MagnetochimicaAA 2011-2012

Marco RuzziMarina Brustolon

3. The exchange spin Hamiltonian

Spin Hamiltonians are Hamiltonians containing only spin operators. The energy values of quantum particles should be obtained as eigenvalues of an Hamiltonian containing spatial and spin operators. In some cases it is very convenient to integrate over all spatial variables, leaving only the spin operators multiplied by numerical parameters. By studying the magnetic properties of the system (e.g. by magnetic resonance methods) one can obtain these numerical parameters and relate them to the property of the system. For example, let us consider two electrons with repulsive coulombic interaction, in two different orbitals. We know that an energy contribution (exchange energy) depends on the relative orientations of the two spins (parallel or antiparallel).

This is due to the Pauli principle, as two identical fermions (as two electrons) must be described by an antisymmetric function, the product of the spin and of the spatial functions must be antisymmetric by exchanging the two electrons.

Therefore the spatial and spin functions are interdipendent as their product must be antisymmetric.

1 2 1 2(1,2) ( , ) ( , )r r S S

1 2 1 2'(1, 2) ( , ) ( , )A Sr r S S

1 2 1 2''(1, 2) ( , ) ( , )S Ar r S S

spatial function

spin function

The symmetric spin function is a triplet

The antisymmetric spin function is a singlet

as we will show.

2

12

'(1, 2) '(1, 2) Triplet

eE

r

The energies of the two wavefunctions are different because the two electrons have a minor electrostatic repulsion in the triplet state thanks to the antisymmetric simmetry of the spatial function.

2

12

''(1, 2) ''(1, 2) Singlet

eE

r

This convenient spin Hamiltonian is:

The energies Etriplet and Esinglet can be referred to the two triplet and singlet spin states A and S.

We will show that by defining a parameter:

we can build a spin Hamiltonian such that, acting on the spin states, it gives the energy difference between the two states due to their different spatial functions.

1 22H JS S

where J is called the exchange integral.

This parameter measures therefore the extent of the electrostatic repulsion between the two electrons depending on the relative orientation of their spins.

SingletTriplet EEJ 2

1

Let obtain the eigenfunctions and eigenvalues of the spin exchange Hamiltonian starting from the functions:

and ,,

We will show that the eigenfunctions of this hamiltonian are the singlet and triplet functions, and we will find that their energy gap obtained from the eigenvalues of the spin Hamiltonian is in fact 2J.

We will do the following:

1. Develop the spin Hamiltonian in its component operators;

2. Build the Hamiltonian matrix;

3. Diagonalize the Hamiltonian matrix finding eigenvalues;

4. Find the spin eigenfunction corresponding to each eigenvalue.

(uncoupled basis)

1. How to develop the spin hamiltonian

In the following slides we will use the symbol I for the angular momenta components.

I+ I- are the so called “raising” and “lowering” operators (operatori scaletta).

1 22H JI I

x y x yI I iI I I iI

1 2 1 2 1 2 1 22 ( ) 2 ( )x x y y z zJ I I J I I I I I I

2 2 x y

I I I II I

i

Let us express the spin components x and y as combinations of raising and lowering operators:

1 2 1 1 2 2 1 2 1 2 1 2 1 2

12 ( ) 2 ( )( ) ( )

4 2x x

JJ I I J I I I I I I I I I I I I

1 2 1 2 1 2 1 2 1 22 ( ) ( )2y y

JJ I I I I I I I I I I

1 2 1 2 1 2 1 22 ( ) ( )x x y yJ I I I I J I I I I

Therefore:

Now, let us apply this hamiltonian to the spin functions for two electrons. We use the uncoupled basis: and ,,

1 2 1 2

1 2 1 2

1 12 2

2 21 1

2 22 2

z z z z

z z z z

JI I J JI I J

JI I J JI I J

1 2 1 2 1 2 1 2

1 2 1 2 1 2

2 ( ) 2 ( )

2 ( )

x x y y z z

z z

J I I J I I I I I I

JI I J I I I I

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

( ) 0 ( )

( ) ( ) 0

I I I I I I I I

I I I I I I I I

In conclusion:

Whereas and are eigenfunctions of -2JI1I2,

and are not eigenfunctions of this spin hamiltonian.

