Magnetic Properties: NMR, EPR, Susceptibility

Transcription

1 Magnetic Properties: NMR, EPR, Susceptibility Part : Selected 5f 1 systems Jochen Autschbach, University at Buffalo, jochena@buffalo.edu J. Autschbach Magnetic Properties 1

2 Acknowledgments: Funding: Current and former graduate students: Ben Pritchard: paramagnetic NMR, EPR studies, Molcas developments new group members: Robert Martin, Alex Marchenko, Tom Duignan: paramagnetic NMR, actinyl species in solution, misc. Support from: Center for Computational Research, SUNY Buffalo Current and former postdocs: Frederic Gendron: f -element studies Fredy Aquino, Prakash Verma: NWChem relativistic magnetic property modules Kamal Sharkas: Molcas developments Collaborators: Boris LeGuennic, Helene Bolvin: f -element projects N. Govind, W. A. de Jong, B. McNamara, H. Cho: NMR & other magnetic properties D. Peng, M Reiher: XC implementation in NWChem S. Patchkovskii: paramagnetic NMR J. Autschbach Magnetic Properties

3 EPR parameters EPR calculations. Example: Kramers doublet φ, φ Zeeman Hamiltonian (g e, neglect rel. corrections to operator) h Z = h(b) h(field-free) = β e (L + S) B It follows a (u {x, y, z}): g ux = Re φ L u + S u φ = Re φ L u + S u φ g uy = Im φ L u + S u φ = Im φ L u + S u φ g uz = Re φ L u + S u φ = Re φ L u + S u φ Similar idea used for SO DFT calculations by van Lenthe et al. (LWA) b Gerloch & McMeeking, Bolvin et al.: c (gg T ) u,v = α L u + S u β β L v + S v α α,β=φ, φ Equivalent strategy for hyperfine coupling (hfc), replacing h(b) by h(m A ) a Abragam & Bleaney (1970). b E. van Lenthe et al., JCP 107 (1997), 488. c H. Bolvin, ChemPhysChem 7 (006), J. Autschbach Magnetic Properties 3

4 EPR parameters Let s first look at some DFT calculations Reminder: LR = linear response. g-factors and hfc calculated via derivative techniques. SO coupling treated as a perturbation to first order. Runs on top of spin-unrestricted (spin polarized) scalar relativistic DFT LWA = (van Lenthe, Wormer, van der Avoird) quasi spin restricted DFT calculations with SO coupling variationally included in the ground state calculation. No spin polarization MA = (magnetic anisotropy) generalized-collinear DFT calculations with different orientations of the spin quantization axis, SO coupling variationally included in the ground state calculation. Includes spin polarization (and spin contamination) implementations available in the ADF and NWCHEM packages J. Autschbach Magnetic Properties 4

5 Actinide hexa-halides [AnX6 ] n electron g-factors: Linear response (SO = perturbation) a ZORA DFT / LR (GIAO) NpF6 UF 6 UCl6 PBE PBE0 Expt Expectation value (SO = variational) b ZORA DFT / LWA (GIAO) NPF6 scalar ZORA SOMO NpF6 UF 6 UCl 6 a b PBE PBE0 Expt B. Pritchard, JA, Theor. Chem. Acc. (P. Pyykko issue), 19 (011), 453 NWChem implementation of LWA method, P. Verma, JA, JCTC 9 (013), 105 J. Autschbach Magnetic Properties 5

6 5, 5 5, 3 5, 1 5, 1 5, 3 5, 5 7, 7 7, 5 7, 3 7, 1 7, 1 7, 3 7, 5 7, 7 5, 5 + ξ ξ ξ , 3 0 ξ ξ ξ , ( 6 + ξ) ξ ξ 3 3 5, ( 6 + ξ) ξ ξ , 3 5ξ ξ ξ ξ , ξ ξ , , ξ ξ , ξ ξ ( 11 + ξ) ξ 0 7, ξ ξ , 1 5ξ ξ , 3 5 ξ 4 10ξ 5 3ξ ( 11 + ξ) 0 0 7, ξ ξ ξ , ξ ξ 11 3 Actinide hexa-halides Other DFT methods for EPR calculations: Variational treatment of SO coupling in EPR calculations using generalized collinear SO DFT with spin quantization direction u ( MA approach a ) g uv = 1 β e S a A,uv = g A β N S i n i i n i B v ϕ u i (B) F(B) ϕ u i (B) m A,v ϕ u i F(m A ) ϕ u i (using GIAOs) HF PBE g / ppt NpF SO scaling factor λ g-shift (ppt) with scaled SO integrals (ZORA). Expt. = -600 (g = 0.6) H(CF+SO) = ζ 5ξ 5ξ 5ξ 3 ξ ξ ξ ξ 3 35ξ 3 ξ ξ 35ξ ξ 0 g iso as a function of SO splitting ζ and CF splitting in O h symmetry a P. Verma and JA, JCTC 9 (013), p. 105 & p See also Schmitt, Jost, & van Wüllen, JCP 134 (011), J. Autschbach Magnetic Properties 6 35ξ

