Report

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 2 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 3 4 Molecular Magnets – possible element in quantum computing Advantages of Molecular It can be in |0> and |1> state simultaneously Magnets 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) Uniform nanoscale size ~1nm Solubility in organic solvents Readily alterable peripheral ligands helps to fine tune the property Device can be controlled by directed assembly or self assembly 5 2-qubit system: Molecular Magnet [Mn12(Rdea)] contains two weakly coupled subsystems M=Methyl diethanolamine M=allyl diethanolamine Subsystem spin should not be identical 6 Ion substitution may be used to redesign MM Cr8 Molecular Ring [1] [2] [3] [4] Cr7Ni Molecular Ring M. Affronte et al., Chemical Communications, 1789 (2007). M. Affronte et al., Polyhedron 24, 2562 (2005). G. A. Timco et al., Nature Nanotechnology 4, 173 (2009). F. Troiani et al., Phys Rev Lett 94, 207208 (2005). 7 To redesign MM we need reliable method to predict magnetic properties H Magnetic H Heisenberg H Anisotropy H Zeeman Heisenberg-Dirac-Van Vleck Hamiltonian ˆ H JS 1 S 2 HDVV E ( ) E ( ) J 2 J = exchange coupling constant Si= spin on magnetic center i Ferromagnetic (F) – when the electrons have Parallel spin Antiferromagnetic (AF) – having Antiparallel spin J>0 indicates antiferromagnetic (anti-parallel ) ground state J < 0 indicates ferromagnetic (parallel) ground state 8 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 state is a functional of electronic density: E [ n ( r )] T [ n ] V ext [ n ] V e e [ n ] n (r )v ext ( r ) dr F HK [ n ] 1 2 V ( r ) eff 2 i i i n(r ) i (r ) 2 (1) Hohenberg-Kohn functional Kohn-Sham equations (2) i Where are KS orbitals, is the system of N effective one-particle equations 9 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: N (r ) i | i (r ) | 2 i 1 Electron interaction accounted for self-consistently via exchange-correlation potential ( V ext 1 2 2 (r ' ) | r r ' | dr ' V xc ) i ( r ) i i ( r ) 10 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 11 DFT+U may reduce self-interaction error U “on-site” electron-electron repulsion From fixed-potential diagonalization (Kohn-Sham response) The +U correction is the one needed to recover the exact behavior of the energy. What is the physical meaning of U? From self-consistent ground state (screened response) We used DFT+U implemented in Quantum Espresso 12 Both metal and ligand need Hubbard term U Idea: Empirically Adjust U parameter on both Metal and the coordinated ligand Complex –Ni4(Hmp) DFT DFT+U(d) DFT+U(p+d) S=0 0.0000 0.00000 0.00000 S=2 0.0011 0.00012 -0.000069 S=4 0.0026 0.00019 -0.000368 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) 13 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 Chemical formula J (cm-1) Plane Wave BS-DFT calculations DFT+U DFT+U Expt metal+ligand metal only [Mn2 (IV)(μO)2 (phen)4]4+ -143.6 -166.6 -131.9 -147.0 [Mn2(IV)(μO)2((ac))(Me4dtne)]3+ -74.9 -87.4 -37.5 -100.0 [Mn2(III) (μO)(ac)2(tacn)2]2+ [Mn2(II) (ac)3(bpea)2]+ [Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+ 5.6 -7.7 -234.0 -3.64 -18.8 -247.6 -40.0 -405 10.0 -1.3 -220 14 (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) Exp BSDFT DFT+U -100 -37 -74.9 We modify the source code of Quantum ESPRESSO to incorporate U on Nitrogen 15 Mn(IV)- no acetate bridge Evaluation of J(cm-1) Exp BSDFT DFT+U -147 -131 -164 [Mn2 (IV)(μO)2 (phen)4]4+ 16 Mn(II) three acetate bridges Mn(III) two acetate bridges [Mn2(II) (ac)3(bpea)2]+ Evaluation of J(cm-1) Exp BSDFT DFT+U -1.5 -8 [Mn2(III) (μO)(ac)2(tacn)2]2+ Exp BSDFT 10 -40 DFT+U 29 17 Mixed valence Mn(III)-Mn(IV) J cm-1 (MnIII-MnIV) [Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+ Exp BSDFT DFT+U -220 -155 -234 18 Löwdin population analysis Atom AFM FM Mn1 Mn2 Oµ1 Oµ2 Oac1 Oac2 N1 N2 N3 N′1 N′2 N′3 3.00 -3.00 0.00 0.00 -0.05 0.05 -0.07 -0.07 -0.07 0.07 0.07 0.07 3.08 3.08 -0.03 -0.03 0.08 0.08 -0.05 -0.05 -0.07 -0.05 -0.05 -0.07 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 Mn cations. 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 Mn cation. As a result, Anderson’s superexchange mechanism, developed for σ-bonding metal-ligand interactions, no longer holds. 