Inorganic Chemistry

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Inorganic Chemistry
Bonding and Coordination Chemistry
Books to follow Inorganic Chemistry by Shriver
Atkins Physical Chemistry Atkins
C. R. Raj C-110, Department of Chemistry
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Bonding in s,p,d systems Molecular orbitals of
diatomics, d-orbital splitting in crystal field
(Oh, Td). Oxidation reduction Metal Oxidation
states, redox potential, diagrammatic
presentation of potential data. Chemistry of
Metals Coordination compounds (Ligands
Chelate effect), Metal carbonyls preparation
stability and application. Wilkinsons catalyst
alkene hydrogenation Hemoglobin, myoglobin
oxygen transport
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CHEMICAL BONDINGA QUANTUM LOOK
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Failure of Classical Mechanics

• Total energy, E ½ mv2 V(x)
• p mv ( p momentum )
• E p2/2m V(x) . . Eq.1

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• Newtons second law is a relation between the
  acceleration d2x/dt2 of a particle and the force
  F(x) it experiences.
• Therefore, v p/m
• Or, p F(x)
• Hit the ball hard, it will move fast
• Hit it soft, it will move slow
• Continuous variation of energy is possible.

Macroscopic World Classical Mechanics - the God
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• Certain experiments done in late 19th century and
  early 20th century gave results, totally at
  variance with the predictions of classical
  physics. All however, could be explained on the
  basis that, classical physics is wrong in
  allowing systems to possess arbitrary amounts of
  energy.
• For example, photoelectric effect.

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A young Max Planck was to give a lecture on
radiant heat. When he arrived he inquired as to
the room number for the Planck lecture. He was
told, "You are much too young to be attending the
lecture of the esteemed professor Planck."
Max Planck E h?
1900 German physicist
Each electromagnetic oscillator is limited to
discrete values and cannot be varied arbitrarily
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Plank had applied energy quantization to the
oscillators in the blackbody but had considered
the electromagnetic radiation to be wave.
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PHOTOELECTRIC EFFECT
J.J. Thomson
Hertz
When UV light is shone on a metal plate in a
vacuum, it emits charged particles (Hertz 1887),
which were later shown to be electrons by J.J.
Thomson (1899).
Light, frequency ?
Vacuum chamber
Collecting plate
Metal plate
I
Ammeter
Potentiostat
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Photoelectric Effect.
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1. No electrons are ejected, regardless of the
   intensity of the radiation, unless its frequency
   exceeds a threshold value characteristic of the
   metal.
2. The kinetic energy of the electron increases
   linearly with the frequency of the incident
   radiation but is independent of the intensity of
   the radiation.
3. Even at low intensities, electrons are ejected
   immediately if the frequency is above the
   threshold.

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Major objections to the Rutherford-Bohr model

• We are able to define the position and velocity
  of each electron precisely.
• In principle we can follow the motion of each
  individual electron precisely like planet.
• Neither is valid.

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Werner HeisenbergHeisenberg's name will always
be associated with his theory of quantum
mechanics, published in 1925, when he was only 23
years.

• It is impossible to specify the exact position
  and momentum of a particle simultaneously.
• Uncertainty Principle.
• ?x ?p ? h/4? where h is Planks Constant, a
  fundamental constant with the value 6.626?10-34 J
  s.

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1879 1955 Nobel prize 1921
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July 1, 1946
Einstein was the father of the bomb in two
important ways 1) it was his initiative which
started U.S. bomb research 2) it was his
equation (E mc2) which made the atomic bomb
theoretically possible.
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Einstein could never accept some of the
revolutionary ideas of quantum mechanics. When
reminded in 1927 that he revolutionized science
20 years earlier, Einstein replied, "A good joke
should not be repeated too often."
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Einstein
h ? ½ mv2 ?

• KE 1/2mv2 h?- ?
• ? is the work function
• h? is the energy of the incident light.
• Light can be thought of as a bunch of particles
  which have energy E h?. The light particles are
  called photons.

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If light can behave as particles,why not
particles behave as wave?
Louis de Broglie The Nobel Prize in Physics 1929
French physicist (1892-1987)
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Louis de Broglie

• Particles can behave as wave.
• Relation between wavelength ? and the mass and
  velocity of the particles.
• E h? and also E mc2,
• E is the energy
• m is the mass of the particle
• c is the velocity.

