Introduction to LMTO methodfolk.uio.no/ravi/FME2011/lectures/Lecture1-7-ravi_lmto.pdf · The radial...

$: Introduction to LMTO methodfolk.uio.no/ravi/FME2011/lectures/Lecture1-7-ravi_lmto.pdf · The radial portion 對R\⠀爀尩 is obtained by solving the atomic DFT equations numerically.$
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method

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Introduction to LMTO method

24 February 2011; V172


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Ab

initio Electronic Structure Calculations in Condensed Matter


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DMol3: Linear Combination of Atomic Orbital

)(

rc jj

iji

Good for molecules, clusters, zeolites, molecular crystals, polymers "open structures"

Rcut

Periodic and a periodic systems

lm

lmnlj YrRr ),()()(

Radial portion atomic DFT eqs. numerically

Angular Portion

Presenter

Presentation Notes

Each MO (molecular orbital) is expanded in terms of basis functions, conventionally called atomic orbitals (MO= LCAO, Linear Combination of Atomic Orbitals). Cij are the MO expansion coefficients j are the atomic basis functions The basis functions are given numerically as values on an atomic-centered spherical-polar mesh, rather than as analytical functions (i.e., Gaussian orbitals). The angular portion of each function is the appropriate spherical harmonic Ylm(θ,φ). The radial portion R(r) is obtained by solving the atomic DFT equations numerically. A reasonable level of accuracy is usually obtained by using about 300 radial points from the nucleus to an outer distance of 10 bohr (~5.3 Å). The use of the exact DFT spherical atomic orbitals has several advantages. For one, the molecule can be dissociated exactly to its constituent atoms (within the DFT context). Because of the quality of these orbitals, basis set superposition effects (Delley, 1990) are minimized and it is possible to obtain an excellent description, even for weak bonds. Atomic basis sets are confined within a cutoff value, rc, appropriate for a particular quality level of DMol3 calculations. This is an important feature of the numerical basis set that can lead to much faster calculations, particularly for solid state systems. DMol3 uses a so-called soft confinement potential, which ensures the strict localization of the basis set within an rc value, without discontinuous derivatives at rc. Geometry optimization is efficient, even with small cutoff values. (PGS)


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8The muffinThe muffin--tin approximationtin approximation

Spherical atoms in a constant interstitial potential


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LMTO MethodAndersen (1975) PRB, 12, 3060Andersen and Jepsen (1984) PRL, 53, 2571

Partitioning of the unit cell into atomic sphere (I) and interstitial regions (II)

MTr ),V(r

I r onstant,C )r(VMT

)r̂(Y),r(u Ll

lll u u)r(V

r)l(l

drdur

drd

r

2

22

11

Inside the MT sphere, an eigen state is better described by the solutions of the Schrödinger equation for a spherical potential:

The function satisfies the radial equation:lu

The only boundary condition: be well defined atlu 0r


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sr),r(N

sr),r(J))(cot()r,()r̂(Yi )r,,(

l

lllL

lLMTOL

sr ,))E(cot(

dd

)r,E( )r(Jl

ll

sr,)r/s(

sr,)l(

)s/r()(P)r,()r̂(Yi )r,(l

l

llL

lMTOL

1122

The basis functions can now be constructed as Bloch sums of MTO:

An LMTO basis function in terms of energy and the decay constant may be expressed as:

Here and represent the Bessel and Neumann functions respectively. lJ lNSince the energy derivative of vanishes at for it leads to:L E ,sr


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)(DD)s()(DD)s( )D(

(r)),(D)(r))(r̂(Yi )r( llm

llLMTO

L

In the atomic sphere approximation (ASA), the LMTO’s can be simplified as :

where is given by :D

derivativearithmiclogD

is chosen such that and its energy derivative matches continuously to the tail function at the muffin-tin sphere boundary.

)(D )r(l

Disadvantages of LMTO-ASA method :

(1) It neglects the symmetry breaking terms by discarding the non-spherical parts of the electron density.

