Molecular Magnetism
Ilaria Ciofini and Claude Daul
Institute of Inorganic and Analytical Chemistry
University of Fribourg
Switzerland
History of magnetism
Lodestone = magnetic core (Fe3O4) was mined in the province of
magnesia:
g
in g
greek writings
g ((800 BC).
)
Thales “the lodestone possesses a soul” (627-547BC).
Descartes postulate the dichotomy of soul and body: modern
science is born (1596-1650).
Langevin
g
explains
p
p
paramagnetism
g
and diamagnetism
g
classically
y
(1905).
Bohr and Van Leeuwen (separately) proved a theorem excluding
magnetism as a classical phenomenon.
Heisenberg explained magnetism as a purely quantum
mechanical effect:exchange.
effect:exchange (1928).
(1928)
Magnetic Exchange Interactions:
In insulators “the spin and magnetic moments whose alignments lead
to magnetic effects are certainly localizable so that
phenomenologically at least, they can be described by a spin
hamiltonian which contains spin operators and exchange terms of
H i
Heisenberg
b
ttype.””
Philip W. Anderson
Type of Interactions:
STRONG
WEAK
diamagnetic
ferro/antiferro magnetic
What is “Molecular Magnetism”?
“Molecular Magnetism deals with the magnetic properties of
isolated molecules and assemblies of molecules.”
molecules.
O. Kahn in Molecular Magnetism
Which are the possible applications?
• Material Science
• Molecular Devices
• Bio-inorganic
Bi i
i S
Systems
t
Molecular Magnetism and Computational Chemistry:
QUALITATIVELY
QUANTITAVELY
Exchange Pathways analysis
Energies of Spin Multiplets
Semi-Empirical Rules
Spin Density analysis
PROBLEMS
SMALL energy differences
Does not give predictions
Spin Contamination
Computational Effort
Illustration1 : ab initio calculation of
singlet-triplet separation (in cm-1)
in cupric acetate hydrate dimer
0th
2nd
order
I
higher
order orders
exp
II
-244
I : JF = 2Kab
II : JAF = -(εg- εu)2 / (Jaa - Jab)
1 P. de Loth, P. Cassoux, J.P. Daudey, J.P . Malrieu,
J. Am. Chem. Soc. 103 (1981) 4007
-286
DFT: Heuristic approach
X-ray diffraction
dr
∫ ρ(r )dr
Cusp
ρ(r )
nuclear positions
# electrons
⎛ ∂ρ
∂ (Rk )⎞
= −2Zk ρ(R0 )
⎜
⎟
⎝ ∂Rk ⎠ R =R
k
ĤΨ
H
Ψ = EΨ
0
General Theory
Exact energy expression
1
2
Eel = -
+
∑
i
∑ ∫
A
∫
ZA
→
→
| R A- r 1|
→
1
2
→
→
→
φ ( r 1 )∇2 φ ( r )d r
i
i
1
1
→
⌠ρ( r 1 )ρ( r 2)
⎮
→
⎮| →
r
⌡ 1 r2 |
→
→
ρ( r 1 ) d r 1
→ →
dr 1 dr 2
+ Exc
Parr R G ;Yang W : Density Functional Theory of Atoms A
Parr,R.G.;Yang,W
Molecules, Oxford University Press, New York 1989
The Kohn-Sham Equation
hksφi = εiφi
hKS =
1∇2
+
2
∑
ZA
→
→
A | R A- r |
1
→
⌠ ρ( r )
→
⎮
2
+
dr 2
→
→
⎮| r - r |
⌡ 1
2
+ VXC
Approximate density functional theories
for exchange and correlation
Xα
Local exchange
LDA
Local exchange +
local correlation
GGA
Local exchange +
local correlation +
gradient corrections
3rd
Generation
of functionals
Xα : Local exchange
functional of the
homogeneous electron gas
LDA: Local exchange
functional + local correlation
functional of the homogeneous electron gas
GGA: Same as LDA + “non-local”
non local
gradient corrections
to exchange and correlation
3rd Generation of functionals: Same as
GGA + instilation of “exact-exchange” and + 2nd derivatives
of the density corrections
Practical
Implementation
Solve Kohn
KohnSham eqs.
