Shruba Gangopadhyay 1,2 & Artëm E. Masunov 1,2,3 1 NanoScience Technology Center

1. First Principle Simulations of Molecular Magnets: Hubbard-U is Necessary on Ligand Atoms for Predicting Magnetic Parameters • Shruba Gangopadhyay1,2 & Artëm E. Masunov1,2,3 1NanoScience Technology Center • 2Department of Chemistry • 3Department of Physics • University of Central Florida Quantum Coherent Properties of Spins - III

3. In this talk • Molecular Magnet as qubit implementation • Use of DFT+U method to predict J coupling • Benchmarking Study • Two qubit system: Mn12 (antiferromagnetic wheel) • Spin frustrated system: Mn9 • Magnetic anisotropy predictions • Future plans

5. Molecular Magnets – possible element in quantum computing • Advantages of Molecular Magnets • Uniform nanoscale size ~1nm • Solubility in organic solvents • Readily alterable peripheral ligandshelps to fine tune the property • Device can be controlled by directed assembly or self assembly It can be in |0> and |1> state simultaneously Molecular Magnet is promising implementation of Qubit • Utilize the spin eigenstates as qubits • Molecular Magnets have higher ground spin states Leuenberger & Loss Nature 410, 791 (2001)

6. 2-qubit system: Molecular Magnet [Mn12(Rdea)] contains two weakly coupled subsystems M=Methyl diethanolamine M=allyldiethanolamine Subsystem spin should not be identical

7. Ion substitution may be used to redesign MM Cr7Ni Molecular Ring Cr8 Molecular Ring • [1] M. Affronte et al., Chemical Communications, 1789 (2007). • [2] M. Affronte et al., Polyhedron 24, 2562 (2005). • [3] G. A. Timco et al., Nature Nanotechnology 4, 173 (2009). • [4] F. Troiani et al., Phys Rev Lett94, 207208 (2005).

8. To redesign MM we need reliable method to predict magnetic properties • Ferromagnetic (F) – when the electrons have Parallel spin • Antiferromagnetic (AF) – having Antiparallel spin Heisenberg-Dirac-Van Vleck Hamiltonian J = exchange coupling constant Si= spin on magnetic center i • J>0 indicates antiferromagnetic (anti-parallel ) ground state • J < 0 indicates ferromagnetic (parallel) ground state

9. (1) (2) Density Functional Theory (DFT)prediction of J from first principles Electronic density n(r)determines all ground state properties of multi-electron system. Energy of the ground stateis a functional of electronic density: Hohenberg-Kohn functional Kohn-Sham equations Where are KS orbitals, is the system of N effective one-particle equations

10. Energy can be predicted for high and low spin states • Density Functional Theory (DFT) • E=E[ρ] • to simplify Kinetic part, total electron density is separated into KS orbitals, describing 1e each: • Electron interaction accounted for self-consistently via exchange-correlation potential

11. Hybrid DFT and DFT+U can be used for prediction of J Pure DFT is not accurate enough due to self interaction error • Broken Symmetry DFT (BSDFT) – Hybrid DFT (The most used method so far) • Unrestricted HF or DFT • Low spin –Open shell  (spin up) β (spin down) are allowed to localized on different atomic centers Representation of J in Broken symmetry terms is now E(HS) - E(BS) = 2JS1S2 • Another alternative for Molecular Magnet DFT+U

12. DFT+U may reduce self-interaction error U “on-site” electron-electron repulsion From fixed-potential diagonalization (Kohn-Sham response) From self-consistent ground state (screened response) We used DFT+Uimplemented in Quantum Espresso The +U correction is the one needed to recover the exact behavior of the energy. What is the physical meaning of U?

13. Both metal and ligand need Hubbard term U Idea: Empirically Adjust U parameter on both Metal and the coordinated ligand Complex –Ni4(Hmp) U parameter on Oxygen not only changing the numerical result It is changing the nature of splitting – preference of ground state C. Cao, S. Hill, and H.-P. Cheng, Phys. Rev. Lett. 100 (16), 167206/1 (2008)

14. Numeric values of U parameters for different atom types are fitted using benchmark set U (Mn)=2.1 eV,U(O)=1.0 eV, U(N)=0.2 eV

15. (Mn(IV))2 (OAc) Computational Details Cutoff 25 Ryd Smearing Marzari-Vanderbilt cold smearing Smearing Factor 0.0008 For better convergence Local Thomas Fermi screening [Mn2(IV)(μO)2((ac))(Me4dtne)]3+ Evaluation of J(cm-1) We modify the source code of Quantum ESPRESSO to incorporate U on Nitrogen

16. Mn(IV)- no acetate bridge Evaluation of J(cm-1) [Mn2 (IV)(μO)2 (phen)4]4+

17. Mn(III) two acetate bridges Mn(II) three acetate bridges [Mn2(II) (ac)3(bpea)2]+ [Mn2(III) (μO)(ac)2(tacn)2]2+ Evaluation of J(cm-1)

