The 37th CMD workshop“Advanced study with ES-opt”

Multi-reference DFTKoichi KusakabeGrad. Sch. Eng. Sci, Osaka University1

Electronic structure of diatomic
molecules
Anti-bonding

Bonding

Non-bonding

aMolecular orbital picture

Heitler-London picture

Known results in DFT-LDA• Molecular orbital picture is given for any inter-atomic distance.
• The stable electronic state in equilibrium is well-reproduced. 2

Molecular orbital given by DFTLDA
0 1s bonding

2 1s anti-bonding

133 3'

2103

Dependence on inter-atomicdistance
3 3'211s,1 & 1s,2
may be constructed.

0Non-bonding
Hitler-London picture

aIs there any concise method to describe this dependence?
4

Correlation effects◼ In DFT-LDA, the Kohn-Sham orbital isdetermined with correlation effects included in
the effective potential.
◼ The correlation effect can behave differently
depending on atomic configuration.
◼ A key is,e2
1Exc n =  drn(r ) dr 'nxc (r, r')2r − r'
Charge fluctuation!
1nxc (r, r ') = n(r ') g n (r, r' ,  ) −  (r − r')d0=    : (nˆ (r ) − n(r ))(nˆ (r ') − n(r ')) :  / n(r )d.
105

H2
Anti-bonding

Non-bonding

BondingaThis is a weak correlation limit!

This is a strong correlation limit!

In real nature & in CI, we have6

Quantum mechanical superposition

O2
This picture is not true!

2s2 2p4a

S=1 ground state!
This is a strong correlation limit!

S=1 ground state!
This is a strong correlation limit!
7

How the multi-reference densityfunctional theory acts?
◼ The Kohn-Sham single-particle description

may be used, but may not always be reliable.
◼ So, if a simplified description allowing a multiSlater determinant is introduced, we may use
the method as another starting point.
◼ A simplest case is a DFT-LDA solution for adegenerate ground state.

S=1, Sz=1

S=0, Sz=0

Determination of GS may bepossible by looking at• charge-charge correlation
or8• variational energy.

Two physical quantities specifying
correlation effects◼ Density-density correlation : FRM(n i = ni + ni − ni − ni2)2◼ Variational energy : DFVT

9

Fluctuation reference method◼ To have a simplified description based onDFT, we need information given by anotheraccurate electronic structure calculation.
◼ Diffusion Monte-Carlo method for electron gas⇒ LDA, GGA◼ Complete-Active-Space ConfigurationInteraction method ⇒ FRM to determine U.
FRM : Fluctuation reference method

10Two Hydrogen systems
H2 Molecule

Hydrogen array11

Local fluctuation on f1s of H2n2i(= ni + ni − ni − ni) as a function of the inter atomic distance
20.7Å

nbonding = 2.0

n2bonding=0n1s = 1.0
21sn= 0 .5

Small U regime

4.0Å

nbonding = 1.0

n1s = 1.0

n bonding = 1.0

n1s = 022Large U regime

We are able to find relevant orbitals with Coulomb suppression by
1. Finding an orbital ( or a set of orbitals ) on which fluctuation becomes 1.
12
2. Using a unitary transformation to have localized orbitals.

Orbital fluctuation in CI calculation
CAS-CI calculation for H2 by S. Yamanaka.

Small U/t
U ～ 0.7eV
Molecular orbital limit

2niHeitler-London limit
Large U/t
U ～ 10eV

In the whole range, mean occupation of 1s is 1.

13

A test calculation of Hydrogensystems◼ Exc[n] is given by LDA of PW91.
◼ Plane-wave expansion with the Troullier-Martins soft-

pseudopotential is introduced for H.
◼ The energy cutoff for the plane-wave is 40Ry.
◼ The solver of single-particle part of the extended KohnSham equation is given by the “opt” code.
◼ Effective many-body problem is solved by the exact
diagonalization with e.g. the Lanczos method.

14

The value of U for H2 determined byFRM
Reference calculation: CASCI method for H2

R=2.0Å
R=3.0Å

R=1.2Å

The 1st method to have U by the fluctuation reference method.
K. Kusakabe, et al. J. Phys.: Condens. Matter (2007).

15

Charge density of H2 given by MRWeak correlation regime
DFT
R=0.6Å
U is notneeded.

