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Methods available in WIEN2k for the treatment
of exchange and correlation effects
F. Tran
Institute of Materials Chemistry
Vienna University of Technology, A-1060 Vienna, Austria
23rd WIEN2k workshop, 4-7 June2016, McMaster University,
Hamilton, Canada
I
W
E
N
2k
Outline of the talk
◮ Introduction
◮ Semilocal functionals:
◮ GGA and MGGA
◮ mBJ potential (for band gap)
◮ Input file case.in0
◮ The DFT-D3 method for dispersion
◮ On-site methods for strongly correlated electrons:
◮ DFT+U
◮ Hybrid functionals
◮ Hybrid functionals
◮ GW
Total energy in Kohn-Sham DFT 1
Et ot =
1
2
.
i
¸
i
s x¸
T
¸
s
2 3
|∇ψ (r)| d r +
2
1
¸ ¸
ρ(r)ρ(r′
) 3 3 ′
s
|r − r′|
d rd r +
¸
en
3
v (r)ρ(r)d r
x s x
E
¸
e
¸
e E
¸
e
¸
n
A,B
Aƒ=
B
2 |RA − RB|
s x
E
¸
n
¸
n
+
1 . ZAZB
+Exc
◮ Ts : kinetic energy of the non-interacting electrons
◮ Eee : electron-electron electrostatic Coulomb energy
◮ Een : electron-nucleus electrostatic Coulomb energy
◮ Enn : nucleus-nucleus electrostatic Coulomb energy
◮ Exc = Ex + Ec : exchange-correlation energy
Approximations for Exc have to be used in practice
=⇒ The reliability of the results depends mainly on Exc!
1
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)
Approximations for Exc (Jacob’s ladder 1)
Exc =
¸
ǫxc (r)d 3r
The accuracy, but also the computational cost, increase when climbing up the ladder
1
J. P. Perdew et al., J. Chem. Phys. 123, 062201 (2005)
The Kohn-Sham Schr¨odinger equations
Minimization of Etot leads to
.
1
.
− ∇2 + vee(r) + ven(r) + vˆxc(r) ψi(r) = ǫiψi(r)
2
Two types of vˆxc:
◮ Multiplicative: vˆxc = δExc/δρ = vxc (KS method)
◮ LDA
◮ GGA
i ,◮ Non-multiplicative: vˆxc = (1/ψi )δExc/δψ∗ = vxc i (generalized KS)
◮ Hartree-Fock
◮ LDA+U
◮ Hybrid (mixing of GGA and Hartree-Fock)
◮ MGGA
◮ Self-interaction corrected (Perdew-Zunger)
Semilocal functionals: trends with GGA
ǫGGA
xc
LDA(ρ, ∇ρ)= ǫx (ρ)Fxc(rs,s)
where Fxc is the enhancement factorand
rs =
1
. 4
3πρ
.1/3
(Wigner-Seitz radius)
s=
|∇ρ|
2(3π2)1/3
ρ4/3
(inhomogeneity parameter)
There are two types of GGA:
◮ Semi-empirical: contain parametersfitted to accurate (i.e.,
experimental) data.
◮ Ab initio: All parametersweredetermined byusing
mathematical conditions obeyed by the exact functional.
Semilocal functionals: GGA
Fx(s) = ǫGGA/ǫLDA
x x
0 0.5 3
0.9
1
1.4
1.3
1.2
1.1
1.5
1.6
1.7
1.8
1 1.5 2 2.5
inhomogeneity parameter s
Fx
LDA
PBEsol
PBE
B88
good for atomization energy of molecules
good for atomization energy of solids
good for lattice constant of solids
good for nothing
Construction of an universal GGA: A failure
Test of functionals on 44 solids1
0 2
0
4
6
8
PBEsol
WC
RGE2 PBE
PBEint
PBEalpha
SG4
0.5 1 1.5
Mean absolute percentage error for latticeconstant
Meanabsolutepercentageerrorforcohesiveenergy
2
 The accurate GGA for solids (cohesive energy/lattice constant).
They are ALL very inaccurate for the atomization of molecules
1
F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
Semilocal functionals: meta-GGA
ǫMGGA
xc
LDA(ρ, ∇ρ, t) = ǫxc (ρ)Fxc(rs,s,α)
where Fxc is the enhancement factorand
◮ α= t−tW
t T F
◮ α= 1 where the electron density is uniform
◮ α= 0 in one- and two-electron regions
◮ α≫1between closed shell atoms
=⇒ MGGA functionals are more flexible
Example: SCAN1 is
◮ as good as the best GGA for atomization energies of molecules
◮ as good as the best GGA for lattice constant of solids
1
J. Sun et al., Phys. Rev. Lett. 115, 036402 (2015)
Semilocal functionals: meta-GGA
0 0.5 2.5 3
1.2
1.1
1
0.9
1.3
1.4
1.5
1 1.5 2
inhomogeneity parameter s
Fx
Fx(s,α) = ǫMGGA/ǫLDA
x x
1.6
LDA
PBE
PBEsol
SCAN (=0)
SCAN (=1)
SCAN (=5)
Semilocal functionals: MGGA MS2 andSCAN
Test of functionals on 44 solids1
0
0
2
4
6
8
PBEsol
WC
PBEint
PBEalpha
RGE2 PBE
SG4
MGGA_MS2
SCAN
Meanabsolutepercentageerrorforcohesiveenergy
 The accurate GGA for solids (cohesive energy/lattice constant).
They are ALL very inaccurate for the atomization of molecules
 MGGA_MS2 and SCAN are very accurate for the atomization of molecules
0.5 1 1.5 2
Mean absolute percentage error for latticeconstant
1
F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
Semilocal potential for band gap: modified Becke-Johnson
◮ Standard LDA and GGA functionals underestimate the band gap
◮ Hybrid and GWaremuchmoreaccurate, but also muchmore
expensive
◮ A cheapalternative is to usethe modified Becke-Johnson(mBJ)
potential: 1
vmBJ
x
BR(r) = cvx (r) + (3c − 2)
1
. .
