Post on 13-Oct-2014
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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
Total orbital angular momentum
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
Total 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
each quantum number the allowed ms are : -
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
each quantum number the allowed ms are : -
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
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 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
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
Total orbital angularmomentum quantumnumber is indicated by letters
: 0 1 2 3 4 S P D F G
T
T
ll
ll
The Term Symbol General procedure for adding the orbital and spin angular momenta in many-electron atoms
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
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 2
The Term Symbol General procedure for adding the orbital and spin angular momenta in many-electron atoms
Shell 1 Shell n
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 2
The sT value of a state isindicated by its spin multiplicity
Spin multiplicity : 2 +1=Number of different
valuesST
ss
mm
TT
The Term Symbol General procedure for adding the orbital and spin angular momenta in many-electron atoms
Shell 1 Shell n
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 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
Example He : 2p d1 13
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
Example He : 2p d1 13
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 :