This section isShown in theright panels.

Strong correlation regime

To reproduce bonding charge incorrelated electron regime, we need toovercome difficulty in the single-reference
description. It is achieved by using themulti-reference description as known inthe Hybrid-DFT.
(Cf. S. Yamanaka, et al.)R=2.0Å

U is need.

16Variational method
DFVT for MR-DFT

DFT model formation

PBE0+GW, PBE0+U will be refined.

-It is possible to evaluate DE[].
Note: If DFVT is used in a space of single Slater
determinant, the Hartree-Fock approx. is given.
This is natural, but not so trivial.
Thanks to Dr. Maruyama & Dr. Friedlich.

17

To understand MR-DFTThis functional contains
the universal energydensity functional F[n]This functional should be
evaluated using a multiSlater description.

18

Heisenberg exchange:
Introduction to the localized spin model◼ Heisenberg and Dirac considered quantum

mechanical origin of magnetic interaction↑↓-J Si ・Sj

J>0 : Parallel spin configuration
Ferromagnetic
J<0 : Anti-parallel spin configuration
Antiferromagnetic

whose order of magnitude can be of the order of
the Coulomb interaction.

Where does J come from?
19

Idea of restriction of the phase space◼ For low-energy states, we may consider onlystates in a restricted phase space.

The 1s orbital ofthe Hydrogen atom

Let’s consider the phase space spannedby spin states |1 s ,↑ , |1 s ,↓ .
20

Atomic orbitals and molecular orbitals◼ Atomic orbitalsr1fa ( r )  (1)†ca 0a◼ Molecular orbitals (in LCAO)

r1

r2ab

1
(fa (r ) + fb (r ) ) (1)21 †(ca + cb† ) 02
21

The Heitler-London theory of H2
A picture in the strong correlation limit!!

r1

r2

r2

r1ababNote that there is no doubly occupied 1s orbital in this theory.

(3)

(4)

22

Energy of spin states

23

Evaluation of J: Direct exchange24Ferromagnetic direct exchange I.

****

25

Ferromagnetic direct exchange II.
◼ Exchange integral of two orthogonal orbitals with S=0

becomes ferromagnetic.
A proof: Consider the exchange integral when S=0.

e2
I =  dr1dr2
fa * (r1 )fb (r1 )fb* (r2 )fa (r2 )r1 − r2

Note that the expression below.

()()

fa* (r1 )fb (r1 ) = Re fa* (r1 )fb (r1 ) + i Im fa* (r1 )fb (r1 )

()

*
e2
I =  dr1dr2
fa * (r1 )fb (r1 ) fa * (r2 )fb (r2 )r1 − r2()

()

()

()

e2
*
*=  dr1dr2
Re fa (r1 )fb (r1 )  Re fa (r2 )fb (r2 )r1 − r2e2**+  dr1dr2
Im fa (r1 )fb (r1 )  Im fa (r2 )fb (r2 )r1 − r2

To show the positivity of I, it is enough to consider an integral of real functions,e2
I =  dr1dr2
(r1 )(r2 )r1 − r226

Ferromagnetic direct exchange III.

-q).

27

Slater determinants for a system with two electrons I.
ab28

Slater determinants for a system with two electrons II.

Multi Slater determinants

29

Spin states in the second quantization30

2-electrons in molecular orbitals

The Heitler-London wf.

Ionic states!

31

The Heitler-London state v.s. the molecularorbitals

The true w.f. is close to a variational state with 0<a<1.
a is dependent on the inter-electron interaction or choice ofthe model (interaction parameters etc.).
32

The Hubbard model : I.
1. Consider a system with atomic sites. Each site is assumed
to have an orbital for the conduction electrons
2. Electrons can hop between neighboring sites by a transfer
integral t.
3. Two electrons coming across at a site feels a repulsive
interaction U.

↑↑ ↓↓

↑↓

Ut↑ ↓↑↓
33

The Hubbard model : II.

34

The Hubbard model : III.

35

The Hubbard model : IV.

36

The Hubbard model : V.

37

The Hubbard model : VI.

38

2-site Hubbard model : I.

t
Uab39

2-site Hubbard model : II.

40

2-site Hubbard model : III.

Crossover from weakcoupling regime to strong
coupling regime41

2-site Hubbard model : IV.
Note: GS is always
a singlet state.

