1. Dynamical Mean-Field Studies of the Actinide Series Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Workshop on Correlated Electron Effects for Anomalous Properties of Elemental Actinides 23 May 2005, STC Conf Room, TA-3-32-Rm 134, Los Alamos

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in the actinide series (Smith-Kmetko phase diagram)

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott Transition in the Actinide Series Lashley et.al.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ??? Mott transition into an open and closed shell systems. Am at room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? G. Kotliar J. ow Temp. Phys 126, 100927. (2002) Single site DMFT, superconductivity must intervene before reaching the Mott insulating state.Capone et. al.Science (2001)

5. Outline The Mott transition across the actinide series and some remarks on DMFT. DMFT studies of Plutonium. S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science 300, 954 (2003). Americium under pressure. New experiments and DMFT results. J. J.C. Griveau J Rebizant G. Lander and G. Kotliar PRL 94, 97002, (2005) S. Murthy Rutgers Ph.D Thesis (2004). S. Savrasov K. Haule and G. Kotliar.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992]. Merge atomic physics and band theory. Atom in a medium. Weiss field. = Quantum impurity model. Solid in a frequency dependent potential.  Realistic combination with band theory: LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Functional formulation. In addition to the density, use functionals of the one electron spectral function. [ R. Chitra and GKPhys. Rev. B62, 12715 (2000). and S. Savrasov PRB (2005) ]. Allows computation of total energy AND one electron spectra. Ideal to treat f electron materials. See work on Cerium, Mc Mahan et. al. (LLNL) Amadon et. al. (CEA) etc…..

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cluster Extensions of Single Site DMFT

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative phase diagram of a frustrated Hubbard model

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Some References Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, (1996). Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu phases: A. Lawson Los Alamos Science 26, (2000) Within LDA fcc Pu has a negative shear modulus. Spin density functional within GGA predicts  Pu to be magnetic with a 5 Bohr magneton moment. oment. Experimentally Pu is not magnetic.

11. Total Energy as a function of volume for Pu Total Energy as a function of volume for Pu W (ev) vs (a.u. 27.2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2004)

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double well structure and  Pu Qualitative explanation of negative thermal expansion[ Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Phys vol.126, 1009 27. (2002)] Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT and the Invar Model A. Lawson et. al. LA UR 04-6008 (LANL)

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

16. DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. Tobin et. al. PHYSICAL REVIEW B 68, 155109,2003

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS K. Haule, Pu- photoemission with DMFT using vertex corrected NCA.

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Approach the Mott point from the right Am under pressure Density functional based electronic structure calculations:  Non magnetic LDA/GGA predicts volume 50% off.  Magnetic GGA corrects most of error in volume but gives m ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 ( 7 F 0 ) Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard”

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in open (right) and closed (left) shell systems. Realization in Am ?? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) J=0 ??? Tc

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Am At room pressure a localised 5f 6 system;j=5/2. S = -L = 3: J = 0 J. Smith & R. Haire, Science (1978) J. Smith, J. Phys. (1979)

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band, (5f) 7

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectrum from 7 F 0 Americium LDA+DMFT Density of States Experimental Photoemission Spectrum (after J. Naegele et.al, PRL 1984) S. Savrasov et. al. Multiplet Effects F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. C. Griveau et. al. (2004)

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS H.Q. Yuan et. al. CeCu2(Si 2-x Ge x ). Am under pressure Griveau et. al. Superconductivity due to valence fluctuations ?

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions and Outlook Motivation: Mott transition in Americium and Plutonium. In both cases theory (DMFT) and experiment suggest a more gradual transformation than postulated in earlier theories. DMFT: Physical connection between spectra and structure. Studied the Mott transition from both ends, Studied open and closed shell cases.. DMFT: method under construction, but it already gives quantitative results and qualitative insights. It can be systematically improved in many directions. Interactions between theory and experiments. Pu: simple picture of alpha delta and epsilon. Interplay of lattice and electronic structure near the Mott transition. Am: Rich physics, mixed valence under pressure ? Superconductivity near the Mott transition.

