Group Theory-Part 12 Correlation Diagrams

8/13/2019 Group Theory-Part 12 Correlation Diagrams

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Correlation Diagrams C734b 2008 1

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

Group Theory-Part 12 Correlation Diagrams

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