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Schroedinger equation, Hydrogen atom, moleculer orbitals, Brillowin zone
and Band theory, Crystal symmetry, Independent-particle picture,
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With better skill, we can do things faster and less mistakes. Try to introducebetter technology (but new technology (e.g, c++) is not necessarily good).
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• Advanced method and theories such as GW (many-body theory).

Monday 15:00~17:00,17:20~19:20
Theory, How it works?, and Practice.
• LECTURE1: PMT method
How to solve one-body problem? PMT= LAPW+LMTO• LECTURE2: Quasiparticle self-consistent GW method
Theory and results until now.
Some kinds of numerical techniques.

• Get Started!
Install ecalj, and go through “ecalj tutorial”.
Observe how it works.
------------------------------------------------------------------------Tuesday: 9:00~17:20, Wednesday 9:00~10:30 (or to 12:10).
•ctrl file and output.
•
Numerical technique and GWinput.
•
Read output (console output, data).
•
Check points (How to get reliable results?)• (For expert course) “Wannier function”, Fat band analysis…

The PMT method: a newlinearized method
PMT =APW + MTO
Muffin-tin

Takao Kotani (tottori-u)

Key point:
1. Iteration cycle.
2. eigenfunctions are expanded withAPWs(augmented plane wave)
and MTOs(muffin-tin orbital)Independent particle picture and total energy
These can be obtained by the density functional theory(DFT) in LDA.
ElectronDensity 𝑛(𝒓)DFT in LDA

Fill up to Fermi energy

One-particle effective potential𝑉(𝒓)iterationcycle  2
 V (r)  i (r)= i  i (r) 2mEigenvalues and Eigenfuncitons

 i , i (r)This iteration cycle until converged = total energy minimization

How to represent density and so on in computer?
How to solve it numerically?
Lists of the Full-potential methods•KKR•Pseudopotential method•PAW•LMTO•LAPWLinearized xxx methods(why do we call them ” linearized” method?
from the view of “exact ” APW method)

Finite basis set•Basis set (finite number of basis) We assume eigenfunctions are given as: p (r )    pj F j (r )j

F j : j  1, 2,...N V Fj ,2m
Overlap matrix Oij  Fi F j
Hamiltonian H ij  FijHO0 ijij 
Finite dimension problem(as the same as LCAO/Gaussian)iteration cycle
previous iteration

One-particle potential V(r)Basis set {Fi (r )}Hamiltonian H ij  Fi
 V Fj ,2m
Overlap matrix Oij  Fi Fj
Diagonalization (H   p O) p  0electron density n(r)Next iteration

How to choose the good basis?
• APW (augmented plane wave)• MTO ( muffin-tin orbital)
are the names of the basis functions.
Both of them are made by “augmentation”.

My conclusion:
To overcome shortcomings in APW basis and MTO basis, weshould use both of APW and MTO together. this means
the PMT method

But wait…
What is the APW and MTO? next page.

MT regions

Interstitial region

http://research.physics.illinois.edu/ElectronicStructure/598SCM-F04/lecture_notes/

Envelope function = PlaneWave(PW)ＡＰＷ
PW in MTMT region

MT region

3-components
PW ＋ Atomic-like part － Counter part:
PW within MT
0th
1st2nd•Good for Na(3s), high energy bands.
•Not so good for Cu(3d), O(2p)•SystematicLinearized APW

fcc Cuplane wave is not
so efficient for 3d
15Ry or more for~1mRy convergence

D.Singh et al PRB49,17424

ＭＴＯ

Atom-centered Hankel function, e.g,exp(  r )for S -channelr

Smooth Hakel function
MT region

MT region

3-components
Hankel ＋ Atomic-like part － counter part of Hankel
0th
1st2nd•Good for localized orbitals such as Cu(3d), O(2p)•Ｎot so good for extended states, surface.
•Not systematic

PMT =APW + MTO
T.K and M.van Schilfgaarde
Phys. Rev. B 81, 125117 (2010)

T.K， H. Kino, and H.Akai
Supercell calculations from H2 through Kr2.
Almost automatic setting of MTOs with APWs
(Energy cutoff 3~4Ry) .
J. Phys. Soc. Jpn. 82, 124714, (2013)
J. Phys. Soc. Jpn. 84, 034702 (2015)

Basis function
MT center at R. Radis R.
a basis F j (r) consists ofF0 j (r ),Envelope function
PW or smHankel

F1 j (r ),F2 j (r )Atomic-like partsfor r  R

Counter part
forr R

Augmentation parts
Cutoff: l  lmax ~ 4, Radial-part expansion

F j (r )  F0 j (r )  F1 j (r  R )  F2 j (r  R )Electron density n 𝒓 and potential 𝑉(𝒓) are expanded in a similar manner.