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Actinides and The Mott Phenomena Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Actinides allow us to probe this physics in ELEMENTS. Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within dynamical mean field theory in the model Hamiltonian context. Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials. Turn the technology developed to solve simple models into a practical quantitative electronic structure method.

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS f7 L=0, S=3.5, J=3.5

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cluster Extensions of Single Site DMFT

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992]. Atom in a medium = Quantum impurity model. Solid in a frequency dependent potential. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] [GK R. Chitra GKPhys. Rev. B62, 12715 (2000). and S. Savrasov cond-matt 0308053]. Allows computation of total energy AND one electron spectra.

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two paths for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). The light, sp (or spd) electrons are extended, well described by LDA.The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting the average energy already contained in LDA. Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, treat with DMFT. In this method U is either a parameter or is estimated from constrained LDA Describes the excitation spectra of many strongly correlated solids..

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional Determine the self energy, the density and the structure of the solid self consistently. By extremizing a functional of these quantities. (Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, PRB 2005). Coupling of electronic degrees of freedom to structural degrees of freedom. Full implementation for Pu. Savrasov and Kotliar Nature 2001. Under development. Functional of G and W, self consistent determination of the Coulomb interaction and the Greens fu

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two paths for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Mott transition, three peak structure and transfer of spectral weigth. Evolution of the one particle spectral function in a frustrated Hubbard model at half filling.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS U/t=4. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS PLUTONIUM.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. C. Lawson et. al. LA UR 04-6008 F(T,V)=Fphonons+Finvar

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Invar model A. C. Lawson et. al. LA UR 04-6008

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Prediction of the Invar Model

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ??? Mott transition into an open and closed (left) shell systems. Am at room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? G. Kotliar J. ow Temp. Phys 126, 100927. (2002) Single site DMFT, superconductivity must intervene before reaching the Mott insulating state.Capone et. al.Science (2001)

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT : What is the dominant atomic configuration,what is the fate of the atomic moment ? Snapshots of the f electron Dominant configuration:(5f) 5 Naïve view Lz=-3,-2,-1,0,1 ML=-5  B,, S=5/2 Ms=5  B Mtot=0 More realistic calculations, (GGA+U),itineracy, crystal fields     ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000) This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce).  Contrast Am:(5f) 6

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ground State Theory of  -Pu S.Y. Savrasov and G. Kotliar PRL 84, (2000). “In conclusion, using a realistic value of the Hubbard U = 4 eV we have been able to describe ground state properties of d-Pu in good agreement with experimental data. This theory correctly predicts the equilibrium volume of the d phase and suggests that nearly complete cancellation occurs between spin and orbital moments. The main shortcoming of the present calculation is the assumed long-range spin and orbital order. This is the essential limitation of the LDA + U approach (or of any static mean field theory): in order to capture the effects of correlations it has to impose some form of long-range order. Static mean field theories are unable to capture subtle many-body effects such as the formation of local moments and their subsequent quenching via the Kondo effect. These deficiencies will be removed by ab initio dynamical mean field calculations for which codes are currently being developed.”

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dual model, Zwignagl and Fulde.Erickson Becker Balatzki and J. Wills. J. Alloys and Compounds 287, (1995) 1-5. Part of the f electrons are in a core like (5f)4 configuration (non magnetic )and 1 5f electron is itinerant. GGA+ Orbital Polarization. Soderlind and Sadigh PRL 92, 1857021. Correct volume of all phases of Pu. Ordered Orbital and Spin moments in all phases of Pu. Disordered Local Moment approach. A. Niklasson, J M. Wills, M I. Katsnelson, I.A. Abrikosov, O. Eriksson, and B. Johansson Phys Rev B 67, 235105 (2003). There are large fluctuating (disordered moments) in Pu. Accounts for the correct volumes and bulk moduli across the actinide series.

