Transcript Hubbard-U is necessary on ligand atom for predicting
First Principle Simulations of Molecular Magnets: Hubbard-U is Necessary on Ligand Atoms for Predicting Magnetic Parameters
Shruba Gangopadhyay 1,2 & Artëm E. Masunov 1,2,3 1 NanoScience Technology Center 2 Department of Chemistry 3 Department 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
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)
Advantages of Molecular Magnets
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 [Mn 12 (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
[1 ] [2] [3] [4]
Cr 8 Molecular Ring Cr 7 Ni Molecular Ring
M. Affronte M. Affronte
et al.
et al.
, Chemical Communications, 1789 (2007).
, Polyhedron
24
, 2562 (2005).
G. A. Timco F. Troiani
et al.
, Nature Nanotechnology
4
, 173 (2009).
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
HDVV
JS 1 S 2
J = exchange coupling constant
E (
)
E (
)
J 2 S
i = 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
]
Hohenberg-Kohn functional
1 2 2
V eff
(
r
)
i
i
i n
(
r
)
i
i
(
r
) 2
(1) (2)
Kohn-Sham equations 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
( 1 2 2
V ext
| (
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) = 2JS 1
S
2 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 –Ni 4 (Hmp)
S=0 S=2 S=4 DFT 0.0000
0.0011
0.0026
DFT+U(d) 0.00000
0.00012
0.00019
DFT+U(p+d) 0.00000
-0.000069
-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 [Mn 2 (IV)(μO) 2 (phen) 4 ] 4+ [Mn 2 (IV)(μO) 2 ((ac))(Me 4 dtne)] 3+ [Mn 2 (III) (μO)(ac) 2 (tacn) 2 ] 2+ [Mn 2 (II) (ac) 3 (bpea) 2 ] + [Mn(III)Mn(IV)(μO) 2 (ac)(tacn) 2 ] 2+ J (cm -1 ) Plane Wave BS-DFT calculations DFT+U metal+ligand DFT+U metal only -143.6
-74.9
5.6
-7.7
-234.0
-166.6
-87.4
-3.64
-18.8
-247.6
-131.9
-37.5
-40.0
-405 Expt -147.0
-100.0
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 [Mn 2 (IV)(μO) 2 ((ac))(Me 4 dtne)] 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 [Mn 2 (IV)(μO) 2 (phen) 4 ] 4+ 16
Mn(II) three acetate bridges Mn(III) two acetate bridges Exp -1.5
[Mn 2 (II) (ac) 3 (bpea) 2 ] + Evaluation of J(cm -1 ) [Mn 2 (III) (μO)(ac) 2 (tacn) 2 ] 2+ BSDFT DFT+U -8 Exp 10 BSDFT -40 DFT+U 29 17
Mixed valence Mn(III)-Mn(IV) [Mn(III)Mn(IV)(μO) 2 (ac)(tacn) 2 ] 2+ J cm -1 (MnIII-MnIV) Exp BSDFT DFT+U -220 -155 -234 18
L
ö
wdin 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
Mn U (ev) O 1 2.1
3 4 6 1 1 1 1 1 N J cm 0.2 -147.77
0.2
-71.92
0.2
-13.84
0.2
0.2
48.76
169.84
-1 2.1
2.1
2.1
3 5 1 0.2
0.2
2.0
-55.27
-50.80
-62.03
20
Failure of BSDFT
Bimetallic complexes with Acetate Bridging ligand
Complexes with Ferromagnetic Coupling
Mix valence complexes Chemical formula [Mn 2 (IV)(μO) 2 ((ac))(Me 4 dtne)] 3+ [Mn 2 (III) (μO)(ac) 2 (tacn) 2 ] 2+ [Mn(III)Mn(IV)(μO) 2 (ac)(tacn) 2 ] 2+ J (cm -1 ) Plane Wave BS-DFT calculations DFT+U metal+ligand DFT+U metal only -74.9
5.6
-234.0
-87.4
-3.64
-247.6
-37.5
-40.0
-405 Expt -100.0
10.0
-220 21
Two qubit system-[Mn 12 (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
Mn1-Mn6΄ Mn1-Mn2 Mn2-Mn3 Mn3-Mn4 Mn4-Mn5 Mn5-Mn6 Mdea Adea Bond Length (Å) 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
J(cm -1 ) PBE B3LYP
+1.2
-6.0
-14.9
+10.9
+9.2
-5.4
-3.5
-5.6
-2.5
+6.3
+5.4
-5.9
B3LYP (Cluster)
+0.04
-2.8
-9.2
+7.0
+8.0
-5.0
DFT+U (X-ray) 4.6
-20.8
-26.8
50.5
56.9
-13.6
DFT+U (Opt) -0.8
-3.7
-23.5
44.0
54.1
-14.2
DFT+U (Opt) -2.38
-23.93
-31.02
57.58
45.89
-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
J
8
H
J
1
( S
1
S
3
S
9
S
7
)
J
2
( S
1
S
2
S
9
S
8
)
J
3
( S
2
S
4
S
8
S
6
)
J
4
( S
3
S
5
S
7
S
5
)
J
5
( S
3
S
4
S
7
S
6
)
J
6
( S
4
S
5
S
6
S
5
)
J
7
( S
2
S
3
S
8
S
7
)
J
8
( S
4
S
6
)
J 6 J 7 J 8 J 1 J 2 J 3 J 4 J 5
Mn-Mn Ǻ 3.35
2.95
3.53
3.43
3.21
3.38
3.46
2.86
J
(cm -1 ) 7.48
-16.87
1.14
25.07
7.92
3.15
4.02
27.32
S=-2(Mn +3 ) S=2 (Mn +3 ) S=5/2(Mn +3 )
Anisotropy –in Molecular Magnet
H
H anisotropy
Magnetic
DS
H
2
Z
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
0 4
4
U(eV) Ce O N 0 0 D (cm -1 169.92
0.5 0.2 8.38
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
J
expt =-0.75 cm-1, D expt = 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 incorrect ferromagnetic ground state ligand atoms leads Anisoptropy prediction needs relativistic pseudopotential For Anisotropy we need good starting wave function ground spin state of the molecule for 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 (r
c
) defines a boundary between the core and valence regions
For
r ≥ r c
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 ≤ r c
, pseudopotential is such that wave functions are nodeless
ε i (at) = ε i (PS)
For Iron 1s 2 2s 2 2p 6 3s 2 3p 6 3d 6 4s 2 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
E LDA
U [ n ( r )]
E LDA [ n ( r )]
E Hub [{ n m I
}]
E DC [{ n I
}]
n(r) is the electronic density
n I
m
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 E Hub and, in some average way, in E LDA .
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