LDA+U: Fundamentals, Open Questions, and Recent Developments

Igor Solovyev

Computational Materials Science Center,National Institute for Materials Science,

Tsukuba, Japan

e-mail: SOLOVYEV.Igor@nims.go.jp

Contents1. Atomic limit

1.1. DFT for fractional particle numbers 1.2. LDA+U and Slater’s transition state1.3. LDA+U and Hubbard model1.4. Rotationally invariant LDA+U 1.5. simple applications

2. LDA+U for solids: postulates and unresolved problems

2.1. choice of basis2.2. charge-transfer energy in transition-metal oxides

3. Other methods of calculation of U : RPA/GW

3.1. U for isolated bands (low-energy models)3.2. LDA+U for metallic compounds -- orbital polarization for itinerant magnets

4. Summary -- Future of LDA+U

Puzzle

×A B

NA NB

ΔNA

• Two separate atoms• no interaction

• but free to exchange electrons

total number of electronsis conserved

However, and are not:

• energy gain

= =

individual electron numbers ( and )may be fractional … and this is precisely the problem …

Other Examples

adatom on surface;chemical reaction, etc…

strongly-correlated systems:weak interactions between atoms(in comparison with on-site energies);the ability of exchange by electronsplays an essential role

I.

III.

stability of atomic configurationsFe[4s 23d 6], Co[4s 23d 7], etc.J.F. Janak, PRB 18, 7165 (1978).

II.

What is wrong ?

• The electron is “indivisible”

• The only (physical) possibility to have fractional populations is the statistical mixture of two (and more) configurations:

where is an integer number.

Then, the energy

is the linear function of

• On the other hand, the system is stable and must have a minimum

a combination of straight line segments

J. P. Perdew, R. G. Parr,M. Levy, and J. L. Balduz,Phys. Rev. Lett. 49, 1691 (1982).

What shall we do ?

The idea is to restore the correct dependence of E on x in LDA

• The absolute values of , , and are O.K., even in LDA (an old strategy of the Xα method)

• In each interval replace the quadratic dependence by the linear one:

where

• LDA+U :

I.V.S, P.H. Dederichs, andV.I. Anisimov, PRB 50, 16861 (1994).

What does it mean ? I.V.S, P.H. Dederichs, andV.I. Anisimov, PRB 50, 16861 (1994).

NA2 NA1 NA NA1

0U

/8

U/2

U/2

ΔV

U

ΔE

U

• ΔEU enforces integer population and penalizes the energy when these populations are fractional

• For integer populations, ΔEU = 0, otherwise ΔEU > 0. Thus,

LDA+U is a constraint-LDA

• The potential exhibits a discontinuity at integer populations.

• The size of this discontinuity is U

LDA+U and Slater’s Transition Stateor meaning of LDA+U eigenvalues

• LDA+U functional

where in each interval

Janak’s theorem

• Slater’s transition state

ionization potential

electron affinity

• Then, and

nothing but LDA+U eigenvalues in the atomic limit

LDA+U and Hubbard model

NA levels,each populatedby 1 electron

1 level populatedby x electrons

• Hubbard model in the mean-field approximation

note that if or

• mimics LDA “smooth” dependence on x and coincides with for integer populations

• LDA+U:

possible extensions: beyond mean-field, ω-dependent self-energy, DMFT

R. Arita(July 31)

V.I. Anisimov, J. Zaanen, andO.I. Andersen, PRB 44, 943 (1991).

note, however, thatthe form of thisdouble-counting isdifferent fromPRB 44, 943 (1991).

Moreover …

Hubbard U

curvature of LDA total energy

Curvature of LDA total energy = Hubbard U

• constraint calculations of U

another possibility (using Janak’s theorem):

P. H. Dederichs et al ., Phys. Rev. Lett. 49, 1691 (1982);V. I. Anisimov and O. Gunnarsson, Phys. Rev. B 43, 7570 (1981);K. Nakamura et al ., Phys. Rev. B 74, 235113 (2006).

Rotationally-Invariant LDA+U and Hund’s rules

• depends on the type of the orbitals

• which orbitals should we use ?

• Strategy:

it depends neither on the form of the basis (i.e., complex versus real harmonics) northe orientation of the coordinate frame

density (population) matrix

matrix of Coulomb interactions

A.I. Liechtenstein, V.I. Anisimov, and J. Zaanen, PRB 52, R5467 (1995);I.V.S., A.I. Liechtenstein,and K. Terakura,PRL 80, 5787 (1998).

• In spherical approximation, is fully specified by

Coulomb ,

exchange , and

“nonsphericity”

controls the number of electrons

control Hund’s rules(at least, in mean-field)

How good is the parabolic approximation for ELDA ?

