Transcript Slide 1

LDA+U: Fundamentals,
Open Questions, and
Recent Developments
Igor Solovyev
Computational Materials Science Center,
National Institute for Materials Science,
Tsukuba, Japan
e-mail: [email protected]
Contents
1. Atomic limit
1.1. DFT for fractional particle numbers
1.2. LDA+U and Slater’s transition state
1.3. LDA+U and Hubbard model
1.4. Rotationally invariant LDA+U
1.5. simple applications
2.
LDA+U for solids: postulates and unresolved problems
2.1. choice of basis
2.2. 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
• Two separate atoms
A
• no interaction
B
×
NA
• but free to exchange electrons
NB
ΔNA
total number of electrons
is conserved
However,
and
• energy gain
=
)
=
individual electron numbers (
and
may be fractional …
and this is precisely the problem …
are not:
Other Examples
III.
adatom on surface;
chemical reaction,
etc…
I.
II.
stability of atomic configurations
Fe[4s 23d 6], Co[4s 23d 7], etc.
J.F. Janak, PRB 18, 7165 (1978).
strongly-correlated systems:
weak interactions between atoms
(in comparison with on-site energies);
the ability of exchange by electrons
plays an essential role
What is wrong ?
• The electron is “indivisible”
J. P. Perdew, R. G. Parr,
M. Levy, and J. L. Balduz,
Phys. Rev. Lett. 49, 1691 (1982).
• 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
What shall we do ?
I.V.S, P.H. Dederichs, and
V.I. Anisimov, PRB 50, 16861 (1994).
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, and
V.I. Anisimov,
PRB 50, 16861 (1994).
U/2
LDA+U is a constraint-LDA
0
• For integer populations, ΔEU = 0,
otherwise ΔEU > 0. Thus,
ΔEU
• ΔEU enforces integer population
and penalizes the energy when
these populations are fractional
U/8
What does it mean ?
ΔV U
• The size of this discontinuity is U
-U/2
• The potential
exhibits a discontinuity at
integer populations.
NA-2 NA-1
NA
NA+1
LDA+U and Slater’s Transition State
or 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
V.I. Anisimov, J. Zaanen, and
LDA+U and Hubbard model O.I. Andersen, PRB 44, 943
(1991).
• Hubbard model in the mean-field approximation
1 level populated
by x electrons
NA levels,
note that
•
if
each populated
by 1 electron
or
mimics LDA “smooth” dependence on x and coincides
with
for integer populations
• LDA+U:
note, however, that
the form of this
double-counting is
different from
PRB 44, 943 (1991).
possible extensions: beyond mean-field,
ω-dependent self-energy, DMFT
R. Arita
(July 31)
Moreover …
curvature of LDA total energy
Hubbard U
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:
A.I. Liechtenstein,
it depends neither on the form of the basis V.I. Anisimov,
(i.e., complex versus real harmonics) nor
and J. Zaanen,
the orientation of the coordinate frame
PRB 52, R5467 (1995);
I.V.S., A.I. Liechtenstein,
density (population) matrix
and K. Terakura,
matrix of Coulomb interactions
PRL 80, 5787 (1998).
• In spherical approximation,
Coulomb
exchange
“nonsphericity”
is fully specified by
,
controls the number of electrons
, and
control Hund’s rules
(at least, in mean-field)
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
Straightforward applications along the original line
atomic impurity levels (Ry)
divalent configurations
I.V.S, P.H. Dederichs, and
V.I. Anisimov,
PRB 50, 16861 (1994).
stable configurations of
3d - impurities in Rb host
monovalent configurations
Fermi level
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)
affinity
ionization
LDA
LDA+U
simply the redefinition of atomic levels,
relevant to the excited-state properties
• solid: interacting levels
after
hybridization
t
t
before hybridization
after
hybridization
position of atomic
levels is important , as
it already contributes
to the ground-state
properties, like
superexchange:
Postulate: LDA+U 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”
“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 …..
… or using mathematical
constructions
M basis orbitals
How
to divide ???
M Wannier functions
but their choice is already
not unique
a naive analogy with
uncertainty principle:
intrinsic uncertainty
of LDA+U
pick up N Wannier orbitals
for localized states
another ill-defined procedure
completeness
of basis
it is impossible to
obtain 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
Δ
UHB
LHB
U : Coulomb interaction
Δ: charge-transfer
O(2p)
energy
U
Superexchange interaction:
T. Oguchi, K. Terakura, and
A.R. Williams, PRB 28, 6443 (1983);
J. Zaanen and G.A. Sawatzky,
Can. J. Phys. 65, 1262 (1987).