(2) The interstitial region is not treated accurately as LMTO replaces the MT spheres by space filling Wigner spheres.


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Linear Augmented plane wave (LAPW) methodLinear Augmented plane wave (LAPW) method

Augmented plane waves:*

( ) *int

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

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

k Gk G l l L L

Li k G r

k G l L LL

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

become smooth linear augmented plane waves:

*

( ) *int

( ) 4 { ( , ) ( , ) } ( ) ( ),

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

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

Li k G r

k G l L LL

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


13Linear Muffin-Tin Orbital (LMTO) method


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KKR partial waves

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

( , ) ( , ),L L l L MT

L l L MT

r E r E a j r r S

r E b h r r S

( , ) ( , )k ikRL L

R

r E e r R E ' ' '

'

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

L

r E j r S b a

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

L L

A r E A r E r

Basic idea of KKR method is to construct a partial wave

Consider its Bloch sum

And demand tail-cancellation:


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16Non-linear KKR Equations

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

L

S E P E A

''' ' ''

0 ''

( ) ( , )k ikR LL L LL L

R L

S E e C h E R

( ) ( , ) ( , )[ ( ) ( )]( )( ) ( , ) ( , )[ ( ) ( )]

hl l l l l l

l jl l l l l l

a E W h h S E D E D EP Eb E W j j S E D E D E

where potential parameters function is

and where KKR structure constants are


17Logarithmic Derivatives

Behavior of Logarithmic Derivative

' ( , )( )( , )l

ll

S S ED ES E

Consider s-wave: 1s has no nodes, 2s has 1 node,…

ornodes=n-l-1. From the point of view of node appearswhen which means that log. derivative diverges!

So logartihmic

derivatives behave as tan(E), they diverge eachtime a new node of radial wave function appears.

( , )l S E( , ) 0l S E


18Wavefunction

as a Function of Energy

r

0( , )l S E

S

E1

E2

E3

( )MTV r

E3

E2

E1 Energy Window for 1s states

Energy Window for 2s states

Energy Window for 3s states


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Logarithmic Derivative as a Function of Energy

r

0( , )l S E

E1

E2

E3

E

0( )lD E

1s 2s 3s 4s

New node of wave functionappears!

( ) 1lD E l

Centers of the nl

band


20Linearized

Solutions

If Dl

(E) can be expanded in Tailor series around some energyEν

, we obtain potential function in a linearized

form

( ) 1 12(2 1)( )

l l

l l l

D E l E ClD E l E V

which solves the band structure problem

1

kl lj

kj l kl lj

w SE C

S

Cl

gives the center of the l-band, wl

gives its width while denominator 1-γS gives additional distortion of the band.


21Energy linearization

Andersen proposed to split energy dependence coming from inside the spheres and from interstitials. Since interstitial region is

small,

Andersen proposed to fix this energy kappa to some value (originally to zero)

Energy Bands

MT-zero V02=E-V0

Average kinetic energy of electron in the interstitial region


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Partial waves of fixed energy tails

0

0 1

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

lL L l L L l L MT

L l L l L MTl

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

r E b h r bY r r Sr

( , ) ( , )k ikRL L

R

r E e r R E '

' ' ''

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

L

r E r Y r S b a ( , ) ( , ) ( )k k k

L L L L kL L

A r E A r E r

Consider as before Bloch sum and demand tail-cancellation:


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2' '{ ( 0) ( )} 0k k

L L L L l LL

S P E A

2 ''' ' '''' 1

0 ''

1( 0) ( )k ikR LL L LL Ll

R L

S e C Y rr

[ ( ) 1]( ) 2(2 1)[ ( ) ]

ll

l

D E lP E lD E l

where potential parameters function is

and where the fixed energy structure constants are

KKR equations become

To minimize the error of fixing the energy, Andersen proposed to enlarge MT spheres to atomic spheres. This method has the name KKR-ASA.


