Fundamentals and Applications of
FLAPW Method: HiLAPW Code

Tamio Oguchi
Institute of Scientific and Industrial Research
Osaka University
CMD Workshop

FLAPW Method• The FLAPW is among the most precise andefficient first-principles methods which are able tosolve density-functional-theory Kohn-Shamequations with the periodic boundary conditions.
- FLAPW =Full-potential Linearized Augmented Plane Wave

• A great number of applications to a variety ofsolid systems!2

1. OUTLINE• Fundamentals of FLAPW Method- One-electron equations and Bloch function- Augmented Plane Wave method- Linear method- Full-potential method• FLAPW Codes- Packages open to public- HiLAPW code and some applications!3

2. OUTLINE• Practical Aspects of FLAPW Methods- First-Principles Calculation – Kohn-Sham Eqs.
- Crystal Structure and Atomic Position- Lattice – Primitive Translation Vector- Space Group- Reciprocal Lattice – Brillouin Zone
- k-point Integration- Eigenvalue Problem- Self-Consistent Field- Mixing of Electron Density!4

Kohn-Sham Equations"!
2h̄∇2 + veﬀ (r) ψj (r) = εj ψj (r)Hψj (r) = −2mn(r) =!

|ψj (r)|2j2

veﬀ (r) = vext (r) + e!

n(r′ )′
dr + µxc (n(r))′|r − r |★ One-electron Kohn-Sham equations are given within
the local density approximation (generalized gradientapproximation) to the density functional theory.
!5Band Theory
How to solve the one-electron equations for particularcondensed-matter systems
Periodic system → Band theory
Bloch Theorem•A quantum-mechanical state in a periodic system can be
specified with a wave number kHψjk (r) = εkj ψjk (r)Dispersion relation: band structure!6Brillouin Zone and Bands

BZ of fcc latticeBand structure of fcc Al

independent quantum number!7

Bloch Function
ψjk (r + R) = eik·R ψjk (r) : Bloch theorem
ψjk (r) = eik·r ukj (r)ukj (r + R) = ukj (r) : periodic function
represented in a
Fourier form!
ψjk (r) = eik·rakj,K eiK·r! K
K: reciprocal latticeki(k+K)·r=
aj,K evectorK!8

Bloch Function• Normalized in a macroscopic volume Ω• k points in BZ are independent and sufficient!
1i(k+K)·r
ψjk (r) = √
ak+K
e
j
Ω K

Hψjk (r) = εkj ψjk (r)!9Secular Equation!
K⟨k + K′ |H − εkj |k + K⟩ak+K=0j

Matrix elements2h̄|k + K|2 δK′ ,K + V (K′ − K)
⟨k + K |H|k + K⟩ =2m′

1V (K − K) =
Ω
′

!

dre

−i(K′ −K)·r

v(r)
!10

Fourier Transform of Potential
1
V (K − K) =Ω′

fcc Al!

dre−i(K′ −K)·r

v(r) ≈!
Rv(r)vatom (|r − R|)

Very slow convergency of FT
due to Coulombic behavior
requires a large number ofK vectors.
!11Orthogonalization to Core Functions
1⟨k + K|φcore ⟩ =
Ω

!

dre

−i(k+K)·r

φcore (r)Very slow convergency of the corefunctions because of localized nature.

!12

★

How to Solve the Problems
1. Contributions from core (nucleus and coreelectron potentials, and orthogonality to coreelectron states) are replaced by a soft (easily
Fourier transformed) potential.
→ pseudopotential
2. PW basis functions are augmented withlocalized functions.
→ augmented basis
3. Green's functions are used as a multiple
scattering problem.
→ Korringa-Kohn-Rostoker method!13

Muffin-tin Potential Approximation
Crystal Potential• Spherical around atoms• Constant in the interstitialv(r) =FCC Cu!

v(|r − R|)vMTZ|r − R| ∈ S|r − R| ∋ S!14

Slater’s Idea
Phys. Rev. 51(1937)846.

