Chemistry 445.

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• Chemistry 445.
• Lecture 3.
• Molecular Orbital Theory

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We start by reminding ourselves of the shapes and
signs of the wavefunction on the atomic orbitals.
Below are the s and three p orbitals, showing
boundary surfaces (HS Fig. 1.9)
Note Pink color indicates sign of wavefunction
opposite to that of the white part of the orbital
.
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The five atomic d-orbitals
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Orbitals and Quantum numbers

• The solution to the Schrödinger wave-equation
  leads to a set of wavefunctions that yields 4
  types of quantum numbers instead of the single
  quantum number yielded by the Bohr model.
• These are
• 1) The principal quantum number, n, which has
  values of 1,2,3, This corresponds to the quantum
  number n in the Bohr model of the atom.

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Quantum numbers (contd.)

• 2) The Azimuthal quantum number l, which has
  values of 0 to (n-1) for each value of n. The
  different values of l correspond to orbital types
  as follows
• l 0 1 2 3
• letter used s p d f
• 3) The magnetic quantum number ml, can have
  values of l through 0 to l for each value of
  l.
  
• Value of l possible values of ml
• 0 0
• 1 -1,0,1
• 2 -2,-1,0,1,2
• 3 -3,-2,-1,0,1,2,3

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• 4) The spin quantum number (ms). This can have
  values of ½ or ½. This means that for each
  value of ml there are two values of ms. It is
  this that leads to the occupation of each orbital
  by two electrons of opposite spin, i.e. with ms
  ½ or ½.
• These quantum numbers lead to the shells
  (different values of n) and subshells (different
  values of l) that lead to our modern
  understanding of chemistry. The number of
  orbitals in each sub-shell (1 for s, 3 for p, 5
  for d, and 7 for f sub-shells) is determined by
  ml, and ms determines that only two electrons of
  opposite spin can occupy each orbital.

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Representation of orbitals
Schrödingers model
z
y
x
s-orbital
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The H-atom compared to many-electron atoms
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The essence of MO theory is that overlap of two
orbitals always occurs in two ways. In one
(bottom), the two 1s orbitals shown here
overlapping have the same sign of the
wavefunction, and so a net overlap occurs. This
produces a lower energy bonding orbital. In the
upper case, the two orbitals are of opposite
sign, and so no net overlap occurs. This produces
a higher energy anti-bonding orbital.
higher energy anti-bonding orbital
Sign of wavefunction is opposite

s1s
1s
1s

s1s
1s
1s
lower energy bonding orbital
sign of wavefunction is the same
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Drawing up a Molecular Orbital (MO) diagram for
H2
energy
arrow represents electron in 1s orbital
energy level of 1s orbital of H-atom
1s atomic orbital of H atom
1s atomic orbital of H atom
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Drawing up a Molecular Orbital (MO) diagram for
H2
s1s anti-bonding molecular orbital in H2 molec
ule
energy
1s atomic orbital of H atom
1s atomic orbital of H atom
s1s bonding molecular orbital in H2 molecule
These are the molecular orbitals of the H2 molecu
le
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Molecular Orbital (MO) diagram for H2 molecule
(bond order 1)
s 1s
asterisk denotes anti-bonding orbital
arrow electron
1s atomic orbital of H
1s atomic orbital of H
atom
atom
s 1s
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Some observations on MO diagrams
A single bond consists of a shared pair of elec
trons (Lewis). In MO theory Bond Order (BO)
(No. of es in bonding levels no. of es in an
ti- bonding levels)/2
BO for H2 (2-0)/2 1
in labeling the molecular orbitals, the type of
overlap is specified (s or p), and the atomic

orbitals involved indicated.
the two arrows are opposite in direction indicat
ing a pair of spin-paired electrons of opposite
spin
because of the Pauli exclusion
Principle each orbital can contain
a maximum of two electrons, which must be of
opposite spin
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Some more observations on MO diagrams
The greater the drop in energy the stronger the
bond. For the H2 molecule the drop is 218
kJ/mol so the enthalpy of dissociation of the H
2 molecule is 436 kJ/mole
In MO theory the reason molecules form is
because the bonding orbitals formed are lower
in energy than the atomic orbitals, and the
electrons are lowered in energy by this amount.
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Even more observations on MO diagrams
Electron excited to anti-bonding level
Photon of Energy hv
BO (1-1)/2 0 for excited state
MO diagrams show how a photon of energy hv
the difference in energy between two MOs, can ca
use an electron to be excited to the higher
energy level MO. In this excited state the bond
order zero and so the H2 molecule can photo-di
ssociate. Whether the transition can occur is
also determined by the parity of the orbitals (g
or u) see later.
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Identification of bonding and non-bonding
molecular orbitals.

• A bonding MO has no nodal plane between the two
  atoms forming the bond, i.e. the electron density
  does not go to zero at a node. An anti-bonding MO
  has a nodal plane where electron-density zero

nodal plane
s(1s) bonding orbital s(1s) anti-bonding
orbital
p(2p) bonding orbital p(2p)
anti-bonding orbital