Atomic

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Dr. F. E. Hernández
Atomic Structure
Chapter IV
Peter Atkins, Physical Chemistry, 7th edition
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Introduction
Lets learn how to use quantum mechanics
Electronic structure of an atom
Hydrogenic atoms Many electron atoms (H, He,
Li2, ..) (Any other)
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n


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Spectroscopy
Detection and analysis of the interaction
matter-electromagnetic radiation. (Absorption
and/or emission)
Spectrum!
Hydrogenic atoms (Structure and spectra)
H
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1st Balmer 2nd Lyman and Paschen
n1 1 (Lyman) n1 2 (Balmer) n1 3 (Paschen)
n2 n11, n12, .. RH 109,677 cm-1 (Rydberg
constant)
The Ritz combination principle
Bohr frequency condition
of any spectral line is the diferrence
between two terms!
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The structure of hydrogenic atoms
We already know the Coulomb potential energy
Z Atomic number V Potential energy of charge
q in the presence of charge q Vacuum
permittivity
The Hamiltonian
One dimensional system
Separation of internal motions!

• Nucleus electron as a whole
• Electron moving relative to nucleus

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Therefore
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c.m coordinates
Relative coordinates
To determine y we separate the wavefunction in
two components (Centrosymmetric )
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Constant
?R/r2
Spherical harmonic!
Radial equation!
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The radial solution
Coulomb potential energy
Centrifugal force
(Angular momentum around the nucleus)
(Field of the nucleus)

• l 0 (Angular momentum ml 0 only exist
  attractive force)
• l ? 0 (Centrifugal term gives a contribution)

Solving the radial equation
a0 Bohr radius 52.917710-12 m
Ln,l Associated Laguerre polynomial!
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Atomic orbitals and their energies
Hydrogenic atomic orbital
n 1, 2, l 0, 1, 2, .., n-1 ml 0, ? 1, ?
2, .., ? l
These parameters define the orbital occupied by
the e-
Electron state!

• n determines the energy of the electron
  (Principal quantum number)
• l defines the magnitude of the angular momentum
  (J) (Momentum quantum number)
• ml gives the projection of l on z (Magnetic
  quantum number)
• ms is known as the spin quantum number

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The energy levels

• Separation in neighboring levels a Z2
• En lt 0 ? Bound state
• En gt 0 ? Unbound state

Ionization energy
Minimum energy required to remove an electron
from the ground state
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Shells and Subshells
n 1 2 3 4 (Shells) K L M N l
0 1 2 3 (Subshells) s p d f
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The s orbitals
1s orbital n 1, l 0, ml 0
r ? 0, V ? 0 ? The electron should be found close
to nucleus!
EK ? , V?
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The probability density

• All s orbital are spherically symmetric. However,
  the radial nodes are different
• 1s ? 0
• 2s ? 1
• 3s ? 2

_at_ polynomial factor equal to zero
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Radial distribution function
(Ring)
Probability density
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The p orbitals
2p orbital n 2, l 1, ml -1, 0, 1
Centrifugal force!
z
The wave function in the plane xy is zero
everywhere at z 0. (Nodal plane)
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Term corresponding to the angular momentum
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The d orbitals
3d orbital n 3, l 2, 1, 0 (d, p, s) d ? ml
-2, -1, 0, 1, 2
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Spectroscopic transitions and selection rules
? Not all the existent transitions are
permissible
The change in the electron angular momentum
should compensate for the spin angular momentum
carried away by the photon (S 1)

• d ? s (l 2 ? l 0)
• s ? s (l 0 ? l 0)

Forbidden!
The selection rule is based on the evaluation of
the transition dipole moment mi,j.
Electric dipole moment operator!
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• If m 0 ? Forbiden
• If m ? 0 ? Allowed

Unless!
lf li ? 1
ml,f ml,i m
Dl ? 1,and Dml 0, ? 1
For hydrogenic atoms
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The structure of many-electron atoms
Complications Electron-electron interaction
We must make approximations to solve the
equation
The orbital approximation
ri this is the vector nucleus-electron i
Each electron is occupying its own orbital

• We will think of hydrogenic-like orbital
• We will consider a modified nuclear charge due to
  the presence of multiple electron

For no electron-electron interaction
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Exact solution!
The helium atom H ? Z 1 ? 1s1 He ? Z
2 ? 1s2
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The Paulis principle
No more than two electron may occupy any given
orbital and, if two do occupy one orbital, then
their spin must be paired. Remember! This
exclusion principle applies to any pair of
identical fermions! ms ?1/2
This principle does not apply to identical bosons
(Integer spin)
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• When labels of fermions are exchanged
• When labels of bosons are exchanged