The Hubbard gap
Energy scale of theAF kinetic exchange
Strong coupling regime(HL)｝Singlet states
Weak coupling regime (MO)Triplet states
42

Concept of the Mott insulator◼ Electron transfer from an orbital to an occupied

orbital is prohibited by the Coulomb repulsion.
This leads to an insulating state.
↓↑↑↑↑e↓↑↓↓↑↓↑Mott gap
orHubbard gap

UJThe half-filled Hubbard modelshows the Mott insulating groundstate.

Formation of the Mott gap inthe single particle excitation.
43

The kinetic exchange◼ Effective interaction at around U=∞P↑Q↑P↑↓↓t↑↓U↑ ↓↓↑↓We have anotherprocess starting fromtransfer of the up
spin moving to theneighboring site.

22t
1
H eff =(S i  S j − )U i, j4

Antiferromagnetic
Heisenberg model.
44Superexchange interaction◼ In the transition-metal oxides, there is a structure of M-O-M(M: transition metal, O: oxygen）. Along this structure, twolocalized spins couples via the superexchane through nonmagnetic oxygen atom.
Often, Mott insulator is formed.
→ M has an open d-shell.
MOM

dx2-y2 @ M
px @ O
dx2-y2 @ M

JAF45

The Kanamori-Goodenough rule
◼ Consider description by non-orthogonalized atomic

orbitals (LCAO picture)◼ The electron transfer is possible from O to M when two
neiboring orbitals are not orthogonal.
◼ The sign of exchange interaction between d and p is,◼ Ferromagnetic if two orbitals are orthogonal with eachother,◼ Antiferromagnetic if two orbitals are not orthogonal◼ Occupation of d-orbitals are determined by the crystal
field splitting and the Hund rule.
Following the above rule, we can determine the sign ofsuperexchange in a qualitative manner.
P.W. Anderson reformulated the rule using the orthogonalized
Wannier basis allowing the second quantization scheme.
46

Exanple of antiferromagnetic
exchange
Non-orthogonal: finite transfer
J: antiferromagneticpp

ps(2,3)
Mn2+ : 6S5/2 d5

Non-perturbative
States in excitation

JAFJF

(1,4)

U(1)
Non-perturbative
ground stateMn2+ : 6S5/2 d47

5MOM

Various types of exchangeinteraction◼ Direct exchange : (Cf. Heitler-London theory)◼ Kinetic exchange : (Cf. the Hubbard model)◼ Super exchange◼ Double exchange◼ RKKY interaction◼ Anisotropic exchange interaction◼ Dzyaloshinskii-Moriya interaction

1. Exchange interactions except for the 1st one
are effective interactions.
2. Effective interactions can be derived from amodel with local interactions.
48

An almighty method◼ Space of DFT models◼ Distance :CASE I
e-vicinity aroundexact solutionni − ni +1 .
◼ Variational principle: DFVT by K.K. (2009).

CASE II
Exact solution

Hartree-Fock
LDAGGA

Gateaux derivativein the model spacedetermines thedirection to follow.

Phase transition
49lines

Super processes
To have an initial state of correlated electron system, we may utilize

The upconversion Hamiltonian1
Ai → Aic  Hˆ + Hˆ Ai → Ai − Hˆ Aic → Ai PˆPB H 2
i = E i .
22
B
Ai → AiAi  Aic 1,iˆ
ˆˆ
H1,i + H 2+ H2−E
which leads us to the convergent model series in the DFT model space.
Cf. K. Kusakabe, I. Maruyama, “Electronic state calculation method, electronic
state calculation device, computer program”, PCT filed No. PCT/JP2011/068589,
date 2011.8.18.
Problems in static state
Magnetism Superconductivity

The superexchange
process

The superpair
hopping
process

Semi-classical &quantum
dynamics

The vander-Waals
interaction…

Quantum
processes in QED!

