The document discusses the use of first principle calculations and the DFT+U method to predict magnetic coupling in organometallic complexes, particularly in molecular magnets that may serve as qubits in quantum computing. Key topics include benchmarking studies of various two-qubit systems, spin frustration, and the need for accurate predictions of magnetic properties through reliable approaches. Future research plans focus on improving computational methods to refine magnetic property predictions and exploring the anisotropy of molecular magnets.

1. Shruba Gangopadhyay
Email: shruba@gmail.com
First principle calculations
of organometallic complexes

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
2

3. Organometallics and applications
3

4. Molecular Magnets – possible element
in quantum computing
4Leuenberger & Loss Nature 410, 791 (2001)
Molecular Magnet is promising
implementation of Qubit
Utilize the spin eigenstates as
qubits
 Molecular Magnets have higher
ground spin states
It can be in |0> and |1> state simultaneously
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 [Mn12(Rdea)]
contains two weakly coupled subsystems
5
M=Methyl diethanolamine M=allyl diethanolamine
Subsystem spin should not be identical

6. Ion substitution may be used to redesign MM
6
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).

7. To redesign MM we need reliable
method to predict magnetic properties
 Ferromagnetic (F) – when the electrons have Parallel spin
 Antiferromagnetic (AF) – having Antiparallel spin
2
J
)(E)(E 
ZeemanAnisotropyHeisenbergMagnetic HHHH


7
Heisenberg-Dirac-Van Vleck Hamiltonian
J = exchange coupling constant
Si= spin on magnetic center i
 21HDVV SJSHˆ
 J>0 indicates antiferromagnetic (anti-parallel ) ground state
J < 0 indicates ferromagnetic (parallel) ground state

8. iiieff rV  





 )(
2
1 2
(1)

i
i rrn
2
)()(  (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
8
Where  are KS orbitals, is the system of N effective one-particle equations

9. Energy can be predicted
for high and low spin states
9
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
2
1
rrVdr
rr
r
V iiixcext 



 
2
1
|)(|)( rr i
N
i
i  


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
10

11. DFT+U may reduce self-interaction error
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)
From fixed-potential diagonalization
(Kohn-Sham response)
U “on-site” electron-electron repulsion
We used DFT+U implemented in Quantum Espresso
11

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
12
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
Chemical formula
J (cm-1)
Plane Wave
calculations
BS-DFT Expt
DFT+U
metal+ligand
DFT+U
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+ 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
13
U (Mn)=2.1 eV, U(O)=1.0 eV, U(N)=0.2 eV

14. (Mn(IV))2 (OAc)
Exp BSDFT DFT+U
-100 -37 -74.9
Computational Details
Cutoff
25 Ryd
Smearing
Marzari-Vanderbilt cold smearing
Smearing Factor
0.0008
For better convergence
Local Thomas Fermi screening
14
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+

15. Mn(IV)- no acetate bridge
Exp BSDFT DFT+U
-147 -131 -164
15
Evaluation of J(cm-1)
[Mn2 (IV)(μO)2 (phen)4]4+

16. Exp BSDFT DFT+U
10 -40 29
16
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+

17. 17
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+

18. 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.
18
Atom AFM FM
Mn1 3.00 3.08
Mn2 -3.00 3.08
Oµ1 0.00 -0.03
Oµ2 0.00 -0.03
Oac1 -0.05 0.08
Oac2 0.05 0.08
N1 -0.07 -0.05
N2 -0.07 -0.05
N3 -0.07 -0.07
N′1 0.07 -0.05
N′2 0.07 -0.05
N′3 0.07 -0.07

19. Dependence of J on U
19
U (ev)
J cm-1Mn O N
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
20
Chemical formula
J (cm-1)
Plane Wave
calculations
BS-DFT Expt
DFT+U
metal+ligand
DFT+U
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+ 5.6 -3.64 -40.0 10.0
[Mn(III)Mn(IV)(μO)2(ac)(tacn)2]2+ -234.0 -247.6 -405 -220

21. Two qubit system-[Mn12(Reda)] complex with
weakly coupled subsystems
21
Predict J for two
coupled sub system
Previous DFT Study predicted J=0
Whereas the J>0 experimentally
Methyl diethanolamine Allyl diethanolamine

22. 22

23. 23
Mdea Adea
Bond Length
(Å)
J(cm-1)
X-ray Opt
PBE 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 -2.38
Mn1-Mn2 3.21 3.21 -6.0 -5.6 -2.8 -20.8 -3.7 -23.93
Mn2-Mn3 3.15 3.18 -14.9 -2.5 -9.2 -26.8 -23.5 -31.02
Mn3-Mn4 3.17 3.17 +10.9 +6.3 +7.0 50.5 44.0 57.58
Mn4-Mn5 3.18 3.15 +9.2 +5.4 +8.0 56.9 54.1 45.89
Mn5-Mn6 3.20 3.21 -5.4 -5.9 -5.0 -13.6 -14.2 -35.48

24. Spin frustrated system –Mn9
24
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
2
21

25. J8
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

26. Anisotropy –in Molecular Magnet
ZeemanAnisotropyHeisenbergMagnetic HHHH


26
2
Zanisotropy 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

27. Prediction of Anisotropy for Ce based
Complex
27
U(eV) J
(cm-1)Ce O N
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
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

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
28

29. Acknowledgements
29
 Prof. Artem Masunov
 Prof. Michael Leuenberger
 Eliza Poalelungi
 Prof. George Christou
 Arpita Pal
 NERSC Supercomputing Facilities (m990)
 ACS Supercomputing Award for Teragrid

30. 30