shruba gangopadhyay 1,2 & artëm e. masunov 1,2,3 1 nanoscience technology center 2 department...
TRANSCRIPT
Shruba Gangopadhyay1,2 & Artëm E. Masunov1,2,3
1NanoScience Technology Center 2Department of Chemistry
3Department of PhysicsUniversity of Central Florida
Quantum Coherent Properties of Spins - III
First Principle Simulations of Molecular Magnets: Hubbard-U is Necessary on Ligand Atoms for Predicting Magnetic
Parameters
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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
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Molecular Magnets – possible element in quantum computing
5Leuenberger & Loss Nature 410, 791 (2001)
Molecular Magnet is promising implementation of QubitUtilize the spin eigenstates as qubits Molecular Magnets have higher ground spin states
It can be in |0> and |1> state simultaneously
Advantages of Molecular MagnetsUniform nanoscale size ~1nmSolubility in organic solvents Readily alterable peripheral ligands helps to fine tune the propertyDevice can be controlled by directed assembly or self assembly
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2-qubit system: Molecular Magnet [Mn12(Rdea)] contains two weakly coupled subsystems
M=Methyl diethanolamine M=allyl diethanolamine
Subsystem spin should not be identical
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Ion substitution may be used to redesign MM
Cr8 Molecular Ring Cr7Ni 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 Lett 94, 207208 (2005).
To redesign MM we need reliable method to predict magnetic properties
Ferromagnetic (F) – when the electrons have Parallel spin Antiferromagnetic (AF) – having Antiparallel spin
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J)(E)(E
ZeemanAnisotropyHeisenbergMagnetic HHHH
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Heisenberg-Dirac-Van Vleck Hamiltonian
J = exchange coupling constant
Si= spin on magnetic center i
21HDVV SJSH
J>0 indicates antiferromagnetic (anti-parallel ) ground stateJ < 0 indicates ferromagnetic (parallel) ground state
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iiieff rV
)(
2
1 2(1)
i
i rrn2
)()( (2)
Kohn-Sham equations
][)()(
][][][)]([
nFdrrvrn
nVnVnTrnE
HKext
eeext
Hohenberg-Kohn functional
Electronic density n(r) determines all ground state properties of multi-electron system. Energy of the ground state is a functional of electronic density:
Density Functional Theory (DFT)prediction of J from first principles
Where are KS orbitals, is the system of N effective one-particle equations
Energy can be predicted for high and low spin states
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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
)()()'|'|
)'(( 2
21 rrVdr
rr
rV iiixcext
2
1
|)(|)( rr i
N
ii
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
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DFT+U may reduce self-interaction error
The +U correction is the one needed to recover the exact behavior of theenergy. What is the physical meaning of U?
From self-consistent ground state (screened response)
From fixed-potential diagonalization(Kohn-Sham response)
U “on-site” electron-electron repulsion
We used DFT+U implemented in Quantum Espresso
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
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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)
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Numeric values of U parameters for different atom types are fitted using benchmark set
Chemical formulaJ (cm-1)
Plane Wave calculations
BS-DFT Expt
DFT+Umetal+ligand
DFT+Umetal 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+ 5.6 -3.64 -40.0 10.0[Mn2(II) (ac)3(bpea)2]+ -7.7 -18.8 - -1.3[Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+ -234.0 -247.6 -405 -220
U (Mn)=2.1 eV, U(O)=1.0 eV, U(N)=0.2 eV
(Mn(IV))2 (OAc)
Exp BSDFT DFT+U
-100 -37 -74.9
Computational DetailsCutoff
25 RydSmearing
Marzari-Vanderbilt cold smearingSmearing Factor 0.0008For better convergence Local Thomas Fermi screening
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Evaluation of J(cm-1)
We modify the source code of Quantum ESPRESSO to incorporate U on Nitrogen
[Mn2(IV)(μO)2((ac))(Me4dtne)]3+
Mn(IV)- no acetate bridge
Exp BSDFT DFT+U
-147 -131 -164
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Evaluation of J(cm-1)
[Mn2 (IV)(μO)2 (phen)4]4+
Exp BSDFT DFT+U
10 -40 2917
Mn(III) two acetate bridges
Evaluation of J(cm-1)
Exp BSDFT DFT+U
-1.5 -8
Mn(II) three acetate bridges
[Mn2(II) (ac)3(bpea)2]+[Mn2(III) (μO)(ac)2(tacn)2]2+
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J cm-1 (MnIII-MnIV)
Exp BSDFT DFT+U
-220 -155 -234
Mixed valence Mn(III)-Mn(IV)
[Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+
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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 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.
