Chem 373- Lecture 21: Term Symbols-I

Lecture 21: Term Symbols-I The material in this lecture covers the following in Atkins.

The Spectra of Complex Atoms 13. 9 Termsymbols and selection rules (a) The total orbital angular momentum (b) The multiplicity (c) The total angular momentum

Lecture on-line Term Symbols (PowerPoint) Term Symbols (PDF)Handouts for this lecture

The Term Symbol

r r

r p

r r L = r ×p

For a single electron moving around a nuclei

The angular momentum

L = r p is conservedwith time

dLdt

=drdt

p +rdpdt

mdpdt

p +r F = 0e

r r r

r rr

r

rr r

×

× ×

= × ×1

0 0

r

r F

r p

r r L = r ×p

Since F is a central forceworking in the same directionas r

r

Total orbital angular momentum

The Term SymbolConsider next two independent(non - interacting) electrons in thesame atom where we neglect theelectron - electron repulsion

r

L1

r1r

L2r

r2r

r p2

p1

When we allow theelctrons to interact this isno longer the case

r

r

L1

r1r

L2r

r2r

r p2 p1

electronrepulsion

electronrepulsion

However the total angularmomentum L will still beconserved

T

It can be used to label a state


Angular momentumpreserved for eachelectron !!!

The Term Symbol

( ) ( ) ,...,( )nn ll mm nn ll mm nn ll mm1 1 1 2 2 2n n

m m mn1 2 m

For a configuration

We have a number of different states (eigenfunctions to the Schrödinger equations)

They are characterized by differentTERM SYMBOLS :

L(llll T)

2ssssT+ 1

jjjjT

Total orbital angularmomentum quantumnumber Tll

Total spin angular quantum number with spin - multiplicity 2

T

T

ssss + 1

Total angular momentum quantumnumber Tjj

As an example 2s p1 22The Term Symbol

Total orbital angularmomentum quantumnumber

: 0 1 2 3 4 S P D F G

T

T

llll

Total spin angular quantum number with spin - multiplicity 2

T

T

ssss + 1

Total angular momentum quantumnumber Tjj

We must now find

The Term Symbol

Total orbital angular momentumquantum number Tll

Total spin - angular momentumnumber Tss

Total angular momentumquantum number Tjj

The Term Symbol

For the orbital - angular momentum


L(i)r

l(i)z

We have seen that we can find common eigenfunctions to

L(i) and L(i) eigenvalues

L(i) : i) i) + 1

L(i) - i), i) - 1, i) - 2,...., i) - 1, i)

2z

2 2

z

ˆ ˆ

ˆ ( (

ˆ : ( ); ( ) : ( ( ( ( (

with

m i m i

h

h

ll ll

ll ll ll ll ll

( )

The Term Symbol Total orbital angular momentum

r

L(i)

r L(j)

LT

r

Consider next two angular momenta L(i) and L(j) with the quantum numbers (i) and (j)

r r

ll ll ll

Their

i j i j i j

sum is a new angular momentum L with the possible quantum numbers

:

TT

T

r

ll

ll ll ll ll ll ll ll( ) ( ); ( ) ( ) ;.....,| ( ) ( ) |+ + − −1

Forvalue

each quantum number the allowed ms are : -

T TT T T T

llll ll ll ll; ;....., ,− −1 1LT

r hmT

Z

(LT)r 2

= h2llllT(llllT + 1)

The Term Symbol

For the spin - angular momentum

Total spin angular momentum

We have seen that we can find common eigenfunctions to

S(i) and S(i) eigenvalues

S(i) : i) i) + 1

S(i) - i), i) - 1, i) - 2,...., i) - 1, i)

2z

2 2

z

ˆ ˆ

ˆ ( (

ˆ : ( ); ( ) : ( ( ( ( (

with

S S

m i m i S S S S SS S

h

h

( )

S(i)r

S(i)z

The Term Symbol Total spin angular momentum

ConsiderS S

next two angular momenta S(i) and S(j) with the S quantum numbers (i) and (j)

v s

TheirS

S S i S j S i S j S i S j

sum is a new angular momentum S with the possible quantum numbers

:

TT

T

v

( ) ( ); ( ) ( ) ;.....,| ( ) ( ) |+ + − −1

For Svalue S S S S


T STT T T T; ;....., ,− −1 1

r

S(i)r

S(j)

ST

r

STr

Z

(ST)r 2

mS T

= h2ST(ST + 1)

The Term Symbol Total angular momentum

ConsiderS

finally a spin angular momenta S(i) with the S quantum numbers (i) and an orbital angular momentum L(i) with the quantum number

v

rll ll

Their

J S i i S i i S i i

sum is a new angular momentum J with the possible J quantum numbers

:

TT

T

v

( ) ( ); ( ) ( ) ;.....,| ( ) ( ) |+ + − −ll ll ll1

r

S(i)r

L(i)

JT

r

For Jvalue J J J J


T JTT T T T; ;....., ,− −1 1

The Term Symbol Total angular momentum

JTr

Z

(JT)r 2

mJ T

= h2JT(JT + 1)

The Term Symbol General procedure for adding the orbital and spin angular momenta in many-electron atoms