19 Dependence of J on U U (ev) Mn O N J cm-1 1 1 0.2 -147.77 2.1 1 0.2 -71.92 3 1 0.2 -13.84 4 1 0.2 48.76 6 1 0.2 169.84 2.1 3 0.2 -55.27 2.1 5 0.2 -50.80 2.1 1 2.0 -62.03 20 Failure of BSDFT Bimetallic complexes with Acetate Bridging ligand Complexes with Ferromagnetic Coupling Mix valence complexes Chemical formula J (cm-1) Plane Wave BS-DFT calculations DFT+U DFT+U Expt metal+ligand metal only [Mn2(IV)(μO)2((ac))(Me4dtne)]3+ -74.9 -87.4 -37.5 -100.0 [Mn2(III) (μO)(ac)2(tacn)2]2+ [Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+ 5.6 -234.0 -3.64 -247.6 -40.0 -405 10.0 -220 21 Two qubit system-[Mn12(Reda)] complex with weakly coupled subsystems Methyl diethanolamine Allyl diethanolamine Predict J for two coupled sub system Previous DFT Study predicted J=0 Whereas the J>0 experimentally 22 23 Mdea Bond Length (Å) Mn1-Mn6΄ Mn1-Mn2 Mn2-Mn3 Mn3-Mn4 Mn4-Mn5 Mn5-Mn6 X-ray 3.46 3.21 3.15 3.17 3.18 3.20 Opt 3.44 3.21 3.18 3.17 3.15 3.21 Adea J(cm-1) PBE B3LYP B3LYP DFT+U DFT+U DFT+U (Cluster) (X-ray) (Opt) (Opt) +1.2 -3.5 +0.04 4.6 -0.8 -2.38 -6.0 -5.6 -2.8 -20.8 -3.7 -23.93 -14.9 -2.5 -9.2 -26.8 -23.5 -31.02 +10.9 +6.3 +7.0 50.5 44.0 57.58 +9.2 +5.4 +8.0 56.9 54.1 45.89 -5.4 -5.9 -5.0 -13.6 -14.2 -35.48 24 Spin frustrated system –Mn9 Experimental Spin Ground state S = 21 2 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 25 J8 H J ( S S S S ) J ( S S S S ) J ( S S S S ) J ( S S S S ) 1 1 3 9 7 2 1 2 9 8 3 2 4 8 6 4 3 5 7 5 J 5( S3S 4 S7S6 ) J 6( S 4S5 S6S5 ) J 7( S 2S3 S8S7 ) J 8( S 4S6 ) Mn-Mn Ǻ J (cm-1) J1 3.35 7.48 J2 2.95 -16.87 J3 3.53 1.14 J4 3.43 25.07 S=2 (Mn+3) J5 3.21 7.92 S=5/2(Mn+3) J6 3.38 3.15 J7 3.46 4.02 J8 2.86 27.32 S=-2(Mn+3) Anisotropy –in Molecular Magnet H anisotropy DS 2 Z H Magnetic H Heisenberg H Anisotropy H Zeeman Relativistic Pseudopotential Resulting from spin–orbit coupling, Produces a uniaxial anisotropy barrier Separating opposite projections of the spin along the axis Non-Collinear Magnetism 27 Prediction of Anisotropy for Ce based Complex U(eV) D Ce O N (cm-1) 0 0 0 169.92 4 0.5 0.2 8.38 4 0.8 0.2 0.16 U(eV) J Ce O N (cm-1) 0 0 0 -359.02 3 0.5 0.2 -12.57 4 0.5 0.2 -4.03 4 0.8 0.2 -3.86 Jexpt=-0.75 cm-1, Dexpt= 0.21 cm-1 28 Summary To predict correct J values we need to include U parameters on both metal and ligand Geometry Optimization of ground state is extremely important for correct prediction of J values Exclusion of U Parameters on ligand atoms leads incorrect ferromagnetic ground state Anisoptropy prediction needs relativistic pseudopotential For Anisotropy we need good starting wave function for ground spin state of the molecule 29 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 30 Acknowledgements Prof. Michael Leuenberger Eliza Poalelungi Prof. George Christou Arpita Pal NERSC Supercomputing Facilities (m990) ACS Supercomputing Award for Teragrid 31 32 34 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 ≥ r the pseudopotential and wave function are required to be the same as for real potential. c Pseudopotential excludes (does not reproduce) core states – solutions are only valence states Inside the sphere r ≤ rc , pseudopotential is such that wave functions are nodeless εi(at) = εi(PS) For Iron 1s2 2s2 2p6 3s2 3p6 3d6 4s2 35 Faliure of bs-dft Bimetallic complexes with Acetate Bridging ligand Complexes with Ferromagnetic Coupling Mix valence complexes 36 Different transition metals in molecular magnets 37 J for other transition metal complexes J cm-1(FeIII-FeIII) Exp BSDFT DFT+U -16 -10 J cm-1(FeIII-FeIII) Exp BSDFT DFT+U -121 -77 -141 38 J cm-1 (CrIII-CrIII) Exp BSDFT DFT+U -15 -10 J cm-1(CrIII-MnIII) Exp BSDFT DFT+U -17 -29 39 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 Biol Inorg Chem (2005) 10: 231–238 40 DFT+U in Quantum Espresso The formulation developed by Liechtenstein, Anisimov and Zaanen, referred as basis set independent generalization ELDAU [ n( r )] ELDA [ n( r )] EHub [{ nmI }] EDC [{ n I }] n(r) is the electronic density nmI 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. 41 Future Plans Compute J for heteroatom (Cr) containing molecular magnetic wheel 42 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 43