03/01/2014
Magnetochimica
AA 2013-2014
5. Unpaired electrons in organic systems
Delocalization of unpaired electrons over several atoms confers extra
stability to the radicals with respect to localized radicals (in general
much more reactive).
Molecular Orbitals (MO) is often expressed in a mixture of valencebond and molecular orbital terms, with typically valence-bond language
σ framework and molecular orbital language used to describe its used
for its π electrons.
E
E
pz
π
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Summary
•
Molecules with π delocalized electrons.
•
MO of π electrons (Hückel method).
•
Proton hyperfine splitting and spin density. McConnell relationship.
Selection of systems
•
Radical anions, cations and neutral radicals
•
Nitroxydes
•
non-Kekulé molecules
•
Photoexcited triplets
•
High spin states
•
Graphene
S = 1/2
S > 1/2
CπS 1
Conjugated π systems
Molecules with π delocalized electrons.
MO of π electrons (Huckel method).
It can be shown that for planar structures, for light elements σ
framework can be treated separately from π electrons.
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CπS 2
Alternant hydrocarbons
An alternant hydrocarbon is one in which the atoms belongs to two groups:
starred and unstarred. By putting a star to non-adjacent atoms, if at the end one
gets only starred atom linked only to unstarred atoms, and viceversa, the
hydrocarbon is said to be alternant.
*
*
.
E
*
*
*
non
alternant
* *
*
alternant
If the number of conjugated atoms is even or odd,
the system is said to equal or even alternat or odd
alternant respectively.
The energy levels are arranged symmetrically
around zero energy; also pairs of OM, one bonding
and antibonding one, corresponding to energy levels
symmetrical, have the coefficients of the linear
combination of each atom equal in absolute value,
but with opposite sign to the starred atoms.
For alternating odd number of atoms marked
(starred) and not starred is different. It then
chooses the stellar C in greater numbers.
*
*
0
NBMO
(SOMO)
Allyl
Odd Alternant
2
*
E
1
4
*
3
LUMO
HOMO
Butadiene
Even alternant
CπS 3
Example:
benzene
σ-structure
Hückel Approximations
π-structure
1. Sii=1, Sij=0
2. All the diagonal elements of
the Hamiltonian Hii have the
same value, α (only for
hydrocarbons)
3. All the off-diagonal elements
of the Hamiltonian, Hij , have the
same value, β , if atoms i and j
are adjacent, otherwise the
value is 0
Energies of the OM
ε1,6 = α +/- 2β
ε2,3 = α + β
ε4,5 = α - β
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π Radicals cations and anions
Molecules with conjugated π electrons with an even number of atoms
belonging to the delocalized MO can give radical anions when reduced
(the unpaired electron goes to the LUMO), and radical cations when
oxidized (an unpaired electron is left in the HOMO).
-2β
β
-β
β
β
LUMO
(SOMO)
2β
β
-2β
β
-β
β
β
HOMO
2β
β
(SOMO)
π Radicals 2
In a first approximation the spin density on each of the C atoms of
a π system radical is given by the square of the coefficient of
its AO in the SOMO (probability of finding the unpaired
electron on that C atom).
An experimental determination of the spin density
distribution on the conjugated frame can be done
by measuring the hyperfine coupling constants of
H atoms linked to the C atoms*.
Electron-nuclear spin interaction
has two contributions:
C
-
H
- Fermi interaction (electron
density is present at the nucleus)
- π-σ
σ Spin polarization mechanism
For undistorted (planar) π radicals only the π-σ spin
polarization mechanism is active
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π-σ spin polarization
The pair of electrons ("spin up" and "spin down“) of the C–H bond have
both a repulsive Coulomb interaction with the unpaired electron. However,
this repulsive interaction is slightly weaker when the electron of the C-H
bond has the same spin state as the unpaired electron, than with the other
one (Pauli principle). The unbalanced spatial distribution of the electron
spin state resulting from the different interaction is called spin
polarization.
Consequently, the spatial distributions of the two electrons of the C-H
bond become slightly distorted; the σ electron with the same spin as the
unpaired electron “moves” towards the C atom, and the other one “moves”
toward the H atom.
In conclusion, on the H atom there will be a negative spin density (i.e. a
spin density with opposite sign with respect to that of the unpaired
electron).
C
C
H
H
On the H nucleus there
is therefore some spin
density of opposite sign
with respect to that on
the 2p orbital.