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Wave Particle Duality

• E mc2 h?
• mc2 h?
• p h /? since ? c/?
• ? h/p h/mv
• This is known as wave particle duality

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Flaws of classical mechanics
Photoelectric effect
Heisenberg uncertainty principle
limits simultaneous knowledge of conjugate
variables
Light and matter exhibit wave-particle
duality Relation between wave and particle
properties given by the de Broglie relations
The state of a system in classical mechanics is
defined by specifying all the forces acting and
all the position and velocity of the particles.
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Wave equation?Schrödinger Equation.

• Energy Levels
• Most significant feature of the Quantum
  Mechanics Limits the energies to discrete
  values.
• Quantization.

1887-1961
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The wave function
For every dynamical system, there exists a wave
function ? that is a continuous,
square-integrable, single-valued function of the
coordinates of all the particles and of time, and
from which all possible predictions about the
physical properties of the system can be obtained.
Square-integrable means that the normalization
integral is finite
If we know the wavefunction we know everything it
is possible to know.
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Time period T, Velocity v, v l/T,
Frequency, n 1/T, v n l
Derivation of wave equation
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An Electron Wave is similar to waves of light,
sound string
Wave motion of a String Amplitude vs. Position
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Displacement y (m)
A
Time t (s)
-A
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1. Maximum displacement A
2. Initial condition

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?
Displacement of a particle in SHM y(x) A sin
2?x/?
A maximum amplitude y amplitude at point x
at t 0 At x 0 , ?/2, ?, 3?/2, 2?, the
amplitude is 0 At x ?/4, 5?/4, 9?/4, the
amplitude is maximum.
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If the wave is moving to the right with velocity
v at time t

• y(x,t) A sin 2?/?(x-vt)
• v/ ?

• y A sin 2?n(x/v - t)
• Differentiating y W.R.T x, keeping t constant
• d2y/dx2 (4p2/ l2) y 0

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• In three dimension the wave equation becomes
• d2y/dx2 d2y/dy2 d2y/dz2 (4p2/l2)y 0
• It can be written as ?2y (4p2/l2)y 0
• We have l h/mv
• ?2y (4p2m2v2/h2) y 0
• E T V or T (E-V) (E total energy)
• V Potential energy, T Kinetic energy
• T 1/2 mv2 m2v2/2m
• m2v2 2m(E-V)

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?2y (8p2m/ h2)(E - V) y 0

• This can be rearranged as
• (- h2/8p2m) ?2 Vy Ey
• Hy Ey
• H (- h2/8p2m)?2 V) Hamiltonian operator

d2y/dx2 (4p2/ l2) y 0
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How to write Hamiltonian for different systems?
(-h2/8?2m)?2 V ? E ?

• Hydrogen atom
• KE ½ m (vx2 vy2 vz2)
• PE -e2/r, (r distance between the electron
  and the nucleus.)
• H (-h2/8?2m) ?2 e2/r
• ?2 ? (8?2 m/h2)(Ee2/r) ? 0
• If the effective nuclear charge is Ze
• H (-h2/8?2m )?2 Ze2/r

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H2 Molecule

• e (x,y,z)
• ra
  rb
• A RAB B

the wave function depends on the coordinates of
the two nuclei, represented by RA and RB, and of
the single electron, represented by r1.
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H2 (-h2/8?2m)?2 V ? E ?

• PE V -e2/ra e2/rb e2/Rab
• H (-h2/8?2m)?2 ( e2/ra - e2/rb e2/Rab)
• The Wave equation is
• ?2 ? (8?2 m/h2) (E e2/ra e2/rb e2/Rab) ?
  0

Born-Oppenheimer approximation
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V -e2/4??01/ra1/rb-1/Rab
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(-h2/8?2m)?2 V ? E ?

• V -2e2/r1 2e2/r2 e2/r12
• H (-h2/8?2m) (?12 ?22) V
• The Wave equation is
• (?12 ?22 )? (8?2 m/h2)(E-V) ? 0

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• PE V ?
• H (-h2/8?2m)(?12 ?22) V
• The Wave equation is
• (?12 ?22 )? (8?2 m/h2)(E-V) ? 0

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V -e2/4??01/ra11/rb1 1/ra2 1/rb2 -1/r12
-1/Rab
attractive potential energy
Electron-electron repulsion
Internuclear repulsion
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Particle in a box
An electron moving along x-axis in a field V(x)
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d2 ? /dx2 8?2 m/h2 (E-V) ? 0 Assume V0
between x0 xa Also ? 0 at x 0 a
d2?/dx2 8?2mE/h2 ? 0
d2?/dx2 k2? 0 where k2 8?2mE/h2
Solution is ? C cos kx D sin kx