Local orbitals(lo): basis functions which are Non-zero only wihtin MTs.
e.g, see http://www.wien2k.at/lapw/index.html
This is also used together.

Nbasis=1081+36×2, 1973+36×２, 3025+36×２

Comparison with Gaussian

ＮＲ：non relativistic

ＧＴＯ：6-311+G(d,p)

Quasiparticle self-consistent GW
Takao kotani(tottori university)• Mean field theory.
What the eigenvalue means?
• GW, and QSGW method
Minimum and previous results.

Mean field theory gives an independent-particle picture• Hartree-Fock theory• Density functional theory
Eigenvalues and Eigenfuncitons  i , i (r)are determined by minimization of E  i (r) , ni The minimization determines
ni = 1 for  i   FERMI (occupied states)ni = 0 for  i   FERMI (unoccupied states) .
NOTE: we can consider
A. configuration of excited states.  QUIZ (picturize this)B. fractional occupancy (this is very theoretical)
We can describe “ground state” and “excited state” easily
by filling electron or removing electron from orbitals  Independent particle picture
But need to know its limitation.

True excited state energy is not so easy;
From a view, you can say “mean-field changes by filling electrons”.

Confusing? But there is a key theorem for mean-field theories.

Janak(Koopman's) theoremE i
ni
when we fill/remove electrons from orbital  i
Left hand side is “True” excitation energy.
(caution: this is “derivative”, not one-electron filled/removed.)Ionization energy of an atom

A.J.Cohen, P. Mori-Sanchez, W.Yang, Science321,792(2008)

Vacuum level
LUMO

LUMO

HOMOHOMO

HFLDA

E E
i ni ni
How about solid (infinite system)?

E E
i 
is satisfied.
ni ni

If

E E
i 
(linear line),ni ni

we have better excitation energies  i   j
How accurate they are?

Conduction band

band gap (𝜀𝐂𝐁𝐌 − 𝜀𝐕𝐁𝐌 )Band gap

Valence band

LDA gives too small
HF(not shown) gives too large
Experimental gap (eV)Key point : meanings of “eigenvalues and eigenfuncions” in the mean-field theory.

GW method and QSGW method(preparation) What is Green’s function?
*Example1: Poisson eq.
𝛥

14𝜋|𝐫 − 𝐫′|𝛥𝜙 = 𝜌(𝐫)= 𝛿(𝐫 − 𝐫′)𝜙(𝐫) =How to convert this to integral equation?

1𝐺 0 = 4𝜋|𝐫 − 𝐫′| is Green function
𝐺0 (𝐫 − 𝐫′)𝜌(𝐫′)𝑑𝐫′𝜕𝜕− 𝐸 f(t) = 𝜌(t)  𝑖 − 𝐸 𝐺0 𝑡 − 𝑡 ′𝜕𝑡𝜕𝑡′′
𝐺0 = 𝜃 𝑡 − 𝑡 ′ 𝑒 −𝑖𝐸(𝑡−𝑡 ) or 𝐺0 = 𝜃 𝑡 ′ − 𝑡 𝑒 −𝑖𝐸(𝑡 −𝑡)(𝛿 𝑡 − 𝑡 ′ means “impact force”.)* Example2: 𝑖′= 𝛿 𝑡 − 𝑡′1* Fourier transformation: 𝜃 𝑡 − 𝑡 ′ 𝑒 −𝑖𝐸(𝑡−𝑡 ) ↔
or …quiz…
𝜔−𝐸+𝑖𝛿
𝛻2For given, 𝐻0 = −+ 𝑉ext 𝐫 + 𝑉H 𝐫 + 𝑉xc 𝐫, 𝐫′ ,2𝑚𝜕
we like to calculate 𝜑( 𝐫, 𝑡) sasisfying 𝑖 − 𝐻0 𝜑( 𝐫, 𝑡) = 𝐹( 𝐫, 𝑡)𝜕𝑡
𝜕𝑖 − 𝐻0 𝐺0 = 𝛿(𝑡 − 𝑡′)𝛿(𝐫 − 𝐫 ′ )𝜕𝑡11⇒ 𝐺0 ==（Note boundary condition）
𝜕𝑖 − 𝐻0 𝜔 − 𝐻0𝜕𝑡24

