Term Symbol-I - Cobalt

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 Total orbital angular momentum
For a single electron
r
moving around a nuclei Since F is a central force
r r
L=r ×p
working in the same direction
as r
r
r
p
The angular momentum
r r r
L = × p is conserved
with time
r
dL
dt = dr r r
r dp
× p+r×
dt
r
dt
1 dp r r
= × p+r×
F=0
m dt
e
0 0
r r r
L=r ×p
r
F
r
p

The Term Symbol
Consider next two
independent
(non - interacting)
electrons in the
same atom where
we neglect the
electron - electron
repulsion
r
p 2
r
r 2
L 1 r
r 1
r
L 2
Angular momentum
preserved for each
electron !!!
Total orbital angular momentum
When we allow the
elctrons to interact this is
no longer the case
r
electron L 1 r
repulsion
L
r
2
p 2 p
r
1
r 2
r
r 1
p 1
electron
repulsion
However the total angular
momentum L T will still be
conserved
It can be used to label a state

The Term Symbol
For a configuration
n
1 1 1 2 2 2
( nlm ) ( nl m ) ,...,( n l m )
n
1 2 m
m m
m n
We have a number of different states
(eigenfunctions to the Schrödinger
equations)
They are characterized by different
TERM SYMBOLS :
Total spin angular quantum
2s + T
1
number sT
with spin - multiplicity
2s T + 1
L(l T
)
Total orbital angular
momentum quantum
j T number l T
Total angular momentum quantum
number j T

The Term Symbol
As an example 2s
1 2
2p
Total orbital angular
momentum quantum
number l T
l T : 0 1 2 3 4
S P D F G
Total spin angular quantum
number sT
with spin - multiplicity
2s + 1
T
Total angular momentum quantum
number j T

The Term Symbol
We must now find
Total orbital angular momentum
quantum number l T
Total spin - angular momentum
number s T
Total angular momentum
quantum number j T

The Term Symbol
Total orbital angular momentum
For the orbital - angular momentum
l(i) z
r
L(i)
We have seen that we can find common eigenfunctions to
L(i) ˆ 2
and L(i) ˆ with eigenvalues
2 2
L(i) ˆ : h l( i) l(
i) + 1
z
L(i) ˆ : hmi
( ); mi ( ): - l( i), l( i) - 1, l( i) - 2,...., l( i) - 1, l(
i)
z
( )

The Term Symbol Total orbital angular momentum
r r
Consider next two angular momenta L(i) and L(j) with the
lquantum numbers l(i) and l(j)
L(j)
r
L(i)
r
r
Their sum is a new angular momentum L
with the possible l quantum numbers
T
T
r
L T
hm T
Z
r
L T
l : l( i) + l( j); l( i) + l( j) −1;.....,| l( i) − l( j) |
T
For each lT
quantum number the allowed m
values are : - l ; l −1;....., l −1,
l
T T T T
T
r 2
(L T ) = h
2 lT (l T + 1)

The Term Symbol
For the spin - angular momentum
Total spin angular momentum
S(i) z
r
S(i)
We have seen that we can find common eigenfunctions to
S(i) ˆ 2
and S(i) ˆ with eigenvalues
2 2
S(i) ˆ : h S( i) S(
i) + 1
z
S(i) ˆ : hm ( i); m ( i): - S( i), S( i) - 1, S( i) - 2,...., S( i) - 1, S(
i)
z
S
( )
S

The Term Symbol
Total spin angular momentum
v s
Consider next two angular momenta S(i) and S(j) with the
S quantum numbers S(i) and S(j)
r
S(j)
v
r
Their sum is a new angular momentum S
S(i)
with the possible S quantum numbers
T
T
r
S T
Z
S : S( i) + S( j); S( i) + S( j) −1;.....,| S( i) −S( j) |
T
m S T
r
S T
For each ST
quantum number the allowed m
values are : - S ; S −1;....., S −1,
S
T T T T
S T
r 2
(S T )
= h 2 S T (S T + 1)

The Term Symbol
Total angular momentum
v
Consider finally a spin angular momenta S(i) with the
S quantum numbers
r
S(i) and an orbital angular
momentum L(i) with the lquantum number l
v
Their sum is a new angular momentum J
with the possible J quantum numbers
T
J : S( i) + l( i); S( i) + l( i) −1;.....,| S( i) − l( i) |
T
r
S(i)
r
J T
r
L(i)
T

The Term Symbol
Total angular momentum
Z
m J T
r
J T
r 2
(J T )
= h 2 J T (J T + 1)
For each JT
quantum number the allowed m
values are : - J ; J −1;....., J −1,
J
T T T T
J T

The Term Symbol General procedure for adding the orbital
and spin angular momenta in many
-electron atoms
Consider a multi - electron
atom with the
electron configuration
n
n2
( nl 1 1m1) ( nl 2 2m2) ........(
nml mm m )
Shell 1 Shell 2 Shell n
1 m
n
2
Ex. He : 1s ; He 1s3d;
2 2 4
O 1s 2s 2p
2 2 6
Cl:1s 2s 2p
;
2 5
3s
3p

The Term Symbol General procedure for adding the orbital
and spin angular momenta in many
-electron atoms
Consider a multi - electron
atom with the
electron configuration
n
n2
( nl 1 1m 1( 2 2 (
m
1) nl m2) ........ nml mm m )
Shell 1 Shell 2 Shell n
1. Add orbital - angular momenta
of electrons pair - wise
# 1+# 2 → # I # I +# 3 →# II
# II+# 4 → ..., L T
Note : add electrons in same shell first
A closed shell contributes zero to l T
n