The spin density transferred by spin polarization on the H is proportional to
that on the corresponding C atom, and it has a sign opposite to that on the
carbon atom.
aH = Q · ρC
This simple relationship (McConnell equation),
with the semiempirical constant Q < 0, allows
to predict the value of the H h.c.c. in
conjugated π systems.
ci2: MO coeff.
The measurement of hyperfine coupling constants in π
radicals has been a very important experimental
benchmark for theoretical approaches to quantum
calculations in the last fifty years. No other
experimental parameter is so directly linked to
electron distribution on a molecule!
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EPR spectrum of Naphtalene radical anion
1 2 −•
5 Gauss
Spectrum like
AX4Y4.
We just observe
the transitions
of A
A quintet of
quintets, each
1:4:6:4:1
Then each line is
further split in 5
lines 1:4:6:4:1
1:4:6:4:1
The Q “constant” varies
approximately in the range
|Q| = 23-30 G (2.3-3.0 mT).
aHi = Q ρc = Q ci2
aHi
ci2
C2=0.180
Hϋckel
calculation
C2=0.069
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Radical
aH/G
exp
ρC
Q/G
CH3•
-23,04
1
-23,04
C5H5•
-5,98
1/5
-29,9
C6H6–•
-3,75
1/6
-22,5
C6H6+•
-4,28
1/6
-25,7
C7H7•
-3,95
1/7
-27,7
C8H8–•
-3,21
1/8
-25,7
McConnell
equation
aH=Q×ρC
Although ESR spectra
cannot give directly
the sign of the
hyperfine coupling
constant (h.c.c.),
there are methods to
obtain this sign. It is
found that all the
h.c.c.’s of the Table
are negative.
MO and Spin density
McConnel’s equation aHi = Q ρc relates the HF interaction with the spin density.
The spin density is related to the fraction of time, or to the charge distribution
of the electron on a particular C atomic orbital.
In odd alternant hydrocarbons, where nodes are expected at the
unstarred atoms, there are some problems that must be solved.
On the basis of Hückel results, the SOMO in odd alternant
hydrocarbons has nodes in correspondence of the unstarred atoms:
The NBMO has
a node on the
unstarred atom
NBMO
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MO and Spin density
•
Therefore the h.c.c. of proton bound to C2 should be zero. However, the
ESR spectrum of allyl radical shows the following aH h.c.c.’s:
The signs of these h.c.c.’s are :
positive
aH=Q×ρC
Q<0
therefore aH<0 means
and viceversa
negative
ρC>0
We have seen that for ρC > 0 we should have an aH< 0 (by π- σ spin
polarization).
A positive sign of aH therefore must correspond to ρC < 0. This is due to
another type of spin polarization, i.e. the π- π spin polarization.
Similarly to the π- σ spin polarization, it is due to spin dependent repulsive
interaction between the unpaired electron and the electrons of the
filled orbitals.
Let’s indicate the spin orbital
with alpha spin in red, and that
with beta spin in blue.
smaller
repulsion
larger
repulsion
Following the monoelectron approach of the simple Huckel theory, the
two alpha and beta spin orbitals should be identical and at the same
energy. But we know that the repulsive electron-electron interaction
will be smaller for alpha-alpha than for beta-alpha spin orbitals.
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π- π spin polarization on the
central C atom (more “blu” spin
than “red”= negative spin
density)
Following the monoelectron approach of the simple Huckel theory, the
two alpha and beta spin orbitals should be identical and at the same
energy. But we know that the repulsive electron-electron interaction
will be smaller for alpha-alpha than for beta-alpha spin orbitals.
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Nitroxides
MNDO/3
C
S
PX
PY
PZ
OM c‘ s
-4E-5
1E-4
-1E-4
0.018
N
S
PX
PY
--1E-5
-6E-5
8E-5
0.651
PZ
C
--5E-5
6E-5
-9E-5
0.018
O
S
PX
PY
--2E-5
1.3E-4
-5E-5
PZ
-0.717
S
PX
PY
PZ
Semiempirical methods use parametrized
overlap integrals and other parameters
obtained from comparison of observables
with experimental values.
Bulky groups around th N-O group
reduces the reactivity. The radicals
can survive even in physiological
conditions for hours.
55%
N-O
45%
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Neutral radicals
Odd alternant hydrocarbons
We have already seen that the conjugated systems with an
odd number of π centers have naturally unpaired electrons.
In alternant odd
molecules we have a
different number of
starred and unstarred
atoms. Let us choose to
have the largest group
starred.
Property of these systems is that the spin density generated by an
unpaired electron in a NBMO-SOMO orbital is zero at the unstarred
carbon positions.