• Δ is an important parameter of electronic
structure of the transition-metal oxides
• How well is the charge-transfer energy described in LDA+U ?
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 meV
Two theoretical parameters:
U and Δ in
One can find parameters of LDA+U potential
by fitting the experimental magnon spectra
I.V.S. and K. Terakura, PRB 58, 15496 (1998).
Magnetic Force Theorem
• For small deviations near the equilibrium,
the total energy change is expressed through
the change of the single-particle energies:
θ
• No need for total energy calculations;
ΔE is expressed through the Kohn-Sham
potential in the ground state.
• Application for the spin-spiral perturbation
θ
rotation of magnetization
• Magnetic interactions:
θ
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).
… 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 LDA+U for MnO, U itself is O.K., but ….
the charge-transfer energy is wrong.
I.V.S. and K. Terakura, PRB 58, 15496 (1998).
(the so-called problem
of the double counting)
Other Methods of Calculation of U :
constraint-LDA versus RPA/GW
Definition: the energy cost of the reaction
constraint-LDA
1. potential
to simulate the charge
disproportionation
RPA/GW
perturbation theory
external potential
→
change of KS orbitals
2. mapping of Kohn-Sham eigenvalues
onto the model
is the number of “d” electrons
3. Fourier transformation
change of charge density
→
→
change of Coulomb potential
→
etc.
screened
bare
Good points of RPA/GW (I)
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
main interband transitions:
(1) O(2p)→V(eg)
(2) O(2p)→V(t2g)
(3) V(t2g)→V(eg)
Intra-Orbital U (eV)
• Construction of model Hamiltonian
for isolated bands
I.V.S., PRB 73, 155117 (2006).
I.V., N. Hamada and K. Terakura, phenomenological
PRB 53, 7158 (1996).
idea
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):
Kohn-Sham (KS) theory
spin polarization
there is no guarantee that ML can be
reproduced at the level of KS - SDFT
spin-magnetization density
EXC=EXC[ρ,m]
charge density
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
• 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
The problem of orbital magnetism in electronic
structure calculations is basically the problem
of local Coulomb correlations
FLAPW potential
from E. Wimmer et al.,
PRB 24, 864 (1981).
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 values
of parameters of the Coulomb interactions. (screening???)
Itinerant Magnets:
LSDA works “reasonably well” for the spin-dependent properties
spin itineracy
atomic picture
for the
orbital magnetism
How to Combine ???
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
Green’s function
Static Approximation
a convolution of
density matrix and
screened 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 GW
full GW for fcc-Ni
‘’model’’ GW for fcc-Ni
40
Im |U |
20
10
0
0
U (eV)
U (eV)
30
Re U
5
10
15 20
ω(eV)
25
M. Springer and F. Aryasetiawan,
Phys. Rev. B 57, 4364 (1998);
F. Aryasetiawan et al.,
Phys. Rev. B 70, 195104 (2004).
Im |U |
Re U
30
35
Takes into account only local Coulomb
interactions between 3d electrons
(controlled by bare u~25eV).
IVS and M.Imada, Phys. Rev. B 71, 045103 (2005).
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
Effective Coulomb Interaction in RPA:
the strong-coupling limit
If
then
effective
Coulomb
interaction
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→∞
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
MS 2.26 2.21 2.20 2.13
ML 0.04 0.05 0.06 0.08
2MS/ML 0.04 0.04 0.05
I.V.S., PRB 73, 155117 (2006).
1.59 1.59 1.59 1.52
0.08 0.10 0.11 0.14
0.10 0.13 0.14 0.13
0.59 0.60 0.60 0.57
0.05 0.05 0.05 0.05
0.17 0.17 0.17 0.19
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).
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&A
• 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 !
• Future (maybe…)
do dot try to equilibrate
too much;
seat down and think
what is next
“energy surface”
LDA+U
(not a stable state…)
“ab initio” models
no adjustable parameters,
but some flexibility with the
choice of the model and
definition of these parameters
fully ab initio:
GW, T-matrix, etc
heavy … at least, today,
but what will be tomorrow?