MagnetochimicaAA 2012-2013

Marco RuzziMarina BrustolonMarina 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

MagnetochimicaAA 2011-2012

Marco RuzziMarina Brustolon

The coupling of Angular Momenta

The coupling of angular momenta 1

22J 2

ˆzJ with eigenvalues

or, considering the two particles together21

,,, 21 JJ mmjj

The wavefunctions of the two particles can be referred to these quantum numbers, therefore:

1,1 Jmj 2

,2 jmj

1

)1( 211 jmjj

12

1ˆ ˆzJJ with eigenvalues

22

22 )1( jmjj

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.

• There are

The coupling of angular momenta 2

states

For example, if the two momenta are two spin ½, there are 2x2 =4 states:

1 2 1 2

1 2 1 2

1 1 1 1 1 1 1 1, ,

2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1,

2 2 2 2 2 2 2 2

We can use a shorter notation, as J1 and J2 are always 1/2:

2121,,,, 21 JJJJ mmmmjj

21,,, 21 JJ mmjj)12)(12( 21 jj

The coupling of angular momenta 3

For the orbital momenta of two p electrons, J1 = J2 =1 , therefore

1 2J Jm m

11 , 10 , 01 , 00 , 1 1 , 11 , 10 , 0 1 , 1 1

1 2(2 1) (2 1) 3 3 9J J states

The states are eigenstates of Jz1, Jz2 and also of Jz , as these three operators commute.

1 2J Jm m

Each of these states is an eigenstate of Jz1 and Jz2. Moreover, they are eigenstates of

1 2

ˆ ˆ ˆz z zJ J J with eigenvalues )(

21 JJJ mmM

The coupling of angular momenta 4

Mtot=

m1+m2

2

1

1

0

0

0

-1

-1

-2

21 ,mm

1,1

0,1

1,0

0,0

1,1

1,1

0,1

1,0

1,1

21J

22J 1z

J2z

J zJ

These states are eigenfunctions of the operators:

with quantum numbers

tot2121 M m m j j

221

22 JJJJ tot

but they are not eigenfunctions of

2Jas does not commute with 1z

J2z

J

The coupling of angular momenta 5 So, we have two choices: either use the basis set of eigenfunctions of:

21J

22J 1z

J2z

JzJ(and ) :

State M =m1+m2

2

1

1

0

0

0

-1

-1

-2

1,1

0,1

1,0

0,0

1,11,1

0,11,0 1,1

Mmmjj ,,,, 2121

or find a basis set of eigenfunctions of:

21J

22J

2J zJ

MJjj ,,, 21

Uncoupled basis

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

-2

-1

-1

0

0

0

1

1

2

M =m1+m2

State

1,1

0,1

1,0

0,0

1,11,1

0,11,0 1,1

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

MJjj ,,, 21

For J = 2 we have five functions::

2,2,1,1

1,2,1,1

0,2,1,1

1,2,1,1

2,2,1,1

The two functions in orange can give two independent linear combinations: one of the two is this coupled function.

The two functions in blue can give two independent linear combinations: one of the two is this coupled function.

The coupling of angular momenta 7

-1

-1

0

0

0

1

1

M =m1+m2

State

0,1

1,0

0,0

1,11,1

0,11,0

J = 2 ,1, 0

For J = 1 there are 3 states, with MJ = 1, 0, -1

MJjj ,,, 21

For J = 1 we have three functions::

1,1,1,1

1,1,1,1

0,1,1,1

The other linear combination is this coupled function.

The other linear combination is this coupled function.

The coupling of angular momenta 8

0

0

0

M =m1+m2

State

0,0

1,11,1

J = 2 ,1, 0

For J = 0 there is 1 state, with MJ = 0

MJjj ,,, 21

For J = 0 we have one function:

0,0,1,1

These three functions can give three independent linear combinations: this one, and the others indicated in the previous two slides.

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:

2121212121 ,,,),,,,(,,,21

mmjjmmJjjCMJjjmm

with M =m1+m2

The C-G coefficients can be obtained with recursion formulae, or can be found in tables.

Valore di Jtot

M

Base disaccoppiataCoefficiente della combinazione lineare

Base disaccoppiata

Coefficienti del tripletto

Coefficienti del singoletto

Tables of Clebsch-Gordan coefficients

For two spins =1/2

Jtot=

Jtot=

1 2 1 2

1 2 1 2

1 1 1 1 1 1 1 1, ,

2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1,

2 2 2 2 2 2 2 2

1 21 2, , ,J Jj j m mTwo spin =1/2, uncoupled basis:

Coupled basis, following the Clebsch-Gordan table:

1 2, , ,j j J M

1 1 1 1 1 1, ,1,1 , , ,

2 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1 1, ,0,0 , , , , , ,

2 2 2 2 2 2 2 2 2 22

Triplet, J=1, M=1

Singlet, J=0,M=0

1

2

etc.

La somma delle loro proiezioni sull’asse z è sempre definita

Due momenti angolari accoppiati

I due momenti sono disaccoppiati, cioè ciascuno può essere sul suo cono di precessione in qualunque posizione indipendentemente dall’altro.

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 .

Due momenti angolari disaccoppiati

s1

s2

S=1MS=+1

MS=-1

S=1

s1

s2

s2

s1

S=1MS=0

2

1

Rappresentazione vettoriale dello stato di tripletto

MS=0

s2

s1

S=0

2

1

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:

........

2,,,

1,,,

,,,

21

21

21

JJjj

JJjj

JJjj

JMJjj ,,, 21 with JJJM J ,...,1,

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

yx

yx

iJJJ

iJJJ

JJ

The effects of raising and lowering operators on a function characterized by J and MJ are the following (see note*):

Using the raising and lowering spin operators 2

For example let us consider the simple pair of spin functions and :

1,))2/12/1)2/3(2/1(

0))2/32/1)2/3(2/1(21

21

21

2121

21

ImIII

II

*We use here different symbols: I instead of J, mI instead of MJ

1,))1()1((,

1,))1()1((,21

21

IIII

IIII

mImmIImII

mImmIImII

Therefore:

I- = I+ = 0 I- = 0 I+ =

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:

• http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CCwQFjAB&url=http%3A%2F%2Fwww.stanford.edu%2Fgroup%2Ffayer%2Flectures%2FChapter15-08.ppt&ei=zK5yULLyEaLg4QS2iID4Cw&usg=AFQjCNEqloSuvqmXYnMWOidt3G-_Wwj4Ag