Plane waves
k+Kφ

1 i(k+K)·r(r) = √ eΩ|r − R| ∋ S|r − R| ∈ S
Augmentation waves!
1
φk+K (r) = √il ak+K
lm Rl (|r − R|; E)Ylm (r − R)Ω lm
Rl (r; E) : radial function for energy E
Ylm (r) : spherical harmonics!15

Augmented Plane Wave
Secular Equation
det ||⟨k + K′ |H − E|k + K⟩|| = 0
"
! 2
h̄
′
|k + K|2 − E δK′ ,K⟨k + K |H − E|k + K⟩ =2m
+ΓAPW
K′ ,K (E)

Matrix elements have non-linear energy
dependence due to logarithmic derivativesof the radial functions.
!16

Problems in APW Method
1.

APW
Energy dependence of the matrix elements ΓK′ ,K (E)requires searching poles of the determinants→ Linear Method by Andersen (1975) andKoelling-Arbman (1975)
LAPW
2. Muffin-tin potential approximation doesn’t work
for less-packing or low-symmetry systems→ Full-potential Method by Weinert (1981)
FLAPW
!17

Linear Method
O.K. Andersen, PRB 12(1975)3060.
D.D. Koelling and G.O. Arbman, JPF 5(1975)2041.

• Remove the energy dependence of radial functions
using Tayler expansion
Rl (r; E) ≈ Rl (r; E0 ) + (E − E0 )Ṙl (r; E0 ) + · · · !
!
dRl (r; E)!!
Ṙl (r; E0 ) =dE
E=E0• Use radial function at E0 and its energy derivative
to represent a radial function with any logarithmic
derivativeRl (r; D) = Rl (r; E0 ) + ω(D)Ṙl (r; E0 )!18

Linear Method• APW → LAPW• KKR → MTO → LMTO
KKR: Korringa-Kohn-Rostoker
MTO: Muffin-Tin Orbital
LMTO: Linear Muffin-Tin Orbital!19

LAPW Method
Augmentation basis!
1
φk+K (r) = √il φk+K
lm (|r − R|)Ylm (r − R)
Ω lm

k+K
k+K
φk+K
(r)
=
A
R
(r;
E
)
+
B
l
l
lm
lm
lm Ṙl (r; El )
! k+K
Alm
determined from thek+K
Blmboundary conditionsElenergy parameter usuallytaken at the center of theoccupied partial band

!20

Why the Linear Method Works Well?
Orthogonality to maximize the variational degree! S
Rl (r; El )Ṙl (r; El )r2 dr = 00Orthogonality to the core functions! S
Rl (r; El )Rcore (r; Ecore )r2 dr = 0
0

! S
0

Ṙl (r; El )Rcore (r; Ecore )r dr = 0
2!21

Why the Linear Method Works Well?
Energy expectation value of the orbital with the exact
logarithmic derivative⟨φl (D)|H|φl (D)⟩S⟨E(D)⟩ =⟨φl (D)|φl (D)⟩S= E + O(E − El )4!22

Coulomb Potential• Coulomb potential for smooth density distributionn(r) =!

eiG·r nGGv C (r) =!

∇2 v C (r) = 4πe2 n(r)CeiG·r vG

G

2
4πe
nG
C
vG =
G2

• Realistic distribution
1
nG =
Ω

!

e−iG·r n(r)dr!23

Full Potential Method
M. Weinert, J. Math. Phys. 22 (1981) 2433.