18. Mixed valence Mn(III)-Mn(IV) [Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+

19. Löwdin population analysis • The oxide dianions (Oµ), and aliphatic N atoms pure σ-donors- have spin polarization opposite to that of the nearest Mn ion, in agreement with superexchange • The aromatic N atoms have nearly zero spin-polarization. • O atoms of the acetate cations have the same spin polarization as the nearest Mncations. This observation contradicts simple superexchange picture and can be explained with dative mechanism. • The acetate has vacant π-orbital extended over 3 atoms, and can serve as π-acceptor for the d-electrons of the Mncation. As a result, Anderson’s superexchange mechanism, developed for σ-bonding metal-ligandinteractions, no longer holds.

20. Dependence of J on U

21. Failure of BSDFT • Bimetallic complexes with Acetate Bridging ligand • Complexes with Ferromagnetic Coupling • Mix valence complexes

22. Two qubit system-[Mn12(Reda)] complex with weakly coupled subsystems Methyl diethanolamine Allyldiethanolamine Predict J for two coupled sub system Previous DFT Study predicted J=0 Whereas the J>0 experimentally

25. Spin frustrated system –Mn9 Experimental Spin Ground state S = Molecules can be divided into two identical part passing through an axis from Mn+2 The Only Possible Combination if one Mn+3 from each half shows spin down orientation

26. 5 J4 J4 3 7 J1 J1 1 9 J6 J6 J5 J5 J2 J7 J7 J2 J8 4 6 J3 J3 8 2 S=-2(Mn+3) S=2 (Mn+3) S=5/2(Mn+3)

27. Anisotropy –in Molecular Magnet Relativistic Pseudopotential • Resulting from spin–orbit coupling, • Produces a uniaxial anisotropy barrier • Separating opposite projections of the • spin along the axis Non-Collinear Magnetism

28. Prediction of Anisotropy for Ce based Complex Jexpt=-0.75 cm-1, Dexpt= 0.21 cm-1

29. Summary • To predict correct Jvalues we need to include U parameters on both metal and ligand • Geometry Optimizationof ground state is extremely important for correct prediction of J values • Exclusion of U Parameterson ligand atoms leads incorrect ferromagneticground state • Anisoptropyprediction needs relativistic pseudopotential • For Anisotropy we need good starting wave function for ground spin state of the molecule

30. Future Work • Prediction of Anisotropy for Mn12 based wheel • Heisenberg Exchange constants • Ion substituted Mn12 wheel • Mn12 cation/anion • Mn12 wheel on the metal surface

31. Acknowledgements • Prof. Michael Leuenberger • Eliza Poalelungi • Prof. George Christou • Arpita Pal • NERSC Supercomputing Facilities (m990) • ACS Supercomputing Award for Teragrid

32. tunneling from macroscopic world to quantumland through the rabbit hole

33. Questions & Suggestions

35. Pseudopotential • Pseudopotentials replace electronic degrees of freedom in the Hamiltonian of chemically inactive electron by an effective potential • A sphere of radius (rc) defines a boundary between the core and valence regions • For r ≥ rcthe pseudopotential and wave function are required to be the same as for real potential. • Pseudopotentialexcludes (does not reproduce) core states – solutions are only valence states • Inside the sphere r ≤ rc, pseudopotential is such that wave functions are nodelessεi(at) = εi(PS) For Iron 1s2 2s2 2p6 3s2 3p63d6 4s2

36. Faliure of bs-dft • Bimetallic complexes with Acetate Bridging ligand • Complexes with Ferromagnetic Coupling • Mix valence complexes

37. Different transition metals in molecular magnets

38. J for other transition metal complexes

40. Application- biocatalysis Polyneuclear – Transition metal centers in the enzyme Important for biocatalysis -Understand the stability of biradical at transition state S Sinnecker, F Neese, W Lubitz, J BiolInorgChem (2005) 10: 231–238

41. DFT+U in Quantum Espresso The formulation developed by Liechtenstein, Anisimov and Zaanen, referred as basis set independent generalization • n(r) is the electronic density • the atomic orbital occupations for the atom I experiencing the “Hubbard” term • The last term in the above equation is then subtracted in order to avoid double counting of the interactions contained both in EHub and, in some average way, in ELDA.

42. Future Plans • Compute J for heteroatom (Cr) containing molecular magnetic wheel

43. Alternative Approach: DFT+U • The DFT+U method consists in a correction to the LDA (or GGA) energy functional to give a better description of electronic correlations. It is shaped on a Hubbard-like Hamiltonian including effective on-site interactions • It was introduced and developed by Anisimov and coworkers (1990-1995) Advantages Over Hybrid DFT • Computationally less expensive • Possibility to treat large systems