⎡ 1
⎤
ρ( r' )
d ' +VXC (ρ( r))⎥Ψi = ε i Ψi (r )
dr'
⎢− ∇ + v(r ) + ∫
r − r'
⎣ 2
⎦
Features:
LCAO expansion:
expansion
waves
STO, GTO, numerical, plane
Coulomb potential:
potential
set
solve Poisson’s eq. or fit ρ(r) to a
of one-center
one center auxilliary functions
Matrixelements: accurate numerical integration in the
Matrixelements
irreducible wedge of the molecule
Methodology based
on
Approximate DFT
MS-X α
1966
DV-X α
1970
ADF
FRIMOL
1994-
1973 -
development in progress
DeMon
1976 -
NUMOL
1982
MS-X α: Make use of partial-waves as basis
37).
(
Relatively fast. Good for ionization potentials and excitation
energies 1( 0). Total energies unreliable39).
(
No geometry optimization. Full use of symmetry.
H relativistic
Has
l ti i ti extension
t
i 53f).
5( 3f) M
Make
k use off muffin-tin
ffi ti
approximation (38). Developed by K.H. Johnson (37) .
DV-Xα: Make use of numerical atomic orbitals or STO's.
Avoids Muffin-tin
Muffin tin approximation by fit of density
45a).
(
Accurate total energies (76d). Relativistic extension (53e).
Numerical integration of matrix elements by Diophantine
integration (40). Developed by Ellis and Painter (40).
Extensive improvements by Delley (D-MOL-program) including
new integration scheme (46c) and geometry optimization.
HFS-LCAO : Make use of STO's . Accurate potentials
41).
(
Full use of symmetry. Relativistic extensions (53a,b). Highly
vectorized (47). Accurate total energies (49). Geometry
optimization (54c). Accurate numerical integration (46b).
Many auxiliary property programs . Pseudo potentials (52a
(52a,d).
d)
Embedding procedures (76h). Energy decomposition scheme
(72). Developed by Baerends,Snijders,Ravenek,Vernooijs and
te Velde (41,53,47,46d)
LCGTO-LSD : Make use of GTO's. Fit of exchange-correlation
and Coulomb potential43).
( Analytical calculation of matrix
elements (48b). Accurate energies. Geometry optimization
(54b,h). Strongly vectorized48b).
(
First developed by
Dunlap (43) as well as Sambe and Felton (42). Extensive
improvements by Salahub and Andzelm (48b)
(D-GAUSS-program) as well as Rösch (74a). Also work by
Pederson (45e) and Painter (45d)
NUMOL : Unique basis
basis-set
set free program
50a,e).
(
Accurate
numerical integration (46a). Efficient generation of
Coulomb potential (50c). Geometry optimization.
Developed by Becke (50 ).
Methodology
É
Post-HF Calculations (as comparison)
É
DFT
Broken Symmetry
Single Determinant
S i P
Spin
Projection
j ti Techniques
T h i
Βroken Symmetry
The electronic symmetry is artificially lowered
Symmetry Equivalent Atoms Become Distinct
BS= a+a' −
a+
a’-
Noodleman L. J. Chem. Phys. 74 (1981) 5737.
E(S)= J/2[S(S+1)] + const.
H=JS1S2
For WEAKLY
Interacting
systems
For MEDIUM
Interacting
systems
[E(S=1)-E(S=0)] =J
E(HS) − E(BS)
J=
2S1S2
E(HS) − E(BS)
J=
2
2S1S2 + α (S)sab
Single determinant
method
th d
Multiplet wave function
Ψi =
Single determinant
Ψi =
α Γ m Γ S mS
∑ Aiμ
i Φμ
μ
Φμ = χ1 χ2 χ3 ...