• Electron density inside the sphere is replaced by asmoothed density with the exact multipole moments.
!
ñ(r) |r − R| ∈ Sn(r) =
n(r) |r − R| ∋ S• Potential outside the sphere is given with thesmoothed density.
• Potential inside the sphere can be solved with thesphere boundary conditions.
!24Full Potential Method• Electron density inside the spheren(r) =!

nlm (r)Ylm (r)!

vlm (r)Ylm (r)lm• Potential function inside the spherev(r) =lm• Matrix element of potential
∆H

K′ ,K

= ⟨k + K |∆v|k + K⟩′

non-spherical part!25

FLAPW Method• Two-dimensional slab models- E. Wimmer, H. Krakauer, M. Weinert and A. J.
Freeman, PRB 24, 864 (1981).
- M. Weinert, E. Wimmer and A. J. Freeman, PRB
26, 4571 (1982).
• Three-dimensional systems- H. J. F. Jansen and A. J. Freeman, PRB 30 , 561(1984).
!26

FLAPW Packages Open to Publicψjk = ψ̃jk +!"ν

• ABCAP• HiLAPW• KANSAIkkψνj− ψ̃νj#• FLEUR• QMD-FLAPW• WIEN!27

HiLAPW• 100% Original Code– LAPW basis functions– LSDA, GGA, Hubbard-U– Scalar relativity, Spin-orbit coupling– All-electron SCF full-potential scheme– BZ integration with tetrahedron method– Group theory– Crystal structure & element database– Total E, forces, DOS, ...
– XAS, Berry phase, dielectric function, ...
optional functionalities!28

HiLAPW• 100% Original Code– Modular executables– fortran90• dynamical memory allocation– BLAS and LAPACK libraries– PSP : PostScript Plot routines– MPI parallelization• Manuals and some useful data
– www.cmp.sanken.osaka-u.ac.jp/~oguchi/HiLAPW/!29

HiLAPW – Executables
executables

contents

xsets

initialization

xlapw

SCF calculation

xdoss

DOS

xnewa

modification k-point data

xwbox

electron density on 3D mesh

xpbox

potential on 3D meshxspin

addition of spin polarization

xsymm

irreducible representation extract

xrept

rearrangement of eigenvalues!30

HiLAPW – Executables
sets.inatomdata
atomdens
spgrdata

xsets

sets.out

wavoutexcodata

wavin

lapw.in

xlapw

lapw.out
dis, ekn, foa, ten, ...

wavoutwavin

post modules!312. OUTLINE• Practical Aspects of FLAPW Methods- First-Principles Calculation – Kohn-Sham Eqs.
- Crystal Structure and Atomic Position- Lattice – Primitive Translation Vector- Space Group- Reciprocal Lattice – Brillouin Zone
- k-point Integration- Eigenvalue Problem- Self-Consistent Field- Mixing of Electron Density!32

First-Principles Calculation• Local Density Approximation to Density Functional
Theory — Kohn-Sham Equations
"
!
2
h̄
k
∇2 + v(r) ψjk (r) = εkj ψjk (r)Hψj (r) = −2mn(r) =occ.
!
j,kv(r) =|ψjk (r)|2

Z

2

e

|r

R |

+e

2

⇥

basic inputn(r )dr + µxc (n(r))|r r |!33

Crystal Structure and Atomic PositionRν = Rlmn + τ α

ν = (lmnα)t2unitcellτα• Lattice Vector
Rlmn = lt1 + mt2 + nt3
⎛

⎞

⎛

x
Rlmn
tx1
⎝ Ry
⎠ = ⎝ ty1
lmn
z
Rlmn
tz1

tx2
y
t2
tz2

Rlmn
⎞
⎛
⎞
tx3
l
y ⎠⎝
t3
m ⎠
tz3
n

Primitive translation vector

O

t1
Rν

!34

Lattice Translation Vector• Primitive Translation Vector (Bravais lattice)
⎛

tx1
⎝ ty1
tz1

tx2
y
t2
tz2

⎞
x

⎛

t3
ax1
y ⎠
y
⎝
t3
a1
=
tz3
az1

ax2
y
a2
az2

⎞
⎛
ax3
T11
y ⎠⎝
T21
a3
T31
az3

T12
T22
T32

⎞

T13
T23 ⎠
T33

• Conventional Translation Vector
⎛

ax1
⎝ ay1
az1

ax2
y
a2
az2

⎞
ax3
y ⎠
a3
⇐ (a, b, c, α, β, γ)az3
Lattice constants!35

Lattice: System and Type
Lattice system
•
⎛
⎞
ax1
⎝ ay1
az1

ax2
y
a2
az2

ax3
c
y ⎠
a3
⇐ (a, b, c, α, β, γ) α bγaz3βsystemcubic
tetragonal
orthorhombic
hexagonal
trigonal(trigonal
monoclinic
triclinic