7 Actinide hexa-halides MA calculations with DFT AnX 6 EPR data: ZORA DFT / LC-PBE0, Expt. in parentheses: g A metal (MHz) A ligand (MHz) NpF (-0.6) -00 (-1994) -51 (-7) UF (-0.8) UCl (-1.1) 19-7 P. Verma, JA, JCTC 9 (013), 105 & 193. Analysis of A Np (MHz) ZORA linear response results for NpF 6 : total PSO SO term (purely relativistic) from Np 5f xyz SOMO F. Aquino, B. Pritchard, JA, JCTC 8 (01), 598. PBE0 functional. J. Autschbach Magnetic Properties 7

8 Actinide hexa-halides CAS-SO based analysis, state component ψ: (dτ : over all but one electron coordinate; u = x, y, or z) ρ(r) = N ψ ψ dτ m u (r) = ψ S u (i)ψdτ j u (r) = i i ψ [ u (i) u (i)] ψdτ m u (r) = p nu p[ϕ u p(r)] 1 ; s(r) = mx + m y + m z Electron density Spin magnetization density Orbital current density N = ρ(r)dv Electron number from ρ S u = 1 mu (r)dv Spin expectation value from m u L u = [r j] u dv Angular momentum from j ρ(r) = p nρ p [φρ p (r)] ; p nρ p = N Natural Orbitals for density p nu p = S u Natural orbitals for spin j u (r) = i p nj p φj p [ u u ] φ j p Non-collinear spin density Per-orbital current density implemented in a Molcas developer s version; B. Pritchard, PhD thesis (014) J. Autschbach Magnetic Properties 8

9 Actinide hexa-halides CAS-SO density and spin density analysis for NpF 6 ' Z 1 W C0:711 'Z W C0:085 'Z 3 W C0:056 'Z 4 W C0:046 'Z 5 W C0:046 ' Z 96 W 0:041 'Z 97 W 0:041 'Z 98 W 0:049 'Z 99 W 0:087 'Z 300 W 0:087 hl i = 1:00 hs i = C0:3 g = 0:7 m x m y m z qm x C m y C m z J. Autschbach Magnetic Properties 9

10 Electronic structure and magnetic properties of neptunyl(vi) complexes For details see F. Gendron et al., Chem. Eur. J. 0 (014), Eisenstein & Pryce, Proc. R. Soc. London A 9 (1955), J. Autschbach Magnetic Properties 10

11 Neptunyl (5f 1 ) Complexes NpO + [NpO (NO 3 ) 3 ] - NpO Cl 4 - Impurities in analogous uranyl (UO + ) host crystals: RbUO (NO 3 ) 3 Cs UO Cl 4 Experimental Ground State g-factors: g = 3.46(4) g = 0.0(0) [NpO (NO 3 ) 3 ] - NpO Cl 4 - g = 1.3 g = 1.30(30) Leung et al. Phys. Rev. 1969, 180, 380, Bleaney et al. Phil. Mag. 1954, 45, 99. J. Autschbach Magnetic Properties 11

12 Gas-phase Structure Optimizations: - DFT Optimizations performed with NWChem - Functional B3LYP - Effective Core Potential + ECP60MWB-SEG valence basis set for Np - aug-cc-pdvz basis set for ligand atoms. DFT (B3LYP) g-factors: VS. Exp: g = 3.36(4) g = 0.0(0) LWA: non- spin-polarized approach MA: spin-polarized Random g-factors with DFT? g = 1.3() g = 1.30(30) Need a model to rationalize the g-factors Spin-Orbit Coupling: Crystal Field: J. Autschbach Magnetic Properties 1