Even alternant hydrocarbons
and non-Kekulè molecules
We expected that even alternant molecules have all the electrons paired in
orbitals, but there are exceptions!
A non-Kekulé molecule is a conjugated hydrocarbon that cannot be
assigned classical Kekulé structures (all the π electrons in double bonds).
Since non-Kekulé molecules have two or more formal radical centers, their
spin-spin interactions can cause electrical conductivity or ferromagnetism
(molecule-based magnets), and applications to functional materials are
expected.
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The smaller molecules are quite reactive and most of them are
easily decomposed or polymerized at room temperature,
strategies for stabilization are needed for their practical use.
Synthesis and observation of these reactive molecules are
generally accomplished by matrix-isolation methods.
The simplest non-Kekulé molecules are biradicals.
Figure S3. Fused -topology. Topological symmetry analyses indicate that bisphenalenyl
S3 and 10 with non-Kekulé structure are in the triplet ground states, while
S4 and 11 with Kekulé structure are in the singlet ground states. Among the physical
properties of phenalenyls, the ground-state spin prediction can be carried out in terms of
the starred-unstarred approach. The spin density distributions of S3 and 11
are calculated using Gaussian 03 program with the UB3LYP/6-31G* level of theory.
Nature Chemistry 3 (2011) 197-204
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Non-kekulè hydrocarbons: a method for
finding the number of unpaired electrons
• A benzenoid structure can be oriented in three different ways
with some of its edges (approx. 1/3) in a vertical direction.
• A benzenoid structure so oriented has peaks (upward pointing
vertices on the upper periphery) denoted by Λ and valleys
(downward pointing vertices on the lower periphery) denoted
by V.
Gordon and Davison have
shown that whenever Λ ≠ V
then the corresponding
benzenoid structure is a
radical.
V -Λ
Λ = 1 for
monoradicals
V -Λ
Λ = 2 for
diradicals,
etc.
Phenalenyl and derivatives
V -Λ
Λ = 1 for
monoradicals
V -Λ
Λ = 2 for
diradicals,
etc.
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Carbenes and nitrenes:
triplet ground states
• As a general rule, when the two interacting electrons
in a neutral organic molecule belong to two half-filled
molecular orthogonal orbitals, the triplet state is
lower in energy with respect to the singlet.
• For example in carbenes and nitrenes the two
unpaired electrons are accommodated in a σ orbital
and in a π orbital respectively on a single C or N atom.
These species can be obtained by photolysis of a
suitable precursor in a glassy matrix or in a crystal,
since in solution they would be non persistent.
N
H
P h e n y lc a r b e n e
P h e n y ln i tr e n e
S/T splitting in the Strong exchange case
ψ (1,2) = ϕ A (1)ϕ B (2) + ϕ B (1)ϕ A (2)
To estimate the singlet-triplet splitting one has to consider the full Hamiltonian
E = ϕ A (1)ϕ B (2) + ϕ B (1)ϕ A ( 2) H Coulomb ϕ A (1)ϕ B (2) + ϕ B (1)ϕ A ( 2)
−
1
1
1
1
1
−
−
−
+
RA1 RB1 RA 2 RB 2 R12
Let us consider in detail the terms contributing to J.
ϕ A (1)ϕ B (2) −
1
1
1
1
1
−
−
−
+
ϕ A (2)ϕ B (1)
R A1 RB1 R A 2 RB 2 R12
Electron-nucleus
interactions
Electron-electron
interaction
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By developing terms of electron-nucleus interaction we obtain:
ϕ A (1)ϕ B (2) −
1
1
ϕ A (2)ϕ B (1) = ϕ A (1) −
ϕ B (1) ϕ B (2) ϕ A (2)
R A1
R A1
Resonance integral
β < 0
Overlap S > 0
By developing terms of electron-electron interaction we obtain:
1
ϕ A (2)ϕ B (1) = K
K > 0
R12
By this analysis in conclusion we find that the exchange energy is
made up by terms like:
ϕ A (1)ϕ B (2) +
J = 2βS + K
negative positive
AF
FM
POSITIVE
The separation between singlet and
triplet is 2J. If J<0 the singlet is lower
in energy.
To have a FM interaction (triplet), the
two two unpaired electrons should have
K > 2βS
NEGATIVE
Photoexcited triplets:
a simple approach for ZFS calculations
For a photoexcited triplet π π* 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 (b) and HOMO (a).