E
Energy
off single
i l determinant
d
i
( )
( )
E Φ μ = E DFT
Φμ
tot
COMPUTED
Single Determinant Method: A
X
B
Considering
g just
j the SOMO as active orbitals
|b>
|φ A>
|φ B>
Four multiplets arise:
|a>
| a2 1A>
| b2 1A>
| ab 1B>
| ab 3B>
|φ X>
|φ 0>
Energy of the Multiplet in Terms of Single Determinant
|b2 1A>
|ab 1B>
Kab/[E(a2)E(b2)]
E(b2)
2E(a↑b↓)-E(a↑b↑)
Kab
||ab 3B>
E(a
( ↑ b↑ )
|a2 1A
A>
E(a2)
E(1B) − E(3B)
=
= E a↑ b ↑ − E a ↑b ↓
2
(
Kab/[E(a2)E(b2)]
where Kab is the exchange integral between the orbitals a and b
) (
)
Spin Projection
Technique
Ψunr,S = aΨS +bΨT
2
b =
a2+b2 =1
1
2
ψ unr,S S ψ unr,S
2
E unr,S = ψ unr,S H ψ unr,S = a 2 ψ S H ψ S + b 2 ψ T H ψ T
ES =
E unr,S − b 2 E T
1− b2
Δ ST = E S − E T =
E unr,S − E T
1− b 2
General Case:
ψ unr,S = a S (S)ψ S + a S (S + 1)ψ S+1 + ... + aS (Smax )ψ S max
B L = (S− )L (S + ) L
L
ψ unr,S BL ψ unr ,S = L! ∑ det[V ( j, j' )]
j,j' =1
αβ β
V( j, j' ) = δ j,j' − ∑ sαβ
jk s j'k nk
k
L−1
ψ(S, M) B L ψ(S, M) = ∏ [S(S + 1) − (M + 1)( M + i + 1)]
i=0
Defining:
P−1
Tk (P,S) = ∏[S(S + 1) − (k + i)(k + i + 1)]
i=0
Tk (0,S) ≡ 1
As (S) + AS (S + 1) + AS (S + 2) + ... + AS (S + L) = 1
TS (1,S +1) AS (S +1) + TS (1,S + 2)AS (S + 2) + ... + TS (1,S + L)AS (S + L) = B1
...
...
TS ( L,S + L) AS (S + L) = BL
E uncor (S) − A S (S + 1)E corr (S + 1) − AS (S + 2)E corr (S + 2) − ...
E corr (S) =
1 − AS (S + 1) − AS (S + 2) − AS (S + 3) − ...
A. A. Ovchinnikov, J. K. Labanowski, Phys. Rev. A (1996) 53, 3946.
H2 Dissociation Energy
gy Curve: an Illustrative Example
p
H2 Dissociation Energy Curve: Triplet and Singlet
2.00
1.50
1.00
Singlet
Triplet
0.50
0.00
-0.50
0 00
0.00
1 00
1.00
2 00
2.00
3 00
3.00
4 00
4.00
5 00
5.00
6 00
6.00
7 00
7.00
8 00
8.00
H2 Dissociation Energy Curve
0 50
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
Full-CI
RHF
RDFT
-0.20
-0.30
-0.40
0 00
0.00
1 00
1.00
2 00
2.00
3 00
3.00
4 00
4.00
5 00
5.00
6 00
6.00
7 00
7.00
Minimal Basis Set
H2 Dissociation Energy Curve: long range behaviour
Single
g Determinant and Broken Symmetry
y
y
Versus full-CI
0.30
0.20
0.10
0.00
0
00
-0.10
Full-CI
RDFT
Single Det
UDFT (BS)
-0.20
-0.30
0.30
-0.40
0 00
0.00
2 00
2.00
4 00
4.00
d(H-H)
6 00
6.00
8 00
8.00
0.20
0.15
0 10
0.10
0.05
0.00
-0
0.05
05
-0.10
-0.15
-0.20
-0.25
-0.30
-0.35
0.00
2.00
4.00
6.00
8.00
d(H-H)
1.00
0.80
0.60
0.40
0.20
0.00
0.00
2.00
4.00
d(H-H)
6.00
8.00
RDFT
UDFT
<S2>(UDFT)
QuickTime™ and a
Animation decompressor
are needed to see this picture.
