Magnetochimica
AA 2011-2012
Marco Ruzzi
Marina Brustolon
4. The Zero Field Splitting spin Hamiltonian
This lecture
• In the previous lecture we have seen an example of
Spin Hamiltonian, i.e. the so called Exchange
Hamiltonian :
•
H  2JS S
1 2
• Now we will analyze a second Spin Hamiltonian, due to
the dipolar interaction between two electrons
magnetic moments.
•This interaction is called Zero Field Splitting
(ZFS) interaction, or also “Fine interaction” to
distinguish it from “Hyperfine interaction”. We
will show that the form of this latter spin
Hamiltonian is :
H  SDS
H ZFS
~
 S  DS
Zero field splitting spin hamiltonian.
~
S is the total spin and D is a symmetric tensor (fine
tensor).
We will see :
a. the physical origin of this Hamiltonian
b. how to obtain it from the classical dipolar interaction
expression.
Physical origin of ZFS
H ZFS
~
 S  DS
This spin hamiltonian corresponds to two different interactions:
1.
electron-electron dipolar interaction
2. mixing between ground and excited electronic states by spinorbit coupling
The two interactions are different and independent from each other,
but they give rise to the same type of spin Hamiltonian.
~
D
Therefore the elements of tensor
are in general due to both
interactions.
However, the spin-orbit coupling is usually predominant for transition
metal ions, and the dipolar interaction for organic triplets.
ZFS as originated from
electron-electron dipolar interaction
Let us suppose to have two unpaired electrons in two different
molecular orbitals (either on the same molecule, or on different
molecules).
E
3M
LUMO
HOMO
Biradical
Triplet
N
O
Between near electrons
there will be a magnetic
interaction, which is
called “dipolar
interaction”, or “fine
interaction”.
z

r
μ2
μ1
The Hamiltonian of the dipolar interaction can be obtained from the
classical dipole-dipole interaction energy:
Edip
 
   
μ 0  μ1  μ 2
(μ1  r )(μ 2  r ) 

3
3


4  r
r5
where r is the distance between the two electrons.
How to pass from the classical energy
expression to the ZFS spin Hamiltonian
In the following we will show how one can obtain the
spin hamiltonian
H ZFS
~
 S  DS
from the classical dipolar interaction expression:
Edip
 
   
μ 0  μ1  μ 2
(μ1  r )(μ 2  r ) 


3
3

4  r
r5
2
r
1
From the classical energy to the spin hamiltonian:
By substituting
H ZFS
μˆ i   g i μ B si
ˆ1  sˆ2
μ0
(sˆ1  rˆ)(sˆ2  rˆ) 
2 s

g1 g 2 μ B  3  3

4
r5
 r

Developing the products this expression becomes:
Cost
μ 0 g1 g2μ 2B 2
H ZFS 
( r s1x s2 x  s1 y s2 y  s1z s2 z 
5
4 r
 3( xs1x  ys1 y  zs1z )( xs2 x  ys2 y  zs2 z ))
and rearranging:
Cost 2
2
2
2
2
2
((
r

3
x
)
s
s

(
r

3
y
)
s
s

(
r

3
z
) s1z s 2 z
1x 2 x
1y 2 y
5
r
 3xy( s1x s 2 y  s1 y s 2 x )  3xz( s1x s 2 z  s1z s 2 x )  3 yz( s1 y s 2 z  s1z s 2 y ))
H ZFS 
It is convenient to express these products in terms of the
total spin S = s1 + s2 and its components.
S x  (s x1  s x 2 )
S x2  (s x1  s x 2 ) 2  s x21  s x22  2s x1s x 2
and analogously for the y and z components. We can write:
1 1
S  ( s x1  s x 2 )    2s x1 s x 2
4 4
2
x
In fact:
s   s x  is y
2
s   s x  is y
s  s
therefore s x 
2
s   s 
1
sx  