5 t(r)
π 6 ρ(r)
xwherev BR is the Becke-Roussel potential, t is the kinetic-energy
density and c is given by
c = α+ β
V
cell
¸
cell
1 |∇ρ(r)|
ρ(r)
3
d r
p
mBJ is a MGGA potential
1
F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)
Bandgapswith
mBJ
0246810121416
0
2
4
6
8
10
12
14
16
Experimentalbandgap(eV)
Theoretical band gap (eV)
Kr
Xe
Si
Ar LiF
LiCl
Ge ScN
GaAs
SiC FeO AlP CdS
C
AlN BN
MgO
ZnOGaN
MnO NiO ZnS
LDA
mBJ
HSE
GW00
GW
How to run a calculation with the mBJ potential?
1. init lapw (choose LDA or PBE)
2. init mbj lapw (create/modify files)
1. automatically done: case.in0modified and case.inm vresp
created
2. run(sp) lapw -i 1 -NI (creates case.r2v and case.vrespsum)
3. save lapw
3. init mbj lapw and chooseoneof the
parametrizations: 0: Original mBJ values1
1: New parametrization2
2: New parametrization for semiconductors2
3: Original BJ potential3
4. run(sp) lapw ...
1
F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)
2
D. Koller et al., Phys. Rev. B 85, 155109 (2012)
3
A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006)
Input file case.in0: keywords for the xc-functional
The functional is specifiedat the 1st line of case.in0.Three different
ways:
1. Specifya global keyword for Ex, Ec, vx, vc:
◮ TOT XC NAME
2. Specify a keyword for Ex, Ec, vx, vc individually:
◮ TOT EX NAME1 EC NAME2 VX NAME3 VCNAME4
3. Specify keywords to use functionals from LIBXC1:
◮ TOT XC TYPE X NAME1 XC TYPE C NAME2
◮ TOT XC TYPE XC NAME
where TYPE is the family name: LDA, GGA or MGGA
1
M. A. L. Marques et al., Comput. Phys. Commun. 183, 2272 (2012)
http://www.tddft.org/programs/octopus/wiki/index.php/Libxc
Input file case.in0: examples with keywords
◮ PBE:
TOT XC PBE
or
TOT EX PBE EC PBE VX PBE VCPBE
or
TOT XC GGA X PBE XC GGA C PBE
◮ mBJ (with LDA for the xc-energy):
TOT XC MBJ
◮ MGGA MS2:
TOT XC MGGA MS 0.504 0.14601 4.0
s x
κ
¸
,c
¸
,b
All available functionals are listed in tables of the UG. and in
$WIENROOT/SRC lapw0/xc funcs.h for LIBXC (if installed)
Dispersion methods for DFT
Problem with semilocal functionals:
◮ They do not include London dispersion interactions
◮ Resultsarequalitatively wrong for systemswhere dispersion
plays a majorrole
Two common dispersion methodsfor DFT:
◮ Pairwise term1:
EPW
c,disp = −
. .
A<B n=6,8,10,...
dampfn (RAB )
CAB
n
Rn
AB
◮ Nonlocal term2:
ENL
c,disp
2
1
¸ ¸
= ρ(r)φ(r, r′)ρ(r′)d3rd3r′
1
S. Grimme, J. Comput. Chem. 25, 1463 (2004)
2
M. Dion et al., Phys. Rev. Lett. 92, 246401 (2004)
The DFT-D3 method1 in WIEN2k
◮ Features of DFT-D3:
◮ Very cheap (pairwise)
◮ CAB
n depend on positions of the nuclei (via coordination
number)
◮ Functional-dependent parameters
◮ Energy and forces (minimization of internal parameters)
◮ 3-body term
◮ Installation:
◮ Not included in WIEN2k
◮ Download and compile the DFTD3 packagefrom
http://www.thch.uni-bonn.de/tc/index.php
copy the dftd3 executable in $WIENROOT
◮ input file case.indftd3 (if not present a default one is copied
automatically)
◮ run(sp) lapw -dftd3 . . .
◮ case.scfdftd3 is included in case.scf
1
S. Grimme et al., J. Chem. Phys. 132, 154104 (2010)
The DFT-D3 method: the input file case.indftd3
Default (and recommended) input file:
damping function f damp
n
the one in case.in0∗
forces
method
func
grad
pbc
abc
cutoff
cnthr
num
bj
def ault
yes
yes
yes
95
40
no
periodic boundary conditions
3-body term
interaction cutoff
coordination number cutoff
numerical gradient
∗default will work for PBE, PBEsol, BLYP and TPSS. For other
functionals, the functional namehasto bespecified(seedftd3.f of
DFTD3 package)
The DFT-D3 method: hexagonal BN1
3 3.2 3.4 3.6 4.4 4.6 4.8 5
0
1.4
1.2
1
0.8
0.6
0.4
0.2
1.6
1.8
2
exp
3.8 4 4.2
Interlayer distance (A˚)
Totalenergy(mRy/cell)
PBE
BLYP
PBE+D3
BLYP+D3
1
F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
Strongly correlated electrons
Problem with semilocal functionals:
◮ They give qualitatively wrong results for solids which contain
localized 3d or 4f electrons
◮ The band gap is too small or even absent like in FeO
◮ The magnetic moments are too small
◮ Wrong ground state
Why?
◮ The strong on-site correlations arenot correctly accountedfor by
semilocal functionals.
