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
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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
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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
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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
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
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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
ELDAU [ 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