a
a
a
a
a
a
a
a
a

b
a
a
b
a
a
a
b
b

c
a
c
c
c
a
c
c
c

α
90
90
90
90
α
90
90
α

β
90
90
90
90
α
90
90
β

γ
90
90
90
120
α
120
γ
γ

a
type
P,I,F
P,I
P,I,F,C
P
R
P)
P,B
P
!36

Lattice: System and Typetype
• Lattice
⎛
⎞

⎛

⎞

1 0 0
0
⎠
P =⎝ 0 1 0 ⎠F = ⎝ 12 0
1
1
0 0 1
0
2
2
⎛
⎞
⎛ 1
⎞
1
1
1 0
0
−2
2
2
1
1 ⎠
1 ⎠
⎝ 0
− 12
A
=
I = ⎝ 12
2
2
2
1
1
1
1
1
0
−
−
2
2
2
2
2
⎛ 1
⎞
⎛ 1
⎞
1
1
0 −2
0
2
2
2
B=⎝ 0 1 0 ⎠C = ⎝ − 21 12 0 ⎠
1
1
0
0
0 1
2
2
1
2

1
2
1
2

!37

Lattice: System and Type• Number of mathematicallyindependent lattice is 14
called Bravais Lattice.
• For example, a facecentered tetragonal lattice
can be represented as
body-centered tetragonal.
• Some of trigonal systems
are represented asrhombohedral R or
hexagonal P.
!38

Atomic Position in a Unitcell
⎛

⎞
x

⎛

τα
ax1
⎝ ταy ⎠ = ⎝ ay1
ταz
az1

ax2
y
a2
az2

⎞
⎛
⎞
ax3
τ1α
y ⎠⎝
τ2α ⎠
a3
τ3α
az3

τα

Atomic positions are represented on
the basis of the conventional lattice
vectors.
!39

Space Group• Symmetry operation{β|vβ } r = βr + vβ⎛

β11
= ⎝ β21β31{β|vβ + Rlmn }non-primitive translation vector

β12
β22
β32

⎞⎛

⎞

⎛

β13
x
β23 ⎠ ⎝ y ⎠ + ⎝
β33
z

vβx
y
vβ
vβz

⎞
⎠
!40

Example: Diamond Structure• fcc a=b=c, α=β=γ=90º• space group Fd-3m (#227)
generators C4[001] + (1/4,1/4,1/4)
C3[111]
I + (1/4,1/4,1/4)• atomic positions (0,0,0); (1/4,1/4,1/4)
International Tables for Crystallography!41

k Group

βk = k + K

The Bloch wave function belongs to an irreducible
representation of the k group.
bcc Cr!42

Space Group Symbol
1,2,3,4,6：rotation axis
m：mirror plane
21：twofold screw with v=1/2
42：fourfold screw with v=2/4
a, b, c：axial glide with v=1/2 along each axis
n：diagonal glide
d：diamond glide-：inversion
4/m：fourfold axis and mirror plane perpendicular to it
4/n：fourfold axis and n-glide plane perpendicular to it!43

Reciprocal Lattice• Definition
R · K = 2πI
lattice vector

I：any integer

Klmn = lb1 + mb2 + nb3
tj × t kbi = 2πti · (tj × tk )• Brillouin zone (BZ) = unitcell of reciprocal lattice• States with k inside BZ are independent→ State sum = k-integration inside BZ!44Brillouin Zone
Unitcell of reciprocal lattice• parallelepiped (b1 , b2 , b3 )easy to treat numerically• Voronoi Polyhedron
Wigner-Seitz cell
bcc

fcc

!45

k Integration
k-point mesh(N1 , N2 , N3 ) division of parallelepiped BZb2n(r) =occ.
!
j,k|ψjk (r)|2• Linear Tetrahedron method• Broadening method

b1

!46

Eigenvalue Problem• Basis function expansionkψj (r) =!