𝑩𝒐𝒖𝒏𝒅𝒂𝒓𝒚 𝒄𝒐𝒏𝒅𝒊𝒕𝒊𝒐𝒏 𝒂𝒍𝒐𝒏𝒈 time-axis
𝜖𝑖 φ∗𝑖 (𝒓)φ𝑖 (𝒓′) +𝐻0 = 𝐻0𝑜𝑐𝑐. + 𝐻0𝑢𝑛𝑜𝑐𝑐. =𝑜𝑐𝑐.

1 G0 ==
𝜔−𝐻0=-+𝑜𝑐𝑐.

𝜖𝑖 φ∗𝑖 (𝒓)φ𝑖 (𝒓′)𝑢𝑛𝑜𝑐𝑐.

φ∗𝑖 (𝒓)φ𝑖 (𝒓′)𝜔−𝜖𝑖 +𝑖𝛿+𝑢𝑛𝑜𝑐𝑐.

φ∗𝑖 (𝒓)φ𝑖 (𝒓′)𝜔−𝜖𝑖 −𝑖𝛿∗′−𝑖𝜀𝑖 (𝑡 ′ −𝑡)𝑜𝑐𝑐. φ𝑖 𝒓 φ𝑖 𝒓 𝜃 𝑡 − 𝑡 𝑒∗ ′′ −𝑖𝜀𝑖 (𝑡−𝑡 ′ )𝑢𝑛𝑜𝑐𝑐. φ𝑖 𝒓 φ𝑖 𝒓 𝜃 𝑡 − 𝑡 𝑒′

(we measure 𝜀𝑖 from 𝜇, or 𝜀𝑖 →all excitation is positive energy)
Quiz: check this 𝐺0 satisfy differential eq at page.24

Why?  this is convenient due to Fermi statistics(adding electron or “adding hole=removing electron”).
25

Many body theory (rough explanation)

𝐻0 is interpreted in the second quantization.

“Fock space.” An excited state can be written as|1,1,0,0, 1,0,1,1,0,1,1,1,1,1,1
Above Ef

Below Ef

Ef
In other words, it can be written as† †𝑎4 𝑎3 𝑎−2 𝑎−5 |1,1,1,1,1,1,1,1,1,1,1† †= 𝑎4 𝑎3 𝑎−2 𝑎−5 |0>

We need to treat a set 𝛀 ={all possible excited+ground states}.
Quiz: Then what the operation 𝐻0 mean? What is 1/(𝜔 − 𝐻0 ) ?
26

1/(𝜔 − 𝐻0 ) 𝑖𝑛 𝑀𝐵𝑃𝑇 (not so accurate)Time-axis

43-2-5tt’

これは(4,3,-2,-5)が時間がたってもそのまま，（4,3, -2, -5)であることを意味する．エネルギーは和になっている．多体理論ではどうなるか？
𝑡1
NOTE: 畳み込み積分 0 f(t−t′)g(t−t′)dt ↔ f(𝜔)g(𝜔)𝜔 − (𝐻0 +(𝐻 − 𝐻0 ))
1
1
1
1
1
1
=+
𝐻 − 𝐻0+(𝐻 − 𝐻0 )(𝐻 − 𝐻0 )𝜔 −𝐻0 𝜔 −𝐻0
𝜔 −𝐻0 𝜔 −𝐻0
𝜔 −𝐻0
𝜔 −𝐻0+…
これはT(exp(-iHt))などと書いたりもする．
𝐻 + 𝑉other 𝐫 = 𝐻0 + (electron-electron interaction) – (𝑉H 𝐫 + 𝑉xc 𝐫, 𝐫′ ) + 𝑉other 𝐫27

Illustration of Many-body perturbation theory(MBPT)

Example:𝑉other 𝐫𝑉xc 𝐫, 𝐫′Time-axis
28

GW approximation
We concentrate on how 𝐺0 is modified by 𝐻0 + (𝐻 − 𝐻0 )We assume 𝐻0 is not so bad as starting point…
𝛻2
𝐻0 = −+ 𝑉ext 𝐫 + 𝑉H 𝐫 + 𝑉xc 𝐫, 𝐫′2𝑚

1.Exchane effect.
(𝐻 − 𝐻0 ) makes jump to another electron.
Effect of Fock term. (and remove self-interaction)
2. Correlation effect.
When a test charge is moving in electron sea.
The test charge is moving
 cause polarization of electrons
 affect force to the test charge.
Particle of 𝐻0  Quasiparticle picture.
29

GW approximation ： １電子波動関数の時間発展
Start from any

1.

e,g.  H LDAH0

21H0  
 Veff  r, r '   G0 2m
  H0G0n(r )  G0 (r, r,0)

2.   i G0  G0 Polarization function
 is density response for one-particle potential.