(Partial) solution to the problem:
◮ Combine semilocal functionals with Hartree-Fock theory:
◮ DFT+U
◮ Hybrid
Even better:
◮ LDA+DMFT (DMFT codesusing WIEN2k orbitals asinput
exist)
On-site DFT+U and hybrid methods in WIEN2k
◮ Forsolids,the hybrid functionals arecomputationally very
expensive.
◮ In WIEN2k the on-site DFT+U 1 and on-site hybrid2,3
methods areavailable. Thesemethodsareapproximations of the
Hartree-Fock/hybrid methods
◮ Applied only inside atomic spheresof selectedatomsand
electrons of a given angular momentum ℓ.
On-site methods → As cheap as LDA/GGA.
1
V. I. Anisimov et al., Phys. Rev. B 44, 943 (1991)
2
P. Nov´ak et al., Phys. Stat. Sol. (b) 243, 563 (2006)
3
F. Tran et al., Phys. Rev. B 74, 155108 (2006)
DFT+U and hybrid exchange-correlation functionals
The exchange-correlation functional is
xc
EDFT+ U/hybrid DFT onsite= Exc [ρ] + E [nmm′ ]
where nmm′ is the density matrix of the correlated electrons
◮ For DFT+U both exchange and Coulomb are corrected:
Eonsite = EHF DFT DFT
x + ECoul −Ex −ECoul
s
corr
¸
ec
¸
tion
x s
double
¸
c
¸
ounting
x
There are several versions of the double-counting term
◮ For the hybrid methods only exchange is corrected:
Eonsite = αEHF − αELDA
x
d
s
.c
¸
o
¸
unt
x
.
x
s
c
¸
or
¸
r.
x
where αis a parameter ∈ [0,1]
How to run DFT+U and on-site hybrid calculations?
1. Create the input files:
◮ case.inorb and case.indm for DFT+U
◮ case.ineecefor on-site hybrid functionals (case.indm created
automatically):
2. Run the job (can only be run with runsp lapw):
◮ LDA+U : runsp lapw -orb . . .
◮ Hybrid: runsp lapw -eece . ..
Foracalculation without spin-polarization (ρ↑ = ρ↓):
runsp c lapw -orb/eece . ..
Input file case.inorb
LDA+U applied to the 4f electrons of atomsNo. 2 and 4:
1 2 0
PRATT, 1. 0
2 1 3
4 1 3
1
0.61 0.07
0.61 0.07
nmod, natorb, i p r
mixmod, amix
iatom, nlorb, lorb
iatom, nlorb, lorb
nsic (LDA+U(SIC) used)
U J (Ry)
U J (Ry)
nsic=0 for the AMF method (lessstrongly correlated electrons)
nsic=1 for the SIC method
nsic=2 for the HMF method
Input file case.ineece
On-site hybrid functional PBE0applied to the 4f electrons of
atomsNo. 2 and 4:
-12.0 2
2 1 3
4 1 3
HYBR
0.25
emin, natorb
iatom, nlorb, lorb
iatom, nlorb, lorb
HYBR/EECE
fraction of exact exchange
SCF cycle of DFT+U in WIEN2k
lapw0 xc,σ ee en→ v D F T
+ v + v (case.vspup(dn), case.vnsup(dn))
orb -up mm′→ v↑
(case.vorbup)
orb -dn mm′→ v↓
(case.vorbdn)
lapw1 -up -orb nk nk
→ ψ↑
, ǫ↑
(case.vectorup, case.energyup)
lapw1 -dn -orb nk nk
→ ψ↓
, ǫ↓
(case.vectordn, case.energydn)
lapw2 -up va l
→ ρ↑
(case.clmvalup)
lapw2 -dn va l
→ ρ↓
(case.clmvaldn)
lapwdm -up mm′
lapwdm -dn mm′
→ n↑
(case.dmatup)
→ n↓
(case.dmatdn)
lcore -up core
lcore -dn core
→ ρ↑
(case.clmcorup)
→ ρ↓
(case.clmcordn)
mixer σ→ mixed ρ and nσ
mm′
Hybrid functionals
◮ On-site hybrid functionals can be applied only to localized electrons
◮ Full hybrid functionals are necessary (but expensive) for solids with
delocalized electrons (e.g., in sp-semiconductors)
Two types of full hybrid functionals available in WIEN2k1:
◮ unscreened:
Exc = EDFT
xc
HF DFT+ α
.
Ex −Ex
.
− λ |r− r′|
|r−r′| |r−r′|
◮ screened (short-range), 1 → e :
Exc = EDFT
xc
SR−HF SR−DFT+ α
.
Ex −Ex
.
screening leads to faster convergence with k-points sampling
1
F. Tran and P. Blaha, Phys. Rev. B 83, 235118 (2011)
Hybrid functionals: technical details
◮ 10-1000 times more expensive than LDA/GGA
◮ k-point and MPI parallelization
◮ Approximations to speed up the calculations:
◮ Reduced k-meshfor the HF potential. Example:
For a calculation with a 12× 12× 12 k-mesh, thereduced
k-meshfor the HF potential can be:
6× 6× 6, 4 × 4× 4, 3 × 3× 3, 2× 2 × 2 or 1× 1× 1
◮ Non-self-consistent calculation of the band structure
◮ Underlying functionals for unscreened and screend hybrid:
◮ LDA
◮ PBE
◮ WC
◮ PBEsol
◮ B3PW91
◮ B3LYP
◮ Use run bandplothf lapw for band structure
Hybrid functionals: input file case.inhf
Example for YS-PBE0 (similar to HSE06 from Heyd, Scuseria and Ernzerhof1
)
0.25
T
0. 165
20
6
3
3
1d-3
fraction α of HF exchange
screened (T, YS-PBE0) or unscreened ( F , PBE0)
screening parameter λ
number of bands for the 2nd Hamiltonian
GMAX
lmax for the expansion of o r b i t a l s
lmax for the product of two o r b i t a l s
r a d i a l i n t e g r a l s below t h i s value neglected
Important: The computational time will dependstrongly on the
number of bands, GMAX, lmax and the number of k-points
1
A. V. Krukau et al., J. Chem. Phys. 125, 224106 (2006)
How to run hybrid functionals?
