group theory-part 12 correlation diagrams
TRANSCRIPT
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Correlation DiagramsCorrelation Diagrams
C734b 2008
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Spin PostulateSpin Postulate
An electron possesses an intrinsic angular momentum in addition to its normal orbitalangular momentum L
S
Electrons exhibit a magnetic moment r
where
=
=
ee
Be m
eS gS g
2
r
h
rr
B where em
mass of the electron and
Bohr magneton
123 JT109274.02
==e
B meh
s =
for electrons and g e
=2.00232
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S is an angular momentum like L with components S x
, Sy
and S z
and associatedself-adjoint operators:
2,,, S S S S z y x which obey similar commutation relationships as2,,, L L L L z y x
2S z y x S S S ,, all commute with but not with each other
Only one component, say zS
can have a common set of eigenfunctions with 2S
Note:Note:2,,, S S S S z y x
f(x, y, z) which means they commute with2,,, L L L L z y x
Total angular momentum of electrons S L J rrr
+= where 2, J J i obey similar
commutation relationships as 2, L Li
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For any angular momentum Ar
Can define raising and lowering operators + A,A as
yx
yx
AiAA
AiAA
=
+=
+
Let the eigenfunctions ofz
2 AandA be aa m, j
with eigenvalues ( ) hh a2aa mand1 j j + respectively.
Can show that: ( ) ( ) 1m, j1mm1 j jm, jA aaaaaaaa += h
am ranges from j a
to +j a
in integer steps.
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Since
2
1s =r
for electrons ( ) hhhh2
1mmand
4
31sss ss
222 ===+=
The two spin eigenvectors are: up)(spin21
,21
ms, s =
and down)(spin21
,-21
ms, s =
121
,21
21
,21
= and 021
,21
21
,21
= m
Spin functions are orthonormal: normalized and orthogonal.
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Spherical Symmetry for many electron atoms (N)Spherical Symmetry for many electron atoms (N)
SLeeo HHHH rr
++=Hamiltonian is given by:
oH
kinetic energy of electrons and the e -
-
nucleus interactions
If o H H
= alone ( ) ( ) ( ) ( ) N21 N,1,2, LL =
with each one e -
state characterized by 4 quantum numbers, n, , m
, m s
eeH
electron
electron
interaction which couples angular momenta of the individualelectrons in 2 possible ways
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LowLow ZZ elements (Z ~< 40)elements (Z ~< 40)
==i
ii
i sSL rr
lrr
Spin-orbit interactions couples SLJformtoSandLrrrrv
+=
RussellRussell --Saunders couplingSaunders coupling
HighHigh ZZ elements (Z > 40)elements (Z > 40)Orbital and spin angular momenta of each electron couples first:
iii s j rlrr
+=
total angular momenta of each individual electron.
These then couple to total =i
i jJrr
j j -- j j couplingcoupling
Note:Note: can have intermediate couplingfor intermediate Z although L-S(Russell-Saunders) scheme is oftenused as a first approximation.
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Note:Note: For low Z S.Lee HH rr>
For high Z 22SLee Z~SLHH vrQrr 2 angular momenta, couple A 1
and A 2
A12
, then A 12
+ A A123
, etc.
The effect of spin-orbit interactions is to split the multiplets
into theircomponents with term symbols: 2S+1 L
J
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Spin-orbit splitting: ( ) ( ) ( )[ ]1SS1LL1JJS)(L,21
E SL +++=rr
where ( )S L,
spin-orbit coupling constant > 0 for < -filed shells
smallest J lies lowest in energy
If
< 0 for > -filled shells largest J lies lowest in energy
HundHund s third rules third rule
Example:Example: (ns) 1(np) 1
configuration 1
= 0; 2
= 1 L = 1s1
= ; s 2
= S = 1, 0 terms are 3P, 1P
When S = 0, L = 1, J = 1 1P1When S = 1, L = 1, J = 2,1,0 3P2,1,0
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-2
-
3P0
3P1
3P2
-H ee
has 2 parts: a Coulomb repulsion J and an exchange interaction K which isnon-classical and is a consequence of the Pauli-Exclusion Principle which requires the totalwave function to be antisymmetric with respect to the interchange of two spin
particles(electrons which are fermions).
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HL.S
3P0
3P1
3P2
1P1
3P
1P(ns)1
(np)1
-K
+K
J
Hee
Ho
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Intermediate Crystal Fields (low Z elements)Intermediate Crystal Fields (low Z elements)
Let H CF
term in the Hamiltonian which describes the electrostatic interaction with thesurrounding ions or ligands.
if > eeCF HH strong crystal field
if >>SLCFee HHH
rr intermediate crystal field
if electron-electron interactionsin ion.
