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Magnetochimica AA 2012-2013 Marco Ruzzi Marina Brustolon
1. The coupling of Angular Momenta 2. EPR in a nutshell 3. The exchange spin Hamiltonian 4. The Zero Field Splitting spin Hamiltonian 5. Radicals with delocalized electron spin density

Magnetochimica AA 2011-2012 Marco Ruzzi Marina Brustolon
The coupling of Angular Momenta

The coupling of angular momenta 1
Two non interacting particles, each with a constant angular momentum, are characterized each by its own eigenvalues of the operators magnitude of the vectors and component along z. with eigenvalues with eigenvalues The wavefunctions of the two particles can be referred to these quantum numbers, therefore: or, considering the two particles together

The coupling of angular momenta 2
There are states For example, if the two momenta are two spin ½, there are 2x2 =4 states: We can use a shorter notation, as J1 and J2 are always 1/2:

The coupling of angular momenta 3
For the orbital momenta of two p electrons, J1 = J2 =1 , therefore states Each of these states is an eigenstate of Jz1 and Jz2. Moreover, they are eigenstates of with eigenvalues The states are eigenstates of Jz1, Jz2 and also of Jz , as these three operators commute.

The coupling of angular momenta 4
These states are eigenfunctions of the operators: with quantum numbers Mtot= m1+m2 2 1 -1 -2 but they are not eigenfunctions of as does not commute with

The coupling of angular momenta 5
So, we have two choices: either use the basis set of eigenfunctions of: (and ) : Uncoupled basis State M =m1+m2 2 1 -1 -2 or find a basis set of eigenfunctions of: Coupled basis 1. The dimensions of the basis sets are the same. 2. The values of J vary between j1+j2 , j1+j2-1,…, | j1+j2| 3. Each function of the coupled basis with a value Mk is a linear combination of the functions of the uncoupled basis with the same Mk value.

The coupling of angular momenta 6
Therefore if j1=1 and j2=1, the possible J values are: J = 2 ,1, 0 For each J value there are 2J+1 states, with MJ = J, J-1,…,-J For J = 2 we have five functions:: State M =m1+m2 2 1 Combination The two functions in orange can give two independent linear combinations: one of the two is this coupled function. 1 Combination The two functions in blue can give two independent linear combinations: one of the two is this coupled function. Combination -1 -1 -2

The coupling of angular momenta 7
J = 2 ,1, 0 For J = 1 there are 3 states, with MJ = 1, 0, -1 For J = 1 we have three functions:: State M =m1+m2 1 Combination The other linear combination is this coupled function. 1 Combination Combination The other linear combination is this coupled function. -1 -1

The coupling of angular momenta 8
J = 2 ,1, 0 For J = 0 there is 1 state, with MJ = 0 For J = 0 we have one function: State M =m1+m2 These three functions can give three independent linear combinations: this one, and the others indicated in the previous two slides. Combination

The coupling of angular momenta 9
The coefficients of the linear combinations of the uncoupled basis to give the coupled one are the Clebsch-Gordan coefficients: with M =m1+m2 The C-G coefficients can be obtained with recursion formulae, or can be found in tables.

Tables of Clebsch-Gordan coefficients
For two spins =1/2 M Jtot= Valore di Jtot Base disaccoppiata Coefficiente della combinazione lineare Jtot= Coefficienti del tripletto Coefficienti del singoletto Base disaccoppiata

Two spin =1/2, uncoupled basis:
Coupled basis, following the Clebsch-Gordan table: Triplet, J=1, M=1 Singlet, J=0,M=0 etc.

Due momenti angolari disaccoppiati
I due momenti sono disaccoppiati, cioè ciascuno può essere sul suo cono di precessione in qualunque posizione indipendentemente dall’altro. La somma delle loro proiezioni sull’asse z è sempre definita Due momenti angolari accoppiati I due momenti sono accoppiati, e la loro somma vettoriale dà il momento totale J. Ciò significa che non sono in una posizione qualsiasi uno rispetto all’altro, ma sono accoppiati in modo da dare sempre come somma vettoriale J. Si noti inoltre che negli stati nei quali è definito J, restano definiti j1 e j2, ma non sono più definiti m1 e m2, ma solo la loro somma M .

S=1 MS=+1 S=1 s2 s2 s2 s1 s1 MS=-1 S=0 s2 MS=0 s1 MS=0 S=1 s1
Rappresentazione vettoriale dello stato di tripletto Rappresentazione vettoriale dello stato di singoletto

Using the raising and lowering spin operators 1
We know that the eigenfunctions of angular momentum operators J2 and Jz are characterized by quantum numbers J and M. For each J value we have a family of functions with different values of M: with The effect of the so called raising and lowering operators: is to transform a function with MJ respectively to the one with MJ+1 and MJ-1

Using the raising and lowering spin operators 2
The effects of raising and lowering operators on a function characterized by J and MJ are the following (see note*): For example let us consider the simple pair of spin functions  and : Therefore: I- =  I+ = 0 I-  = 0 I+  =  *We use here different symbols: I instead of J, mI instead of MJ

Exercise: Obtain the spin functions of the coupled basis from an uncoupled basis for two electron spins (or any other angular momentum with J=1/2), by using the raising and lowering operators.

So, which basis of eigenfunctions for two or more angular momenta should be used?
Coupled or uncoupled? The answer stays in the type of spin Hamiltonian, as we will see.

Nel sito WEB della Stanford University con questo indirizzo trovate una utile serie di slides sui momenti angolari:

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