LDA U: Fundamentals,

1
LDAU Fundamentals, Open
Questions, and
Recent Developments
Igor Solovyev
Computational Materials Science Center, National
Institute for Materials Science, Tsukuba, Japan
e-mail SOLOVYEV.Igor_at_nims.go.jp
2
Contents

• 1. Atomic limit
• 1.1. DFT for fractional particle numbers
• 1.2. LDAU and Slaters transition state
• 1.3. LDAU and Hubbard model
• 1.4. Rotationally invariant LDAU
• 1.5. simple applications
• LDAU for solids postulates and unresolved
  problems
• 2.1. choice of basis
• 2.2. charge-transfer energy in transition-metal
  oxides
• Other methods of calculation of U RPA/GW
• 3.1. U for isolated bands (low-energy models)
• 3.2. LDAU for metallic compounds --
• orbital polarization for itinerant magnets

3
Puzzle

• Two separate atoms

• no interaction

A
B

• but free to exchange electrons

total number of electrons is conserved
NA
NB
?NA
However, and are not

• energy gain

individual electron numbers ( and
) may be fractional and this is precisely the
problem
4
Other Examples
III.
adatom on surface chemical reaction, etc
I.
strongly-correlated systems weak interactions
between atoms (in comparison with on-site
energies) the ability of exchange by
electrons plays an essential role
II.
stability of atomic configurations Fe4s 23d 6,
Co4s 23d 7, etc. J.F. Janak, PRB 18, 7165
(1978).
5
What is wrong ?
J. P. Perdew, R. G. Parr, M. Levy, and J. L.
Balduz, Phys. Rev. Lett. 49, 1691 (1982).

• 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
6
I.V.S, P.H. Dederichs, and V.I. Anisimov, PRB 50,
16861 (1994).
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 Xa method)

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

where

• LDAU

7
I.V.S, P.H. Dederichs, and V.I. Anisimov, PRB
50, 16861 (1994).
What does it mean ?

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

U/8

• For integer populations, ?EU 0,
• otherwise ?EU gt 0. Thus,

?EU
0
LDAU is a constraint-LDA
U/2

• The potential
• exhibits a discontinuity at
• integer populations.

?V U
-U/2

• The size of this discontinuity is U

NA-2
NA-1
NA
NA1
8
LDAU and Slaters Transition State or meaning
of LDAU eigenvalues

• LDAU functional

where in
each interval
Janaks theorem

• Slaters transition state

ionization potential
electron affinity

• Then, and

nothing but LDAU eigenvalues in the atomic limit
9
V.I. Anisimov, J. Zaanen, and O.I. Andersen, PRB
44, 943
(1991).
LDAU and Hubbard model

• Hubbard model in the mean-field approximation

1 level populated by x electrons
NA levels, each populated by 1 electron
note that if or

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

note, however, that the form of
this double-counting is different from PRB 44,
943 (1991).

• LDAU

R. Arita (July 31)
possible extensions beyond mean-field,
?-dependent self-energy,
DMFT
10
Moreover
curvature of LDA total energy
Hubbard U
Curvature of LDA total energy Hubbard U

• constraint calculations of U

another possibility (using Janaks 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).
11
Rotationally-Invariant LDAU and Hunds rules

• depends on the
  type of the orbitals

• which orbitals should we use ?

• Strategy

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).
it depends neither on the form of the basis
(i.e., complex versus real harmonics) nor the
orientation of the coordinate frame
density (population) matrix
matrix of Coulomb interactions

• In spherical approximation, is
  fully specified by
• Coulomb ,
• exchange ,
  and
• nonsphericity

controls the number of electrons
control Hunds rules (at least, in mean-field)
12
How good is the parabolic approximation for ELDA ?
T(2)
I.V.S. and P.H. Dederichs, Phys. Rev. B 49, 6736
(1994).
T(1)
d - impurities in alkali host (Rb)
T(2) divalent configuration
T(1) monovalent configuration
localized levels in free-electron gas
d
13
Straightforward applications along the original
line
I.V.S, P.H. Dederichs, and V.I. Anisimov, PRB
50, 16861 (1994).
divalent configurations
stable configurations of 3d - impurities in Rb
host
atomic impurity levels (Ry)
monovalent configurations
Fermi level
broken lines the levels which are supposed to
be empty
solid lines the levels which are supposed to be
occupied
14
LDAU for atoms and for solids

• pure atomic limit
• (no hybridization)

affinity
ionization
LDA LDAU
simply the redefinition of atomic
levels, relevant to the excited-state properties

• solid interacting levels

position of atomic levels is important , as it
already contributes to the ground-state
properties, like superexchange
t
t
after hybridization
after hybridization
before hybridization
15
Postulate LDAU functional for solids
(double-counting)
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
density matrix
number of localized electrons
16
Kohn-Sham equations in LDAU
where
is a non-local operator
The final answer depends on the choice of the
basis
an obvious, but very serious problem
17
Is There Any Solution ?
The basic problem is ..
or using mathematical
constructions
How to divide ???
M basis orbitals
M Wannier functions but their choice is
already not unique
a naive analogy with uncertainty principle
pick up N Wannier orbitals for localized states
another ill-defined procedure
intrinsic uncertainty of LDAU
it is impossible to obtain the exact solution
within LDAU
completeness of basis
18
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).
19
Other problems charge-transfer energy in TMO
?
U Coulomb interaction
UHB
LHB
? charge-transfer energy
O(2p)
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).
Superexchange interaction

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

• How well is the charge-transfer energy described
  in LDAU ?