13 5f M L = 0 5f M L = 1 5f M L = 5f M L = 3 Np(VI) = 5f 1 Hund s Rules: L = 3, S = ½, J = L+S = 7, J = L-S = 5 g = 3.36(4) g = 0.0(0) Importance of the Crystal Field g = 1.3() g = 1.30(30) Refer to handout re. definitions of the j, m j spinors and matrix elements for L, S, SO J. Autschbach Magnetic Properties 13 1/

14 /u Φ7/u Relative Energy (cm -1 ) u Φ u 3/u Φ5/u 3/u 100δ 88φ + 1δ Φ 5/u E 1/ E1/ 78φ + 17φ1 + 4δ E1/u E1/u 50φ φ 5/u 88δ + 1φ 100φ Φ7/u Φ = A' E1/ E1/ 37δ1 + 37δ + 6φ1 Φ = Eu E3/u 43δ + 36δ 1 + 0φ E3/u Φ 1 = A' 1 = E'' E 1/ E 3/ E 3/ E1/ = Bu 1 = B 1u 50δ1 + 50δ 57φ 1 + 1φ + 1δ E3/u E3/u E3/u 56δ δ + 10φ 34φ φ + 31δ E 3/u Dinf h D * inf h D 3h D * 3h D 4h D * 4h SF SO 1st order SO nd order SF SO 1st order SO nd order SF SO 1st order SO nd order NpO + [NpO (NO 3 ) 3 ] - [NpO Cl 4 ] - Energies and assignments of low-energy electronic states according to CAS(1,4)SCF (ROHF, basically). CAS(7,10)SCF and PT gives the same state ordering and similar magnetic properties. J. Autschbach Magnetic Properties 14

15 Free Neptunyl NpO + : D h Symmetry Molecular Diagram: Crystal Field Splitting: NpO + u m l = u m l = 3 Crystal Field Hamiltonian: SO coupling mixes with Basis set for Hamiltonian Model: and J. Autschbach Magnetic Properties 15

16 Model Hamiltonian: H m = H SO + H CF Λ = splitting of δ and φ 5f orbitals due to linear CF ζ = SO coupling constant λ = Λ/ζ define H m /ζ in l, m l basis (from 6th order terms in CF) H m /ζ 3, + 1, 1 3, + 1 3, λ Eigenfunctions = Kramers doublet ψ, ψ transforming as spin-1/ pseudo-spin eigenfunctions,, projection along z: g = ψ L z + S z ψ g = Re ψ L x + S x ψ Time reversal: K ψ = ψ ; K ψ = ψ Zeeman Hamiltonian: β e (L + S) B (nonrelativistic, but commonly used) J. Autschbach Magnetic Properties 16

17 Solving the eigenvalue problem gives real coefficients A, B with A + B = 1. In the basis 5, 5, 7, 5, we have coefficients a, b. One finds for the g-factors: NpO + ψ = A 3, B, 1 g = (4A + 6B ) = a 5, 5 + b 7, 5 = ( 30 7 a b ab) ψ = A 3, 1 B, + 1 g = 0 = a 5, + 5 b 7, + 5 Choice of ψ based on the negative sign of g for [NpO (NO 3 ) 3 ] (see below). At the spinor level, the CF mixes j = 5/ with j = 7/. Due to linear symmetry, m j remains a good quantum number. J. Autschbach Magnetic Properties 17

18 NpO + : g versus strength and sign of the CF ±g 5.0 NpO Λ Ζ from ab-initio data: (CAS(1,4)SCF) ζ 36 Λ 135 A B a b g 4.18 g Λ = 0: g = 30/7 = 4.9 corresponding to 5, 5 Λ + : Φ with L, S opposite sign Λ + : with L, S same sign ab-initio: g = 4.40 (from CAS(7,10)PT) J. Autschbach Magnetic Properties 18

19 Nature of the Ground State: Natural Spin Orbitals NpO + NSO u : + contrib. to < S z > NSO u : - contrib. to < S z > Single-State CAS(7,10)PT-SO: = 1183 cm -1, = 38 cm -1 g ıı = 4.40 GS = 88% + 1% Using the CF Model: g ıı = 4.4 A = 0.94 and B = GS = 89% + 11% J. Autschbach Magnetic Properties 19

