Download CHAPTER 9 Beyond Hydrogen Atom

4/26/2010
CHAPTER 9
Beyond Hydrogen Atom
Multi-electron Atom
Pauli Exclusion Principle
Atomic Structure and the Periodic
Table
Total Angular Momentum
Anomalous Zeeman Effect
Dimitri Mendeleev
What distinguished Mendeleev was not only genius, but a passion for the elements.
They became his personal friends; he knew every quirk and detail of their behavior.
- J. Bronowski
Independent Particle Approximation
The potential of an electron depends on its distance from the
nucleus as well as the its distance from other electron. The
independent Particle approximation simply assumes that the
potential for each particle is the same as that of a single electron
orbiting
biti the
th nucleus
l
lik
like that
th t in
i th
the H
Hydrogen
d
atom.
t
Multi-electron atoms
When more than one electron is involved, the potential and the
wave function are functions of more than one position:
 

V  V (r1 , r2 ,..., rN )
 

   (r1 , r2 ,..., rN , t )
Solving
g the Schrodinger
g Equation
q
in this case can be very
y hard.
But we can approximate the solution as the product of singleparticle wave functions:
 




 (r1 , r2 ,..., rN , t )  1 (r1 , t )  2 (r2 , t )   N (rN , t )
And it turns out that, for electrons (and other spin ½ particles), all
the i’s must be different. This is the Pauli Exclusion Principle.
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Pauli Exclusion Principle
To understand atomic spectroscopic data, Pauli proposed his
exclusion principle:
No two electrons in an atom may have the same set of
quantum numbers (n, ℓ, mℓ, ms).
It applies to all particles of half-integer spin, which are called fermions,
and particles in the nucleus are fermions.
The periodic table can be understood by two rules:
The electrons in an atom tend to occupy the lowest energy
levels available to them.
The Pauli exclusion principle.
Symmetry of the Wave function
Two electron Schrodinger equation

2 2  
2 2  
 
 
1 (r1 , r2 ) 
 2 ( r1 , r2 )  V ( r1 , r2 )  E (r1 , r2 )
2m 2
2m 2
The probability density does not change if we interchange the
particles dP   (r , r ) 2
  2
  2
1 2
 ( r1 , r2 )   ( r2 , r1 )
dV
The total wave function has two possibilities
 
 
 ( r , r )   ( r , r ) symmetric
s
1
2
s
 
2
1
 
 A (r1 , r2 )   A (r2 , r1 ) Antisymmetric
Symmetry of the Wave function
If a and b are the four quantum numbers of the two electrons then we
can construct linear combinations
Symmetric
 