The Term Symbol General procedure for adding the orbital
and spin angular momenta in many
-electron atoms
Consider a multi - electron
atom with the
electron configuration
n
n2
( nl 1 1m1) ( nl 2 2m2) ........(
nml mm m )
Shell 1 Shell 2 Shell n
1 m
Total orbital angular
momentum quantum
number l T is indicated
by letters
l T : 0 1 2 3 4
S P D F G
n

The Term Symbol General procedure for adding the orbital
and spin angular momenta in many
-electron atoms
Consider a multi - electron
atom with the
electron configuration
n
n2
( nl 1 1m 1( (
m
1) nl 2 2m2) ........ nml mm m )
Shell 1 Shell 2 Shell n
1. Add spin - angular
momenta
of electrons
pair - wise
# 1+# 2 →# I # I +# 3 →# II
# II+# 4 → ..., s T
Note : add electrons in same shell first
A closed shell contributes zero to s T
n

The Term Symbol General procedure for adding the orbital
and spin angular momenta in many
-electron atoms
Consider a multi - electron
atom with the
electron configuration
n
n2
( nl 1 1m1) ( nl 2 2m2) ........(
nml mm m )
Shell 1 Shell 2 Shell n
1 m
The s T value of a
state is
indicated by its
spin multiplicity
Spin multiplicity :
2 sT
+1=
Number of different
m values
ST
n

The Term Symbol General procedure for adding the orbital
and spin angular momenta in many
-electron atoms
Consider a multi - electron
atom with the
electron configuration
n
n2
( nl 1 1m1) ( nl 2 2m2) ........(
nml mm m )
Shell 1 Shell 2 Shell n
1 m
r
S
r
J T
ˆ 2 2
J T ; h ( j T + 1)
j T
where :
j T = sT + l T,
s + l - 1,.. | s - l |
T
r
L
T T T
n
r
Finally add L
v
and S
T
T

The Term Symbol
Example He : 2p
1 1
3d
We have l (1) = 1; s( 1) = 1 2
We have l (2) = 2; s( 2) = 1 2
There is (2l(1) + 1)(2s(1) + 1) =
3 x 2 = 6
2p spin orbitals
The total Hamiltonian is
2
2
h h Z Z
Ĥ=- ∇ - ∇ − − +
2m 2m r r r
e
1 2 2 2 1
e 1 2 12
Omitting at the moment
electron - electron repulsion
2
2
h h Z Z
Ĥ o = - ∇1 2 - ∇2 2 − −
2m 2m r r
e
e 1 2
There is (2l(2) + 1)(2s(2) + 1) =
5 x 2 = 10
3d spin orbitals
They can be combined in
6x10 = 60 ways
1 1
The 2p 3d configuration
has 60 different states
Without electron - electron
repulsion all 60 states
would have the same energy
ε + ε
E= 2p 3d

The Term Symbol
Example He : 2p
1 1
3d
We have l (1) = 1; s( 1) = 1 2
We have l (2) = 2; s( 2) = 1 2
Thus combining orbital
angular momenta
l T = 2+
1; l T = 2+ 1−1;
l T = 2−1;
3 2 1
The total Hamiltonian is
2
2
h h Z Z
Ĥ=- ∇ - ∇ − − +
2m 2m r r r
e
1 2 2 2 1
e 1 2 12
When electron - electron
repulsion is included
states with different L
and S T will have different
energy
T
Next combining spin -
angular momenta
1 1
s T = +
2 2 ; s 1 1
T = −
2 2 ;
1
0

The Term Symbol
Example He : 2p
1 1
3d
We have l (1) = 1; s( 1) = 1 2
We have l (2) = 2; s( 2) = 1 2
Thus combining orbital
angular momenta
l T = 2+
1; l T = 2+ 1−1;
3 2
l T = 2−1;
1
Next combining spin -
angular momenta
1 1
s T = +
2 2 ; s 1
T = −
2
1 0
1
2 ;
+ 1
( LT, ST) 2S T
L( LT) (2ST
+ 1)
×
(2L
T + 1)
Number of
States
3
(3,1) F 21
1
(3, 0) F 7
3
(2,1) D 15
1
(2, 0) D 5
3
(1,1) P 9
1
(1, 0) P 3
Total 60

The Term Symbol
States with different spin -
( T, T 2S T + 1
L S ) L( LT) (2ST
+ 1)
× multiplicity will differ in energy.
(2L
T + 1)
The state withthe higher
Number of spin - multiplicity will
States be lower in energy. The energy
3
(3,1) F 21 willdecrease with increasing
1
(3, 0) F 7 spin - multiplicity
3
(2,1) D 15
1
(2, 0) D 5 States with different L T
3
(1,1) P 9 quantum numbers will
1
(1, 0) P 3 have differentenergies.
Total 60
The higher the L T
quantum number the
lower the energy

What you must learn from this lecture
For a configuration
n n
( nlm) 1( n l m ) 2,...,( nmm
l mm)
n m
11 1 2 2 2
Be able to construct
term symbols :
be able to evaluate :
1. Total spin angular quantum
number sT
and spin - multiplicity
2s
+ 1
T
2.
Total orbital angular
momentum quantum
number l
T
2s T
+ 1
L(l T
)
j T
3.
Total angular momentum quantum
number j
T
NB : remember closed shell adds
up to S = 0 and L = 0
T
T