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comparison of theory and experiment. Good agreement over the majority of the Brillouin zone, is significant. The phonon frequencies depend on the forces acting on the atoms as a result of their displacement. Ability to compute forces, is a first step to derive potentials, and do molecular dynamics. Discrepancies along (111) are significant. Role of temperature ? Improve the impurity solver ? Non local corrections, and deviations from DMFT. Spectral Density Functional. Connection between spectra and bonding. Microscopic theory of Pu, connecting its anomalies to the vicinity of a Mott point. Combining theory and experiment we can more than the sum of the parts. Next step in Pu, much better defined problem, discrepancy in (111 ) zone boundary, may be due to either the contribution of QP resonance, or the inclusion of nearest neighbor correlations. Both can be individually studied.

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field View of Pu ( Savrasov Kotliar and Abrahams, Nature 2001) Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Is the natural consequence of earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP ( Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000))

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The delta –epsilon transition The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transverse Phonon along (0,1,1) in epsilon Pu in self consistent Born approximation.

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. Tobin et. al. PHYSICAL REVIEW B 68, 155109,2003

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  Pu and delta Pu differ electronically by the distribution of spectral weight in the resonance and the Hubbard band. U/W is not so different in alpha and delta The specific heat of delta Pu, is only twice as big as that of alpha Pu. The susceptibility of alpha Pu is in fact larger than that of delta Pu. The resistivity of alpha Pu is comparable to that of delta Pu and near the Mott limit.

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity PRL 91,061401 (2003)

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How to track the origin of the resonance ?Turn to Optics! Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior. See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Americium

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Americium under pressure (Lindbaum et. al. PRB 2003)

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Am Equation of State: LDA+DMFT Predictions (Savrasov Kotliar Haule Murthy 2005) LDA+DMFT predictions:  Non magnetic f 6 ground state with J=0 ( 7 F 0 )  Equilibrium Volume: V theory /V exp =0.93  Bulk Modulus: B theory =47 GPa Experimentally B=40-45 GPa Theoretical P(V) using LDA+DMFT Self-consistent evaluations of total energies with LDA+DMFT. Accounting for full atomic multiplet structure using Slater integrals: F (0) =4.5 eV, F (2) =8 eV, F (4) =5.4 eV, F (6) =4 eV New algorithms allow studies of complex structures. Predictions for Am II Predictions for Am IV Predictions for Am III Predictions for Am I

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS New picture of the electronic structure of Am. The traditional picture of Am, views its f electrons as a closed shell (5f) 6 As a result the spd electrons are free electron-like. This resulted in the early prediction that Am should be a superconductor. Theoretical calculations and experiments shows that Am is very close to a mixed valence situation that can be induced by a small amount of pressure!! At larger pressures a Mott transition and a Tc vs V with a dome- like shape results.

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. C. Griveau et. al. (2004)

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in open (right) and closed (left) shell systems. S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ??? Tc

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Griveau et.al. (2004)

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions DMFT produces non magnetic state, around a fluctuating (5f)^5 configuration with correct volume the qualitative features of the photoemission spectra, quasiparticle resonance and Hubbard band, and a double minima structure in the E vs V curve. Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions account for delta-epsilon transition. Anomalous phonons in epsilon Pu. Calculations can be refined, include multiplets, better impurity solvers, frequency dependent U’s, electronic entropy. User friendly interfaces.

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Experiments and Theory are needed to test the different pictures of the electronic structure of Pu Model of Erickson and Wills : 4 (5f) electrons are core-like and 1 is delocalized. DMFT picture: all the 5 (5f) electrons are equivalent, they are localized over short time scales and itinerant over long time scales resulting in Hubbard band and quasiparticle resonance in the spectra. Both pictures require strong correlations in the delta phase but how to differentiate between them experimentally ? Focus on the alpha phase. Resonant Photoemission Probe unoccupied states. Upper Hubbard band, BIS. Optics. X ray absortion. Etc.. Fermi Surface Probes. Different Fermi surfaces.

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Search for ordered phases and DLM On the possibility of magnetic moments in plutonium J. C. Lashley, A. Lawson, R. J. McQueeney, and G. H. Lander. (2004) Elastic Neutron Scattering. Inelastic Neutron Scattering. Magnetic Susceptibility. Specific heat measurements in a magnetic field. No indication whatsoever of ordered or disordered moments in either alpha or delta Pu.

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998). Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Reviews: Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar (2002)