Multielectron Atoms
Eisberg & Resnick, Ch 10
Exchange Symmetry
SO AS FAR AS SPATIAL DISTANCES ARE CONCERNED:
Indistinguishable SYM spatial wavefns
are closer together in space
Indistinguishable ASYM spatial wavefns
are farther apart in space
than for Distinguishable particles
Helium Summary
exchange-related
Repulsion
(spin)
ℓ-related
repulsion
spin-singlet
spin-triplet
H-basis
functions
1P
1
3P
0,1,2
1s2p
n-related
repulsion
spin-singlet
spin-triplet
1s2s
1S
0
3S
1
1s2*
1s2*
*=s,p
spin-singlet
x
1s1s
1s1s
1s1s
1S
0
Hartree-Fock Methods
Choose initial shape
For Coulomb Potl V(r)
Loop until V(r) doesn’t change much
Solve Schro Eqn
for En Yn
Insert fine structure
corrections
Build atom according to
This set of orbital energies En
Use the collection of Yn*Yn to
Get new electron charge distrib
Use Gauss’ Law to
get new V(r) shape
V (r )  k
Z eff e
r
2
Note: This shows how the
orbitals shift as viewed
from the perspective
of an s-orbital.
r2 ~ n2 ao / Zeff
En ~ (Zeff2/n2) ( -13.6 eV )
Hartree-Fock
Effective Charge
Effects
Hierarchy of Interactions
• Hartree-Fock Central Coulomb
• Residual Coulomb
– Stot ordering
– Ltot ordering
• Spin-Orbit
• …
Impact upon a 3d4p basis config
( sample )
Zeeman
Weak Field Zeeman
• Hartree-Fock Coulomb
• Residual Coulomb
– stot effects
– ℓtot effects
• Fine Structure
– spin-orbit ( jtot becomes important )
– relativistic
• Zeeman
H’Zeeman = - mtot * Bext
Weak Field Zeeman
mtot

mtot

 ml tot

 m s tot
e 
g
Ltot
2m

e 
gs
Stot
2m

e 

( Ltot  2 Stot )
2m
Weak Field Zeeman
Jtot
mtot
project average mtot onto Jtot
g
m tot  mtot cos g
 mtot
onto
J
mtot
 
μ J
mJ

 
e 

( L  2S )  ( L  S )
2m
mtot j ( j  1)
Weak Field Zeeman
Jtot
monto J
projection of
Bext
mtot onto J onto B
monto J  monto J
onto B
Jz
J
H’Zeeman = - mtot * Bext
e
()( )
2m


 
( L  2S )  ( L  S )
j ( j  1)
mj
j ( j  1)
Bext
stot=0
Strong Field Zeeman
• Hartree-Fock Coulomb
• Residual Coulomb
– stot effects
– ℓtot effects
• Zeeman
• Fine Structure
– spin-orbit
– relativistic
H’Zeeman = - mtot * Bext
Strong Field Zeeman
Ltot
Stot

mtot
Bext

e 

( Ltot  2 Stot )
2m
H’Zeeman = - mtot * Bext
EZeeman
strong

e
(m tot  2 ms tot ) Bext
2m