H2 Dissociation Energy Curve:
• DFT in the Local Approximation generally OVERBINDING
• Restricted DFT and HF Approaches do not give the
Correct Long Range Behaviour
• The Single Determinant Method yields the correct
behaviour at far distances
• Allowing the spin α and β wave functions to have different
spatial components (Unrestricted) yields a Localised (BS)
state and includes non dynamical correlation (correct
asymptotic behaviour)
Application
s
• Model Systems:
H-He-H
• Organic
g
Radicals:
BVD
Nitroxides
• Metal Complexes:
Organic Radical and Dia/Para magnetic Metal Ions
Binuclear Metal Complexes
Singlet-triplet separation (cm -1)a computed for H −He−H at
different H−He distances.
-1
Singlet -Triplet separation (cm ) computed for H-He-H
Method of
calculation
AB INITIO:
CAS (2,2)
OVB-MP2(2,2)
CAS(4,3)
OVB-MP2(4,3)
QCISD(T)
Res-BS
Unres-BS
S
full CI
DFT
Xα BS
Xα-BS
LDA-BS
BP-BS
Xα-SD
Xα
SD
LDA-SD
BP-SD
H−He Distances ( )
1.25
1.625
2.00
4204
4358
4294
4530
4928
4298
4580
4860
476
420
484
492
790
526
554
544
48
26
46
48
202
60
60
50
6004
12432
10529
5631
9050
7799
646
760
1268
745
1553
1170
58
158
134
64
183
123
The reported values are E(3Σu)- E(3Σg). POSITIVE values indicate a SINGLET G
a T he reported values are E(3Σu)-E(1Σg). Positive values indicate
that the singlet is the ground state.
H2NO-H
NO H
Singlet-triplet separation (cm-1)a computed for H2NO–H with an O-H distance
of 1.9 Å.
Method of Calculation
E(3B1)-E(1B1)
a The following
ab initio:
abbreviations are
CAS(2,2)/ROHF
-884
used here: RMP2 =
RMP2
-2149
E(3B1) and E(1B1)
CAS(6,12)
-972
obtained from
SOCI(6,4,8)
-970
restricted open shell
f ll CI
full
-962
962
MP2 calculations;
DFT:
SOCI(n,a,e) =
-931
Xα-SD
second order CI
LDA SD
LDA-SD
-840
840
calculation on an
BP-SD
-668
active space of a
active orbitals, e empty orbitals with n active electrons (this corresponds to a
CAS(n,a) calculation followed by a CISD calculation including e virtual orbitals.
[(acac)2Cu-ONH2]
Ψ
Si l t t i l t separation
Singlet-triplet
ti (cm
( -1) computed
t da for
f [(acac)
[(
)2Cu−ONH
C ONH2].
]
Method of Calculation
E(3A’’)-E(1A”)
ab initio:
CAS(2,2)/ROHF
-52.4
SOCI(0,2,94)
-48.5
-50.3
SOCI(8,2,8)b
DFT:
-180
Xα -SD
LDA-SD
-171
BP-SD
-169
Experiment
-10 ÷ -70
a ψ = 90° was used in all the calculations.
b The 18 orbitals included in the SOCI calculations are the valence 3d orbitals on
copper and the valence 2p orbitals of the oxygen atom of the radical.
Singlet-triplet separation (cm-1) computed for [(acac)2Cu−ONH2] at different ψ
angles.
angles
ψ (deg)
Method of calculation
ab initio:
CAS(2 2)/ROHF
CAS(2,2)/ROHF
DFT:
Xα-SD
LDA-SD
BP-SD
0°
45°
-53.0
53 0
-49.2
49 2
-184
-174
-173
-190
-179
-178
[Cu2Cl6]2-
Variation of ϕ :
0° : Antiferromagnetic
45°:: Ferromagnetic
45
70°: Antiferromagnetic
Singlet - triplet separation for [Cu2Cl6]2- in cm-1
ϕ (deg)
Method of
Calculation
0°
45°
70°
-36
58
-87
-122
-362
-256
41
-160
35
0 ÷ -40
40
309
246
200
437
374
402
80 ÷ 90
-72
-191
-109
106
-45
145
----
1
ab initio :
DFT :
Xα -BS
LDA-BS
BP-BS
Xα -SD
LDA-SD
BP-SD
E
Experiment
i
t
1 O.
O 1 Castell,
Castell J.
J Miralles,
Miralles R.