20
LDAU for the transition-metal oxides what we
have and what should be?
21
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 meV
Two theoretical parameters U and ? in
One can find parameters of LDAU potential by
fitting the experimental magnon spectra
I.V.S. and K. Terakura, PRB 58, 15496 (1998).
22
Magnetic Force Theorem

• 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
• ?E is expressed through the Kohn-Sham
• potential in the ground state.

?

• Application for the spin-spiral perturbation

rotation of magnetization

• 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).
23
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).
in LDAU for MnO, U itself is O.K., but . the
charge-transfer energy is wrong.
(the so-called problem of the double counting)
I.V.S. and K. Terakura, PRB 58, 15496 (1998).
24
Other Methods of Calculation of U
constraint-LDA versus RPA/GW
Definition the energy cost of the reaction
constraint-LDA
RPA/GW
potential
1.
perturbation theory
to simulate the charge disproportionation
external potential ?
change of KS orbitals ?
change of charge density ?
mapping of Kohn-Sham eigenvalues
2.
change of Coulomb potential ?
onto the model
etc.
screened
bare
is the number of d electrons
3.
Fourier transformation
25
Good points of RPA/GW (I)

• Construction of model Hamiltonian
• for isolated bands

problem to solve screening of 3d electrons by
the same 3d electrons
F. Aryasetiawan (this workshop) I. V. S.
(symposium)
Example isolatedt2g band in SrVO3
Intra-Orbital U (eV)
main interband transitions
(1) O(2p)?V(eg) (2) O(2p)?V(t2g) (3)
V(t2g)?V(eg)
I.V.S., PRB 73, 155117 (2006).
I.V., N. Hamada and K. Terakura, PRB 53, 7158
(1996).
phenomenological idea
26
Good points of RPA/GW (II) LDAU for
itinerant systems
Example Orbital Magnetism in Metallic Compounds
27
Orbital Magnetism and Density-Functional Theory
spin-magnetization density

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

EXCEXC?,m
Kohn-Sham (KS) theory
charge density
spin polarization
ML should be a basic variable ? we need an
explicit dependence of EXC on ML EXCEXC?,m,
ML
there is no guarantee that ML can be reproduced
at the level of KS - SDFT

• the concept of orbital functionals and orbital
  polarization

28
Some Phenomenology

• orbital magnetism is driven by relativistic
• spin-orbit interaction
  
• (a gradient of electrostatic potential)

the main effect comes from small core region

• does not commute with

is not an observable, except the same core
region where is nearly spherical
FLAPW potential from E. Wimmer et al., PRB 24,
864 (1981).
The problem of orbital magnetism in
electronic structure calculations is basically
the problem of local Coulomb correlations
29
Several empirical facts about LDAU for
itinerant compounds
if U0.7 eV
General consensus the form of LDAU functional
is meaningful, but ... ... provided that we can
find a meaningful explanation also for the small
values of parameters of the Coulomb interactions.
(screening???)
30
Itinerant Magnets LSDA works reasonably well
for the spin-dependent properties
atomic picture for the orbital magnetism
spin itineracy
How to Combine ???
31
Screened Coulomb interactions for itinerant
magnets elaborations and justifications
bare interaction

• RPA screening

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

• polarization

• self-energy within GW approximation

one-electron Greens function
32
Static Approximation
a convolution of density matrix and screened
Coulomb interaction
like in LDAU
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).
33
A toy-model for GW
full GW for fcc-Ni
model GW for fcc-Ni
40
30
U (eV)
U (eV)
Im U
Im U
20
10
Re U
Re U
0
0 5 10 15 20 25 30 35
?(eV)
M. Springer and F. Aryasetiawan, Phys. Rev. B
57, 4364 (1998) F. Aryasetiawan et al., Phys.
Rev. B 70, 195104 (2004).
Takes into account only local Coulomb
interactions between 3d electrons (controlled
by bare u25eV).
IVS and M.Imada, Phys. Rev. B 71, 045103 (2005).

• Local Coulomb interactions reproduce the main
  features of full GW calculations
• asymptotic behavior U(??8)
• position of the kink of ReU and the peak of
  ImU
• strong-coupling regime for small ?, where UP-1
  and does not depend on bare u

34
Effective Coulomb Interaction in RPA the
strong-coupling limit
If
effective Coulomb interaction
then
35
Static Screening of Coulomb Interactions in RPA
I.V.S., PRB 73, 155117 (2006).
Effective Coulomb (U) and exchange (J)
interactions versus bare interaction u
Conclusion for many applications one can use the
asymptotic limit u?8
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 LDAU )
36
Ferromagnetic Transition Metals
I.V.S., PRB 73, 155117 (2006).
0.59 0.60 0.60 0.57 0.05 0.05 0.05
0.05 0.17 0.17 0.17 0.19
MS 2.26 2.21 2.20 2.13 ML 0.04 0.05 0.06
0.08
1.59 1.59 1.59 1.52 0.08 0.10 0.11
0.14 0.10 0.13 0.14 0.13
2MS/ML 0.04 0.04 0.05
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).
37
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).
38
Summary -- Future of LDAU

• many successful applications, but many
  obstacles

QA

• 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 ?

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

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

39

• Future (maybe)

do dot try to equilibrate too much seat down and
think what is next
LDAU (not a stable state)
energy surface
ab initio models
fully ab initio GW, T-matrix, etc
no adjustable parameters, but some flexibility
with the choice of the model and definition of
these parameters
heavy at least, today, but what will be
tomorrow?