• Applying Boundary conditions
• ? 0 at x 0 ? C 0
• ? ? D sin kx

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• ? D sin kx
• Applying Boundary Condition
• ? 0 at x a, ? D sin ka 0
• sin ka 0 or ka n?,
• k n?/a
• n 0, 1, 2, 3, 4 . . .
• ?n D sin (n?/a)x
• k2 8?2m/h2E or E k2h2/ 8?2m
• E n2 h2/ 8ma2 k2 n2 ?2/a2
• n 0 not acceptable ?n 0 at all x
• Lowest kinetic Energy E0 h2/8ma2

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An Electron in One Dimensional Box

• ?n D sin (n?/a)x
• En n2 h2/ 8ma2
• n 1, 2, 3, . . .
• E h2/8ma2 , n1
• E 4h2/8ma2 , n2
• E 9h2/8ma2 , n3

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Characteristics of Wave Function
He has been described as a moody and impulsive
person. He would tell his student, "You must not
mind my being rude. I have a resistance against
accepting something new. I get angry and swear
but always accept after a time if it is right."
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Characteristics of Wave Function What Prof.
Born Said

• Heisenbergs Uncertainty principle We can never
  know exactly where the particle is.
• Our knowledge of the position of a particle can
  never be absolute.
• In Classical mechanics, square of wave amplitude
  is a measure of radiation intensity
• In a similar way, ?2 or ? ? may be related to
  density or appropriately the probability of
  finding the electron in the space.

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The wave function ? is the probability amplitude
Probability density
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The sign of the wave function has not direct
physical significance the positive and negative
regions of this wave function both corresponds to
the same probability distribution. Positive and
negative regions of the wave function may
corresponds to a high probability of finding a
particle in a region.
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Characteristics of Wave Function What Prof.
Born Said

• Let ? (x, y, z) be the probability function,
• ?? d? 1
• Let ? (x, y, z) be the solution of the wave
  equation for the wave function of an electron.
  Then we may anticipate that
• ? (x, y, z) ? ?2 (x, y, z)
• choosing a constant in such a way that ? is
  converted to
• ? (x, y, z) ?2 (x, y, z)
• ? ??2 d? 1

The total probability of finding the particle is
1. Forcing this condition on the wave function is
called normalization.
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• ??2 d? 1 Normalized wave function
• If ? is complex then replace ?2 by ??
• If the function is not normalized, it can be done
  by multiplication of the wave function by a
  constant N such that
• N2 ??2 d? 1
• N is termed as Normalization Constant

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Acceptable wave functions
The wave equation has infinite number of
solutions, all of which do not corresponds to any
physical or chemical reality.

• For electron bound to an atom/molecule, the wave
  function must be every where finite, and it must
  vanish in the boundaries
• Single valued
• Continuous
• Gradient (d?/dr) must be continuous
• ? ?d? is finite, so that ? can be normalized
• Stationary States
• E Eigen Value ? is Eigen Function

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Need for Effective Approximate Method of Solving
the Wave Equation

• Born Oppenheimer Principle.
• How can we get the most suitable approximate wave
  function?
• How can we use this approximate wave function to
  calculate energy E?

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Operators
For every dynamical variables there is a
corresponding operator
Energy, momentum, angular momentum and position
coordinates
Symbols for mathematical operation
Operators
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Eigen values

• The permissible values that a dynamical variable
  may have are those given by
• ?? a?
• - eigen function of the operator ? that
  corresponds to the observable whose permissible
  values are a
• ? -operator
• ? - wave function
• a - eigen value

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?? a?
If performing the operation on the wave function
yields original function multiplied by a
constant, then ? is an eigen function of the
operator ?
? e2x and the operator ? d/dx
Operating on the function with the operator d
?/dx 2e2x constant.e2x
e2x is an eigen function of the operator ?
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• For a given system, there may be various possible
  values.
• As most of the properties may vary, we desire to
  determine the average or expectation value.
• We know
• ?? a?
• Multiply both side of the equation by ?
• ??? ?a?
• To get the sum of the probability over all space
• ? ??? d? ? ?a? d?
• a constant, not affected by the order of
  operation

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Removing a from the integral and solving for a
a ? ??? d?/ ? ?? d?
? cannot be removed from the integral.
a lt? ?? ?? gt/ lt? ?? gt
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Variation Method Quick way to get E

• H? E?
• ? H? ? E? E? ?
• If ? is complex,
• E ? ?H ? d?/ ? ? ?d?
• E?? ?H ??? /?? ??? (4)
• Bra-Ket notation

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What does E ?? ?H ??? /?? ??? tell us ?