k k
φi Ciji• Secular equation
HC = SCE• Matrix elements
Hij = ⟨φki |H|φkj ⟩Sij = ⟨φki |φkj ⟩!47

Basis Functions• Plane waves- pseudopotential- simple, fast, extendable, transferability• Plane waves + Augmentation functions- all-electron scheme- robust, precise, complicated• Local orbitals- minimal- real space → O(N)!48

Self-Consistent Field{Zν , Rν }nin (r)v(r)HC = SCE
nout (r)⟨|nin (r) − nout (r)|⟩ > δE({Zν , Rν }) n(r)⟨|nin (r) − nout (r)|⟩ ≤ δ!49

Mixing of Electron Density• Simple Method
(i+1)
(i)
(i)
nin
= (1 − α)nin + αnout• Extended Anderson Method
(i+1)
(i)
(i)
nin
= (1 − α)n̄in + αn̄out(i)n̄in =i!

(j) (j)β nin(i)n̄out =

min
β

(i)
(i)
n̄in − n̄out(j) (j)β nout

j=i−M

j=i−M

! "

i
!

#2

dr

i
!

j=i−M

β (j) = 1!50

Mixing of Electron Density!"|nin − nout |2 dr

#1/2

!51

Convergency to SCF
fcc CuPW
SW€€|nin (r) − nout (r)|2 dr#1/2[−∆ + v[nin ]] ψjk (r) = εkj ψjk (r)occ.
!
nout =|ψjk |2j,k

Etotal[−∆ + v[nout ] + vext ] ψjk (r)

Eext
€

!"

= εkj ψjk (r)vext = v[nin ] − v[nout ]!
Eext = vext nout dr!52€

Appendix• Logarithmic Derivatives• Density of States• Precision of FLAPW• All-Electron vs. Pseudopotential• Murnaghan’s Equation of State!53

Logarithmic Derivative!
!
d=ln Rl (r; E)!!
Ll (E) =Rl (S; E)
drRl′ (S; E)

r=S

The APW eigenfunction satisfies the boundary
conditions (logarithmic derivatives) on thespheres among the general solutions.

SW

PW
S

r

!54

Single-MT Problem
Radial Equation in Rydberg units
"
!
2
2 d
l(l + 1)d+
+ v(r) − E Rl (r; E) = 0
− 2−
2
dr
r dr
r
! S
2
2
R
(r;
E)r
dr = 1
Normalizationl0

Radial function
Pl (r; E) = rRl (r; E)"!
2l(l + 1)d
+ v(r) − E Pl (r; E) = 0
− 2+
2
dr
r
! S
Pl2 (r; E)dr = 1
0!55

Energy Dependence of Radial Function!

"l(l + 1)d Pl (r; E)=+ v(r) − E Pl (r; E)
2
2
dr
r
2

EL < Eatom < EU!56

Logarithmic Derivative
Rl (S; E)dLl (E) ==ln Rl (r; E)
Rl (S; E)
drr=S

EL < Eatom < EU
Rl (EL ) > Rl (Eatom ) > Rl (EU )′′′|Rl (EL )| < |Rl (Eatom )| < |Rl (EU )|0 > Ll (EL ) > Ll (Eatom ) > Ll (EU )!57

Logarithmic Derivative
Ll (E)

bcc W-d

Eatom!

Rl (EA ) = 0
Rl′ (EA ) ̸= 0!

Rl (EB ) ̸= 0
Rl′ (EB ) = 0!58

Logarithmic Derivative!

Rl (EA ) = 0
Rl′ (EA ) ̸= 0

Antibonding
state

!

