Andersen’s LMTO methodsavrasov.physics.ucdavis.edu/Works/Talks/lect18.pdf · Andersen’s LMTO method S. Y. Savrasov New Jersey Institute of Technology O. K. Andersen, PRB 12, 3060

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Andersen’s LMTO methodAndersen’s LMTO method

S. Y. Savrasov New Jersey Institute of Technology

O. K. Andersen, PRB 12, 3060 (1975)O. K. Andersen, O. Jepsen, PRL 53, 2571 (1984)

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q Solving Schroedinger’s equation for solids

q Basis sets & variational principle

q Envelope functions

q Linear muffin-tin orbitals

q Tight-binding LMTO representation

q Advanced topics

ContentContent

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Material ResearchMaterial Research

Computations of properties of materials

Ground state:• Density, total energy, magnetization• Volume & crystal structure• Lattice dynamics, frozen magnons

Excitations:• One-electron spectrum• Photoemission & Optics• Superconductivity• Transport

Ab initio Material Design

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Solving Schroedinger equation for solidsSolving Schroedinger equation for solids

2( ( ) ) ( ) 0DFT kj kjV r E rψ−∇ + − =

Density Functional Theory (Hohenberg, Kohn, 1964Kohn, Sham, 1965)

( )kj rψ describe one-electron Bloch states

( ) ( )ikRkj kjr R e rψ ψ+ =

due to periodicity of the potential

( ) ( )DFT DFTV r R V r+ =

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Solving Solving Schroedingers Schroedingers equation for solidsequation for solids

Many-Body Theory (Hartree-Fock, GW, DMFT, etc.)2 ( ) ( , ', ) ( ') ' ( )kj kj kj kjr r r r dr E rω ω ω ωψ ω ψ ψ−∇ + Σ =∫

( , ', )r r ωΣwhere is a non-hermitian complex frequency-dependent self-energy operator.

( ) ( )ikRkj kjr R e rω ωψ ψ+ =

due to periodicity of the self-energy:

Bloch property is retained

( , ' , ) ( , ', )r R r R r rω ωΣ + + = Σ

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2( ( ) ) ( ) 0kj kjV r E rψ−∇ + − =Solution of differential equation is required

Properties of the potential ( )V r

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Properties of the solutions: , ( )kj kjE rψ

2Zer

−

Core Levels

Energy Bands

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Basis Sets & Variational PrincipleBasis Sets & Variational Principle

Solving differential equation using expansion

( ) ( )kj kkj r A rα α

α

ψ χ= ∑( )k rαχwhere is a basis set satisfying Bloch theorem

( ) ( )k ikR kr R e rα αχ χ+ =

Two most popular examples:

• Plane waves, ,

• Linear combinations of local orbitals

( )( )k i k G rr eαχ +→ Gα →

( ) ( )k ikR

R

r e r Rα αχ χ= −∑

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Variational principle leads us to solve matrix eigenvalueproblem

2| |k k kjkjV E Aα β β

β

χ χ⟨ −∇ + − ⟩ =∑

where

2| |k k kH Vαβ α βχ χ= ⟨ −∇ + ⟩

( ) 0k k kjkjH E O Aαβ αβ β

β

− =∑

|k k kOαβ α βχ χ= ⟨ ⟩

is hamiltonian matrix

is overlap matrix

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Linear combinations of local orbitals will be considered.

( ) ( )k ikR

R

r e r Rα αχ χ= −∑


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Smart choice of is important.( )rαχ

Linear Combination of Atomic Orbitals (LCAO)

ˆ( ) ( , ) ( )ll nlm lmr r E i Y rαχ ϕ=

( ) ( )k ikRnlm nlm

R

r e r Rχ χ= −∑ to be used in variational principle

Atomic potential nlmE


NJITMuffin-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

= <

= = >

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

0V

NJIT Muffin-tin sphere MTS


E

Solve radial Schroedinger equation inside the sphere ˆ( , ) ( )l

l lmr E i Y rϕ2( ( ) ) ( , ) 0rl sph lV r E r Eϕ−∇ + − =

Solve Helmholtz equation outsidethe sphere

2 20

20

( ) ( , ) 0rl lV E r

E V

ϕ κ

κ

−∇ + − =

= −

2 ˆ( , ) ( )ll lmr i Y rϕ κ

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MTSMTS−

Solution of Helmholtz equation outsidethe sphere

2( , ) ( ) ( )l l l l lr a j r b h rϕ κ κ κ= +

where coefficients provide smoothmatching with

,l la b( , )l r Eϕ

, ,

, ' '

l l l

l l l

a W jb W hW f g f g g f

ϕϕ

==

= −

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Linear combinations of local orbitals should be considered.

