Magnetochimica AA 2012-2013 Marco Ruzzi Marina Brustolon
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Transcript of Magnetochimica AA 2012-2013 Marco Ruzzi Marina Brustolon
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