3.

 (r, r ', t  t ')＝ n(r, t ) V (r ', t ')W   v  1   v  vW in the RPA
11G0G0

e 2 (time-dependent Hartree approx.)v (r, r ') r r'Self-energy（Many-body effect is downfolded into one-body propagetor）
4.
W  i G0W Self-energy

が「交換相関」を与える。

G0
interpretation１：exchange hopping＋electron moving is other electrons
2: screened Coulomb interaction + Coulomb holeG0  n VH also2
H  r, r ',      V ext  r  +V H  r     r, r ',  21G HNOTE : In  dr ' dt ' H  r, r ', t  t '   (r ', t '),V ext  r  is takes as V ext  r   (r  r ')

  dependence of   r, r ',   is very problematic

31

QSGW: How to determine better H0 ?
𝐻(𝜀𝑖 )|𝜑𝑖 (𝐫) = 𝜀𝑖 |𝜑𝑖 (𝐫)   Vext  VH  ( i )  i (r )   i i (r) 2mIt is not so easy to treat energy-dependent potential!

(r, r ',  )   -independent Vxc (r, r ')H0 
 Vext  VH  Vxc (r, r ')  (r, r ',  )2m
An average procedure(not shown here)

This idea is supported by Quasiparticle picture (Landau-Silin’s).

GW and the “RPA total energy”

We start from H LDA   i , i (r)E  E0k  E0ext  EH  EX  ECOCC.
  ii   i Vext (r ) i  EH  EX  EC
2mii
OCC.

EH  EX  EC

𝐺0 =…

E

Ei   (r) Vext  VH  (i )  (r)
i 2mi
ni
This is one-shot GW

1で展開
߱−𝐻0

  i   (r ) ( i )  VXCi

LDA i (r)Historically, Ei   i  i (r ) ( Ei )  VXC LDA i (r ) ,(but Ei   i  i (r ) ( i )  VXC LDA i (r ) is better) 33

Non-local potential term (as Fock exchange term)
is important.
I. Localized electrons  LDA+U type effect(Onsite non-locality. self-interaction included).
It can break time-reversal symmetry  No orbital moment.

II. Extended electrons  GW type effect for semiconductor.
Important to describe band gap. Off-site non-locality.
(required to distinguish “bonding orbital”
and “anti-bonding” orbitals. ) next page34

Schematic explanation on off-site exchange
IfBonding

ＨＯＭＯ

Anti-Bonding, ＬＵＭＯ,

square of them are the same.

 bonding V (r )  bonding   anti-bonding V (r )  anti-bonding .
Local potential can not distinguish
LUMO and HOMO.

35

sp bonded systems

LDA: broken blue
QSGW: green
GLDAWLDA: Dotted red
O: Experiment

NaGaAsm* (QSGW) = 0.073
m* (LDA) = 0.022
m* (expt) = 0.067
Gap too large by ~0.3 eVBand dispersions ~0.1 eVNa bandwidth reduced by 15%
Ga d level well described36

Calculated(eV)BAND GAP
LDA, one-shot GW

QSGW(we see systematicover estimation).

With D.Deguchi and K.Sato (Osaka-u)金属的なものＱＳＧＷでＯＫ．
ハイブリッド法では場合によりうまくいかない。

38

Physics in QSGWLDA（GGA） → homogeneous gas OK! （Physics)
Hartree-Fock → H-atom OK!
（chemistry）“True results” may be between its middle;

１３
Hybrid methd（B3LYP, HSE…)
×HF＋ ×LDA４４•Problem：the mixing ratio may be dependent on materials.

–QSGW： Instead of bare Coulmb ｖ,––

we use “Dynamically Screened Coulom interaction W”.
W is determined self-consistently.
Independent-particle theory where we take into accountthe charge fluctuation at the RPA self-consistently.
39