1. init lapw
2. Recommended: run(sp) lapw for the semilocal functional
3. save lapw
4. init hf lapw (this will create/modify input files)
1. adjust case.inhf according to your needs
2. reduced k-meshfor the HF potential? Yes or no
3. specify the k-mesh
5. run(sp) lapw -hf (-redklist) (-diaghf) ...
SCF cycle of hybrid functionals in WIEN2k
lapw0 -grr → v DFT (case.r2v), αEDFT (:AEXSL)
x x
lapw0 xc
→ vDFT + vee + ven (case.vsp, case.vns)
lapw1 nk nk
→ ψDFT, ǫDFT (case.vector,case.energy)
lapw2 nk
→
.
n k ǫDFT (:SLSUM)
hf
lapw2 -hf
→ ψnk, ǫnk (case.vectorhf, case.energyhf)
→ ρval (case.clmval)
lcore
mixer
→ ρcore (case.clmcor)
→ mixed ρ
Calculation of quasiparticle spectra from many-body theory
◮ In principle the Kohn-Shameigenvalues should beviewed as
mathematical objects and not compareddirectly to experiment
(ionization potential and electron affinity).
◮ The true addition and removal energiesǫiarecalculated from the
equation of motion for the Green function:
2
¸ , ¸
1
− ∇2
+ ven(r) + vH(r) + Σ(r, r′
, ǫi)ψi(r′
)d3
r′
= ǫiψi (r)
◮ The self-energy Σ is calculated from Hedin’s self-consistent
equations1:
Σ(1, 2) = i
¸
G(1, 4)W (1+
, 3)Γ(4, 2, 3)d(3, 4)
W (1, 2) = v(1, 2) +
¸
v(4, 2)P(3, 4)W (1, 3)d(3, 4)
P(1, 2) = −i
¸
G(2, 3)G(4, 2)Γ(3, 4, 1)d(3, 4)
δG(4,5)
Γ(1, 2, 3) = δ(1, 2)δ(1, 3) +
¸ δΣ(1, 2)
G(4, 6)G(7, 5)Γ(6, 7, 3)d(4, 5, 6, 7)
1
L. Hedin, Phys. Rev. 139, A769 (1965)
The GW and G0W0 approximations
◮ GW: vertex function Γ in Σ set to 1:
Σ(1, 2) = i
¸
G(1, 4)W (1+, 3)Γ(4, 2,3)d(3, 4) ≈ iG(1, 2+)W (1, 2)
2π
∞
− ∞
Σ(r, r′, ω)=
i
¸
G(r, r′, ω+ ω′)W(r, r′, ω′)e−iδω′
dω′
′
G(r, r ,ω) =
∞
. i i
∗ ′ψ(r)ψ (r )
i=1
ω− ǫi − iηi
W (r, r′, ω) =
¸
v(r, r′′)ǫ−1(r′′, r′, ω)d3r′′
◮ G0W0 (one-shot GW ):
Gand W arecalculated using the Kohn-Shamorbitals and
eigenvalues. 1st order perturbation theory gives
i
ǫGW K S K S K S K S= ǫi + Z (ǫi )(ψi |ℜ(Σ(ǫi
K S)) − vxc|ψi )
A few remarks on GW
◮ GW calculations require very large computational ressources
◮ Gand W dependon all (occupied and unoccupied) orbitals (up
to parameter emax in practice)
◮ GWis the state-of-the-art for the calculation of (inverse)
photoemissionspectra, but not for optics sinceexcitonic effects
are still missing in GW (BSE code from R. Laskowski)
◮ GW is more accurate for systems with weak correlations
FHI-gap: a LAPW GW code1
◮ Based on the FP-LAPW basis set
◮ Mixed basis set to expand the GW-related quantities
◮ Interfaced with WIEN2k
◮ G0W0, GW0 @LDA/GGA(+U )
◮ Parallelized
◮ http://www.chem.pku.edu.cn/jianghgroup/codes/fhi-gap.html
1
H. Jiang et al., Comput. Phys. Comput. 184, 348 (2013)
Flowchart of FHI-gap
How to run the FHI-gap code?
1. Run a WIEN2k SCF calculation (in w2kdir)
2. In w2kdir, executethe script gapinit to preparethe input files for
GW:
gap init -d <gwdir> -nkp <nkp> -s 0/1/2 -orb -emax <emax>
3. Eventually modify gwdir.ingw
4. Execute gap.x or gap-mpi.x in gwdir
5. Analyse the results from:
1. gwdir.outgw
2. the plot of the DOS/band structure generated by gap analy
Parameters to be converged for a GW calculation
◮ Usual WIEN2k parameters:
◮ Size of the LAPW basis set (RKmax)
◮ Number of k-points for the Brillouin zone integrations
◮ GW-specific parameters:
◮ Size of the mixed basis set
◮ Number of unoccupiedstates (emax)
◮ Number of frequencies ωfor the calculation of Σ =
¸
GWd ω
Band
gaps
0246810121416
0
2
4
6
8
10
12
14
16
Experimentalbandgap(eV)
Theoretical band gap (eV)
Kr
Xe
Si
Ar LiF
LiCl
Ge ScN
GaAs
SiC FeO AlP CdS
C
AlN BN
MgO
ZnO GaN
MnO NiO ZnS
LDA
mBJ
HSE
GW00
GW
Some recommendations
Before using a method or a functional:
◮ Read a few papers concerning the method in order to know
◮ why it has been used
◮ for which properties or types of solids it is supposedto be
reliable
◮ if it is adapted to your problem
◮ Do you have enough computational ressources?
