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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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

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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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