Why we need ＱＳＧＷ？
We need good independent-particle picture in order tocalculate linear responses (magnetic optical, transport…).
H0 (eigenvalue and eigenfunctions) are required.
＊ Band gap, Effective mass
＊Relative position of levels.
ＬＵＭＯ－ＨＯＭＯ, Fermi energy…

Metal

Fermi energy

Molecule
ＬＵＭＯ Molecule on
top of metal.
ＨＯＭＯ
40

(12a.u.)3cell

LUMO
LUMO
HOMO
HOMO

計算の収束性は不明．．．

41

ZnO

Black:QSGW
Red:LDA

3.87eV
0.71eV

Green:GLDAWLDA (Z=1,Offdiagonal included) 3.00eV
Blue：e-only self-consistency
3.64eV
Experiment(+correction) 3.60eV
42
Kotani et al PRB76,165106(2007)

d systems“No Pushing down 3d-band’’ in GW.
43

NiO

Black:QSGW Red:LDA Blue: e-only44

Black:t2g Red:eg

NiO MnO dos

Red(bottom): expt

45

NiO MnO dielectricBlack:Im eps with LFC Red:expt46

Spin wave dispersion based on QSGW

47

srtio3QSGW

S. A. Chambers et al, Surface Sci 554,81-89 (2004)
48

Electron configuration
lift of degeneracy of 3d orbitals due tocrystal fieldeg

x2-y2
∆𝐸 eg

3z2-r2[3d]
Cu atom

t2g

xy

cubic
(regularoctahedron)yz, zxtetragonal(elongatedoctahedron)Around Ef, we have two eg bands.

Band structure: Two eg orbitals．
Band width and ∆𝐸 eg
La2CuO4(VWN)

doi:10.1038/srep12050

La2CuO4(QSGW)∆𝐸 eg

HgBa2CuO4(VWN)

HgBa2CuO4(QSGW)

50

Wannier function method. Extract tight-binding parameters
La2CuO4(VWN)

La2CuO4(QSGW)

HgBa2CuO4(VWN)

HgBa2CuO4(QSGW)Miyake’s Maxloc Wannier code is re-implemented(TK and H.Kino）．

Effective interaction 𝑈(𝜔) for𝑑𝑧 2 and 𝑑𝑥 2 −𝑦252

Impact ionization rate(auger process)
Unoccupied bandOccupied band

Fermi’s Golden rult  transition rate. Sum up for all
final states. Matrix element of transition is＜1,3｜electron-electron interaction｜4,2＞（Energy conservation and Momentum conservation） 53

InAs impact ionization rate(=auger process= high energy electron lose energy with e-h pair)direct gap 0.38eV

QSGW

123eV

45Present model: Sano and Yoshii JAP77 2020 (1995)

YH3(fcc structure 6x6x6)

GGA

L

QSGW

Γ

X

L

Γ

X

CuGaSe2 (eight atoms in cell, 2x2x2. ~3hours per iteration by single core)

1shotGW

T

Γ

QSGW

NT

Band gap GGA:0.4eV, GW(1shot noZ) 1.37 eV,QSGW(1.79eV)
Exper. 1.63, 1.67, 1.73eV

ΓN

Band gap for CuAB2 （eV)BA SSeAl

3.62(3.49)

2.91(2.67)

Ga

2.83(2.43)

1.69(1.68)

In

1.22(u=0.22) ---1.48(u=0.23) (1.04)(1.53)

Experiments in (…)

CuInS2 band gap（eV)

0.22 0.23

uJ.Vidal, PRL 104, 056401 (2010)

3x3x3 calculation(need check)57

PdO(tetragonal , four atoms per cell 3x3x2 calculation , 1hour per iteration )

GGA

Γ

X

M

QSGW

Γ ZRAZ

Γ

X

M

Γ ZRA

Z

Band gap about 0.7 eVLaNiO3M.J.Han, H.Kino, T.K (arXiv:1402.4915)

59

60

61

summary• Mean field theory and excitation energy
LDA, Hartree-Fock and QSGWJanak theorem, finite vs.infinite system• RPA total energy
QSGW method• Some Results
Band gap, GaAs and Na, Cu
NiO, Spin Wave, ZnO, LaMnO3,InAs (impact ionization), YH3, CuGaSe2,PdO• How QSGW works for atoms and molecules?
See F.Bruneval J.Chem.Phys 136,194107(2012)