◮ hybrid functionals and GW require (substantially) more
computational ressources (and patience)

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Methods available in WIEN2k for the treatment of exchange and correlation effects

  • 1. Methods available in WIEN2k for the treatment of exchange and correlation effects F. Tran Institute of Materials Chemistry Vienna University of Technology, A-1060 Vienna, Austria 23rd WIEN2k workshop, 4-7 June2016, McMaster University, Hamilton, Canada I W E N 2k
  • 2. Outline of the talk ◮ Introduction ◮ Semilocal functionals: ◮ GGA and MGGA ◮ mBJ potential (for band gap) ◮ Input file case.in0 ◮ The DFT-D3 method for dispersion ◮ On-site methods for strongly correlated electrons: ◮ DFT+U ◮ Hybrid functionals ◮ Hybrid functionals ◮ GW
  • 3. Total energy in Kohn-Sham DFT 1 Et ot = 1 2 . i ¸ i s x¸ T ¸ s 2 3 |∇ψ (r)| d r + 2 1 ¸ ¸ ρ(r)ρ(r′ ) 3 3 ′ s |r − r′| d rd r + ¸ en 3 v (r)ρ(r)d r x s x E ¸ e ¸ e E ¸ e ¸ n A,B Aƒ= B 2 |RA − RB| s x E ¸ n ¸ n + 1 . ZAZB +Exc ◮ Ts : kinetic energy of the non-interacting electrons ◮ Eee : electron-electron electrostatic Coulomb energy ◮ Een : electron-nucleus electrostatic Coulomb energy ◮ Enn : nucleus-nucleus electrostatic Coulomb energy ◮ Exc = Ex + Ec : exchange-correlation energy Approximations for Exc have to be used in practice =⇒ The reliability of the results depends mainly on Exc! 1 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)
  • 4. Approximations for Exc (Jacob’s ladder 1) Exc = ¸ ǫxc (r)d 3r The accuracy, but also the computational cost, increase when climbing up the ladder 1 J. P. Perdew et al., J. Chem. Phys. 123, 062201 (2005)
  • 5. The Kohn-Sham Schr¨odinger equations Minimization of Etot leads to . 1 . − ∇2 + vee(r) + ven(r) + vˆxc(r) ψi(r) = ǫiψi(r) 2 Two types of vˆxc: ◮ Multiplicative: vˆxc = δExc/δρ = vxc (KS method) ◮ LDA ◮ GGA i ,◮ Non-multiplicative: vˆxc = (1/ψi )δExc/δψ∗ = vxc i (generalized KS) ◮ Hartree-Fock ◮ LDA+U ◮ Hybrid (mixing of GGA and Hartree-Fock) ◮ MGGA ◮ Self-interaction corrected (Perdew-Zunger)
  • 6. Semilocal functionals: trends with GGA ǫGGA xc LDA(ρ, ∇ρ)= ǫx (ρ)Fxc(rs,s) where Fxc is the enhancement factorand rs = 1 . 4 3πρ .1/3 (Wigner-Seitz radius) s= |∇ρ| 2(3π2)1/3 ρ4/3 (inhomogeneity parameter) There are two types of GGA: ◮ Semi-empirical: contain parametersfitted to accurate (i.e., experimental) data. ◮ Ab initio: All parametersweredetermined byusing mathematical conditions obeyed by the exact functional.
  • 7. Semilocal functionals: GGA Fx(s) = ǫGGA/ǫLDA x x 0 0.5 3 0.9 1 1.4 1.3 1.2 1.1 1.5 1.6 1.7 1.8 1 1.5 2 2.5 inhomogeneity parameter s Fx LDA PBEsol PBE B88 good for atomization energy of molecules good for atomization energy of solids good for lattice constant of solids good for nothing
  • 8. Construction of an universal GGA: A failure Test of functionals on 44 solids1 0 2 0 4 6 8 PBEsol WC RGE2 PBE PBEint PBEalpha SG4 0.5 1 1.5 Mean absolute percentage error for latticeconstant Meanabsolutepercentageerrorforcohesiveenergy 2  The accurate GGA for solids (cohesive energy/lattice constant). They are ALL very inaccurate for the atomization of molecules 1 F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
  • 9. Semilocal functionals: meta-GGA ǫMGGA xc LDA(ρ, ∇ρ, t) = ǫxc (ρ)Fxc(rs,s,α) where Fxc is the enhancement factorand ◮ α= t−tW t T F ◮ α= 1 where the electron density is uniform ◮ α= 0 in one- and two-electron regions ◮ α≫1between closed shell atoms =⇒ MGGA functionals are more flexible Example: SCAN1 is ◮ as good as the best GGA for atomization energies of molecules ◮ as good as the best GGA for lattice constant of solids 1 J. Sun et al., Phys. Rev. Lett. 115, 036402 (2015)
  • 10. Semilocal functionals: meta-GGA 0 0.5 2.5 3 1.2 1.1 1 0.9 1.3 1.4 1.5 1 1.5 2 inhomogeneity parameter s Fx Fx(s,α) = ǫMGGA/ǫLDA x x 1.6 LDA PBE PBEsol SCAN (=0) SCAN (=1) SCAN (=5)
  • 11. Semilocal functionals: MGGA MS2 andSCAN Test of functionals on 44 solids1 0 0 2 4 6 8 PBEsol WC PBEint PBEalpha RGE2 PBE SG4 MGGA_MS2 SCAN Meanabsolutepercentageerrorforcohesiveenergy  The accurate GGA for solids (cohesive energy/lattice constant). They are ALL very inaccurate for the atomization of molecules  MGGA_MS2 and SCAN are very accurate for the atomization of molecules 0.5 1 1.5 2 Mean absolute percentage error for latticeconstant 1 F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