I.V.S. and P.H. Dederichs, Phys. Rev. B 49, 6736 (1994).

d - impurities in alkali host (Rb)

d

localized levels in“free-electron gas”

T(1+)

T(2+): divalent configuration

T(2+)

T(1+): monovalent configuration

Straightforward applications along the original line

divalent configurations

monovalent configurations

I.V.S, P.H. Dederichs, andV.I. Anisimov, PRB 50, 16861 (1994).

stable configurations of3d - impurities in Rb host

Fermi level

atom

ic im

purit

y le

vels

(Ry)

broken lines: the levels which are supposed to be empty

solid lines: the levels which are supposed to be occupied

LDA+U for atoms and for solids• pure atomic limit (no hybridization)

ionization

affinity

LDA LDA+U

simply the redefinition of atomic levels,relevant to the excited-state properties• solid: interacting levels

before hybridization

after hybridization

after hybridization

position of atomic levels is important , as it already contributes to the ground-state properties, likesuperexchange:

tt

Postulate: LDA+U functional for solids

The same as for atoms, but the “subsystem of localized electrons” is defined by means of projections onto some basis (typically, of atomic-like) orbitals:

(double-counting)

density matrix

number of “localized electrons”

“Kohn-Sham” equations in LDA+U

where

is a non-local operator

The final answer depends on the choice of the basis

an obvious, but very serious problem ………

Is There Any Solution ?

The basic problem is …..

How to divide ???

M basis orbitals

M Wannier functionsbut their choice is already

not unique

pick up N Wannier orbitalsfor localized states

another ill-defined procedure

… or using mathematical constructions

a naive analogy withuncertainty principle:

intrinsic uncertaintyof LDA+U

completeness of basis

it is impossible toobtain the exact solution within LDA+U

Example: construction of “Hubbard model” for fcc-Ni

exact (LMTO) bands

canonical 3d bands

canonical 4s bands

• in total, there are 6 bands (five 3d + one 6s) near the Fermi Level (zero energy)

• is it possible to describe them it terms of only 5 Wannier functions ?

• Yes, but only with some approximations

Wannier bands

I.V.S and M. Imada, PRB 71, 045103 (2005).

Other problems: charge-transfer energy in TMO

U

Δ

U : Coulomb interaction

Δ: charge-transfer energy

Superexchange interaction:

• Δ is an important parameter of electronic structure of the transition-metal oxides

• How well is the charge-transfer energy described in LDA+U ?

T. Oguchi, K. Terakura, and A.R. Williams, PRB 28, 6443 (1983);J. Zaanen and G.A. Sawatzky,Can. J. Phys. 65, 1262 (1987).

O(2p)

LHB UHB

LDA+U for the

transition-metal oxides:what we have

and what should be?

Magnetic Interactions in MnO: phenomenology

experimental spin-wave dispersion:M. Kohgi, Y. Ishikawa, and Y. Endoh,Solid St. Commun. 11, 391 (1972).

J1

J2

Two experimental parameters:

J1 = -4.8 meV, J2 = -5.6 meVTwo theoretical parameters:U and Δ in

One can find parameters of LDA+U potentialby fitting the experimental magnon spectra

I.V.S. and K. Terakura, PRB 58, 15496 (1998).

Magnetic Force Theorem

θ

θ

θ

• For small deviations near the equilibrium, the total energy change is expressed through the change of the single-particle energies:

• No need for total energy calculations; ΔE is expressed through the Kohn-Sham potential in the ground state.

• Application for the spin-spiral perturbation

• Magnetic interactions:

A.I. Liechtenstein et al., JMMM 67, 65 (1987);I.V.S. and K. Terakura, PRB 58, 15496 (1998);P. Bruno, PRL 90, 087205 (2003).

rotation of magnetization

… And … The Answer Is ……

MnO

Many thanks to Takao Kotani for OEP:T. Kotani and H. Akai, PRB 54, 16502 (1996);T. Kotani, J. Phys.: Condens. Matter 10, 9241 (1998).

I.V.S. and K. Terakura, PRB 58, 15496 (1998).

in LDA+U for MnO, U itself is O.K., but ….the charge-transfer energy is wrong.

(the so-called problemof the double counting)

Other Methods of Calculation of U :constraint-LDA versus RPA/GW

Definition: the energy cost of the reaction

constraint-LDA RPA/GWpotential

to simulate the charge disproportionation

1.

2. mapping of Kohn-Sham eigenvalues onto the model

is the number of “d” electrons

3. Fourier transformation

perturbation theory

external potential →

change of KS orbitals →change of charge density → change of Coulomb potential →

etc. barescreened

Example: isolatedt2g band in SrVO3

main interband transitions:

(1) O(2p)→V(eg)

(2) O(2p)→V(t2g)

(3) V(t2g)→V(eg)

Intr

a-O

rbita

l U (

eV)

Good points of RPA/GW (I)

• Construction of model Hamiltonian for isolated bands

problem to solve:screening of 3d electronsby “the same” 3d electrons

F. Aryasetiawan (this workshop);I. V. S. (symposium)

I.V.S., PRB 73, 155117 (2006).

