1. 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

2. 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

3. Puzzle × AB NANA NBNB ΔNAΔNA Two separate atoms no interaction but free to exchange electrons total number of electrons is conserved However, and are not: energy gain == individual electron numbers ( and ) may be fractional … and this is precisely the problem …

4. 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 electrons plays an essential role I. III. stability of atomic configurations Fe[4s 2 3d 6 ], Co[4s 2 3d 7 ], etc. J.F. Janak, PRB 18, 7165 (1978). II.

5. 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).

6. 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 X α 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).

7. What does it mean ? I.V.S, P.H. Dederichs, and V.I. Anisimov, PRB 50, 16861 (1994). NA2NA2NA1NA1 NANA NA1NA1 0 U/8 U/2  U/2 Δ V U ΔEUΔEU Δ E U enforces integer population and penalizes the energy when these populations are fractional For integer populations, Δ E U = 0, otherwise Δ E U > 0. Thus, LDA+U is a constraint-LDA The potential exhibits a discontinuity at integer populations. The size of this discontinuity is U

8. 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

9. LDA+U and Hubbard model N A levels, each populated by 1 electron 1 level populated by 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, and O.I. Andersen, PRB 44, 943 (1991). note, however, that the form of this double-counting is different from PRB 44, 943 (1991).

10. 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).

11. 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) nor the 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)

12. How good is the parabolic approximation for E LDA ? 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

13. Straightforward applications along the original line divalent configurations monovalent configurations I.V.S, P.H. Dederichs, and V.I. Anisimov, PRB 50, 16861 (1994). stable configurations of 3d - impurities in Rb host Fermi level atomic impurity levels (Ry) broken lines: the levels which are supposed to be empty solid lines: the levels which are supposed to be occupied

14. 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, like superexchange: tt

15. 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”

16. “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 ………

17. Is There Any Solution ? The basic problem is ….. How to divide ??? M basis orbitals M Wannier functions but their choice is already not unique pick up N Wannier orbitals for localized states another ill-defined procedure … or using mathematical constructions a naive analogy with uncertainty principle: intrinsic uncertainty of LDA+U completeness of basis it is impossible to obtain the exact solution within LDA+U

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 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

20. LDA+U 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). J1J1 J2J2 Two experimental parameters: J 1 = -4.8 meV, J 2 = -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).

22. 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; Δ E 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

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). 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 problem of the double counting)

24. Other Methods of Calculation of U : constraint-LDA versus RPA/GW Definition: the energy cost of the reaction constraint-LDARPA/GW potential 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. bare screened

25. Example: isolatedt 2g band in SrVO 3 main interband transitions: (1) O(2p) → V(e g ) (2) O(2p) → V(t 2g ) (3) V(t 2g ) → V(e g ) Intra-Orbital U (eV) 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) 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): “LDA+U” for itinerant systems Example: Orbital Magnetism in Metallic Compounds

27. Orbital Magnetism and Density-Functional Theory in the spin-density-functional theory (SDFT): charge density spin-magnetization density E XC =E XC [ρ,m] spin polarization Kohn-Sham (KS) theory there is no guarantee that M L can be reproduced at the level of KS - SDFT M L should be a basic variable ⇒ we need an explicit dependence of E XC on M L : E XC =E XC [ ρ, m, M L ] the concept of orbital functionals and orbital polarization

28. 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 electronic structure calculations is basically the problem of local Coulomb correlations

29. 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???)

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 RPA screening: bare interaction polarization polarization: self-energy within GW approximation: one-electron Green’s function L. Hedin, Phys. Rev. 139, A796 (1965); F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998).

32. 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).

33. A toy-model for GW full 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 Re U 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) 0 5 10 15 20 25 30 35 0 10 20 30 40 ω(eV) Re U Im |U | Re U

34. Effective Coulomb Interaction in RPA: the strong-coupling limit If then effective Coulomb interaction

35. 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 )

36. 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 2M S /M L : P. Carra et al., Phys. Rev. Lett. 70, 694 (1993). M S 2.26 2.21 2.20 2.13 M L 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 0.59 0.60 0.60 0.57 0.05 0.05 0.17 0.17 0.17 0.19 2 M S / M L 0.04 0.04 0.05 I.V.S., PRB 73, 155117 (2006).

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 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 !

39. Future (maybe…) “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? LDA+U (not a stable state…) do dot try to equilibrate too much; seat down and think what is next “energy surface”