ( ) ( )L L l Lj r h r j r

where S_alpha plays a role of (screened) structure constantsand we introduced screened Bessel functions

Screened structure constants are short ranged:( ) ( )

( )

( )

/( )

S I S S

S S I S

For s-electrons, transofrming to the k-space2

( ) 2

( ) 1/

( ) ( ) /( ( )) 1/( )

S k k

S k S k I S k k

Choosing alpha to be negative constant, we see that it playsthe role of Debye screening radius. Therefore in the real space screened structure constants decay exponentially

while bare structure constants decay as

( ) ( ) exp( / )S R R

( ) 1/S R R

Screening parameters alpha have to be chosen from the condition of maximum localization of the structure constantsin the real space. They are in principle unique for any given structure. However, it has been found that in many cases there exist canonical screened constants alpha(details can be found in the literature).

Since in principle the condition to choose alpha is arbitrarywe can also try to choose such alpha’s so that the resultingLMTO becomes (almost) orthogonal! This would leadto first principle local-orbital orthogonal basis.

In the literature, the screened mostly localized representationis known as alpha-representation of TB-LMTOs. The Representaiton leading to almost orthogonal LMTOs is known as gamma-representation of TB-LMTOs. If screeningconstants =0, we return back to original (bare/unscreened) LMTOs

Tight-Binding LMTOTight-Binding LMTO

Since mathematically it is just a transformation of thebasis set, the obtained one-electron spectra in all representations (alpha, gamma) are identicalwith original (long-range) LMTO representation.

However we gain access to short-range representationand access to hopping integrals, and building low-energytight-binding models because the Hamiltonian becomesshort-ranged:

' ' ' ' ( )k ikRL L L L

R

H e H R

Advanced Topics: FP-LMTO MethodAdvanced Topics: FP-LMTO Method

2' ' | | 0k k kj

L kj L LL

V E A

Problem: Representation of density, potential, solution of Poissonequation, and accurate determination of matrix elements

with LMTOs defined in whole space as follows

' ''

int

( ) ( ) ( ) ( ),

( ) ( , ),

k h j kL L L L L MT

L

k ikRL L

R

r r r S r

r e h r R r

FP-LMTO MethodFP-LMTO Method

Ideas:

Use of plane wave Fourier transformsWeirich, 1984, Wills, 1987, Bloechl, 1986, Savrasov, 1996

Use of atomic cells and once-center spherical harmonicsexpansions Savrasov & Savrasov, 1992

Use of interpolation in interstitial region by Hankel functions Methfessel, 1987

At present, use of plane wave expansions is most accurate

int

ˆ( ) ( ) ( ),

( ) ,

lL L MT

L

iGrG

G

r r i Y r r S

r e r

To design this method we need representation for LMTOs

' ''

( )int

ˆ( ) ( ) ( ),

( ) ( ) ,

k k lL LL L MT

L

k i k G rL L

G

r r i Y r r S

r k G e r

Problem: Fourier transform of LMTOs is not easy since

int( ) ( , ),k ikRL L

R

r e h r R r

Solution: Construct psuedoLMTO which is regualt everywhere

and then perform Fourier transformation

int

( ) ,

( ) ( ) ( , ),

kL MT

k k ikRL L L

R

r smooth r S

r r e h r R r

( )L r

( , )Lh r

The idea is simple – replacethe divergent part inside the spheresby some regular function which matchescontinuously and differentiably.

What is the best choice of these regularfunctions?

The best choice would be the one when the Fourier transform is fastlyconvergent.

The smoother the function the faster Fourier transform.

( )L rWeirich proposed to use linearcombinations

This gives

Wills proposed to match up to nth order

This gives

with optimum n found near 10 to 12

( ) ( )l l l la j r b j r

4( ) ~ 1/L k G G

( ) ( ) ( ) ( ) ...l l l l l l l la j r b j r b j r c j r 3( ) ~ !!/ n

L k G n G

Another idea (Savrasov 1996, Methfessel 1996)Smooth Hankel functions

22 2( ) ( ) ( ) ( )l l r ll L Lh r i Y r r e i Y r

Parameter is chosen so that the right-hand side isnearly zero when r is outside the sphere.

Solution of the equation is a generalized error-like functionwhich can be found by some recurrent relationships.

It is smooth in all orders and gives Fourier transformdecaying exponentially

2 2| | / 4( ) ~ k GL k G e

2 2' ' ' '

2 2' ' ' '

| | | |

| | | |

MT

MT

k k k kL L V L L

k k k kL L V L L

V V

V V

Finally, we developed all necessary techniques to evaluatematrix elements

where we have also introduced pseudopotential

( )int

( ) ,

( ) ( ) ,

MT

i k G rG

G

V r smooth r S

V r V r V e r

Exact LMTOs

LMTOs are linear combinations of phi’s and phi-dot’s insidethe spheres, but only phi’s (Hankel functions at fixed ) inthe interstitials.

Can we construct the LMTOs so that they will be linear inenergy both inside the spheres and inside the interstitials(Hankels and Hankel-dots)?

Yes, Exact LMTOs are these functions!

Advanced TopicsAdvanced Topics

Let us revise the procedure of designing LMTO:

Step 1. Take Hankel function (possibly screened) as an envelope.

Step 2. Replace inside all spheres, the Hankel function by linear combinations of phi’s and phi-dot’s with the conditionof smooth matching at the sphere boundaries.

Step 3. Perform Bloch summation.SSSS

Design of exact LMTO (EMTO):

Step 1. Take Hankel function (possibly screened) as an envelope.

Step 2. Replace inside all spheres, the Hankel function by only phi’swith the continuity conditionat the sphere boundaries.

The resulting function is no longer smooth!

SSSS( )L LRK r R K

Step 3. Take energy-derivative of the partial wave

So that it involves phi-dot’s inside the spheres and Hankel-dot’sin the interstitials.

Step 4. Consider a linear combination

where matrix M is chosen so that the whole construction becomessmooth in all space (kink-cancellation condition)

This results in designing Exact Linear Muffin-Tin Orbital.

LRK

' ' ' '' '

( ) ( ) ( )kLR kLR LRL R L RL R

r K r M K r

Non-linear MTOs (NMTOs)

Do not restrict ourselves by phi’s and phi-dot’s, continueTailor expansion to phi-double-dot’s, etc.

In fact, more useful to consider just phi’s at a set ofadditional energies, instead of dealing with energy derivatives.

This results in designing NMTOs which solve Schroedinger’sequation in a given energy window even more accurately.

Advanced TopicsAdvanced Topics

MINDLab SoftwareMINDLab Software

http://www.physics.ucdavis.edu/~mindlab

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