Open-shell and Magnetic Systems with CRYSTAL Tools Klaus Doll Molpro Quantum Chemistry Software Institute of Theoretical Chemistry, D-70569 Stuttgart, Germany MSSC 2018, Turin, September 2018
Contents Introduction Hints for the input Examples
Magnetism Considered here: Ferro-, Antiferro-, Ferrimagnetism typically in periodic systems with: d- or f-electrons but also with p-electrons defects Molecules: molecules with unpaired electrons Questions: Strength : exchange interaction J magnetic moments, spin and charge densities, electrical field gradient dependence on pressure
Types of exchange interaction, I Direct (between neighbouring atoms) Indirect (superexchange): via bridging ion here: antiferromagnetic
Types of exchange interaction, II Double exchange: ferromagnetic exchange interaction which occurs because the magnetic ion can show mixed valency Indirect exchange: around defect, mediated by conduction electrons
Types of exchange interaction, III Itinerant magnetism: 3d metals (Fe, Co, Ni) with partially filled shells different description with models which take into account kinetic energy
Computational methods for periodic systems and analogously KS, UKS (Kohn-Sham)
An atom: oxygen Atomic configuration: 1s2 2s2 2p4 Ground state: 3P2 (Hund s rules) 1) maximise spin: 2 unpaired electrons => spin 1 (2*ħ/2), 2S+1 =3 2) maximise L: (3 up-electrons: Lup=0, 1 down electron, Ldown=1) 1 => P J = L+ S => 2 3) more than half full p-shell: Eigenvalues: S2 : S(S+1) =2 L2 : L(L+1) = 2
O with Molpro (www.molpro.net) Input file, part 1: gprint,basis,orbital basis=vdz geometry={o} {rhf,throrb=0; occ,2,1,1,0,1;closed,2,0,0,0,1 wf,8,4,2 } i=1 meth(i) = 'RHF' e(i)= energy {multi;noextra occ,2,1,1,0,1;closed,2,0,0,0,1;frozen,2,0, 0,0,1 wf,8,4,2 expec2,lxx,lyy,lzz} i=i+1 meth(i) = 'MULTI - 1 state' e(i)= energy part 2 (copy+paste below part 1): {multi occ,2,1,1,0,1;closed,2,0,0,0,0;frozen,0,0, 0,0,0 wf,8,4,2; wf,8,6,2; wf,8,7,2; expec2,lxx,lyy,lzz} i=i+1 meth(i) = 'MULTI-stateaveraged' e(i)= energy {uhf,throrb=0; occ,2,1,1,0,1;closed,2,0,0,0,1 wf,8,4,2 } i=i+1 meth(i) = 'UHF' e(i)= energy table,meth,e;
O with Molpro Output: Energies: METH RHF MULTI - 1 state MULTI-stateaveraged UHF E -74.78751307-74.78751307-74.78618804-74.79216606 RHF: symmetry-broken: 2 pz electrons, 1 px, 1 py electron Not spherically symmetric, no L2 eigenstate! <1.4 L**2 1.4> 2.005004608348 MCSCF state-averaged: 3 states, one with 2 px electrons, one with 2 py, one with 2pz!MCSCF expec <1.4 L**2 1.4> 2.000000000000!MCSCF expec <1.6 L**2 1.6> 2.000000000000!MCSCF expec <1.7 L**2 1.7> 2.000000000000 => spherically symmetric! UHF: symmetry-broken, not an eigenstate of S2 EXPECTATION VALUE OF S**2: 2.00436680