• Given any ?, E can be calculated.
• If the wave function is not known, we can begin
  by educated guess and use Variation Theorem.
• ?1 ? E1
• ?2? E2

If a trial wave function is used to calculate
the energy, the value calculated is never less
than the true energy Variation Theorem.
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• ?1 ? E1
• ?2? E2
• The Variation Theorem tells that
• E1 , E2? Eg, Eg true energy of the ground state
• IF, E1 ? E2,
• Then E2 and ?2 is better approximation to the
  energy and corresponding wave function ?2 to the
  true wave function

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Variation Method The First Few Steps

• We can chose a whole family of wave function at
  the same time, like trial function with one or
  more variable parameters C1, C2, C3,.
• Then E is function of C1, C2, C3 .etc.
• C1, C2, C3 . etc. are such that E is minimized
  with respect to them.
• We will utilize this method in explaining
  chemical bonding.

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Chemical Bonding

• Two existing theories,
• Molecular Orbital Theory (MOT)
• Valence Bond Theory (VBT)
• Molecular Orbital Theory
• MOT starts with the idea that the quantum
  mechanical principles applied to atoms may be
  applied equally well to the molecules.

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MOT We can write the following principles

• Describe Each electron in a molecule by a certain
  wave function ? - Molecular Orbital (MO).
• Each ? is defined by certain quantum numbers,
  which govern its energy and its shape.
• Each ? is associated with a definite energy
  value.
• Each electron has a spin, ½ and labeled by its
  spin quantum number ms.
• When building the molecule- Aufbau Principle
  (Building Principle) - Pauli Exclusion Principle.

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Simplest possible moleculeH2 2 nuclei and 1
electron.

• Let the two nuclei be labeled as A and B wave
  functions as ?A ?B.
• Since the complete MO has characteristics
  separately possessed by ?A and ?B,
• ? CA?A CB?B
• or ? N(?A ? ?B)
• ? CB/CA, and N - normalization constant

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This method is known as Linear Combination of
Atomic Orbitals or LCAO

• ?A and ?B are same atomic orbitals except for
  their different origin.
• By symmetry ?A and ?B must appear with equal
  weight and we can therefore write
• ?2 1, or ? 1
• Therefore, the two allowed MOs are
• ? ?A ?B

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For ?A ?B we can now calculate the energy

• From Variation Theorem we can write the energy
  function as
• E ??A?B ?H ??A?B?/??A?B ??A?B?

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Looking at the numerator E ??A?B ?H
??A?B?/??A?B ??A?B?

• ??A?B ?H ? ?A?B? ??A ?H ??A?
• ??B ?H ??B?
• ??A ?H ??B?
  
• ??B ?H ??A?
• ??A ?H ? ?A? ??B ?H ??B? 2??A?H ??B?

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ground state energy of a hydrogen atom. let us
call this as EA
??A ?H ? ?A? ??B ?H ??B? 2??A?H ??B?

• ??A ?H ? ?B? ??B ?H ??A? ?
• ? resonance integral

? Numerator 2EA 2 ?
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Physical Chemistry class test answer scripts will
be shown to the students on 3rd March (Tuesday)
at 530 pm in Room C-306 Sections 11 and 12
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Looking at the denominator E ??A?B ?H
??A?B?/??A?B ??A?B?

• ??A?B ??A?B? ??A ??A?
• ??B ??B?
• ??A ??B?
• ??B ??A?
• ??A ??A? ??B ??B? 2??A ??B?

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??A ??A? ??B ??B? 2??A ??B?
?A and ?B are normalized, so ??A ??A? ??B ??B?
1
??A ??B? ??B ??A? S, S Overlap integral.
? Denominator 2(1 S)
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Summing Up . . . E ??A?B ?H ??A?B?/??A?B
??A?B?
Numerator 2EA 2 ?
Denominator 2(1 S)
E (EA ?)/ (1 S) Also E- (EA - ?)/ (1 S)
E EA ?
S is very small ? Neglect S
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Energy level diagram
EA - ?
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Linear combination of atomic orbitals
Rules for linear combination
1. Atomic orbitals must be roughly of the same
energy.
2. The orbital must overlap one another as much
as possible- atoms must be close enough for
effective overlap.
3. In order to produce bonding and antibonding
MOs, either the symmetry of two atomic orbital
must remain unchanged when rotated about the
internuclear line or both atomic orbitals must
change symmetry in identical manner.
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Rules for the use of MOs When two AOs mix,
two MOs will be produced Each orbital can
have a total of two electrons (Pauli principle)
Lowest energy orbitals are filled first
(Aufbau principle) Unpaired electrons have
parallel spin (Hunds rule) Bond order ½
(bonding electrons antibonding electrons)
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Linear Combination of Atomic Orbitals (LCAO)
The wave function for the molecular orbitals can
be approximated by taking linear combinations of
atomic orbitals.
?A
?B
c extent to which each AO contributes to the MO
?AB N(cA ?A cB?B)
?2AB (cA2 ?A2 2cAcB ?A ?B cB2 ?B 2)
Overlap integral
Probability density
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Constructive interference
bonding
?g
cA cB 1
?g N ?A ?B
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?2AB (cA2 ?A2 2cAcB ?A ?B cB2 ?B 2)
density between atoms
electron density on original atoms,
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The accumulation of electron density between the
nuclei put the electron in a position where it
interacts strongly with both nuclei.
Nuclei are shielded from each other
The energy of the molecule is lower
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Destructive interference Nodal plane
perpendicular to the H-H bond axis (en density
0) Energy of the en in this orbital is higher.
?A-?B
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• The electron is excluded from internuclear region
  ? destabilizing

