Download Atomic structure

Atomic structure
3.1. Structure and spectra of hydrogenic atoms
3.1.1 The structure of hydrogenic atoms
3.1.2 Atomic orbitals and their energies
3.1.3 Spectroscopic transitions and selection rules
3.2. The structure of many electron atoms
3.2.1 The orbital approximation
3.2.2 The spectra of complex atoms
3.2.3 Singlet and triplet states
3.2.4 Spin-orbit coupling
Spherical coordinates 1
x = r sin cos
 (r , ,  )  R(r ) Y ( ,  )
y = r sin sin
z = r cos
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Radial 1
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Radial 2
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Average distance
r   * rˆ d   r |  |2 d
Spherical coordinates
d  r 2 dr sin  d d
  2
r     r Rn2,l | Yn,l |2 r 2 dr sin  d d
0 0 0
r
n ,l
 1  l (l  1)  a0
 n 2 1  1 

2
n  Z
 2
For a given n, the mean radius
follows the order d < p < s
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p-orbitals 1
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p-orbitals-2
pz
py
Y(θφ)
px
R(r)Y(θφ)
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d-orbitals
In the shell n=3, there are 3 subshells:
 l= 0  ml= 0
 1 s-orbital
 l= 1  ml= -1, 0, 1
 3 p-orbitals
 l= 2  ml= -2, -1, 0, 1, 2  5 d-orbitals
From a linear combination of these
orbitals, we can build real orbitals that
are also solutions of the SE for the
hydrogenic atom: dxy, dyz, dzx, dx2-y2, dz2
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Effective charge
2s
1s
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Ordering of orbitals as f. of Z
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Electron configurations
Pauli exclusion principle:
“No more than 2 electrons may occupy any
given orbital. If 2 electrons occupy one orbital,
then their spin must be paired (antiparallel: ).”
Hund’s rule:
“An atom in its ground state adopts a
configuration with the greatest number of
unpaired electrons”
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Spin correlation
two electrons with different spin (antiparallel) have
a certain probability to be at the same position.
The electrostatic repulsion is increased: the system
in its singlet state is destabilized
when 2 electrons in 2 different orbitals have the
same spin (parallel), they are repelled to each other
thanks to their spin.
that prevents an additional electrostatic repulsion.
The Triplet state (2 spins parallel) is more stable
than the singlet state (2 spins antiparallel)
this is the explanation for Hund’s rule.
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He-atom
l= 1, ml = 0, 1
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usefull
ˆ   E,

2
2
ˆ    d V

2m dx 2
ˆ   E

2
2
ˆ    d V

2m dx 2
Ĥ  T  V


  dxdydz  1

O   Oˆ  d
Oˆ   O
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