O with CRYSTAL Input file, part 1: Oxygen Atom MOLECULE 1 1 8 0. 0. 0. END 86 0 0 9 2.0 1.0 11720.000000 0.000710 1759.000000 0.005470 400.800000 0.027837 113.700000 0.104800 37.030000 0.283062 13.270000 0.448719 5.025000 0.270952 1.013000 0.015458 0.302300-0.002585 0 0 9 2.0 1.0 11720.000000-0.000160 1759.000000-0.001263 400.800000-0.006267 113.700000-0.025716 37.030000-0.070924 13.270000-0.165411 5.025000-0.116955 1.013000 0.557368 0.302300 0.572759 part 2 (copy+paste below part 1): 0 0 1 0.0 1.0 0.302300 1. 0 2 4 4.0 1.0 17.700000 0.043018 3.854000 0.228913 1.046000 0.508728 0.275300 0.460531 0 2 1 0.0 1.0 0.275300 1.000000 0 3 1 0.0 1.0 1.185000 1.000000 99 0 END UHF ATOMSPIN 1 11 SPINLOCK 23 MAXCYCLE 60 FMIXING 70 PPAN END spin-polarised atom with spin initial spin state (2 more up electrons than down electrons, fix for 3 cycles)
O with CRYSTAL Output: Energies: ATOMIC WAVEFUNCTION(S) NUCLEAR CHARGE 8.0 SYMMETRY SPECIES S P N. ELECTRONS 8.0 NUMBER OF PRIMITIVE GTOS 19 5 NUMBER OF CONTRACTED GTOS 3 2 NUMBER OF CLOSED SHELLS 2 0 OPEN SHELL OCCUPATION 0 4 ZNUC SCFIT TOTAL HF ENERGY KINETIC ENERGY VIRIAL THEOREM ACCURACY 8.0 12-7.478618804E+01 7.478582785E+01-2.000004816E+00 3.1E-06 Identical to Molpro, state-averaged (-74.78618804) == SCF ENDED - CONVERGENCE ON ENERGY Identical to Molpro, UHF (-74.79216606) E(AU) -7.4792165768403E+01 CYCLES 7
Exchange interaction: mathematically Triplet: Singlet: [ ](Φa (1) Φb (2) Φb (1)Φ a (2)) [ ](Φa (1) Φb (2)+Φb (1) Φa (2)) compute energy difference Esinglet-Etriplet fit models: 1 S 2) Heisenberg: H= J ( S Ising: H= J (S1 S 2)
Calculations for solids: only Sz Eigenstate feasible, fit Ising model ferromagnet versus antiferromagnet: energy difference: njsz2 n: neighbors energy scale: ~J (~TN) order of mev total energy: H-atom 13.6 ev ; Ni atom ~ 40000 ev => code must be accurate (little numerical noise)
Keywords general: look at test cases, tutorials for examples Always: UHF or DFT SPIN Geometry: often a SUPERCEL is necessary Usually: ATOMSPIN Often: SPINLOCK (keep number of cycles smaller than number necessary to converge otherwise artificial solution) Sometimes: EIGSHIFT LEVSHIFT MODISYMM
NiO: band gap with various functionals LDA: K. Terakura, T. Oguchi, A. R. Williams, J. Kübler, Phys. Rev. B 30, 4734 (1984) B3LYP: I. Moreira et al, Phys. Rev. B 65, 155102 (2002) Hartree-Fock: M. D. Towler et al, Phys. Rev. B 50, 5041 (1994)
Band gaps: difficult with density functional theory GaAs NiO ZnO TiO2 LDA 0.1 0.2 0.9 1.8 B3LYP 1.5 4.1 3.2 3.4 HF 15.1 11.7 10 Expt. (ev) 1.4 4.0 3.4 3.0 Hartree-Fock: correct self-interaction, but Coulomb-repulsion unscreened LDA: self-interaction error (e.g. hydrogen: 1 electron interacts with itself) B3LYP: mixed Hartree-Fock and LDA, is in between Practical approach: use several methods, compare, this gives approximate result Luckily not all properties are so difficult to compute! Geometries, frequencies, charges usually match well occupied bands, unoccupied bands match well, but not the gap
Superexchange calculation for NiO FM: -43043.1899 ev AF2: -43043.2065 ev energy difference: 0.0166 ev = 3JS2 => J = 5.5 mev exp.: J=20 mev M. D. Towler, N.L. Allan, N.M.Harrison, V.R. Saunders, W. C. Mackrodt, E. Aprà, Phys. Rev. B 50, 5041 (1994) today: ~ 1 minute CPU time for NiO
Superexchange calculation for KMnF3 J=0.1 mev J ~ d-14 exp.:j=0.3 mev d -12 +/- 2 J. M. Ricart, R. Dovesi, C. Roetti, V.R. Saunders, Phys. Rev. B 52, 2381 (1995) R. Dovesi, F. Freyria-Fava, C. Roetti, V. R. Saunders, Faraday Discuss. 106, 173 (1997)