Atom AFM FM
Mn1 3.00 3.08Mn2 -3.00 3.08Oµ1 0.00 -0.03Oµ2 0.00 -0.03
Oac1 -0.05 0.08Oac2 0.05 0.08
N1 -0.07 -0.05N2 -0.07 -0.05N3 -0.07 -0.07
N′1 0.07 -0.05N′2 0.07 -0.05N′3 0.07 -0.07
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Dependence of J on U
U (ev)J cm-1Mn O N
1 1 0.2 -147.772.1 1 0.2 -71.923 1 0.2 -13.844 1 0.2 48.766 1 0.2 169.84
2.1 3 0.2 -55.272.1 5 0.2 -50.802.1 1 2.0 -62.03
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Failure of BSDFT
Bimetallic complexes with Acetate Bridging ligand Complexes with Ferromagnetic Coupling Mix valence complexes
Chemical formulaJ (cm-1)
Plane Wave calculations
BS-DFT Expt
DFT+Umetal+ligand
DFT+Umetal only
[Mn2(IV)(μO)2((ac))(Me4dtne)]3+ -74.9 -87.4 -37.5 -100.0
[Mn2(III) (μO)(ac)2(tacn)2]2+ 5.6 -3.64 -40.0 10.0[Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+ -234.0 -247.6 -405 -220
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Two qubit system-[Mn12(Reda)] complex with weakly coupled subsystems
Predict J for two coupled sub system
Previous DFT Study predicted J=0Whereas the J>0 experimentally
Methyl diethanolamine Allyl diethanolamine
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Mdea Adea
Bond Length (Å)
J(cm-1)
X-ray OptPBE B3LYP B3LYP
(Cluster) DFT+U(X-ray)
DFT+U(Opt)
DFT+U(Opt)
Mn1-Mn6΄ 3.46 3.44 +1.2 -3.5 +0.04 4.6 -0.8 -2.38Mn1-Mn2 3.21 3.21 -6.0 -5.6 -2.8 -20.8 -3.7 -23.93Mn2-Mn3 3.15 3.18 -14.9 -2.5 -9.2 -26.8 -23.5 -31.02Mn3-Mn4 3.17 3.17 +10.9 +6.3 +7.0 50.5 44.0 57.58Mn4-Mn5 3.18 3.15 +9.2 +5.4 +8.0 56.9 54.1 45.89Mn5-Mn6 3.20 3.21 -5.4 -5.9 -5.0 -13.6 -14.2 -35.48
Spin frustrated system –Mn9
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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
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J1
J2
J3
J4
J5
J 6
J 7
J8
J6
J4
J1
J 2
J3
J 5 J7
1
2
3
4
5
6
79
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S=-2(Mn+3)
S=2 (Mn+3)
S=5/2(Mn+3)
)SS(J)SSSS(J)SSSS(J)SSSS(J
)SSSS(J)SSSS(J)SSSS(J)SSSS(JH
648783275654667435
57534684238921279311
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
J5 3.21 7.92
J6 3.38 3.15
J7 3.46 4.02
J8 2.86 27.32
Anisotropy –in Molecular Magnet
ZeemanAnisotropyHeisenbergMagnetic HHHH
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2Zanisotropy DSH
Resulting from spin–orbit coupling, Produces a uniaxial anisotropy barrier Separating opposite projections of the spin along the axis
Relativistic Pseudopotential
Non-Collinear Magnetism
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Prediction of Anisotropy for Ce based Complex
U(eV)J
(cm-1)Ce O N 0 0 0 -359.023 0.5 0.2 -12.574 0.5 0.2 -4.034 0.8 0.2 -3.86 U(eV)
D(cm-1)Ce O N
0 0 0 169.92
4 0.5 0.2 8.38
4 0.8 0.2 0.16
Jexpt=-0.75 cm-1, Dexpt= 0.21 cm-1
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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
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Prediction of Anisotropy for Mn12 based wheel Heisenberg Exchange constants
Ion substituted Mn12 wheel Mn12 cation/anion Mn12 wheel on the metal surface
Future Work
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Acknowledgements
Prof. Michael Leuenberger Eliza Poalelungi Prof. George Christou Arpita Pal NERSC Supercomputing Facilities (m990) ACS Supercomputing Award for Teragrid
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tunneling from macroscopic world
to quantumland through the
rabbit hole
Questions &
Suggestions
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PseudopotentialPseudopotentials 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 ≥ rc the pseudopotential and wave function are required to be the same as for real potential.
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
Faliure of bs-dft
Bimetallic complexes with Acetate Bridging ligand
Complexes with Ferromagnetic Coupling Mix valence complexes
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Different transition metals in molecular magnets
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J for other transition metal complexes
J cm-1(FeIII-FeIII)
Exp BSDFT DFT+U
-121 -77 -141
J cm-1(FeIII-FeIII)
Exp BSDFT DFT+U
-16 -10
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J cm-1 (CrIII-CrIII)
Exp BSDFT DFT+U
-15 -10
J cm-1(CrIII-MnIII)
Exp BSDFT DFT+U
-17 -29
Application- biocatalysis
Polyneuclear – Transition metal centers in the enzyme
Important for biocatalysis -Understand the stability of biradical at transition state
40S Sinnecker, F Neese, W Lubitz, J Biol Inorg Chem (2005) 10: 231–238
DFT+U in Quantum Espresso
The formulation developed by Liechtenstein, Anisimov and Zaanen, referred as basis set independent generalization
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}]n[{E}]n[{E)]r(n[E)]r(n[E IDC
ImHubLDAULDA
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.
Imn
Future Plans
Compute J for heteroatom (Cr)
containing molecular magnetic
wheel
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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
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