2
2
s   s 
1
sx  


2
2
s  s
sy 
2i
s   s 
1

4
4
s   s 
1
2
sx  


4
4
s  
2
x

The effect of the operator s2x is always to give ¼; therefore we can
substitute ¼ to the operator.
From:
S x2  ( s x1  s x 2 ) 2 
1 1
  2s x1 s x 2
4 4
we obtain:
S y2  ( s y1  s y 2 ) 2 
1 1
  2 s y1 s y 2
4 4
s x1 s x 2 
S z2  ( s z1  s z 2 ) 2 
1 1
  2 s z1 s z 2
4 4
By a similar procedure:
From:
1 2 1
Sx 
2
4
etc.
we obtain:
S x S y  S y S x  2(s x1s y 2  s y1s x 2 )
s x1s y 2  s y1s x 2  1 2 (S x S y  S y S x )
S x S z  S z S x  2(s x1 s z 2  s z1 s x 2 )
s x1s z 2  s z1s x 2  1 2 (S x S z  S z S x )
S y S z  S z S y  2(s y1s z 2  s z1s y 2 )
s y1s z 2  s z1s y 2  1 2 (S y S z  S z S y )
Cost 2
H ZFS  5 ((r  3x 2 ) s1x s 2 x  (r 2  3 y 2 ) s1 y s 2 y  (r 2  3z 2 ) s1z s 2 z
r
 3xy( s1x s 2 y  s1 y s 2 x )  3xz( s1x s 2 z  s1z s 2 x )  3 yz( s1 y s 2 z  s1z s 2 y ))
By replacing the single spin operators with the total spin
operator we obtain:

μ 0 g 2 μ 2B 1 2
2
2
2
2
2
2
2
2
H
(
r

3
x
)
S

(
r

3
y
)
S

(
r

3
z
)
S
x
y
z
4 r 5 2
 3xy( S x S y  S y S x )  3xz( S x S z  S z S x )  3 yz( S y S z  S z S y )

This operator contains both spatial operators and spin
operators. The spatial operators are given by products
of the vectorial components of the vector joining the
positions of the two electrons. To obtain a spin
Hamiltonian, we must find the average values of the
spatial operators over the spatial wavefunction
representing the two electrons:
 (1,2) r 2  3 x 2  (1,2)  r 2  3 x 2
etc.
Finally, the products of spin operators can be put in a convenient
tensorial form:
H ZFS
 r 2  3x 2

5
r


μ 0 g 2μ 2B

[
S
S
S
]
x
y
z 
8 r 5




H ZFS
~
 S  DS
 3xy
r5
r2  3y2
r5
 3xz
r5
 3 yz
r5
r 2  3z 2
r5


 Sx 
 
 S y 
  Sz 
 


H ZFS
~
 S  DS
~
D
è simmetrico e senza traccia
 X
Nel sistema di assi principali:
D


2
2
2
H   XS  YS  ZS
ZFS
x
y
0
Y
0 
0 

 Z 
z
E’ conveniente riscrivere l’Hamiltoniano per lo stato di tripletto
usando due parametri ottenuti dai precedenti:
YX
E
2
3
D Z
2
Con questi parametri l’Hamiltoniano
si può riscrivere come:
1
H ZFS D( S z2  S 2 )  E( S x2  S y2 )
3
e più
convenientemente:
1
1
H ZFS D( S  S ( S  1))  E( S2  S2 )
3
2
2
z
Magnetic energy levels and EPR spectra
1
1
H ZFS D( S z2  S ( S  1))  E( S2  S2 )
3
2
The eigenfunctions and eigenvalues of this hamiltonian are:
Eigenfunctions
X = 1/2{ -}
Y = 1/2{ +}
Z
= 1/2{ + }
Eigenvalues
1
EX  D  E
3
1
EY  D  E
3
2
EZ  - D
3
E
X
Y
Z
Now we consider also the presence of an external
magnetic field. Let us assume an isotropic g factor.
0
ZFS and Zeeman interactions
1
1
2
2
H  gμ B B S  D( S  S ( S  1))  E( S   S  )
3
2
2
z
The eigenfunctions of
this term are the
triplet functions


S M s  11
1
2
(  )
1 1
10
H SM S  g B B0 S z SM S 
 M S g B B0 SM S
The eigenfunctions of
this term are the
linear combinations of
the triplet functions
X 
Y 
11  1  1
2
11  1  1
2
Z  10
When both terms of the spin Hamiltonian are present, the
eigenfunctions are a mixing (on one of the two basis). Depending on the
relative magnitudes of the two interactions, is more convenient to use a
basis or the other one.
1
1
H  gμ B B S  D( S  S ( S  1))  E( S 2  S 2 )
3
2
2
z
On the basis of the ZFS eigenfunctions:
X
X
Y
Z
Y
Z
1
DE
3
gμ B BlZ
 igμ B BlY
gμ B BlZ
1
DE
3
gμ B BlX
gμ B BlX
2
 D
3
igμ B BlY
The elements out of
diagonal depends on the
direction cosines of the
magnetic field in the XYZ
axes of the fine tensor as
lx, ly, lz.
We can find easily the roots to diagonalize this secular determinant
for B//X,Y,Z (this could correspond to a triplet in a single crystal
oriented in the magnetic field).
For example, for B//Z we have: lx =0 ly = 0 lz=1
therefore:
X
Y
Z
X
1
DE
3
gμ B B
0
Y
gμ B B
1
DE
3
0
Z
0
0
2
EZ   D
3