  • 12. Semilocal potential for band gap: modified Becke-Johnson ◮ Standard LDA and GGA functionals underestimate the band gap ◮ Hybrid and GWaremuchmoreaccurate, but also muchmore expensive ◮ A cheapalternative is to usethe modified Becke-Johnson(mBJ) potential: 1 vmBJ x BR(r) = cvx (r) + (3c − 2) 1 . . 5 t(r) π 6 ρ(r) xwherev BR is the Becke-Roussel potential, t is the kinetic-energy density and c is given by c = α+ β V cell ¸ cell 1 |∇ρ(r)| ρ(r) 3 d r p mBJ is a MGGA potential 1 F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)
  • 13. Bandgapswith mBJ 0246810121416 0 2 4 6 8 10 12 14 16 Experimentalbandgap(eV) Theoretical band gap (eV) Kr Xe Si Ar LiF LiCl Ge ScN GaAs SiC FeO AlP CdS C AlN BN MgO ZnOGaN MnO NiO ZnS LDA mBJ HSE GW00 GW
  • 14. How to run a calculation with the mBJ potential? 1. init lapw (choose LDA or PBE) 2. init mbj lapw (create/modify files) 1. automatically done: case.in0modified and case.inm vresp created 2. run(sp) lapw -i 1 -NI (creates case.r2v and case.vrespsum) 3. save lapw 3. init mbj lapw and chooseoneof the parametrizations: 0: Original mBJ values1 1: New parametrization2 2: New parametrization for semiconductors2 3: Original BJ potential3 4. run(sp) lapw ... 1 F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009) 2 D. Koller et al., Phys. Rev. B 85, 155109 (2012) 3 A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006)
  • 15. Input file case.in0: keywords for the xc-functional The functional is specifiedat the 1st line of case.in0.Three different ways: 1. Specifya global keyword for Ex, Ec, vx, vc: ◮ TOT XC NAME 2. Specify a keyword for Ex, Ec, vx, vc individually: ◮ TOT EX NAME1 EC NAME2 VX NAME3 VCNAME4 3. Specify keywords to use functionals from LIBXC1: ◮ TOT XC TYPE X NAME1 XC TYPE C NAME2 ◮ TOT XC TYPE XC NAME where TYPE is the family name: LDA, GGA or MGGA 1 M. A. L. Marques et al., Comput. Phys. Commun. 183, 2272 (2012) http://www.tddft.org/programs/octopus/wiki/index.php/Libxc
  • 16. Input file case.in0: examples with keywords ◮ PBE: TOT XC PBE or TOT EX PBE EC PBE VX PBE VCPBE or TOT XC GGA X PBE XC GGA C PBE ◮ mBJ (with LDA for the xc-energy): TOT XC MBJ ◮ MGGA MS2: TOT XC MGGA MS 0.504 0.14601 4.0 s x κ ¸ ,c ¸ ,b All available functionals are listed in tables of the UG. and in $WIENROOT/SRC lapw0/xc funcs.h for LIBXC (if installed)
  • 17. Dispersion methods for DFT Problem with semilocal functionals: ◮ They do not include London dispersion interactions ◮ Resultsarequalitatively wrong for systemswhere dispersion plays a majorrole Two common dispersion methodsfor DFT: ◮ Pairwise term1: EPW c,disp = − . . A<B n=6,8,10,... dampfn (RAB ) CAB n Rn AB ◮ Nonlocal term2: ENL c,disp 2 1 ¸ ¸ = ρ(r)φ(r, r′)ρ(r′)d3rd3r′ 1 S. Grimme, J. Comput. Chem. 25, 1463 (2004) 2 M. Dion et al., Phys. Rev. Lett. 92, 246401 (2004)
  • 18. The DFT-D3 method1 in WIEN2k ◮ Features of DFT-D3: ◮ Very cheap (pairwise) ◮ CAB n depend on positions of the nuclei (via coordination number) ◮ Functional-dependent parameters ◮ Energy and forces (minimization of internal parameters) ◮ 3-body term ◮ Installation: ◮ Not included in WIEN2k ◮ Download and compile the DFTD3 packagefrom http://www.thch.uni-bonn.de/tc/index.php copy the dftd3 executable in $WIENROOT ◮ input file case.indftd3 (if not present a default one is copied automatically) ◮ run(sp) lapw -dftd3 . . . ◮ case.scfdftd3 is included in case.scf 1 S. Grimme et al., J. Chem. Phys. 132, 154104 (2010)
  • 19. The DFT-D3 method: the input file case.indftd3 Default (and recommended) input file: damping function f damp n the one in case.in0∗ forces method func grad pbc abc cutoff cnthr num bj def ault yes yes yes 95 40 no periodic boundary conditions 3-body term interaction cutoff coordination number cutoff numerical gradient ∗default will work for PBE, PBEsol, BLYP and TPSS. For other functionals, the functional namehasto bespecified(seedftd3.f of DFTD3 package)
  • 20. The DFT-D3 method: hexagonal BN1 3 3.2 3.4 3.6 4.4 4.6 4.8 5 0 1.4 1.2 1 0.8 0.6 0.4 0.2 1.6 1.8 2 exp 3.8 4 4.2 Interlayer distance (A˚) Totalenergy(mRy/cell) PBE BLYP PBE+D3 BLYP+D3 1 F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
  • 21. Strongly correlated electrons Problem with semilocal functionals: ◮ They give qualitatively wrong results for solids which contain localized 3d or 4f electrons ◮ The band gap is too small or even absent like in FeO ◮ The magnetic moments are too small ◮ Wrong ground state Why? ◮ The strong on-site correlations arenot correctly accountedfor by semilocal functionals. (Partial) solution to the problem: ◮ Combine semilocal functionals with Hartree-Fock theory: ◮ DFT+U ◮ Hybrid Even better: ◮ LDA+DMFT (DMFT codesusing WIEN2k orbitals asinput exist)