Rl (EB ) ̸= 0
Rl′ (EB ) = 0

Bonding
state!59Density of Statesbcc Crd

D(E) =

!
j,k

p

=

!
j,k

⟨ψjk |ψjk ⟩δ(E − εkj )

s

total

δ(E − εkj )2Xm|mihm| = 1

X X
2
k
4
=
h j |mi (E
m

=

!
m

j,k

Dm (E)

3"kj )5

partial DOS !60

Wave Functions• LAPW Basis

1 i(k+K)·r
φ̃(r) = √ e
Ω
#
!"
k+K
l
φk+K (r) =
Ak+K
R
(r
)
+
B
Ṙ
(r
)
i
Ylm (r̂α )
l α
l α
αlm
αlm
k+K

αlm

• Degrees of Variational Freedom
Kmax

lmax

• Choice of MT Sphere Radius!61

Electron Density and Potentialñ(r) =

ṽ(r) =nαLM (rα )iL YLM (r̂α )

G

!

αLM

!

!

vαLM (rα )iL YLM (r̂α )!

eiG·r nG

eiG·r vGn(r) =

v(r) =GαLM• Accuracy of Expansion
Gmax

Lmax

variational parameters?

• Choice of MT Sphere Radius!62

Perturbative Consideration• Second-Order Perturbation
2|⟨k+
K|H|k⟩|∆εk = −
|k + K|2 − εk• Variational Parameters of the Wave Functions|k + K| ≤ Kmax

lmax
!63

Muffin-Tin Sphere Radius
When a sufficient lmax is assumed,• In case of large MT sphere radius, because of smaller
volume in the interstitial region fewer PW expansion
is needed.

• In case of small MT sphere radius, because of larger
volume in the interstitial region more PW expansion
is needed.

• A variational dimensionless parameter
RKmax
!64

Wave Functions and Electron Densityn(r) =!
k,n|ψnk (r)|2Gmax ≥ 2Kmax
Lmax ≥ 2lmax• Convergency of the electron density and potential
expansion should be checked, especially when GGA is
used.

• For small MT spheres used, higher Gmax may be required
to represent pseudized charge density.
!65Muffin-Tin Sphere Radius• Non-overlapping spheres
A margin should be considered if the atomic positions
are changed, for example in a structural optimization
calculation.

• Negligible penetration of the core functions outside
Within both assumptions with sufficient lmax and Lmax,
the accuracy does not depend on the choice of MT
sphere radius but does on RKmax.
!66

Total Energy vs. Muffin-Tin Radius
R=0.9Å

R=1.0Å

R=1.1Å

!67

Precision of FLAPW Method• Wave Functions

RKmax• Electron Density and Potential
Gmax ≥ 2Kmax

Lmax ≥ 2lmax• Choice of MT Sphere Radius• Over-completeness of APW Basis Functions
Since the PW basis is a complete set in all the space, an
APW basis with the excessive number of PW results inindefinite solutions.
!68All-Electron vs. Pseudopotential
Δ-gauge

Science 351, aad3000 (2016)!69

Murnaghan’s Equation of State
B0
p= 0
B

"✓

◆ B0

#

⌦
1
⌦0
✓
◆
✓
◆
dp
1
dB
0
B0 =
⌦
B =
⌦
d⌦ 0
B0
d⌦ 0
"
#
✓ ◆ B0B0 ⌦1⌦
0E(⌦) =+
1+E
B 0 B 0 1 ⌦0<latexit sha1_base64="5RUYa/K8i7D1ceGjIJMngmXZisY=">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</latexit><latexit sha1_base64="CBSfYAjUuTC7pbAN6Yt+RMeFndA=">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</latexit><latexit sha1_base64="FwcgfXKOANbf5FAxKag+3AxYy5A=">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</latexit><latexit sha1_base64="Gc2nX0R6Uo4Ouqo4+0r7n2jwtdU=">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</latexit>

1 a.u. in pressure = 1.47108 × 104 GPa
F. D. Murnaghan, Proc. Natl. Acad. Sci. USA 30, 244 (1944).

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