2
D
3
1
2
2 1/ 2
E   D  E  ( gμ B B ) 
3
The EPR allowed transitions are those with Ms= ±1.
B // X
X
Y
Z
B// Y
B
B
B// Z
B
B
B
B
B
X
B0
Single crystal
analysis
B0 // X
Y
B  3 X gμ B
1000
Z
2000
Y
3000 4000
B0 (gauss)
5000
B
Z
B0 // Y
X
B  3Y gμ B
1000
2000
Z
3000 4000
B0 (gauss)
B
X
B0 // Z
Y
5000
B  3Z gμ B
1000
2000
3000 4000
B0 (gauss)
5000
1 G = 10-4 T
EPR of triplets in a disordered matrix
Disordered
matrices
B
1000
2000
3000 4000
B0 (gauss)
5000
X
B0
Y
Z
Y
Z
X
Z
X
B0
Y
The principal values of the
ZFS tensor can be obtained
also from the features of the
spectra in disordered phases.
Disordered
matrices
3XZFS
3YZFS
3ZZFS
2500 3000 3500 4000 4500
Field (G)
assorbimento
Disordered
matrices
Powder EPR spectrum
of Naphtalene triplet
B0
(E. Wasserman et al.
J. Chem. Phys. 41, 1763 (1964))
dI/dB0
B0
B0
Some examples of ZFS parameters
D / cm-1
|E|/cm-1
Benzene
0.158
0.0065
Naphtalene
0.104
Anthracene
±0.072
0.007
Biphenyl
±0.109
0.004
C60
0.0115
0.00047
C70
0.0052
0.00069
Fullero
pyrrolidine
0.0090
0.0014
Triplets
• 1. Ground state triplets: e.g. non
kekulè molecules.
• 2. Thermally accessible triplets
(J~kT)
• Photoexcited triplets
Photoexcited triplets:
a simple approach for ZFS
calculations
• For a photoexcited triplet  * we can
assume the simple model of an electron in
HOMO and another one in LUMO.
• Therefore we can calculate the ZFS
parameters as deriving from the dipolar
interaction between the spin distributions in
LUMO and HOMO.
Example: triplet of Butadiene
LUMO
HOMO

μ1

μ2
We can separate the total dipolar interaction in the
interactions between pairs of spin densities on atoms.
Osservazioni:
1. L’interazione tra le densità di
spin sullo stesso atomo di
carbonio non c’è, perché per il
principio di Pauli due elettroni
con lo stesso spin non possono
stare sullo stesso orbitale
atomico.
2. L’interazione dipolare sarà tanto più grande quanto più
piccolo è il volume nel quale sono distribuiti i due spin.
3. I tripletti del tipo n * hanno in generale
un’interazione dipolare più grande rispetto ai tripletti
* perché contribuiscono le distribuzioni di spin sullo
stesso atomo, quindi molto vicine (il principio di Pauli in
questo caso non si oppone!).
Time resolved EPR (TR-EPR) spectra of
photoexcited triplets
Photoexcited triplets are born with non Boltzmann
populations of the spin states. Therefore if the EPR
spectrum is obtained shortly after the laser pulse (1 s)
the EPR lines are part in enhanced absorption and part in
emission.
The parameters obtained are D and E , and the spin
populations. The latter ones tell how the triplet was born
(for example by ISC from the excited singlet).
TR-EPR spectra of triplets
3XZFS
3YZFS
3ZZFS
A
2500
3000
3500
4000
4500
EPR spectrum
of a non polarized
triplet
Field (G)
A
E
2500
3000
A absorption
3500
4000
4500 Field (G)
E Emission
TR-EPR spectrum
of a photoexcited
polarized triplet
ZFS as originated from
spin-orbit coupling
The Hamiltonian to be considered as a perturbation in
addressing the ZFS in this case is the same as for g
tensors:
H  geμ B S  B  μ B L  B  L  S
The second and third terms represent the Zeeman
interaction of the orbital angular momentum and the
spin-orbit coupling.
The terms in L couple electronically excited states
into the ground state and contribute in second order.
There are four contribution to the second order
energy. One of these is the term in 2, second order
in S and independent of B. This gives rise to a ZFS
~
spin hamiltonian as in the previous slide: H  S  D
S
ZFS
Quartet states
Quintet states