NJIT
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)
NJIT
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
NJIT
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
NJIT
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+ =
NJIT
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
NJIT
Solving Solving Schroedingers Schroedingers equation for solidsequation for solids
2( ( ) ) ( ) 0kj kjV r E rψ−∇ + − =Solution of differential equation is required
Properties of the potential ( )V r
NJIT
Solving Solving Schroedingers Schroedingers equation for solidsequation for solids
Properties of the solutions: , ( )kj kjE rψ
2Zer
−
Core Levels
Energy Bands
NJIT
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= −∑
NJIT
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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
NJIT
Linear combinations of local orbitals will be considered.
( ) ( )k ikR
R
r e r Rα αχ χ= −∑
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
NJIT
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
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
NJITMuffin-tin sphere MTS
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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
NJIT
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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
==
= −
NJIT
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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
= − <
= >
NJIT
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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:
NJIT
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
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
NJIT
Basis Sets & Variational PrincipleBasis Sets & Variational Principle
( , )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!
NJIT
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.)
NJIT
Envelope FunctionsEnvelope Functions
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
NJIT
Envelope FunctionsEnvelope Functions
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
NJIT
Envelope FunctionsEnvelope Functions
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+
+
=
+∑
NJIT
Envelope FunctionsEnvelope Functions
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
NJIT
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.
Envelope FunctionsEnvelope Functions
Implicit energy dependence complicates the problem!
NJIT
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
−
= + −
= + −′=
&& &
NJIT
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
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
∑
∑
&
NJIT
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
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 = −
NJIT
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
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
= + +
+∑&
&
NJIT
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.
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
NJIT
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
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
NJIT
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
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.
NJIT
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
∑
∑
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
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= ∑
NJIT
Linear 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 (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.
NJIT
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= −
=
∑
∑
NJIT
TightTight--Binding LMTOBinding 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 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
NJIT
TightTight--Binding LMTOBinding LMTO
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.
NJIT
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
NJIT
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
NJIT
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!
Advanced TopicsAdvanced Topics
SSSS( )L LRK r R Kκ κ− =
NJIT
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 = + ∑ &
Advanced TopicsAdvanced Topics
NJIT
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
NJIT
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
∑
∑
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO 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
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
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
∑
∑
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
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∑%%
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
( )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.
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
( )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 ++%
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
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 − ++%
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
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∑%% %
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
Computer Programs available:
LmtART
(ASA-LMTO & FP LMTO methods)
http://physics.njit.edu/~savrasov
NJIT
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:
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
NJIT
Advanced Topics: FPAdvanced Topics: FP--LMTO MethodLMTO Method
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
NJIT
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.