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2' '

[ ( ) 1]det{ ( 0) 2(2 1) } 0[ ( ) ]

k llm m m m

l

D E lS lD E l

Canonical band structures (Andersen , 1973)At the absence of hybridization, a remarkable consequence ofKKR ASA equations is canonical energy bands:

2 [ ( ) 1]( 0) 2(2 1)[ ( ) ]

k llj

l

D E lS lD E l

For a given l block, one can diagonalize

the structureconstants and obtain (2l+1) non-linear equations

whose solutions give rise to band structures E(kj), so calledcanonical band structures.


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Canonical d-band for fcc material

Cl

wl


26Comparison with bands of Cu


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Energy Linearization (Andersen, 1973)Energy Linearization (Andersen, 1973)

General idea to get rid of E-dependence: use Tailor 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 ED E S S E S E

Before doing that, consider one more useful construction:envelope function.In fact, concept of envelope functions is very general. By choosing appropriate envelope functions, such as plane waves, Gaussians, spherical waves (Hankel

functions) we will generate various

electronic structure methods (APW, LAPW, LCGO, LCMTO, LMTO, etc.)


28Envelope FunctionsEnvelope Functions

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

( )i k G re

( )

*4 (| | ) ( ) ( )

i k G r

l L LL

ej 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


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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

Construction of Augmented Spherical Wave


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Envelope FunctionsEnvelope Functions

Algorithm, in terms of which we came up with the augmented spherical wave (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 r

{ ( , ) ( , )}/L l L lr E a j r b

( , ) ( , )k ikRL L

R

r E e r R E

( , )L r E

( , )Lh r

( , )kL r E


31Linearization over EnergyLinearization over Energy

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 resolved the energy dependence!

The basis functions can be used in the variational principle.

General idea to get rid of E-dependence: use Tailor series and 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 ED E S S E S E


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Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals

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

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 S

provided we also fix to some number (say 0)0E V


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Bloch sum should be constructed and one center expansionused:

0

' ''

( )

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( )

ikRL

RikR

l L l l L l LR

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

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 S


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In more compact notations, LMTO is given by

' ''

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

L

r r r S where we introduced radial functions

( ) ( , ) ( , )

( ) ( , ) ( , )

h h hL l L l l L lj j jL l L l l L l

r a r E b r E

r a r E b r E

which match smoothly to Hankel

and Bessel functions.



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Summary of LMTO method


36Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals

Accuracy and Atomic Sphere Approximation:

LMTO is accurate to first order with respect to (E-Eν

) withinMT spheres.

LMTO is accurate to zero order (k2

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 accurate method which eliminates interstitial region and increases the accuracy. Works well for close packed structures, for open structures needs empty spheres.


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LMTO definition (k dependence is highlighted):

' ''

int

( ) ( ) ( ) ( ),

( ) ( , ),

k h j kL L L L L MT

Lk ikRL L

R

r r r S r

r e h r R r

VariationalVariational

EquationsEquations

which should be used as a basis in expanding

Variational principle gives us matrix eigenvalue problem.

2' ' | | 0k k kj

L kj L LL

V E A

( ) ( )kj kkj L L

L

r A r


38TightTight--Binding LMTOBinding 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

Rk ikR

L L L LR

r e r R

H e H R


39TightTight--Binding LMTOBinding LMTOAny 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


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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 Sα

coefficients are to be determined.


41Demand 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 Sα

plays a role of (screened) structure constants

and we introduced screened Bessel functions


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Screened structure constants are short ranged:( ) ( )

( )

( )

/( )

S I S S

S S I S

For s-electrons, transforming to the k-space2

( ) 2

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

S k kS 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


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Screening parameters alpha have to be chosen from the condition of maximum localization of the structure constants in 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 theliterature).

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


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TightTight--Binding LMTOBinding LMTOSince 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


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Introduction to LMTO methodfolk.uio.no/ravi/FME2011/lectures/Lecture1-7-ravi_lmto.pdf · The radial...

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