( , ) ( , )k ikRL L

R

r E e r R Eχ χ= −∑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

χ ϕ

χ κ κ

= <

= + >is a bad choice 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

χ ϕ κ

χ κ

= − <

= >

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Bloch sum:

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

χ χ

ϕ κ κ

ϕ κ κ κ

ϕ κ κ δ

≠

= − =

− + − =

− + =

+ −

∑

∑

∑

∑

''' ' ''

0 ''

( ) ( , )k ikR LL L LL L

R L

S e C h Rκ κ≠

= ∑ ∑where structure constants are:

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A single L-partial wave

' ' ''

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

L

r E r E j r S b aχ ϕ κ κ δ= + −∑is not a solution:

2( ( ) ) ( , ) 0kMT LV r E r Eχ−∇ + − ≠

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

L L

A r E A r E rχ ϕ ψ= =∑ ∑However, a linear combination can be a solution

Tail cancellation is needed

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

L

S b E a E Aκ δ− =∑which occurs at selected , kj

kj LE A

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( , )kL r Eχ

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 Aχ χ⟨ −∇ + + − ⟩ =∑

( ) ( , )kj kkj L L kj

L

r A r Eψ χ= ∑

which solves the entire problem sufficiently accurate.Unfortunately, drawback: implicit E-dependence!

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Envelope FunctionsEnvelope Functions

General idea to get rid of E-dependence: use Taylor seriesand get LINEAR MUFFIN-TIN ORBITALS (LMTOs)

1

( , ) ( , ) ( ) ( , )

( , ) ( , ) ( ) ( , )

( ) ( , ) / ( , )

l l l l l l

l l l l l l l

l l l

r E r E E E r E

r D r E D D D r E

D E S S E S E

ν ν ν

ν ν ν ν

ϕ ϕ ϕ

ϕ ϕ ϕϕ ϕ

−

= + −

= + −′=

&& &

Before doing that, consider one more useful construction:envelope function.In fact, concept of envelope functions is very general. By choosing appropriate envelope functions, suchas plane waves, Gaussians, spherical waves (Hankel functions)we will generate various electronic structure methods(APW, LAPW, LCGO, LCMTO, LMTO, etc.)

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Algorithm, in terms of which we came up with the MUFFIN-TIN ORBITAL construction:

Step 1. Take a Hankel function

Step 2. Augment it inside the sphereby linear combination:

Step 3. Construct a Bloch sum

0( , ) ( , )L Lh r E V h rκ− =

( , ) ( , )/L l L lr E a j r bϕ κ−

( , ) ( , )k ikRL L

R

r E e r R Eχ χ= −∑

( , )L r Eχ

( , )Lh rκ

( , )kL r Eχ

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Why take Hankel function as an envelope?

Step 1. Take ANY functionwhich has one center expansionin terms of

Step 2. Augment it inside the sphereby linear combination:

Step 3. Construct a Bloch sum

( )L rαΘ

( , ) ( , )k ikRL L

R

r E e r R Eα αχ χ= −∑

( , ) ( )l L l La r E b rα α αϕ − Ξ

'( )L rβΞ

( , )L r Eχ

( , )kL r Eχ

( )L rαΘ

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Envelope functions can be Gaussians or Slater-type orbitals.They can be plane waves which generates augmentedplane wave method (APW)

( )i k G re +

( )

*4 (| | ) ( ) ( )

i k G r

l L LL

e

j k G r Y r Y k Gπ

+ =

+ +∑S S S S

*

( , )

4 ( , ) ( ) ( )k G

k Gl l L L

L

r E

r E a Y r Y k G

χ

π ϕ+

+

=

+∑

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Condition of augmentation – not necessarily smooth like withHankel functions. APWs are not smooth but continuous.We can require that linear combination of APWs is smooth:

( , ) ( )kjk G k G kj kj

G

A r E rχ ψ+ + =∑which solves the problem and delivers spectrum

Another option – use APWs in the variational principle whichtakes into account discontinuity in the derivative of the basisfunctions (Slater, 1960)

In all cases so far implicit energy dependence is present!

, ( )kj kjE rψ

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We learned how envelope functions augmented inside the spheres generate good basis functions.

The basis functions can be continuous and smooth like MTOs orsimply continuous like APWs, in all cases either condition of tail cancellation or requirement of smoothness leads us to a set of equation which delivers the solution of the Schroedingerequation with MT potential.

Variational principle can be used for a full potential case.If basis functions are not smooth additional terms in thefunctional have to be included.


Implicit energy dependence complicates the problem!

