Correlation in correlated materials (mostly transition metal oxides) Lucas K. Wagner University of Illinois at Urbana-Champaign

Transcription

1 Correlation in correlated materials (mostly transition metal oxides) Lucas K. Wagner University of Illinois at Urbana-Champaign

2 Understanding of correlated materials is mostly phenomenological FN- DMC (Slater- Jastrow): good energetics, but due to cancellation of errors Use reduced density matrices and accurate wave functions to calculate electron correlations

3 Transition metal oxides and agempts to describe them

4 Strongly correlated materials High Tc superconductivity Heavy fermions MoG insulators Collossal magnetoresistance Metal- insulator transitions After C. Hartinger

5 Phenomenological models of the physics: Hubbard model Valence bond theory First principles calculations: DFT (hard to include correlation) Quantum chemistry (hard to apply to solids) Quantum Monte Carlo (fixed node?) GW (perturbation theory) Phenomenological models + first principles: DFT+U DFT+DMFT

6 NiO: the band gap Experiment LDA+DMFT Jiang et al. PRB (2010) Ren et al. PRB (2006)

7 Reaction enthalpies U helps, but hard to use predictively Wang, Maxisch, and Ceder. PRB 73, (2006)

8 Problems with DFT+Hubbard: Lots of (too much) flexibility: Double- counting scheme Level of lagice model treatment Basis on which to apply lagice model How we choose the value of U Do we include intersite terms? LiGle a priori guidance! QMC methods to give guidance?

9 QMC calculations on transition metal oxide materials (a very brief summary)

10 Early work: TMO molecules E(DMC, B3LYP orbitals)-e(dmc, HF orbitals) (ev) ScO TiO VO CrO MnO σ 1 σ 1 δ 1 σ 1 δ 2 σ 1 δ 2 π 1 σ 1 δ 2 π 2 Wagner & Mitas, Chem. Phys. LeG (2003) Wagner & Mitas, J. Chem. Phys. 126, (2007)

11 FN- DMC(SJ) energetics performance is preny good General note: d- p hybridization needs to be properly described for the best results. After that, Cohesive energies in good agreement w/experiment: MnO, FeO, NiO, BaTiO 3, ZnO, V x O y Equations of state: FeO Band Gap: FeO, MnO, BaTiO 3, ZnO Needs and Towler Int. J. Mod. Phys. B (2003) Wagner, J. Phys.: Condens. MaGer (2007) Kolorenc and Mitas, Phys. Rev. LeG. 101, (2008) Kolorenc, Hu, and Mitas, Phys. Rev. B 82, (2010) Ertekin, Wagner, and Grossman, (in preparation) Bande and Luchow, Phys. Chem. Chem. Phys., 2008, 10, 3371

12 Signs of trouble: TMO molecules Bond length (angstroms) Experiment DMC LDA UCCSD(T) Binding Energy of MO (ev) Experiment DMC LDA UCCSD(T) ScO TiO VO CrO MnO ScO TiO VO CrO MnO Bond length and binding energy are very close to experiment when using B3LYP orbitals But the dipole moment still changes a lot with multiple determinants!

13 Despite encouraging energetic performance, there are indications that the wave function is not very good at the DMC(SJ) level! Going forward Guess wave functions (hope they work) General wave function expansion (but the space is exponential!) Understand the physics (too hard?)

14 Analyzing electron correlation using reduced density matrices

15 Understand the physics This has two purposes: Get beger accuracy in QMC Help understand the physics of TMO materials

16 Strong correlations? Physicists: Strong electron- electron term in effective low- energy Hamiltonian (Hubbard U) Chemists: Large static correlation (left- right correlation) Electron in place A à another electron not in place A

17 Correlation in electron gas Ĥ = 1 2 X r 2 i + X i i<j 1 r ij High density: kinetic energy dominates Low density: Coulomb energy dominates

18 Left- right correlation: H 2 molecule Electron- nucleus and kinetic energy dominate Electron- nucleus smaller, kinetic energy smaller: Electron on A à other electron on B

19 Introduction to density matrices 1(r) 2(r) Ansa^: (r 1,r 2 )=a 1 1 (r 1 ) 1 (r 2 ) a 2 2 (r 1 ) 2 (r 2 )

20 (r 1,r 2 )=a 1 1 (r 1 ) 1 (r 2 ) a 2 2 (r 1 ) 2 (r 2 ) 1RDM ij = h c i c j i 1RDM = a a 2 2 2RDM ij,ij = h c i c j c ic j i a RDM d = 0 a 2 2 c ij = 2RDM ij,ij 1RDM ii 1RDM jj c = a 2 1 a 4 1 a 2 1a 2 2 a 2 1a 2 2 a 2 2 a 4 2

21 General case (r 1,r 2 )= 1 (r 1 ) 1 (r 1 ) 1RDM = 2RDM d = a a 2 2 a a 2 2 1RDM = 2RDM d = c = a 2 1 a 4 1 a 2 1a 2 2 a 2 1a 2 2 a 2 2 a 4 2 c =

22 2- RDM diagonal 1(r) 2(r) 2RDM d = a a 2 2 (r 1,r 2 ) ' 1 (r 1 ) 1 (r 2 ) c 2 (r 1 ) 2 (r 2 )

23 Relation to Hubbard picture If we change basis: a1 a 2 a 1 a 2 c = 1 2 a 1 a 2 a 1 a 2 c = a 2 1 a 4 1 a 2 1a 2 2 a 2 1a 2 2 a 2 2 a 4 2

24 c Positive Only on- site repulsion: More complex structure:

25 c = a 2 1 a 4 1 a 2 1a 2 2 a 2 1a 2 2 a 2 2 a 4 2 Most sets of four orbitals give rise to real- space repulsion with this pagern Including combinations of localized and diffuse orbitals. a1 a 2 a 1 a 2 c = 1 2 a 1 a 2 a 1 a 2 Localized sets of orbitals give this pagern

26 Evaluating electron correlation in transition metal oxides

27 Change in occupation number MnO 2 convergence up down ScO TiO VO CrO MnO MnO 2 FeO 2 CoO 2 Use small molecules Expand in Slater determinants, only doubles Converge the 1- RDM of the wave function with respect to determinants Cutoff threshold Orbital #

28 Electronic structure of the TM monoxides Doubly occupied Singly occupied Filling order Virtual orbitals

29 Diagonal of the 1- RDM Majority (up) Minority (down)

30 Doubly Singly Virtual orbitals Minority spin Majority spin c

31 s s dp dp dp d d d dp dp Nominal Large prob s dp

32 Localized orbitals: MnO Oxygen centered TM- centered Very clearly left- right correlation!

33 Rho^c for all 5 TM monoxides ScO TiO VO CrO MnO

34 TM dioxides: the angle Kulik & Marzari, J. Chem. Phys. 134, (2011) DMC(SJ) geometries: results DFT DMC(SJ) Exp MnO / /- 5 FeO / /- 10 CoO /

35 TM- O2: electronic structure Singly occupied states Unoccupied states FeO2 doubly occupies CoO2 doubly occupies Doubly occupied states

36 Majority (up) Minority (down)

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38 Posi tive Symmetric left- right correlation Enabled by partial d occupation

39 Special thanks: David Ceperley and his group UIUC Physics department Taub campus cluster NSF XSEDE computer resources