Therefore we must find now the eigenvalues and eigenfunctions of -2JI1I2 on the basis and .

1 2 1 2 1 22 ( )z zJI I J I I I I

1

2J

1

2J J

1

2J J

1

2J

2., 3. How to build and diagonalize the matrix of the Hamiltonian

To find the eigenfunctions of a Hamiltonian on a given complete functions basis we can build the matrix on that basis, and then diagonalize it.

H H H H

H H H H

H H H H

H H H H

0

0 0

0 0

0

0

0 0 0

In the present case and are eigenfunctions of the hamiltonian, but the other two functions are not.

If a function of the basis is an eigenfunction of the Hamiltonian, all its out diagonal integrals are equal to zero. We say that the function is not “mixed” with other functions by the operator.

1

2H H J

1

2H H J

H H J

10 0 0

21

0 02

10 0

21

0 0 02

J

J J

J J

J

We must find the two linear combinations of the functions and which diagonalize this block. To do this before we have to find the eigenvalues of the hamiltonian.

We obtain the Hamiltonian matrix elements:

Therefore:

To diagonalize the 2x2 block we must find the energy eigenvalues by solving the secular determinant:

1

2 01

2

J E J

J J E

221

02J E J

2 23 2

2 2

J J J J JE

1

2

3

21

2

E J

E J

Now we will find the linear combinations of and corresponding to each of these eigenvalues.

4. How to find the spin eigenfunctions

1 2

1 2

10

2

10

2

i ii

i ii

J E c Jc

Jc J E c

By normalization we obtain:

Then we can now solve the equations for the coefficients corresponding to the two eigenstates:

1( )

2 Singlet

1( )

2 Triplet

For

1 11 2

c c

singlet

1

3

2E J

For

triplet

2 21 2 c c

2

1

2E J

In conclusion

• The eigenfunctions of the spin exchange operator are the coupled functions for two spins ½:

1

21

2

, SS M

1,1

0,0

1,0

1, 1

1 2

1 1 3( ) 2 ( )

22 2JS S J

1 2

1 1 1( ) 2 ( )

22 2JS S J

1 2

12

2JS S J

1 2

12

2JS S J

Singlet or Triplet?

The energy of a system of two electrons, described by different spatial wavefunctions, and near enough to feel a substantial electrostatic inter-electronic repulsion, depends on the relative orientation of their spins.

The spins can be parallel (triplet, three states) or antiparallel (a singlet). The energy gap is equal to 2J, where J is called exchange integral. The energy order of the spins state depends on the sign of J.

E E

2J> 00 0 2J< 0

E E

2J> 00 0 2J< 0

When J>0 the ground state is a triplet

When J<0 the ground state is a singlet

The spin coupling for two interacting electrons is said to be:

0 0 JJ

Ferromagnetic

Antiferromagnetic

Warning!

• Some authors use a spin exchange Hamiltonian with a J’=-2J

• Sometimes the triplet-singlet energy

separation is called 2K

21' SSJH

When the exchange integral is relevant in determining the electronic properties of a pair of electron spins?

1. Two electrons in same molecule2. Biradicals

4. Near paramagnetic centers S=1/2 in a solid

3. Complexes of pairs of paramagnetic transition ions

1. Two electrons in same molecule

Usually in molecules all the molecular orbitals are doubly occupied. Therefore in the ground state each pair of electrons is in singlet state, and the molecule is diamagnetic.

Molecules in an excited state with the two electrons occupying two different orbitals can be in a singlet or in a triplet state. Usually for the states with the same spatial configuration of orbitals the triplet is at lower energy than the singlet, i.e. J>0.