Two &multiphoton
processes

50

Exchange scattering
Feynman diagram of exchange
Ce 4f1

Evaluation of c-f exchange(JCe 4f1
VBCB

Ce 4f2
Ce 4f

Ce 4f2)eff − cf c fj i

Ce 4fVBCB

MB @ EFMB @ EFAF effective exchange⇒ The Kondo Screening

Upconversion Hamiltonian with super processes
1Ai → Aic  Hˆ + Hˆ Ai → Ai − Hˆ Aic → Ai PˆPH
 = E i ,22
B
 1,i
 i
ˆ + Hˆ Ai → Ai + Hˆ Ai  Aic − E B 2
H1,i22


2

3
†
3
ˆ
ˆ
ˆ
ˆ
ˆ
()
()
()
()
H1,i = T + V = −drlim
r'D
r+drvrnr+E g  ,r'→rreff
2m   iveff (r ) = e 2

33 n(r )n(r ')+ Ee i n +  d 3 rvext (r )n(r ).
  d rd r 'n(r )  2r − r'

51

A unified picture of high-Tc materials
Isolated center band

Embedded center band

EE

EF

Correlation enhanced

Band structure of Tl-cuprates

EFCorrelation enhanced

Empirical rule :1. Embedded type preferable
2. Larger band width of center band preferable
3. Correlation induced by localized Wannier orbital preferable
52

Evaluation of matrix elements
4x4x1 meshMaterialBloch diag.

Bl off-diag.

On-siten.n. off-site

Hg-comp.

18.6 eV at G

12.0 eV etc.

10.2 eV

6.4 eV

Tl-comp.

18.6 eV at G

5.4 eV etc.

8.3 eV

5.1 eV

Process A, C : A correlator as energy increment by fluctuation in the number
density of particles in the correlated band allows an approximated form via
introduction of approximated Green’s functions (resolvents).
Process B : A gap in un-correlated bands allows an approximated form as,53

Super process as two-particle G.F.
Symmetric form of the bareCoulomb interaction

Effective interaction diagram leading
the Bethe-Salpeter equation (BSE)

Bubble diagram by thepolarization function withvertex correction (red part)

Ladder diagram54

Derivation of screened interaction
Bare interaction term

Polarization function

Screening in c-RPA leｖel : Vertex correction can be considered.

Screened interaction term

55

Super process in general forms
General operator expression

Phase-space dependent expression

A matrix element in an approximated expression corresponding to a 3rd order diagram

56

Electron-hole interaction insemiconductors

57

The GW schemeKSTˆ + Vˆext + VˆHartree + Vˆxc   iKS= e iKS,s  i ,s .
 ,sKohn-Sham equation( )ˆ e QP   QP = e QP  QP .
Tˆ + Vˆext + VˆHartree + i , i ,
i ,
i ,


Quasi-particle equation

QPˆ ( )  QP = dr ' r,s ˆ ( ) r ',s r,s i ,
s ,s ' r ',s '  i ,

Self-energy expressions'=  dr ' s = ( r, r '; ) iQP, ( r ' ) .

s ( r, r '; ) =id ' e i ' Gs ( r, r '; +  ' )W ( r, r '; ' ) d '.
2p

Gs ( r, r '; ) = 0KS
 iKS,s ( r ) ( i ,s ( r ' ) )*The 0-th order Green function

 − e iKS,siW ( r, r '; ) =  e ((KSWGG' ( q ; ) ei G ' + q )r 'The GW self-energy− i G + q )r ',Screened interaction

GG 'KS
2
WGG' ( q ; ) = −4p e1
1
KS −1e GG,' ( q ; )G+q
G '+ qKS2e GG' ( q ; ) =  G ,G − 4p e

1
1KS − 0
 GG,' ( q ; )G+qG '+ q

Screened interaction matrix
Dielectric function
58

GW as an approximation
1Ai → Aic  Hˆ + Hˆ Ai → Ai − Hˆ Aic → Ai PˆPB H 2
i = E i .
22
B
Ai → AiAi  Aic
 1,i
ˆ
ˆ
ˆ
H1,i + H 2+ H2−E iQP= c†,i ,  0

i

Ac → A j− Hˆ 2 jPB

1Aic → Acjˆˆ
H1, i  i , j + H 2−   , EPB Hˆ 2Ai → Ai  iQP,
cW j , j , ;i , i , ( ) G ( j ,  ; i ,  ;  ) c†j , ci ,  iQP,
QP
QP
QPQP⇒ Tˆ + Vˆext + VˆHartree + ˆ (e i , )  i , = e i ,  i , .

59

Thank you for your attention.

K. Kusakabe60