Consider a multi - electron atom with theelectron configuration

( ( (1 1n

2 2n

m mn1 mnn ll mm nn ll mm nn ll mmmm1 2

2) ) ........ )

Shell 1 Shell 2 Shell n

Ex. He : 1s ; He 1s3d;

O 1s 2s 2p

Cl : 1s 2s 2p

2

2 2 4

2 2 6

;

3 32 5ss pp



( ( (1 1n

2 2n


2) ) ........ )

Shell 1 Shell 2 Shell n1. Add orbital - angular momenta of electrons pair - wise

# 1+# 2 # I → # I +# 3 # II→

# II+# 4 ..., LT→

NNootteell

: add electrons in same shell firstA closed shell contributes zero to T



( ( (1 1n

2 2n


2) ) ........ )

Shell 1 Shell 2 Shell n

Total orbital angularmomentum quantumnumber is indicated by letters

: 0 1 2 3 4 S P D F G

T

T

ll

ll


Shell 1 Shell n

# I +# 3 # II→

# II+# 4 ..., T→ ss

NNootteess

: add electrons in same shell firstA closed shell contributes zero to T

# 1+# 2 # I →

1. Add spin - angular momenta of electrons pair - wise


( ( (1 1n

2 2n


2) ) ........ )

Shell 2


Shell 1 Shell n


( ( (1 1n

2 2n


2) ) ........ )

Shell 2

The sT value of a state isindicated by its spin multiplicity

Spin multiplicity : 2 +1=Number of different

valuesST

ss

mm

TT


Shell 1 Shell n


( ( (1 1n

2 2n


2) ) ........ )

Shell 2

ˆ )J ; ( where :

= + ,+ - 1, .. | - |

T2 2

T T

T T T

T T T

h jj jj

jj ss llss ll ss llTT

+ 1

r

Sr

L

JTr

Finally add L

and ST

T

r

v

The Term Symbol

Example He : 2p d1 13

We have (1) = 1; 1) =

12

ll ss(

We have (2) = 2; 2) =

12

ll ss(

There is (2l(1) +1)(2s(1) +1) = 3 x 2 = 62p spin orbitals

There is (2l(2) +1)(2s(2) +1) = 5 x 2 = 103d spin orbitals

The total Hamiltonian is

They can be combined in6x10 = 60 ways

The 2p 3d configurationhas 60 different states

1 1

H = -

2m-

2mZr

Zr r

2

e

2

e 1 2 12

h h∇ ∇ − − +12

22 1

Omitting at the momentelectron - electron repulsion

H = -

2m-

2mZr

Zro

2

e

2

e 1 2

h h∇ ∇ − −12

22

Without electron - electronrepulsion all 60 stateswould have the same energyE = 2p 3dε ε+

The Term Symbol


We have (1) = 1; 1) =

12

ll ss(

We have (2) = 2; 2) =

12

ll ss(

Thus combining orbitalangular momenta ll T = +2 1;

3 ll T = + −2 1 1;

2 ll T = −2 1;

1

Next combining spin -angular momenta

ssT = +1

212

; ssT = −1

212

;

1 0

H = -

2m-

2mZr

Zr r

2

e

2

e 1 2 12

h h∇ ∇ − − +12

22 1

The total Hamiltonian is

When electron - electronrepulsion is includedstates with different Land S will have differentenergy

T

T

The Term Symbol


We have (1) = 1; 1) =

12

ll ss(

We have (2) = 2; 2) =

12

ll ss(

ll T = +2 1;3

ll T = + −2 1 1;2

ll T = −2 1;1

Next combining spin -angular momenta

ssT = +1

212

; ssT = −1

212

;

1 0

Thus combining orbitalangular momenta

( , L( (2 (2 Number of States

T T2S

T T

T

TLL SS LL SSLL

) ) ))

+ + ×+

1 11

(3,1) F 21 3

(2,1) D 15 3

(2, 0) D 51

(1,1) P 93

(1, 0) P 31

Total 60

(3, 0) F 71

( , L( (2 (2 Number of States

T T2S

T T

T

TLL SS LL SSLL

) ) ))

+ + ×+

1 11

(3,1) F 21 3

(2,1) D 15 3

(2, 0) D 51

(1,1) P 93

(1, 0) P 31

Total 60

(3, 0) F 71

The Term Symbol States with different spin -multiplicity will differ in energy. The state withthe higher spin - multiplicity willbe lower in energy. The energy willdecrease with increasing spin - multiplicity

States with different quantum numbers will have differentenergies. The higher the quantum number the lower the energy

T

T

LL

LL

For

n l m n l m n l m

ss

l

j

NB

a configuration

be able to evaluate :

1. Total spin angular quantum number and spin - multiplicity 2

Total orbital angularmomentum quantumnumber

Total angular momentum quantumnumber

remember closed shell addsup to S and L = 0

n nm m m

n

TT

T

T

T T

1 2 m( ) ( ) ,..., ( )

.

.

:

11 1 2 2 2

1

2

3

0

+

=

L(llll T)

2ssssT+ 1

jjjjT

What you must learn from this lecture

Be able to construct term symbols :

Chem 373- Lecture 21: Term Symbols-I

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