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Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals

Introduction of phi-dot function gives us an idea that wecan always generate smooth basis functions by augmentinginside every sphere a linear combinations of phi’s and phi-dot’s

The resulting basis functions do not solve Schroedinger equation exactly but we got read of the energy dependence!

The basis functions can be used in the variational principle.

General idea to get rid of E-dependence: use Teilor seriesand get read off the energy dependence.

1

( , ) ( , ) ( ) ( , )

( , ) ( , ) ( ) ( , )

( ) ( , ) / ( , )

l l l l l l

l l l l l l l

l l l

r E r E E E r E

r D r E D D D r E

D E S S E S E

ν ν ν

ν ν ν ν

ϕ ϕ ϕ

ϕ ϕ ϕϕ ϕ

−

= + −

= + −′=

&& &

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Augmented plane waves:*

( ) *int

( , ) 4 ( , ) ( ) ( ),

( , ) 4 (| | ) ( ) ( ),

k Gk G l l L L

L

i k G rk G l L L

L

r E r E a Y r Y k G r S

r E e j k G r Y r Y k G r

χ π ϕ

χ π

++

++

= + ∈

= + + ∈Ω

∑

∑

become smooth linear augmented plane waves:

*

( ) *int

( ) 4 ( , ) ( , ) ( ) ( ),

( ) 4 (| | ) ( ) ( ), ,

k G k Gk G l l l l l l L L

L

i k G rk G l L L

L

r r E a r E b Y r Y k G r S

r e j k G r Y r Y k G r S r

υ υχ π ϕ ϕ

χ π

+ ++

++

= + + ∈

= = + + ∈ ∈Ω

∑

∑

&

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Consider local orbitals.

Energy-dependent muffin-tin orbital defined in all space:

becomes energy-independent

ˆ( , ) ( , ) ( )/ ( ),

ˆ( , ) ( ) ( ),

lL l l l l L MT

lL l L MT

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

r E h r i Y r r S

χ ϕ κ

χ κ

= − <

= >

ˆ( , ) ( , ) ( , ) ( ),

ˆ( , ) ( ) ( ),

lL l l l l l l L MT

lL l L MT

r E a r E b r E i Y r r S

r E h r i Y r r Sν νχ ϕ ϕ

χ κ

= + <

= >

&

provided we also fix to some number (say 0)0E Vκ = −

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Bloch sum should be constructed and one center expansionused:

0

' ''

( )

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( )

ikRL

R

ikRl L l l L l L

R

kl L l l L l L L L

L

e r R

a r E b r E e h r R

a r E b r E j r S

ν ν

ν ν

χ

ϕ ϕ κ

ϕ ϕ κ κ≠

− =

+ + − =

+ +

∑

∑

∑

&

&Final augmentation of tails gives us LMTO:

' ' ' ' ' ' ''

( ) ( , ) ( , )

( , ) ( , ) ( )

k h hL l L l l L l

j j kl L l l L l L L

L

r a r E b r E

a r E b r E Sν ν

ν ν

χ ϕ ϕ

ϕ ϕ κ

= + +

+∑&

&

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In more compact notations, LMTO is given by

' ''

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

L

r r r Sχ κ= Φ + Φ∑where we introduced radial functions

( ) ( , ) ( , )

( ) ( , ) ( , )

h h hL l L l l L l

j j jL l L l l L l

r a r E b r E

r a r E b r Eν ν

ν ν

ϕ ϕ

ϕ ϕ

Φ = +

Φ = +

&&

which match smoothly to Hankel and Bessel functions.


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Another way of constructing LMTO.Consider envelope function as

Inside every sphere perform smooth augmentation

( ) ( , )k ikRL L

R

r e h r Rχ κ= −∑%

' ''

' ''

( ) ( , ) ( , ) ( )

( ) ( ) ( ) ( )

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

χ κ κ κ

χ κ

= +

⇓

= Φ + Φ

∑

∑

%

which gives again LMTO construction.

SSSS

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We could do the same trick for a single Hankel function

SSSS

( ) ( , )L Lr h rχ κ=%

Inside every sphere perform one-Center expansion

0

0 ' ''

( ) ( , )

(1 ) ( , ) ( )L L R

R L L LL

r h r

j r R S R

χ κ δ

δ κ

= +

− −∑%

and augmentation

0

0 ' ''

( ) ( )

(1 ) ( ) ( )

hL L R

jR L L L

L

r r

r R S R

χ δ

δ

= Φ +

− Φ −∑ Bloch summation is trivial.

