1. Unit-IV
Many Electron Atoms

2. Identical Particles
 Identical Particles
Example: all electrons have the same
mass, electrical charge, magnetic
properties…
Therefore, we cannot distinguish one
electron from another.
 Quantum theory then restricts the kinds
of states for electrons

3. Two Families of Fundamental
Particles
 Fermions
Enrico Fermi
1901-1954
 Bosons
Satyendranath Bose
1894-1974

4. Fermions
 electrons, neutrinos, protons, quarks...
 At most one particle per quantum state
Bosons
 photons, W, Z bosons, gluons,4He
(nucleus and atom), 16O(nucleus), 85Rb
(atom),…
 Possibly many particles in the same
quantum state

5. Fermions versus Bosons
 Quantum theory tells us that angular
momentum (amount of spin) = n (h/2)
 Experimental fact:
Fermions have n = 1, 3, 5, …
Bosons have n = 0, 2, 4, …

6. Many Particle Systems
 Schrodinger Equation for a many particle system
with xi being the coordinate of particle i (r if 3D)
 the Hamiltonian has kinetic and potential energy
 if only two particles and V just depends on separation
then can treat as “one” particle and use reduced mass
 in QM, H does not depend on the labeling. And so if
any i  j and j i, you get the same observables or
state this as (for 2 particles)
H(1,2)=H(2,1)
t
x
x
x
i
x
x
x
H n
n



)
,
(
)
,
( 2
1
2
1





)
,
(
)
2
1
2
1
( 2
1
2
2
2
2
2
1
2
n
x
x
x
V
x
m
x
m
H 

 








7. Exchange Operator
 Using 2 particle as example. Use 1,2 for both space coordinates
and quantum states (like spin)
 the exchange operator is
 We can then define symmetric and antisymmetric states for 2 and
3 particle systems
1
:
)
2
,
1
(
)
1
,
2
(
))
2
,
1
(
(
)
1
,
2
(
)
2
,
1
(
12
12
12
12












s
eigenvalue
u
u
P
u
P
P
u
u
P
))
1
,
2
(
)
2
,
1
(
(
1
))
1
,
2
(
)
2
,
1
(
(
1





 



N
N
A
S
))
2
,
3
,
1
(
)
2
,
1
,
3
(
)
1
,
2
,
3
(
)
1
,
3
,
2
(
)
3
,
1
,
2
(
)
3
,
2
,
1
(
(
1
))
2
,
3
,
1
(
)
2
,
1
,
3
(
)
1
,
2
,
3
(
)
1
,
3
,
2
(
)
3
,
1
,
2
(
)
3
,
2
,
1
(
(
1


























N
N
A
S

8. Particles in a Spin State
 If we have spatial wave function for 2 particles in a box which
are either symmetric or antisymmetric
 there is also the spin. assume s= ½
 as Fermions need totally antisymmetric:
spatial Asymmetric + spin Symmetric (S=1)
spatial symmetric + spin Antisymmetric (S=0)
 if Boson, need totally symmetric and
so symmetric*symmetric or antisymmetric*antisymmetric
)
(
2
1
)
0
(
)
1
(
)
(
2
1
)
0
(
)
1
(

















z
A
z
s
z
s
z
s
S
S
S
S



 S=1
S=0

9. Spectroscopic Notation
n 2S + 1Lj
 This code is called spectroscopic or term
symbols.
 For two electrons we have singlet states
(S = 0) and triplet states (S = 1), which
refer to the multiplicity 2S + 1.

10. Term Symbols
 Convenient to introduce shorthand notation to label energy levels that
occurs in the LS coupling regime.
 Each level is labeled by L, S and J: 2S+1LJ
◦ L = 0 => S
◦ L = 1 => P
◦ L = 2 =>D
◦ L = 3 =>F
 If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or
terms, 2P3/2 and 2P1/2
 2S + 1 is the multiplicity. Indicates the degeneracy of the level due to
spin.
◦ If S = 0 => multiplicity is 1: singlet term.
◦ If S = 1/2 => multiplicity is 2: doublet term.
◦ If S = 1 => multiplicity is 3: triplet term.

12. Many-Electron Atoms
 Consider two electrons outside a closed shell.
 Label electrons 1 and 2.
 Then
J = L1 + L2 + S1 + S2
 There are two schemes, called LS coupling
and jj coupling, for combining the four angular
momenta to form J.
 The decision on which scheme to use
depends on the relative strength of the
interactions.
 JJ coupling predominates for heavier
elements.

13. LS, or Russell-Saunders, Coupling
 The LS coupling scheme is used for
most atoms when the magnetic field is
weak.
L = L1 + L2
S = S1 + S2
 The L and S combine to form the total
angular momentum:
J = L + S
Use upper case for
many - electron atoms.

14. Allowed Transitions for LS
Coupling
DL = ± 1 DS = 0
DJ = 0, ± 1 (J = 0 J = 0 is
forbidden)

15. jj Coupling
J1 = L1 + S1
J2 = L2 + S2
J = S Ji
Allowed Transitions for jj
Coupling
Dj1 = 0,± 1 Dj2 = 0
DJ = 0, ± 1 (J = 0 J = 0 is
forbidden)

16. Helium Atom Spectra
 One electron is presumed to be in the ground state,
the 1s state.
 An electron in an upper state can have spin anti
parallel to the ground state electron (S=0, singlet
state, parahelium) or parallel to the ground state
electron (S=1, triplet state, orthohelium).
 The orthohelium states are lower in energy than the
parahelium states.
 Energy difference between ground state and lowest
excited is comparatively large due to tight binding of
closed shell electrons.
 Lowest Ground state (13S) for triplet corresponding
to lowest singlet state is absent.

17. DL=±1,
DS = 0,
DJ =0, ± 1

18. Sodium Atom Spectrum
 Selection Rule: Dl=±1
 Various Transitions:
 Principal Series (np 3s, n=3,4,5,…)
 Sharp Series (ns 3p, n=3,4,5,…)
 Diffuse Series (nd 3p, n=3,4,5,…)
 Fundamental Series (nf 3d, n=3,4,5,…)

20. Sodium Atom Fine Structure
 Selection Rule: Dl=0, ±1