NiO: spin density plot Hartree-Fock magn. moment J Ni O 1.9 0.1-5.4 mev B3LYP 1.8 0.2-29 mev LDA 1.6 0.4-94 mev exp.: -20 mev
Some more examples Hartree-Fock B3LYP LDA exp. NiO -5.5 mev -29 mev -94 mev -20 mev KNiF3-2.6 mev -15 mev La2CuO4-36 mev -130 mev MnF2 0.2 mev 0.05 mev -0.07 mev -0.3 mev J1 J2-9 mev systematic deviation; LDA too delocalized; HF too localized prediction possible for 180º angle
Fe(pyrimidine)2Cl2:a molecule based magnet canted =>weakly ferromagnetic Orientation of the magnetic moment from Mössbauer spectroscopy and calculation (electrical field gradient) R. Feyerherm, A. Loose, T. Ishida, T. Nogami, J. Kreitlow, D. Baabe, F. J. Litterst, S. Süllow, H.-H. Klauss, K. Doll, Phys. Rev. B 69,134427 (2004)
Exchange interaction FePM2Cl2 NiPM2Cl2 J exp. th. (B3LYP) -0.03 mev -0.08 mev -0.25 mev -0.6 mev Increase of J when compressing by 5%: ~15% (Experiment and theory)
Bulk modulus exp.: very soft: 15 GPa calculated: 18 GPa What happens under pressure? a;c Fe-N Fe-Cl N-C C-H 7.0972; 19.840 7.5331; 20.623 2.14 2.26 2.46 2.45 1.34 1.35 1.08 1.09 C-C (Å) 1.38 1.38 A. U. B. Wolter, H.-H. Klauss, F. J. Litterst, T. Burghardt, A. Eichler, R. Feyerherm, S. Süllow, Polyhedron 22, 2139 (2003) J. Kreitlow, D. Menzel, A. U. B. Wolter, J. Schoenes, S. Süllow, K. Doll, Phys. Rev. B 72, 134418 (2005)
Rb4O6: magnetism in p-shell most favorable solution: Rb4(O2-)2 (O22-) O22- (peroxide) nonmagnetic, O2- (superoxide) magnetic moment ~1 μb 1 peroxide, bond length 1.56 Å 2 superoxides, bond length 1.36 Å symmetry becomes I-42d (in case of ferromagnetic order) J. Winterlik, G. H. Fecher, C. A. Jenkins, C. Felser, C. Mühle, K. Doll, M. Jansen, L. M. Sandratskii, J. Kübler, Phys. Rev. Lett. 102, 016401 (2009)
Raman spectra Mn3O4: Raman Tetragonal at room temperature Becomes ferrimagnetic below 42 K red: Oxygen green: Mn, octahedral site (6-fold coordinated) sky blue: Mn, tetrahedral site (4-fold coordinated) simulated as ferromagnetic (all Mn up spin) T. Larbi, K. Doll, T. Manoubi, J. Alloys Compd. 688, 692 (2016)
Raman spectra Mn3O4: ferrimagnetic red: Oxygen green: Mn, octahedral site, up spin dark blue: Mn, octahedral site, down spin sky blue: Mn, tetrahedral site, up spin symmetry reduced in simulation, becomes orthorhombic magnetic states discussed in A. Chartier, Ph. D Arco, R. Dovesi, V. R. Saunders, Phys. Rev. B 60, 14042 (1999)
Symmetry change of modes Mn3O4: Raman http://www.cryst.ehu.es/cryst/gsrelraman.html Space group, high symmetry: 141 Point group: Oh (4/mmm) Point group, low symmetry: mmm
Example for f-elements: GdN LDA: metallic, very good agreement with previous calculations, e.g. P. Larson and W. R. L. Lambrecht, Phys. Rev. B 74, 085108 (2006) B3LYP: only points of band structure touch Fermi surface K. Doll, J. Phys.: Cond. Matt. 20, 075214 (2008)
Conclusion Methods: UHF, spin-polarised DFT magnetism: systematic deviation for exchange interaction J, hybrid functionals give relatively good agreement with experiment vibrational spectra well reproduced unusual magnetism (p-electrons involved)