R Caballol,
Caballol Chem.
Chem Phys
Phys. 179 (1994) 377
ϕ = 45 °
ϕ = 0°
ϕ = 70°
Ti(CatNSQ)2
Computed Multiplet Structure for Ti(CatNSQ)2 and
Sn(CatNSQ)2.
ΔE
State
3A
2
1A
2
1A
1
1A
1
Ti
0.
Sn
0.
57 [56(1)]
57.
58 [23(1)]
58.
9510.
11934.
9571
9571.
11991
11991.
a The energies are computed as differences from the ground 3A2 state.
The experimental values are in squ are brackets.
-1) for Ti(CatNSQ)2 and Sn(CatNSQ)2
Electronic Transitions Energies (cm
Ti
Transition
8a 2 → 19b2
States
3
A2→3B2
Computed
Experimental
8078
80
8
9800
A2→3A2
A2→ B1
13645
16990
14200
17300/18600
A2→ A2
26868
23900
A2→ B1
3
A2→3A2
31096
28200/30400
39772
35900
States
Computed
Experimental
A2→ B1
5768
9900
18b2 → 19b2
3
7a 2 → 19b1
3
8a 2 → 9a2
3
8a 2 → 20b1
18b1 → 20b1
3
3
3
3
Sn
Transition
9 2 → 20b1
9a
3
19b2 → 20b1
3
A2→ A2
11038
14800
8a 2 → 20b1
20b1 → 10a
10 2
A2→ B1
3
A2→3B1
17018
17900/19200
31968
26000
A2→3A2
35371
27400/30900
9a 2 → 10a2
3
3
3
3
3
35100
BVD
(biverdazyl diradical)
Experimental Data:
• Ground State S=0 Anti ferromagnetic Coupling
• ΔE(T-S)= 760 cm-1(solution); 887 cm-1(crystal)
QuickTime™ and a
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Animation decompressor
are needed to see this picture.
Free Rotor Model.
Exchange Coupling Constant ( J=E(T)-E(S) )
J (cm-1) vs αJ
1600
1400
1200
1000
Jcry
s.
B3LYP(vacuo)
800
Jsol.
600
400
200
0
0 00
0.00
BLYP
20 00
20.00
40 00
40.00
60 00
60.00
80 00
80.00
100 00
100.00
B3LYP(sol)
(E0 − E j )/ kT ]
[
∑ j ΔJ j e
Averaging over the vibrational motion
J
T
= Jmin +
j
(E0 − E j )/ kT ]
[
∑e
j
In Vacuo
CHCl3
34.6
566
585
721
26.8
757
774
844
αmin
Jmin
<Jmin> (0 K)
<Jmin> (298 K)
Experimental:
Js=760 cm-1;
Jc=887cm-1
For all the calculation: UB3LYP/6-31G*
J in cm-1; Angles in degree.
Perspectives
TD-DFT Linear Response Theory
DMRG Density Matrix Renormalisation Group
TD-DFT Linear Response Theory
TD KS eqs.