Hückel approach:
Slater determinant
D
E
= 3ψ
D
ψ
3
E
ψ = (| ...s sab | + | ...s sab |) / 2 = {ab}
3
MO = ∑i ciMO ϕi
ψ
3
D
E
ψ =
3
D
1
(cia c bj − cib c aj )cra*csb* ϕi (1)ϕ j (2) ϕ r (1)ϕ s (2)
∑
E
2 ijrs
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ϕi (1)ϕ j (2)
−
3g 2 β 2
4hca03 Rγδ3
D
E
Approximate expression by
S.A. Boorstein, M. Gouterman J. Chem. Phys. 41 (1964) 2776.
ϕ r (1)ϕ s (2) =


αi α j αr αs

2
2
+
+
[(α
α
)/2]
[(α
α
)/2]
 i

r
j
s
5/4

 
1
α j αsr
 
- 1 + 9/r0 ' 2 -225ω/4r0 ' 4  
αi αr
 ⋅ P(r0 ' )
⋅ exp −
rir2 −
rjs2  ⋅ 
2
4 
(α j + αs )   cos(2φγδ )  
- 1 + 5/r0 ' -105ω/4r0 '  
 (αi + αr )
 ωr ' 3 / 2 + (15ω / 2 − 4) r0 ' +(75ω/2 - 18)/r0 ' +225ω/2r0 ' 3 
+ Q(r0 ' ) 03
3
ωr0 ' / 2 + (7ω / 2 − 4 / 3) r0 ' +(35ω/2 - 10)/r0 ' +105ω/2r0 ' 
r0 '
P(r0 ' ) = (1/2π )1/2 ∫ exp( −t 2 /2)dt
− r0 '
ω=
Q(r0 ' ) = (1/2π )1/2 exp( − r0 '2 /2)
r0 ' =
2
2(αi + αr )(α j + αs )Rγδ
(αi + α j + αr + αs )
Rγδ =| rγ − rδ |
rγ =
rδ =
2
4(αi + αr )(α j + α s )
(αi + α j + α r + α s ) 2
cos(2φγδ ) =
Yγδ2 − X γδ2
Yγδ2 + X γδ2
Rγδ2 = X γδ2 + Yγδ2
αi ri + αr rr
αi + α r
α j r j + αs rs
α j + αs
The ZFS parameters for trans-nitrostilbene
measured in a frozen toluene
|X|=6 mT
|Y|=57 mT,
|Z|=63 mT
Fully planar:
Tilted –NO2
(charge transfer):
X=-11 mT
Y=-27 mT,
Z= 38 mT
X=-18 mT
Y=-40 mT,
Z= 58 mT
Phys. Chem. Chem. Phys., 2004, 6, 2396–2402
…One can also make still black-box
calculations with packages like Gaussian
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Quartet states
Quintet states
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Graphene
Not properly an organic molecule
but we start with a bottom-up
contruction of graphene, starting
from organic molecules
Graphene: a ….magnetic material
?
?
Curie-Weiss with
Néel T of 18 and
36 K
?
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Bottom-up approach to graphene
Synthetic organic spin chemistry for structurally welldefined open-shell graphene fragments
Nature Chemistry 3 (2011) 197-204
Yasushi Morita, Shuichi Suzuki, Kazunobu Sato & Takeji Takui
.
Phenalenyl — a triangular neutral radical consisting of three adjacent benzene
rings — and π-conjugated derivatives based on the same motif, can be viewed as
'open-shell graphene fragments'. This Perspective discusses their electronicspin structures, the properties that arise from their unpaired electrons, and
highlights their potential applications for molecular spin devices.
Bottom-up approach to graphene
Edge states: a molecular approach
Arm-chair
termination
Zig-zag
termination
Zig-zag terminations are normally often locate LOCALIZED electrons, in the
sense that they do not extend over the entire molecule.
The MO has densities that are not spread over the entire structure, in fact it
is localized at the EDGE of the structure.
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T. Enoki et al, Phys. Rev. B 81 (2010) 115408
Why looking for organic
ferromagnetism
• The development of new organic ferromagnetic
materials is a challenge, and is being pursued
vigorously owing to their useful and attractive
properties, such as:
• being lightweight;
• their solubility in organic solvents giving rise to the
possibilities of liquid magnets, colloidal dispersions
and Langmuir- Blodgett films;
• their transparency in many spectral regions making
them suitable for photomagnetic switches and optical
data storage;
• and, the possibility of perpendicular magnetic
ordering leading to higher density of data storage.
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