Lets consider
?- ?-
Possibilities for the spin
?- ?-
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The following three spin possibilities do not
change the sign of the total function when labels
are exchange
(Symmetric)
?-
?-
(Antisymmetric)
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Lets assume
r1 r2
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Penetration and shielding
H ? 1s1 He ? 2s2
Li ? 1s22s1
L shell 2s1 (Valence electron)
A complete shell reduces the effective nuclear
charge
Complete shell K
(Penetration shielding)
s Shielding constant
Note Because the s orbital electrons are more
likely to be found close to the nucleus, they
penetrate more than p orbital which probability
to be at the nucleus is zero. Therefore, s
electron experience less shielding.
Z 6 Orbital s 1s 0.3273 2s 2,7834 2p 2.8642
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The building-up principle
Proposes an order for the hydrogenic orbitals
that account for the ground-state configuration
of neutral atoms.
1s-2s-2p-3s-3p-4s-3d-4p-
This order is in correlation with the energy of
the orbital (E ) ? lower total energy
Note electrons occupy different orbitals of a
given shell before doubly occupying any one of
them
He
An atom in its ground state adopts a
configuration with the greates of unpaired
electrons. This reduces the energy because the
interaction electron-electron becomes smaller
when they are separated.
Hunds rule!
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The building-up principle
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The periodic table
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Ionization energy and electron affinity

• Na 1s2-2s2-2p6
• O2- 1s2-2s2-2p6

Removing electrons from the valence shell we can
generate ions
Ne
IE Minimum energy necessary to remove an
electron from an atom
I1 lt I2 lt
M(g) ? M(g) e-
Element I1 (KJ/mol) I2 (KJ/mol) H
1312 --- He 2372 5251 Mg 738 1451 Na
496 4562
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EEA This is the energy released when an
electron attaches to a gas phase atom
Element EEA (KJ/mol) Cl 349 F 322 H
73 O 141
The spectra of complex atoms
As the of electrons increases the spectrum
becomes more complicated
(This is for all transitions)
The spectra gives information about electrons
energies. We should also take into account
electron-electron interaction.
Quantum defects Penetration shielding
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Singlet and triplet states
1s1- 2s1
Ms 0 (Singlet) ? s- Ms 0,?1(Triplet)
He 1s2
Ms 1 Ms 0 Ms -1
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Spin-orbit coupling
High energy
Low energy
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There is a total angular momentum J, mj
J l ½ J l ½

• For l 0 ? J 1/2
• For l 1 ? J 3/2 , 1/2
• For l 2 ? J 5/2, 3/2

In order to get the energies of the levels
knowing quantum s n, l, and J
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-. B Orbital angular momentum of the electron, B
a l. -. m Electron spin, m a s.
Now,
The strength of the spin-orbit coupling depends
on the nuclear charge
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Term symbols and selection rules

• The letter The total orbital angular momentum
  quantum number L.
• The left superscript Gives the multiplicity of
  the term through the total spin quantum number S.
• The right subscript Represents the value of the
  total angular momentum quantum number J.

Contribution to the energies, and splitting
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Total orbital angular momentum
Tell us the magnitude of the angular momentum
through L(LL)1/2
It has 2L1 orientations distinguished by L,
L1,..,-1
Same apply to S ? Ms J ? Mj
For multiple electrons
e.g.
Ne3s1 ? l 0 ? L0. S He2s2, 2p2 ? l1
1, l21 ? L 2, 1, 0. D, P, S

• The orbital angular momentum is zero for closed
  shell

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Total multiplicity
For multiple-electron system we need to access
the total spin angular quantum number.
The multiplicity is 2s1
Hunds rule The greater the multiplicity the
lower the energy
e.g
(Singlet) (Doublet) (Triplet)

• s 0 ? 2s1 2 (0) 1 1
• s 1/2 ? 2s1 2 (1/2) 1 2
• s1 s2 ½ ½ ? 2s1 2 (1) 1 3

Note There is not net spin for closed shell
because all electrons are
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Total angular momentum
J tell the relative orientation of the spin and
orbital angular momenta of 1 electron.
The total J is for several electron.
e.g.
For several electrons, we should consider the
coupling of all the spin and orbital angular
momenta
Ne3s1 ? L 0 ? S, j1/2 ? 2S1/2 He2s2,
2p1 ? L 1? P ? 2P3/2, 2P1/2
Russell-Saunder coupling
(For weak spin-orbit coupling)
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The selection rules

• No change of the overall spin
• DS 0
• - The orbital angular momentum of an individual
  electron must change DL 0, ? 1, .. (Dl ? 1)
• - DJ 0, ? 1, ..

They arise from the conservation of angular
momentum during a transition and from the fact
that a photon has spin equal to 1.
Note For heavy atoms DS ? 1
Effect of magnetic fields
Orbital and spin angular momentum ? Magnetic
field
Modified the atomic spectra The degeneracy is
removed
1 0 -1
p
s
The Zeeman effect!
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