vext (r,t
, ) = vstat ( r) + vt (r ) ⋅ f (t )
The noninteracting system is defined by
⎛ ∇2
⎞
∂
⎜
⎟
i ϕ i (r,t ) = −
+ v KS ( r,t ) ϕ i (r, t )
∂t
⎝ 2
⎠
v KS (r,t ) = vext (r,t ) + ∫ d 3 r'
ρ (r' ,t ) δAXC [ρ ]
+
r − r''
δ (r,tt )
δρ
In the adiabatic limit (low frequency):
AXC [ρ ] = ∫ dt E XC [ρ ]ρ =ρ (r,t )
Time-Dependent Response
⎡
⎤
ρ (1) (r'',t' )
δ 2 EXC
3
3
(1)
ρ (r,t ) = ∫ d r dt χ KS(t,t' ,r,r' ) vt (r' ) f (t' ) + ∫ d r' '
+ ∫ d r'' (1)
ρ
(
r'',t'
)
⎥⎦
⎢⎣
r'−r''
δρ (r') δρ (1) (r'')
(1)
3
(1)
⎡
⎤
δ2EXC
3 ρ (r'',ω)
3
(1)
ρ (r,ω) =∫ d r χKS(ω,r,r') vt(r') f (ω)+∫ d r''
+∫ d r'' (1)
ρ (r'',ω)
(1)
⎢⎣
⎥⎦
r'−r''
' ''
δ (r'')δρ
δρ
δ (r'''')
(1)
3
To find a self-consistent solution for ρ(1) (r,ω) we use the parametrisation:
ρ(1)(r,ω) = ∑[Piaσ (ω)ϕ*aσ(r)ϕiσ (r)+Piaσ (ω)ϕaσ (r)ϕi*σ(r)]
iaσ
The kernel or generalized susceptibility χKS(ω,r,r') is given as (cf. Casida)
χ KS,σ ,σ ' (ω , r,r' ) = δ σσ '
(vt )iaσ
⎡ϕ i*σ (r )ϕ aσ (r)ϕ i σ ( r' ) ϕ *aσ (r' ) ϕ iσ (r) ϕ a*σ (r )ϕ i*σ (r' )ϕ aσ ( r' ) ⎤
−
∑
ω
−
ε
−
ε
ω + (ε aσ − ε iσ )
ia ⎢
(
)
⎣
⎦⎥
aσ
iσ
= ∫ d 3 r ϕ i*σ ( r) vt (r) ϕ aσ (r )
[δ στ δ ijδ ab (ε aσ − ε iσ + ω ) + Kiaσ ,jbτ ]Pjbτ + K iaσ ,bjτ Pbjτ = − (vt )iaσ
[δ στ δ ijδ ab (ε aσ − ε iσ − ω ) + Kiaσ ,bjτ ]Pbjτ + K iaσ , jbτ Pjbτ = −(vt )iaσ
K klσ ,nmτ
∂vklCoul
∂vklXCσ
σ
=
+
∂Pnmτ ∂Pnmτ
= ∫ d rd
3
3
r' ϕ *kσ
⎛ 1
⎞ *
δ 2 E XC
⎟ ϕ nτ ( r' )ϕ mτ ( r' )
+ (1)
(r )ϕ l σ ( r) ⎜
(1)
r
−
r'
δρ
r'
δρ
r'
'
⎝
( )
( )⎠
Pseudo-Eigenvalue Problem: (M. Casida)
Ω(ω ) FI = ω I2 FI
ΩI: Excitation Energy
FI: Intensity
(
)(
)
Ω ijσ ,klτ (ω ) = δ στ δ ik δ jl (ε l τ − ε kτ ) + 2 ni σ − n jσ ε jσ − ε i σ K ijσ ,klτ (ω ) (nlτ − nk τ )(ε lτ − ε k τ )
2
K klσ ,nmτ
∂vklCoul
∂vklXCσ
σ
=
+
∂Pnmτ ∂Pnmτ
= ∫ d rd
3
3
r' ϕ *kσ
⎛ 1
⎞ *
δ 2 E XC
⎟ ϕ nτ ( r' )ϕ mτ ( r' )
+ (1)
(r )ϕ l σ ( r) ⎜
(1)
r
−
r'
δρ
r'
δρ
r'
'
⎝
( )
( )⎠
DMRG Density Matrix Renormalisation Group
H = ∑ Tijc iσ c jσ +
i, j,σ
1
∑ Vijkl c iσ c jσ' ckσ' c lσ
2 ijklσσ'
DMRG
Unoccupied
LUMO
HOMO
Occupied
Test Calculation: H2O
Software:
• ADF
• Gaussian94
• Home
o e made
ade So
Software
t ae
Acknowledgements:
Prof. V. Barone (Uni. of Naples)
Prof. V. Bencini (Uni. of Florence)
Dr K.
Dr.
K Doclo (Uni.
(Uni Fribourg / ETHZ)
Prof. P. Fantucci (Uni. of Milan)
F. Totti (Uni. of Florence)
J L B
J.-L.
Barras (Uni.
(U i off F
Fribourg)
ib
)
R. Bruyndonckx (Uni. of Fribourg)
M. Buchs (Uni. of Fribourg)
F. Mariotti (Uni. of Fribourg)
Swiss National Science Foundation and COST Action D9.