Antibonding
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Molecular potential energy curve shows the
variation of the molecular energy with
internuclear separation.
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Looking at the Energy Profile

• Bonding orbital
• called 1s orbital
• s electron
• The energy of 1s orbital
• decreases as R decreases
• However at small separation, repulsion becomes
  large
• There is a minimum in potential energy curve

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H2
11.4 eV 109 nm
LCAO of n A.O ? n M.O.
Location of Bonding orbital 4.5 eV
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The overlap integral

• The extent to which two atomic orbitals on
  different atom overlaps the overlap integral

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S gt 0 Bonding
S lt 0 anti
Bond strength depends on the degree of overlap
S 0 nonbonding
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Homonuclear Diatomics

• MOs may be classified according to
• (i) Their symmetry around the molecular axis.
• (ii) Their bonding and antibonding character.
• ?1s? ?1s? ?2s? ?2s? ?2p? ?y(2p) ?z(2p)
  ??y(2p) ??z(2p)??2p.

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dx2-dy2 and dxy
97
B
g- identical under inversion
A
u- not identical
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Place labels g or u in this diagram
su
pg
pu
sg
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First period diatomic molecules
?1s2
Bond order 1
Bond order ½ (bonding electrons antibonding
electrons)
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Diatomic molecules The bonding in He2
?1s2, ?1s2
Bond order 0
Molecular Orbital theory is powerful because it
allows us to predict whether molecules should
exist or not and it gives us a clear picture of
the of the electronic structure of any
hypothetical molecule that we can imagine.
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Second period diatomic molecules
?1s2, ?1s2, ?2s2
Li
Li
Li2
Bond order 1
2?u
2s
2s
2?g
Energy
1?u
1s
1s
1?g
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Diatomic molecules Homonuclear Molecules of the
Second Period
Be
Be
Be2
2?u
?1s2, ?1s2, ?2s2, ?2s2
2s
2s
2?g
Energy
Bond order 0
1?u
1s
1s
1?g
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Simplified
105
Simplified
106
MO diagram for B2
Diamagnetic??
107
Li 200 kJ/mol F 2500 kJ/mol
108
Same symmetry, energy mix- the one with higher
energy moves higher and the one with lower energy
moves lower
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MO diagram for B2
Paramagnetic
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C2
Diamagnetic
X
Paramagnetic ?
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General MO diagrams
O2 and F2
Li2 to N2
112
Orbital mixing Li2 to N2
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Bond lengths in diatomic molecules
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Summary
From a basis set of N atomic orbitals, N
molecular orbitals are constructed. In Period 2,
N8.
The eight orbitals can be classified by symmetry
into two sets 4 ? and 4 ? orbitals.
The four ? orbitals from one doubly degenerate
pair of bonding orbitals and one doubly
degenerate pair of antibonding orbitals.
The four ? orbitals span a range of energies, one
being strongly bonding and another strongly
antibonding, with the remaining two ? orbitals
lying between these extremes.
To establish the actual location of the energy
levels, it is necessary to use absorption
spectroscopy or photoelectron spectroscopy.
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Distance between b-MO and AO
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Heteronuclear Diatomics.

• The energy level diagram is not symmetrical.

• The bonding MOs are closer to the atomic
  orbitals which are lower in energy.
• The antibonding MOs are closer to those higher in
  energy.

c extent to which each atomic orbitals
contribute to MO
If cA?cB the MO is composed principally of ?A
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HF
120
HF
1s 1 2s, 2p 7
? c1 ?H1s c2 ?F2s c3 ?F2pz
Largely nonbonding
2px and 2py
1?2 2?21?4
Polar