Photoexcitation takes a molecule from its ground singlet state to an excited singlet, but spin-orbit coupling can produce the intersystem-crossing with the two spins passing from a singlet to a photoexcited triplet state at lower energy.

Se gli stati di singoletto e di tripletto hanno dei livelli vibrazionali molto vicini, in presenza di perturbazioni dovute all’accoppiamento spin-orbita (più grande per atomi pesanti) uno degli spin si gira e si passa dallo stato di singoletto a quello di tripletto che è più basso in energia. Questo fenomeno si chiama “Intersystem Crossing” o ISC.

Dallo stato di tripletto la molecola può subire una emissione spontanea di un fotone (fosforescenza) passando allo stato fondamentale.

Photoexcited triplet states

Diagramma di Jablonski

Energy difference

for antracene 2 eV

Triplet ground states

H

P h e n y lc a r b e n e

N

P h en y ln itre n e

In carbenes and nitrenes the two unpaired electrons are accommodated in a orbital and in a 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.

Energy difference 2J

for phenylnitrene

0.35 eV

N

O

CH3

CH3

CH3

CH3

Radicali nitrossido

Biradicali nitrossido

CH3

N

CH3

O

CH3

CH3

CH3

CH3

Energy difference 2J

10-6-10-8 eV

0.1 eV

Inorg. Chem., 2012, 51 (15), pp 7966–7968

The two Cu(II) ions (S=1/2), are coupled antiferromagnetically. J depends on the bridge ion.

A radical nitronyl nitroxide with its spin density distribution The structure

of the radical crystal

The pattern of exchange interactions between pairs of radicals at different positions

Polyhedron 24 (2005) 2368–2376

The triplet states are degenerate?

2J> 0

E

0

E

0 2J< 0

The three triplet states,

corresponding to Ms= +1, 0,

-1 (as far as the spin exchange Hamiltonian H= -

2J S1S2 is concerned), should

have the same energy -1/2 J.

The energies of the three states are different, and depend on the

Ms quantum number, thanks to magnetic energy contributions:

a. in presence of a magnetic field B, for the Zeeman interaction between the two spins magnetic moments and the field;

b. for the dipolar interaction between the magnetic moments of the two electrons. As this interaction is always present, also without an external magnetic field, it is called Zero Field Splitting (ZFS). This will be treated in the following lectures.

BUT:

Spin Hamiltonian for two interacting electron spins in a magnetic field

Let us consider first the spin Hamiltonian given by the sum of the Zeeman terms and of the exchange term. In an

magnetic field B0, is the following:

21022011 2 SSJBSgBSgH BB Consider B0//z. Then:

21022011 2 SSJBSgBSgH zBzB

If g1g2 the eigenfunctions of the Zeeman term and of the exchange term are respectively those of the uncoupled and of the coupled basis. This means that neither of the two basis diagonalizes the Hamiltonian matrix, and the eigenfunctions of H will be a specific linear combination depending on the relative values of the difference in Zeeman energies and the exchange parameter J. This combinations will be found by diagonalizing the Hamiltonian matrix.

In conclusion• We have seen that an energy contribution due to electrostatic electron-electron

repulsion can be represented by a spin hamiltonian, -2JS1S2

• The eigenfunctions of the exchange Hamiltonian are the triplet and singlet functions.

• If J>0 the triplet lies lower in energy than the singlet; the absolute value of energy difference is 2J.

• The value of J depends on the extent of electrostatic interaction between the two electrons; approximately it decreases linearly on increasing the distance r between the electrons.

• J is “large” for electrons on the same molecule, as in photoexcited triplets for molecules with a ground singlet state, or in molecules with ground triplet states, as carbene and nitrene. In these cases all the magnetic interactions are smaller than the spin exchange term (the spin coupled basis is a good basis).

• J is “small” for biradicals, which are usually two stable radicals linked together by a non-conjugated chain. The energy difference 2J can be of the same order as kT, and the populations of the triplet and singlet in this case depend on the temperature. The spin eigenfunctions depend on the relative values of the various terms of the spin Hamiltonian.