  • 22. On-site DFT+U and hybrid methods in WIEN2k ◮ Forsolids,the hybrid functionals arecomputationally very expensive. ◮ In WIEN2k the on-site DFT+U 1 and on-site hybrid2,3 methods areavailable. Thesemethodsareapproximations of the Hartree-Fock/hybrid methods ◮ Applied only inside atomic spheresof selectedatomsand electrons of a given angular momentum ℓ. On-site methods → As cheap as LDA/GGA. 1 V. I. Anisimov et al., Phys. Rev. B 44, 943 (1991) 2 P. Nov´ak et al., Phys. Stat. Sol. (b) 243, 563 (2006) 3 F. Tran et al., Phys. Rev. B 74, 155108 (2006)
  • 23. DFT+U and hybrid exchange-correlation functionals The exchange-correlation functional is xc EDFT+ U/hybrid DFT onsite= Exc [ρ] + E [nmm′ ] where nmm′ is the density matrix of the correlated electrons ◮ For DFT+U both exchange and Coulomb are corrected: Eonsite = EHF DFT DFT x + ECoul −Ex −ECoul s corr ¸ ec ¸ tion x s double ¸ c ¸ ounting x There are several versions of the double-counting term ◮ For the hybrid methods only exchange is corrected: Eonsite = αEHF − αELDA x d s .c ¸ o ¸ unt x . x s c ¸ or ¸ r. x where αis a parameter ∈ [0,1]
  • 24. How to run DFT+U and on-site hybrid calculations? 1. Create the input files: ◮ case.inorb and case.indm for DFT+U ◮ case.ineecefor on-site hybrid functionals (case.indm created automatically): 2. Run the job (can only be run with runsp lapw): ◮ LDA+U : runsp lapw -orb . . . ◮ Hybrid: runsp lapw -eece . .. Foracalculation without spin-polarization (ρ↑ = ρ↓): runsp c lapw -orb/eece . ..
  • 25. Input file case.inorb LDA+U applied to the 4f electrons of atomsNo. 2 and 4: 1 2 0 PRATT, 1. 0 2 1 3 4 1 3 1 0.61 0.07 0.61 0.07 nmod, natorb, i p r mixmod, amix iatom, nlorb, lorb iatom, nlorb, lorb nsic (LDA+U(SIC) used) U J (Ry) U J (Ry) nsic=0 for the AMF method (lessstrongly correlated electrons) nsic=1 for the SIC method nsic=2 for the HMF method
  • 26. Input file case.ineece On-site hybrid functional PBE0applied to the 4f electrons of atomsNo. 2 and 4: -12.0 2 2 1 3 4 1 3 HYBR 0.25 emin, natorb iatom, nlorb, lorb iatom, nlorb, lorb HYBR/EECE fraction of exact exchange
  • 27. SCF cycle of DFT+U in WIEN2k lapw0 xc,σ ee en→ v D F T + v + v (case.vspup(dn), case.vnsup(dn)) orb -up mm′→ v↑ (case.vorbup) orb -dn mm′→ v↓ (case.vorbdn) lapw1 -up -orb nk nk → ψ↑ , ǫ↑ (case.vectorup, case.energyup) lapw1 -dn -orb nk nk → ψ↓ , ǫ↓ (case.vectordn, case.energydn) lapw2 -up va l → ρ↑ (case.clmvalup) lapw2 -dn va l → ρ↓ (case.clmvaldn) lapwdm -up mm′ lapwdm -dn mm′ → n↑ (case.dmatup) → n↓ (case.dmatdn) lcore -up core lcore -dn core → ρ↑ (case.clmcorup) → ρ↓ (case.clmcordn) mixer σ→ mixed ρ and nσ mm′
  • 28. Hybrid functionals ◮ On-site hybrid functionals can be applied only to localized electrons ◮ Full hybrid functionals are necessary (but expensive) for solids with delocalized electrons (e.g., in sp-semiconductors) Two types of full hybrid functionals available in WIEN2k1: ◮ unscreened: Exc = EDFT xc HF DFT+ α . Ex −Ex . − λ |r− r′| |r−r′| |r−r′| ◮ screened (short-range), 1 → e : Exc = EDFT xc SR−HF SR−DFT+ α . Ex −Ex . screening leads to faster convergence with k-points sampling 1 F. Tran and P. Blaha, Phys. Rev. B 83, 235118 (2011)
  • 29. Hybrid functionals: technical details ◮ 10-1000 times more expensive than LDA/GGA ◮ k-point and MPI parallelization ◮ Approximations to speed up the calculations: ◮ Reduced k-meshfor the HF potential. Example: For a calculation with a 12× 12× 12 k-mesh, thereduced k-meshfor the HF potential can be: 6× 6× 6, 4 × 4× 4, 3 × 3× 3, 2× 2 × 2 or 1× 1× 1 ◮ Non-self-consistent calculation of the band structure ◮ Underlying functionals for unscreened and screend hybrid: ◮ LDA ◮ PBE ◮ WC ◮ PBEsol ◮ B3PW91 ◮ B3LYP ◮ Use run bandplothf lapw for band structure
  • 30. Hybrid functionals: input file case.inhf Example for YS-PBE0 (similar to HSE06 from Heyd, Scuseria and Ernzerhof1 ) 0.25 T 0. 165 20 6 3 3 1d-3 fraction α of HF exchange screened (T, YS-PBE0) or unscreened ( F , PBE0) screening parameter λ number of bands for the 2nd Hamiltonian GMAX lmax for the expansion of o r b i t a l s lmax for the product of two o r b i t a l s r a d i a l i n t e g r a l s below t h i s value neglected Important: The computational time will dependstrongly on the number of bands, GMAX, lmax and the number of k-points 1 A. V. Krukau et al., J. Chem. Phys. 125, 224106 (2006)
  • 31. How to run hybrid functionals? 1. init lapw 2. Recommended: run(sp) lapw for the semilocal functional 3. save lapw 4. init hf lapw (this will create/modify input files) 1. adjust case.inhf according to your needs 2. reduced k-meshfor the HF potential? Yes or no 3. specify the k-mesh 5. run(sp) lapw -hf (-redklist) (-diaghf) ...
