Complex Atoms;
The Exclusion Principle and the Periodic System
In order to understand the electron distributions in atoms, another
principle is needed. This is the Pauli exclusion principle:
No two electrons in an atom can occupy the same quantum state.
The quantum state is specified by the four quantum numbers; no
two electrons can have the same set.
Wolfgang Pauli
The Nobel Prize in Physics 1945
"for the discovery of the Exclusion Principle,
also called the Pauli Principle"
Question 1: why can we use 1-electron orbitals in this problem
Question 2: why do l-values for the same n have differing energies
The potential energy in a multi-electron atom:
Z
Ze2
V 
4 0 i 1

1
e2
1


 
ri 4 0 i  j ri  r j
attraction
to nucleus

repulsion between
electrons
V is a non-central potential
Central Field Approximation:
 
ri 
The overall effect of V 


 i 

Vc r 
Ze2
for r  0

4 0 r

e2
4 0 r
for r  
Then V can be written in terms of an
“effective screened nuclear charge”
Vcfa r   
Z eff r e 2
4 0 r
is centrally directed toward the nucleus
This is a central potential !!
Separation of variables in the central field approximation
Product wave function
 




r1, r2 ,  , rZ   1r1  2 r2    Z rZ 
1. Angular part of the wave function
is the same, hence angular functions
li mi i , i 
Potential and eigenenergy
V
Z

Vc ri 
E
i 1
Z

Ei
i 1
Insertion of trial yields a set of
equations:
2. Radial part of the wave function
 Z eff r   2  1
2 d

rRnl  

 Rnl  Enl Rnl
2
2mr dr
2mr 
 r
Energy Enl depends on 


2 2 

i ri   Vc ri ri   Ei ri 
2m
The potential function is not Coulombic
i.e. is not 1/r but

Z eff r 
r
Energy Enl does not depend on m
Wave functions for single “orbital”

 i ri   Rnl ri li mi i , i  , 
Radial part of the wave function
 Z eff r   2  1
2 d

rRnl  

 Rnl  Enl Rnl
2
2mr dr
2mr 
 r
Energy Enl depends on 
Energy Enl does not depend on m
High 
“classical picture”
Low 
s-states have higher density
near the nucleus
For low  values (and same n)
electron comes closer to the nucleus
More Coulomb attraction
More binding energy
Wave functions for single “orbital”

 i ri   Rnl ri li mi i , i  , 
Lowering of low
 quantum states as an effect of screening
Complex Atoms; the Aufbau Principle
This chart shows the occupied – and some unoccupied – states in
He, Li, and Na.
The Periodic Table of the Elements
Each value of is
given its own letter
symbol.
Electron configurations are written by
giving the value for n, the letter code
for , and the number of electrons in
the subshell as a superscript.
For example, here is the groundstate
configuration of sodium:
1s22s22p63s1
The Periodic Table of the Elements
subshells
Aufbau principle for multi-electron atoms
Eigenfunctions in multi-electron atom
 




r1, r2 ,  , rZ   1r1  2 r2    Z rZ 
1) Electrons fill the one-electron orbitals

 i ri   Rnl ri li mi i , i  , 
into a “configuration”:
3) For filled shells


m i  0  Ltot  0
shell


s i  0  Stot  0
shell

 i ri 
i
Degeneracy 2n2 structures
the Periodic System
2) Pauli principle dictates: single
occupancy
Note:
this is about ground states of the atoms
Electron spin and the atom
Additional quantum number
nlm  nlsml ms
Wave functions

nlm r , t   Rnl r lm  ,  eiEnt /  , 
No effect on energy levels,
except for magnetic coupling to
other angular momenta
Degeneracy
22  1
2n 2
Selection rules
s is not a spatial operator,
so no effect on dipole selection rules
Ground state orbital configurations for multi-electron atoms
Noble gases
He configuration 1s  ; closed shell
2
2
2
6
Ne conf. 1s  2s  2 p  ; closed shell
Ar conf.
1s 2 2s 2 2 p 6 3s 2 3 p 6; closed shell
Alkali metals
Li
Na
K
one open shell electron
1s 2 2s 
1s 2 2s 2 (2 p)6 3s  one open shell electron
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s  ....
Earth alkali metals
Mg
Ca
1s 2 2s 2 (2 p)6 3s 2 closed
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2
Binding energies of the
one-electron orbitals
vary with Z:
Screening effects
“Term Value”
Irregularities and degeneracies
(2S+1)L
J
Transition metals; effect of 3d orbitals:
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 2
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 3
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 3d 5
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 5
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 6
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 7
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 8
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 3d 10
1s 2 2s 2 (2 p)6 3s 2 3 p 6 4s 2 3d 10
ground term
1S
0
2D
3/2
3F
2
4F
3/2
7S
3
6S
5/2
Near degeneracy
of 3d and 4s orbitals
Competition in allocation
of electrons
5D
4
4F
9/2
3F
4
2S
1/2
1S
0
Coupling of the
Angular momenta
4d electrons
2p electrons
3d electrons
3p electrons
5d
5f
6f
One–electron systems
Alkali metals
Li(2s), Na(3s),K(4s), Rb (5s)
Two-electron systems
Earth-alkali metals
Filled shells
He(1s2), Ne(2p6)
Noble gases