 s (r1 , r2 )  C  a (r1 ) b (r2 )  b (r1 ) a ( r2 ) 
Antisymmetric
 




 A (r1 , r2 )  C  a (r1 ) b (r2 )  b (r1 ) a (r2 )
For
o two
oe
electrons
ec o s with sy
symmetric
e c wave
a e function
u c o the
e two
o pa
particles
c es ca
can
exist simultaneously with a=b, however the anitsymmetric wave
function is zero if a=b, the electrons cannot be in the same state.
So according to quantum mechanics the anitsymmetric wave function
is consistent with the Pauli's exclusion principle.
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Example
9.1 An atom with four electrons is in an excited state. One electron is
in energy level 1.04 eV, one electron is in energy level 4.16 eV and
two are in energy level 9.36 eV. When the atom returns to its ground
state what would the possible energies of the photons.
Atomic Structure
Hydrogen: (n, ℓ, mℓ, ms) = (1, 0, 0, ±½) in ground state.
In the absence of a magnetic field, the state ms = ½ is degenerate with
the ms = −½ state.
Helium: (1, 0, 0, ½) for the first electron.
(1, 0, 0, −½) for the second electron.
Electrons have anti
anti-aligned
aligned (ms = +½ and ms = −½)
½) spins.
spins
The principle quantum number also has letter codes.
n=
1 2 3 4...
Letter =
K L M N…
n = shells (eg: K shell, L shell, etc.)
Electrons for H and He
nℓ = subshells (e.g.: 1s, 2p, 3d)
atoms are in the K shell.
Total number of electrons in any shell is
H: 1s2
He: 1s1 or 1s
n 1
N n  2 (2l  1)  2(1  3  5  ...n  1)  2n 2
l 0
Atomic Structure
How many electrons may be in each subshell?
Total
For each mℓ: two values of ms
2
For each ℓ: (2ℓ + 1) values of mℓ
2(2ℓ + 1)
Recall:
ℓ = 0 1 2 3 4 5 …
letter = s p d f g h …
ℓ = 0, (s state) can have two electrons.
ℓ = 1, (p state) can have six electrons, and so on.
The lower ℓ values have more elliptical orbits than
the higher ℓ values.
Electrons with higher ℓ values are more
shielded from the nuclear charge.
Electrons lie higher in energy than those
with lower ℓ values.
4s fills before 3d.
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Examples
9.2 write down all possible sets of quantum numbers for an electron
in a 4f and 2p subshels.
9.3 Suppose the outer electron in a potassium atom is in a state with
l=2. Compute the magnitude of L. What are the possible values of j
and the possible magnitudes of J?
9.4 write down the electron configuration of Carbon.
9.5 what element has this ground state electron configuration
1s22s22p63s22p2
9.6 which of the following atoms would you expect to have its ground
state split by the spin orbit interaction: Li, B, Na
The
Periodic
Table
Groups and
Periods
Groups:
Vertical columns.
Same number of
electrons in an ℓ
orbit.
orbit
Can form similar
chemical bonds.
Periods:
Horizontal rows.
Correspond to
filling of the
subshells.
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The Periodic Table
Inert Gases:
Last group of the periodic
table
Closed p subshell except
helium
Zero net spin and large
ionization energy
Their atoms interact weakly
with each other
Fig a) Closed shell at 2, 10,
18, 36 and 54
Fig b) Increase in atomic
radius
The Periodic
Table
Halogens:
Need one more electron to fill outermost subshell
Form strong ionic bonds with the alkalis
More stable configurations occur as the p subshell
is filled
Transition Metals:
Three rows of elements in which the 3d, 4d, and 5d
are being filled
Properties primarily determined by the s electrons,
rather than by the d subshell being filled
Have d-shell electrons with unpaired spins
As the d subshell is filled, the magnetic moments,
and the tendency for neighboring atoms to align
spins are reduced
Lanthanides (rare earths):
Have the outside 6s2 subshell
completed
As occurs in the 3d subshell, the
electrons in the 4f subshell have
unpaired electrons that align
themselves
The large orbital angular momentum
contributes to the large
ferromagnetic effects
The Periodic
Table
Actinides:
Inner subshells are being filled while
the 7s2 subshell is complete
Difficult to obtain chemical data
because they are all radioactive
Have longer half-lives
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Coupling of Angular Momenta
Corrections to Energy Levels
Orbital – Orbital Coupling, spin-spin coupling and spin orbit
coupling, jj coupling
Orbital-Orbital Coupling
Important because coulomb repulsion between electrons means
the electron probability density is dependent on L.
L Atoms with low
Z have the largest effect on energies
For 2 electron system:
States with larger value of l have lower energy
  
L  L1  L2
allowed values of l  l1  l2 , l1  l2  1,...l1  l2
L2  l (l  1) 2
Coupling of Angular Momenta
Spin-Spin Coupling
Magnetic moment between electrons is weak, but for atoms with
low Z values, the Pauli exclusion principle is important.
For 2 electron system
If spins are parallel the energy levels are further apart then when
spins are antiparallel. Total wave function is antisymmetric so if
the sping
p gp
part is symmetric,
y
, the space
p
p
part must be
antisymmetric and thus Pauli principle requires the energy level
to be further apart. One of the energy level is lower than normal
1
s1  s2 
2
  
S  S1  S 2 total intrinsics angular momentum quantum number s
be zero or one.
S 2  s( s  1) 2
Coupling of Angular Momenta
Spin-Orbit Coupling
Much smaller effect on energy levels if Z is not very large. Often
called LS coupling or Russell-Saunders coupling.
  
J  LS
allowed values of j  l  s , l  s  1,,...l  s
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Coupling of Angular Momenta
J-J Coupling
For large Z values the spin-orbit coupling is much stronger than the
spin-spin and orbital-orbital coupling. In this case we first find the
spin-orbit coupling correction and do the other two next.
  
J1  L1  S1

 
J 2  L2  S2
j1 and j2 are the two total angular momentum quantum numbers.
  
J  J1  J 2
Many-Electron Atoms
Hund’s rules:
The total spin angular momentum S should be maximized to the
extent possible without violating the Pauli exclusion principle.
Insofar as rule 1 is not violated, L should also be maximized.
For atoms having subshells less than half full, J should be minimized.
For a two-electron atom
There are LS coupling and jj coupling to combine four angular
momenta J.
Examples
9.7 Consider a system of two electron each with l=1 and s=1/2. a)
what are possible values of the quantum number L for the total
angular momentum? b) what are the possible values of the quantum
number S for the total spin? c) find the possible quantum numbers j
for the combination J=L+S. d) what are the possible quantum number
j1 and j2 for the total angular momentum of each particle? e) find the
possible values of j from the combination j1 and j2
p
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