20 Crystal Field [NpO (NO 3 ) 3 ] - : D 3h Symmetry Crystal Field Splitting: = A' A m l = +/-3 [NpO (NO 3 ) 3 ] - u u 1 = A' 1 = E'' A 1 m l = +/-3 D inf h SF D 3h SF E m l = +/-3 Two phenomenological CF parameters: Λ = splitting of δ and φ 5f orbitals due to axial CF (now negative) Γ = splitting of the degeneracy of φ 5f orbitals due to equatorial CF J. Autschbach Magnetic Properties 0

21 Model Hamiltonian: H m = H SO + H CF Λ = splitting of δ and φ 5f orbitals due to axial CF (now negative) Γ = splitting of the degeneracy of φ 5f orbitals due to equatorial CF ζ = SO coupling constant λ = Λ/ζ, γ = Γ/ζ define H m /ζ in l, m l basis as H m /ζ 3, + 1, 1 3, γ 3, + 1 3, λ 0 3, γ 0 3 Compared to NpO +, there is an additional term in ψ from 3, + 1, to describe the CF splitting of the φ orbitals and the corresponding (partial) quenching of the orbital angular momentum. At the spinor level, the eq. CF mixes 7, + 7 with 5, 5, 7, 5 because m j ceases to be a good quantum number J. Autschbach Magnetic Properties 1

22 Solving the 3 3 eigenvalue problem gives real coefficients A, B, C with A + B + C = 1. In the basis 5, 5, 7, 5, 7, + 7, we have coefficients a, b, c. One finds for the g-factors: [NpO (NO 3 ) 3 ] ψ = A 3, B, 1 + C 3, + 1 g = 4A 6B + 8C = a 5, 5 + b 7, 5 + c 7, + 7 = 30 7 a 40 7 b ab + 8c ψ = A 3, 1 B, + 1 C 3, 1 g = (4AC + 6BC) = a 5, + 5 b 7, c 7, 7 = ( 6 ac + 16 bc) 7 7 Experimentally, the sign of the product, g 1 g g 3, of the principal g-factors can be measured. For [NpO (NO 3 ) 3 ], it is negative. a g is negative a M. H. L. Pryce, Phys. Rev. Lett. 3 (1959), 375. J. C. Eisenstein, M. H. L. Pryce, J. Res. Natl. Bur. Stand. Sect. A 69 (1965), 17. J. Autschbach Magnetic Properties

23 1 0 1 ±g Λ 0 Λ 5 3 NpO NO3 3 4 Λ 10 Neptunyl(VI) 5 Λ Γ Ζ ±g 0.5 NpO NO Λ 10 Λ 5 Λ 0 Λ Γ Ζ from ab-initio data: (CAS(1,4)SCF) [NpO (NO 3 ) 3 ] ζ 399 Λ -697 Γ 4588 A B C a 0.97 b c g g 0.46 ab-initio: g = 3.75, g = 0. (from CAS(1,4)SCF) g = 3.49, g = 0.3 (from CAS(7,10)PT) expt. g = 3.405(8), g = 0.05(6) (CF model improves when π orbitals are included and a second set of CF parameters is introduced) J. Autschbach Magnetic Properties 3

24 [NpO (NO 3 ) 3 ] natural orbitals for spin-z magnetization: Single-State CAS(7,10)PT-SO: = -396 cm -1, = 5610 cm -1, = 38 cm -1 g ıı = and g = 0.30 GS = 80% + 0% Using the CF Model: g ıı = -3.4 and g = A = 0.86, B = and C = -0.1 GS = 81% + 19% Spin Magnetization Component m Z : < L ıı > = -.346, < S ıı > = < L > = 0.317, < S > = J. Autschbach Magnetic Properties 4

25 Crystal Field NpO Cl 4 - : D 4h Symmetry Crystal Field Splitting: = A' = E u NpO Cl 4 - u 1 = A' 1 = B u u = E'' 1 = B 1u D inf h SF D 3h SF D 4h SF Two phenomenological CF parameters: Λ = splitting of δ and φ 5f orbitals due to axial CF (< 0 as for [NpO (NO 3 ) 3 ] ) Γ = splitting of the degeneracy of δ 5f orbitals due to equatorial CF J. Autschbach Magnetic Properties 5