I.V., N. Hamada and K. Terakura, PRB 53, 7158 (1996).

phenomenologicalidea

Good points of RPA/GW (II):

“LDA+U” for itinerant systems

Example: Orbital Magnetism in Metallic Compounds

Orbital Magnetism and Density-Functional Theory

• in the spin-density-functional theory (SDFT):

charge density

spin-magnetization density

EXC=EXC[ρ,m]

spin polarization

Kohn-Sham (KS) theory

there is no guarantee that ML can be reproduced at the level of KS - SDFT

ML should be a basic variable ⇒we need an explicit dependence of EXC on ML: EXC=EXC[ρ,m, ML]

• the concept of orbital functionals and orbital polarization

Some Phenomenology

FLAPW potential from E. Wimmer et al., PRB 24, 864 (1981).

• orbital magnetism is driven by relativistic

spin-orbit interaction

(a gradient of electrostatic potential)

• does not commute with

is not an observable, except the same

core region where is nearly spherical

the main effect comes from small core region

The problem of orbital magnetism in electronicstructure calculations is basically the problem of local Coulomb correlations

Several empirical facts about LDA+U for itinerant compounds

if U=0.7 eV

General consensus: the form of LDA+U functional is meaningful, but ... ... …provided that we can find a meaningful explanation also for the small valuesof parameters of the Coulomb interactions. (screening???)

Itinerant Magnets:

LSDA works “reasonably well” for the spin-dependent properties

atomic picture for the

orbital magnetismspin itineracy

How to Combine ???

Screened Coulomb interactions for itinerant magnets:elaborations and justifications

• RPA screening:bare interaction

polarization

• polarization:

•self-energy within GW approximation:

one-electronGreen’s function

L. Hedin,Phys. Rev. 139, A796 (1965);F. Aryasetiawan and O. Gunnarsson,Rep. Prog. Phys. 61, 237 (1998).

Static Approximation

a convolution of density matrix andscreened Coulomb interaction

like in LDA+U

Philosophy:expected be good for -integrated (ground state) properties,but not for -resolved (spectral) properties. (???)

V.I Anisimov, F. Aryasetiawan, and A.I Lichtenstein, J. Phys.: Condens. Matter 9, 767 (1997).

Other “static approximations”:

M. van Schilfgaarde, T. Kotani, and S. Faleev, PRL 96, 226402 (2006).

A toy-model for GWfull GW for fcc-Ni

M. Springer and F. Aryasetiawan, Phys. Rev. B 57, 4364 (1998); F. Aryasetiawan et al., Phys. Rev. B 70, 195104 (2004).

‘’model’’ GW for fcc-Ni

Takes into account only local Coulomb interactions between 3d electrons (controlled by bare u~25eV).

Local Coulomb interactions reproduce the main features of full GW calculations:• asymptotic behavior U(ω∞);• position of the kink of ReU and the peak of ImU; • strong-coupling regime for small ω, where U~P-1 and does not depend on bare u

IVS and M.Imada, Phys. Rev. B 71, 045103 (2005).

U (

eV

)

U (

eV

)

0 5 10 15 20 25 30 350

10

20

30

40

ω(eV)

Re U

Im |U | Im |U |

Re U

Effective Coulomb Interaction in RPA: the strong-coupling limit

If

then

effectiveCoulombinteraction

Static Screening of Coulomb Interactions in RPA

Effective Coulomb (U) and exchange (J) interactions versus bare interaction u

Conclusion: for many applications one can use the asymptotic limit u→∞

I.V.S., PRB 73, 155117 (2006).

The screening in solids depends on the symmetry: U and J are generally different for different representations of the point group (beyond the spherical approximation in LDA+U )

Ferromagnetic Transition Metals

Spin (blue area), orbital (red area), and total (full hatched area) magnetic moments. The experimental data (neutron scattering) are summarized in: J. Trygg et al., Phys. Rev. Lett. 75, 2871 (1995);CMXD and sum rules for 2MS/ML: P. Carra et al., Phys. Rev. Lett. 70, 694 (1993).

MS 2.26 2.21 2.20 2.13

ML 0.04 0.05 0.06 0.08

1.59 1.59 1.59 1.520.08 0.10 0.11 0.140.10 0.13 0.14 0.13

0.59 0.60 0.60 0.570.05 0.05 0.05 0.050.17 0.17 0.17 0.192MS/ML 0.04 0.04 0.05

I.V.S., PRB 73, 155117 (2006).

Uranium Pnictides and Chalcogenides: UX

Spin (blue area), orbital (red area), and total (full hatched area) magnetic moments.The experimental data are the results of neutron diffraction.

I.V.S., PRB 73, 155117 (2006).

Summary -- Future of LDA+U

• many successful applications, but … many obstacles

• Q: is it really ab initio or not ?

A: probably “not”, mainly because of its basis dependence

• Q: is it possible to overcome this problem ?

Q&A

A: ..................................................................................(please, fill it yourself)

• Probably, good method to start… However, do not steak to it forever !

• Future (maybe…)

“ab initio” models

no adjustable parameters,but some flexibility with thechoice of the model and definition of these parameters

fully ab initio:GW, T-matrix, etc

heavy … at least, today,but what will be tomorrow?

LDA+U (not a stable state…)

do dot try to equilibratetoo much;seat down and think what is next

“energy surface”