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LMTO definition (κ 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

κ

κ

χ κ

χ κ

= Φ + Φ ∈Ω

= − ∈ Ω

∑

∑


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 Aκ κ κκ

χ χ⟨ −∇ + − ⟩ =∑

( ) ( )kj kkj L L

L

r A rκ κκ

ψ χ= ∑

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Accuracy and Atomic Sphere Approximation:

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

LMTO is accurate to zero order (κ2 is fixed) in the interstitials.

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

ASA is a fast, accurate method which eliminates interstitial region and increases the accuracy.

Works well for close packed structures, for open structures needsempty spheres.

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TightTight--Binding LMTOBinding LMTO

Tight-Binding LMTO representation

LMTO decays in real space as Hankel function which depends on κ2=E-V0 and can be slow.

Can we construct a faster decaying envelope?

Advantage would be an access to the real space hoppings:

' ' ' '

( ) ( )

( )

k ikRL L

R

k ikRL L L L

R

r e r R

H e H R

κ κ

κ κ κ κ

χ χ= −

=

∑

∑

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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 κ2=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 Rα κ κ= −∑

1/ lLZ r +

' 1' /

lLM r +

( ) ~ 0scrV r

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Once problem of screening is solved (Andersen, Jepsen, 1984)screened Hankel functions can be used as envelope functionsand this leads us to so called:

Tight-Binding LMTO Representation.

Since mathematically it is just a transformation of thebasis set, the obtained one-electron spectra 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.

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Exact LMTOs

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

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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

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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 Kκ κ− =

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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.

LRKκ&

' ' ' '' '

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

r K r M K rκχ = + ∑ &


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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.


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Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method

2' ' | | 0k k kj

L kj L LL

V E Aκ κ κκ

χ χ⟨ −∇ + − ⟩ =∑

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

κ

κ

χ κ

χ κ

= Φ + Φ ∈Ω

= − ∈Ω

∑

∑

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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

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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

ρ ρ

ρ ρ

= <

= ∈Ω

∑

∑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

κ κ

κ κ

χ χ

χ χ +

= <

= + ∈Ω

∑

∑

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Problem: Fourier transform of LMTOs is not easy since

int( ) ( , ),k ikRL L

R

r e h r R rκχ κ= − ∈ Ω∑

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 rκ

κ κ

χ

χ χ κ

= <

= = − ∈Ω∑%%

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( )L rκχ%

( , )Lh rκ

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.

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( )L rκχ%Weirich 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 rκ κ+ &4( ) ~ 1/L k G Gκχ +%

( ) ( ) ( ) ( ) ...l l l l l l l la j r b j r b j r c j rκ κ κ κ+ + + +& && &&&3( ) ~ !!/ n

L k G n Gκχ ++%

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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 rηκ κ −−∇ − =%

Parameter η 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 η

κχ − ++%

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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

κ κ κ κ

κ κ κ κ

χ χ χ χ

χ χ χ χΩ

Ω

⟨ −∇ + ⟩ = ⟨ −∇ + ⟩ +

⟨ −∇ + ⟩ − ⟨ −∇ + ⟩% %% % % %

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+

= <

= = ∈Ω∑%% %

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Computer Programs available:

LmtART

(ASA-LMTO & FP LMTO methods)

http://physics.njit.edu/~savrasov

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q Multiple-kappa LMTO basis sets and multi-panel technique.q LSDA together with GGA91 and GGA96. q Total energy and force calculationsq LDA+U method for strongly correlated systems.q Spin-orbit coupling for heavy elements.q Finite temperatures q Full 3D treatment of magnetization in relativistic calculations.q Non-collinear magnetizm. q Tight-binding regime. q Hopping integrals extraction regime. q Optical Properties (e1,e2, reflectivity, electron energy loss spectra)

LmtART features:


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Computer Programs available:

MindLab

(Material Information & Design Laboratory)

Educational edition

MS Windows based software freely available on CDs

during our Workshop, http://physics.njit.edu/~savrasov

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ConclusionConclusion

• LMTO, TB-LMTO, EMTO, NMTO is economical localized orbitalbasis set to solve the Schroedinger equation.

• It is physically transparent and allows readily to analyzethe nature of bonding in solid-state and molecular systems.

• It is one of the most popular basis sets and widely used in density functional total energy calculations

• It provides a minimal basis for building correct models describedby many-body Hamiltonians and further developments ofcombining electronic structure techniques with many-bodydynamical mean-field methods.

Documents

Andersen’s LMTO methodsavrasov.physics.ucdavis.edu/Works/Talks/lect18.pdf · Andersen’s LMTO method S. Y. Savrasov New Jersey Institute of Technology O. K. Andersen, PRB 12, 3060