Consider the effect on free ion electron configurationsfree ion electron configurations and deduce states and theirdegeneracies
Later: will correlate strong
to intermediate CFs.
In O h
symmetry 5d orbitals t2g
(dxy
, dyz
, dxz
) + e g
(dz2
, d x2-y2
) and E(t 2g
) < E(e g
)since these orbitals point
at the ligands.
Opposite scenario occurs in T d
symmetry where E(e g
) < E(t 2g
)
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To determine states in a strong field, use BetheBethe s method of Descending Symmetrys method of Descending Symmetry
Method based on:(i)
Electrostatic fields dont affect spin
ji
(ii) If (1,2) = i(1) j(2) and i(1) forms a basis for i
and j(2) forms a basis for j
means i(1) j(2) forms a basis for the direct product
Due to Pauli Exclusion Principle:(1)
2 electrons in the same orbital generates a singlet state only.
(2)
2 electrons in different orbitals generates a singlet and a triplet state.
E lE l
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Example:Example:Find all states that forma d 2
configuration in a strong field of O h
symmetry. Correlatewith those of the free ion and those of an ion in an intermediate field.
Configurations are (t 2g
)2, (t 2g
)1(eg
)1
and (e g
)2
Parity of d 2
terms
g.
Simply use O character table to reduce direct products
O character table:
11103T
11103T
00212E
11111A
11111A
6C6C3C8CEO
2
1
2
1
'2423
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Consider (e g
)2
6 states:2 singlets when the 2 electrons in the same e g
orbital, and 1 singlet and 1 tripletwhen they are in different e
g
orbitals
6 state functions are contained in the direct product of gg ee
gg ee reduces in O to eaa 21
Dont know which are singlets and triplets.
Bethess
method is to lower the symmetry until all representations in the direct productin the direct productare one-dimensional
Examine the Correlation Tables
In D 4h
symmetry: A 1g
A1g
; A 2g
B1g
; E g
A1g
+ B 1g
e1g a1g
+ b 1g
orbitals; that is, they split.
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1g12
1g
1g
3
1g
11
1g
1
1g
1g12
1g2g
A bB,B ba
Aae
Since the electrostatic field does not affect spin means that the 3B1g
state in D4h
must havecome from 3A2g
state in O h
.All other states must be singlets.
in O h
d2
1A1g
, 3A2g
, 1Eg
Total degeneracy = 6 as expected ( = (2S+1)x state degeneracy = 1x1 + 3x1 +1x2)
Next:Next: do t 2g2
configuration in O h
Reduce in C 2h
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2g1gg1g2g2g tteatt =
ga
g
g
b
a
g
g
g
b
b
a
g
g
g
b
a
a
( ) ( )2hggggggh2g2g C baa baaOtt
P t i 2 l t
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Put in 2 electrons:
g12
g
g12
g
g3
g11
g1g
g3
g11
g1g
g3
g11
g1g
g12
g
A bA(2)a
B,B(2)ba
B,B(1)baA,A(2)(1)aa
A(1)a
We are looking for triplet state(s) in C 2h
that transform as g3
g3
g3 BBA
In O h
this mustmust be the T 1g
state
Therefore, states are: h2g1
1g3
g1
1g1 OinTTEA
Total degeneracy = 1x1 + 1x2 + 3x3 + 1x3 = 15
Lastly:Lastly: t2g1eg1.
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ast y:y 2g g
Method of descending symmetry is not necessary since both singletsand triplets are allowed (t 2g
, eg
are different orbitals)
h2g1gg2g Ointtet
Therefore, states are: gggg T T T T 23
21
13
11 ,,,
Total degeneracy = 1x3 + 3x3 + 1x3 + 3x3 = 24
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Correlation DiagramsCorrelation Diagrams
Connects strong field states to intermediate field states
Rules:Rules:
1.) Non-crossing rule: states of same symmetry and spin multiplicity may
not cross.
2.) Hunds rules: a) states of highest spin multiplicity lie lowest in energy
b) terms with the same S, the one with highest orbital L lies lowest.
Rules apply strictly only to ground states.
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8
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Next:Next: consider an nd 8
configuration
Need to observe 2 new principles.
1.) doubly occupied orbitals contribute A 1g
to direct products and 0 to S and M S
due to thePauli Exclusion Principle.
2.) Account for spin-pairing energy. Electrons in degenerate levels tend to have unpairedspins whenever possible.
It is an empirical fact that spin-pairing in eg
orbitals requires moremore
energy than in the t2gorbitals
) ) )210:are pairse#
etEetEetE:d
g
4g
42g
3g
52g
2g
62g
8
Q
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) 22
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) 22g2gg2g tEteEeE