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Jim Greer

Spin-spin coupling in N@C 60 and P@C 60. Jim Greer. Review computed properties of group V atoms in C 60 Calculations of spin-spin interactions Next steps and calculations. Outline. Computational Methods. B3-LYP Density functional theory (unrestricted)

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Jim Greer

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  1. Spin-spin coupling in N@C60 and P@C60 Jim Greer

  2. Review computed properties of group V atoms in C60 Calculations of spin-spin interactions Next steps and calculations Outline

  3. Computational Methods • B3-LYP Density functional theory (unrestricted) • ROHF Restricted open-shell Hartree Fock theory • VTZ+P Valence TZ basis set with polarization functions. • Charges and densities with DZ+P

  4. Quartet spin ——› Experimental ESR signal Theoretical spin purity N@C60 <S2> = 3.753 P@C60 <S2> = 3.751 Charge transfer Experimental‡ upper limit 0.1 electrons Mulliken analysis N@C60 0.020 electrons transferred to the cage P@C60 0.025 ——"—— Atomic Nature of N and P in C60 Pure spin quartet <S2> = 3.75 ‡A. Weidinger, M. Waiblinger, B. Pietzak, T. Almeida Murphy, Appl. Phys.66 (1998) 259

  5. Calculated Vibrational Frequencies • Ih is a global minimum • Lowest frequency is of t1u symmetry • N@C60 110 cm-1 • P@C60 156 cm-1 • Remaining frequencies corresponds to C60 • N@C60 1.2 cm-1 • P@C60 3.3 cm-1 • Agrees with experimental estimates Vibrational motion of endohedral atom Average deviation to pristine C60

  6. P- P C60 C60+ P@C60 0 9t1u (LUMO) -5 6hu (HOMO) Energy [eV] -10 3p -15 7ag 3s -20 -25 Energy levels for P, C60, and P@C60

  7. Unchanged C60 energy levels • Except 7ag in P@C60 which is shifted up by 761 meV • The corresponding shift in N@C60 is 69 meV • Average deviation from C60 • N@C60 5 meV • P@C60 30 meV

  8. Contraction of the endohedral atoms’ electron density • Larger hyperfine interaction • 50% increase for N@C60 • 150% for P@C60 • Calculated difference density ρ(P@C60) - [ρ(C60) + ρ(P)]

  9. Experiment and theory Exothermic decay at moderate temperatures Pure quartet spin No charge transfer Charge contraction Theoretical details Buried open-shell orbitals Slight lowering of endo-hedral energy levels Destabilization of highest occupied ag orbital ——› Trapped Model of Bonding in N and P@C60

  10. Atomic Nitrogen Exp Calc 2D 2.384 2.624 2P 3.576 3.805 Endohedral Nitrogen Calculated 2.867 eV 3.968 eV Lower tolerance due to system size

  11. z y

  12. S. Melchor, J. Dobado, J.A. Larsson and J.C. Greer, JACS 2003

  13. P@C60 Ar@C60 N@C60 Ne@C60

  14. C60 is a dielectric shell Electric field shielding is 70 to 80% Not significantly influenced by charge transfer from endohedral dopants (1e- Li, 3 e- La) Hence long decoherence lifetimes! Shielding

  15. Review computed properties of group V atoms in C60 Calculations of spin-spin interactions Next steps and calculations Outline

  16. Difficulties converging unrestricted DFT calculations Two open shells (more in some cases of reduced symmetry) Buried open shells Small energy differences Very large basis sets

  17. The Full CI wavefunction Contains all possible Slater Determinants (SD), or more generally Configuration State Functions (CSF) with a given spin and symmetry that can be generated for a certain basis set. It is often written with reference to the Hartree-Fock wavefunction Or simply Configuration Interaction

  18. - antisymmetrization operator εP is the sign of the permutation P SDs are not necessarily pure eigenstates of but will also mix in other values of s. This is rectified by the use of CSFs, through the use of the spin-projection operator where k = N/2, (N/2)-1,…,0 or 1/2 except for k = s.

  19. Exact spin coupling – no spin contamination, no need for messy open shell coupling coefficients Slight Jahn-Teller distortion Ih to D2h STO-3G basis set on carbon atoms (small!) aug-cc-pVTZ basis set on nitrogen, phosphorus (large!) Perform neutral N,P@C60 Hartree-Fock calculation, occupy virtual (diffuse) state in many body state, then apply spin projector E = E(;) – E(;), negative implies naïve Hund’s rule rules

  20. Spin-spin coupling in N@C60 Exact spin coupling with a single CSF E = -6 meV

  21. Spin-spin coupling in P@C60 Exact spin coupling with a single CSF E = -3 meV

  22. Spin-spin coupling in N@C60 Exact spin coupling with correlation (approx. 1000 CSFs) E = +134 meV

  23. Spin-spin coupling in P@C60 Exact spin coupling with correlation (approx. 1000 CSFs) E = -455 meV

  24. Single state calculations show very weak spin-spin coupling, favor parallel spins, and coupling is stronger in N@C60 Correlated calculations indicate 1 to 2 orders of magnitude stronger coupling, however coupling changes sign between N@C60 and P@C60, and is stronger for P@C60 Uh Oh!

  25. Review computed properties of group V atoms in C60 Calculations of spin-spin interactions Next steps and calculations Outline

  26. Correlation is important Need to fully converge (many more CSFs) calculations to insure a balanced treatment between spin up and down Increase in coupling strength with correlation and change in sign very interesting- if true! Need to look at role of 7ag in P@C60 Check role of Jahn-Teller in correlated calculations Compare with another computational method Coupling calculations

  27. Next: Bardeen Tunnel Matrix Elements M1 M2 • Transisition matrix elements give electron jump probability • Need to resolve issues with basis sets and correlation for molecular systems (fullerenes, contacts) • Matrix elements to be computed at the same level (basis sets, # of CSFs) as spin-spin coupling

  28. p Left reservoir | Device region | Right reservoir Quantum Transport (r1,r2,…,rn)  f(q,p) QM  device modelling

  29. Solution: the Wigner function f(q,p) of Y(r1,r2,…,rN). f(q,p) -- phase space picture of quantum mechanics <Y, T Y> =  p2/2m f(q,p) dq dp Used previously for scattering bc for single-electron wavefunction transport with phonons Right physics at this level of theory

  30. Incoming electrons have a distribution f(p) fixed by the nature of the contacts Constrain incoming electrons

  31. Incoming electrons have a distribution f(p) fixed by the nature of the contacts The molecule chooses to reflect or transmit as many as it wants to minimise the total energy <Y,H Y> Vary Y while keeping incoming electrons fixed: constrained minimisation problem At zero bias, I = (h/2im) [ Y* Y - Y Y*] = 0 Apply a finite bias H = H0 + U(x,y,z) = H0 + e E z obtain nonzero I

  32. Apply a finite bias H = H0 + U(x,y,z) = H0 + e E z Get Y which minimises < Y | H | Y > subject to Normalisation < Y | Y > =1 Grid of Wigner constraints < Y | Fi | Y > = f i , i = 1,n Calculate I(E) = (h/2im) [ Y* Y - Y Y*] : (V,I) Iterate for other applied bias values

  33. Experiment Our recent result Problem common to all previously applied theoretical methods: Magnitude of the current is two to three orders of magnitude too high

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