26 Model Hamiltonian: H m = H SO + H CF Λ = splitting of δ and φ 5f orbitals due to axial CF (now negative) Γ = splitting of the degeneracy of δ 5f orbitals due to equatorial CF ζ = SO coupling constant λ = Λ/ζ, γ = Γ/ζ define H m /ζ in l, m l basis as H m /ζ 3, 1, + 1, , 1 3 ε, λ γ, + 1 ε 1 γ 1 + λ Compared to NpO +, there is an additional term in ψ from, + 1, to describe the CF splitting of the δ orbitals Also, there is an apparent coupling εζ between 3, 1 and, + 1 because we have left out the remaining 5f orbitals. J. Autschbach Magnetic Properties 6

27 Solving the 3 3 eigenvalue problem gives real coefficients A, B, C with A + B + C = 1. One finds for the g-factors: [NpO Cl 4 ] ψ = A 3, 1 + B, C, + 1 g = 4A + 6B C = 30 7 a b ab c ψ = A 3, B, 1 + C, 1 g = (4BC + 6AC) = ( 8 ac bc) 7 7 Note: Based on the transformation properties of a spin-1/ under symmetry operations according to the D 4h double group, and based on the structure of the magnetic moment matrices obtained from the ab-initio calculations versus those of the pseudo-spin Zeeman Hamiltonian, g of [NpO Cl 4 ] is assigned to be positive. J. Autschbach Magnetic Properties 7

28 4 3 Λ 0 Λ 5 Λ 10 1 Λ 5 NpO Cl4 ±g 0 Λ Γ Ζ Λ Λ 5 ±g Λ 0 Λ 5 Λ NpO Cl Γ Ζ from ab-initio data: (CAS(1,4)SCF) [NpO Cl 4 ] ζ 395 Λ Γ 867 εζ -15 A B C 0.30 a b c 0.30 g g ab-initio: g = 3.47, g = 0.9 (from CAS(1,4)SCF) g = 3.7, g = 1.7 (from CAS(7,10)PT) expt. g = 1.3(), g = 1.30(30) J. Autschbach Magnetic Properties 8

29 [NpO Cl 4 ] Free Complex Ion Neptunyl(VI) Single-State CAS(7,10)PT-SO: g ıı = 3.7 and g = 1.73 GS = 67% + 33% Expectation Values: < L ıı > = 1.984, < S ıı > = < L > = 0.961, < S > = Free Complex: Ground State mainly Φ in character J. Autschbach Magnetic Properties 9

30 [NpO Cl 4 ] Embedded Complex Ion Neptunyl(VI) Single-State CAS(7,10)PT-SO: g ıı = and g = GS = 44% + 55% Expectation Values: < L ıı > = 0.860, < S ıı > = < L > = 1.11, < S > = -0.9 Embedded Ion Complex: Ground State mainly Δ in character J. Autschbach Magnetic Properties 30

31 Neptunyl(VI) NpO Cl4 Experimental data from complex embedded in uranyl host crystal, Rb counter ions near-degeneracy of and Φ states equatorial CF splits degeneracy of 5fδ orbitals axial yl-oxygens split degeneracy between fδ and fφ [NpO Cl4 ] expt. CASPT gas phase CASPT embedded gk g 1.3() (30) A tough problem for approximate KS-DFT... F. Gendron, D. Paez-Hernandez, F. P. Notter, B. Pritchard, H. Bolvin, JA, Chem. Eur. J. 0 (014), J. Autschbach Magnetic Properties 31

32 Actinyl-tris-carbonate complexes (isostructural with nitrate) Environmentally important NMR data available Using Bertini / Bleaney equations for axial system: 3cos Θ δ dip = 1 1πr χ 3 ax (3 S(S + 1) cos θ 1) ; χ ax = χ χ ; χ i = µ 0 µ B g i 3kT assumption: small contact spin densities at carbons ( note: this assumes a point magnetic ion) g > g negative dipolar eq. 13 C shifts Π 4 Θ F. Gendron, B. Pritchard, H. Bolvin, JA, Inorg. Chem. 53 (014), J. Autschbach Magnetic Properties 3

33 Actinyl-tris-carbonate complexes a U(V): GS of parentage, Φ very low T /K g g δ dip ( 13 C) Expt. PNMR shift relative to free carbonate: a -6 ppm Np(VI): GS of parentage; Φ low T /K g g δ dip ( 13 C) Expt. PNMR shift relative to free carbonate: b -94 ppm a Mizuoka, Grenthe, Ikeda (005). b Clark, Hobart, Neu (1995). F. Gendron, H. Bolvin, JA, Inorg. Chem. 53 (014), J. Autschbach Magnetic Properties 33