  • 32. SCF cycle of hybrid functionals in WIEN2k lapw0 -grr → v DFT (case.r2v), αEDFT (:AEXSL) x x lapw0 xc → vDFT + vee + ven (case.vsp, case.vns) lapw1 nk nk → ψDFT, ǫDFT (case.vector,case.energy) lapw2 nk → . n k ǫDFT (:SLSUM) hf lapw2 -hf → ψnk, ǫnk (case.vectorhf, case.energyhf) → ρval (case.clmval) lcore mixer → ρcore (case.clmcor) → mixed ρ
  • 33. Calculation of quasiparticle spectra from many-body theory ◮ In principle the Kohn-Shameigenvalues should beviewed as mathematical objects and not compareddirectly to experiment (ionization potential and electron affinity). ◮ The true addition and removal energiesǫiarecalculated from the equation of motion for the Green function: 2 ¸ , ¸ 1 − ∇2 + ven(r) + vH(r) + Σ(r, r′ , ǫi)ψi(r′ )d3 r′ = ǫiψi (r) ◮ The self-energy Σ is calculated from Hedin’s self-consistent equations1: Σ(1, 2) = i ¸ G(1, 4)W (1+ , 3)Γ(4, 2, 3)d(3, 4) W (1, 2) = v(1, 2) + ¸ v(4, 2)P(3, 4)W (1, 3)d(3, 4) P(1, 2) = −i ¸ G(2, 3)G(4, 2)Γ(3, 4, 1)d(3, 4) δG(4,5) Γ(1, 2, 3) = δ(1, 2)δ(1, 3) + ¸ δΣ(1, 2) G(4, 6)G(7, 5)Γ(6, 7, 3)d(4, 5, 6, 7) 1 L. Hedin, Phys. Rev. 139, A769 (1965)
  • 34. The GW and G0W0 approximations ◮ GW: vertex function Γ in Σ set to 1: Σ(1, 2) = i ¸ G(1, 4)W (1+, 3)Γ(4, 2,3)d(3, 4) ≈ iG(1, 2+)W (1, 2) 2π ∞ − ∞ Σ(r, r′, ω)= i ¸ G(r, r′, ω+ ω′)W(r, r′, ω′)e−iδω′ dω′ ′ G(r, r ,ω) = ∞ . i i ∗ ′ψ(r)ψ (r ) i=1 ω− ǫi − iηi W (r, r′, ω) = ¸ v(r, r′′)ǫ−1(r′′, r′, ω)d3r′′ ◮ G0W0 (one-shot GW ): Gand W arecalculated using the Kohn-Shamorbitals and eigenvalues. 1st order perturbation theory gives i ǫGW K S K S K S K S= ǫi + Z (ǫi )(ψi |ℜ(Σ(ǫi K S)) − vxc|ψi )
  • 35. A few remarks on GW ◮ GW calculations require very large computational ressources ◮ Gand W dependon all (occupied and unoccupied) orbitals (up to parameter emax in practice) ◮ GWis the state-of-the-art for the calculation of (inverse) photoemissionspectra, but not for optics sinceexcitonic effects are still missing in GW (BSE code from R. Laskowski) ◮ GW is more accurate for systems with weak correlations
  • 36. FHI-gap: a LAPW GW code1 ◮ Based on the FP-LAPW basis set ◮ Mixed basis set to expand the GW-related quantities ◮ Interfaced with WIEN2k ◮ G0W0, GW0 @LDA/GGA(+U ) ◮ Parallelized ◮ http://www.chem.pku.edu.cn/jianghgroup/codes/fhi-gap.html 1 H. Jiang et al., Comput. Phys. Comput. 184, 348 (2013)
  • 38. How to run the FHI-gap code? 1. Run a WIEN2k SCF calculation (in w2kdir) 2. In w2kdir, executethe script gapinit to preparethe input files for GW: gap init -d <gwdir> -nkp <nkp> -s 0/1/2 -orb -emax <emax> 3. Eventually modify gwdir.ingw 4. Execute gap.x or gap-mpi.x in gwdir 5. Analyse the results from: 1. gwdir.outgw 2. the plot of the DOS/band structure generated by gap analy
  • 39. Parameters to be converged for a GW calculation ◮ Usual WIEN2k parameters: ◮ Size of the LAPW basis set (RKmax) ◮ Number of k-points for the Brillouin zone integrations ◮ GW-specific parameters: ◮ Size of the mixed basis set ◮ Number of unoccupiedstates (emax) ◮ Number of frequencies ωfor the calculation of Σ = ¸ GWd ω
  • 40. Band gaps 0246810121416 0 2 4 6 8 10 12 14 16 Experimentalbandgap(eV) Theoretical band gap (eV) Kr Xe Si Ar LiF LiCl Ge ScN GaAs SiC FeO AlP CdS C AlN BN MgO ZnO GaN MnO NiO ZnS LDA mBJ HSE GW00 GW
  • 41. Some recommendations Before using a method or a functional: ◮ Read a few papers concerning the method in order to know ◮ why it has been used ◮ for which properties or types of solids it is supposedto be reliable ◮ if it is adapted to your problem ◮ Do you have enough computational ressources? ◮ hybrid functionals and GW require (substantially) more computational ressources (and patience)