4. Failure of Classical Mechanics
• Total energy, E = ½ mv2 + V(x)
• p = mv ( p = momentum )
• E = p2/2m + V(x) ……… . . Eq.1
“ A moving ball
I know it all ”
5. • Newton’s 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”
6. Max Planck E = hν
1900 German physicist
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."
“Each electromagnetic oscillator is limited to discrete
values and cannot be varied arbitrarily”
7. Plank had applied energy quantization
to the oscillators in the blackbody but
had considered the electromagnetic
radiation to be wave.
8. Hertz J.J. Thomson
PHOTOELECTRIC EFFECT
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).
Classical expectations
Vacuum Light, frequency ν
As intensity of light increases, force
chamber
Collecting increases, so KE of ejected electrons
Metal should increase.
plate
plate
Electrons should be emitted whatever
the frequency ν of the light.
Actual results:
I Maximum KE of ejected electrons is
Ammeter independent of intensity, but dependent on ν
Potentiostat For ν<ν0 (i.e. for frequencies below a cut-
off frequency) no electrons are emitted
11. (i) No electrons are ejected, regardless of the intensity of the
radiation, unless its frequency exceeds a threshold value
characteristic of the metal.
(ii) The kinetic energy of the electron increases linearly with
the frequency of the incident radiation but is independent
of the intensity of the radiation.
(iii) Even at low intensities, electrons are ejected immediately
if the frequency is above the threshold.
12. 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.
13. Werner Heisenberg
Heisenberg'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/4π where h is Plank’s
Constant, a fundamental constant with
the value 6.626×10-34 J s.
14. Einstein
h ν = ½ mv2 + φ
• KE 1/2mv2 = hν- φ
∀ φ is the work function
• hν is the energy of the incident light.
• Light can be thought of as a bunch of
particles which have energy E = hν. The
light particles are called photons.
15. 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)
16. Wave Particle Duality
• E = mc2 = hν
• mc2 = hν
• p = h /λ { since ν = c/λ}
∀ λ = h/p = h/mv
• This is known as wave particle duality
17. 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.
18. Wave equation?
Schrödinger Equation.
• Energy Levels
• Most significant feature of the Quantum
Mechanics: Limits the energies to
discrete values.
• Quantization.
1887-1961
19. The wave function
For every dynamical system, there exists a wave function Ψ
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.
20. Derivation of wave equation
Time period = T, Velocity = v, v = λ/T,
Frequency, ν = 1/T, v = ν λ
21. y
x
If the wave is moving to the right with velocity ‘v’ at time ‘t’
y(x,t) = A sin 2π/λ(x-vt)
λ= v/ ν
• y = A sin 2πν(x/v - t)
• Differentiating y W.R.T x, keeping t
constant A wave eqn.
• δ 2y/δx2 + (4π 2 / λ 2 ) y = 0 is born
22. • In three dimension the wave equation
becomes:
• δ 2 ψ/δx2 + δ 2 ψ/δy2 + δ 2 ψ/δz2 + (4π 2 /λ 2 )ψ = 0
· It can be written as ∇ 2ψ + (4π 2 /λ 2 )ψ = 0
· We have λ = h/mv
• ∇ 2ψ + (4π 2 m2v2/h2 ) ψ = 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)
23. ∇ 2ψ + (8π 2 m/ h2 )(E -
V) ψ = 0
• This can be rearranged as
• {(− h2/8π 2 m) ∇ 2 + V}ψ = Ε ψ
· Hψ = Ε ψ
• Η = [(− h2/8π 2 m)∇ 2 + V) Hamiltonian
operator
{(-h2/8π 2m)(∂ 2/∂x2 + ∂ 2/∂y2 + ∂ 2/∂z2) + V} Ψ = E Ψ
δ 2y/δx2 + (4π 2 / λ 2 ) y = 0
24. How to write Hamiltonian for different
systems?
{(-h2/8π 2m)∇ 2 + V} Ψ = E Ψ
-e
• Hydrogen atom: r
• KE = ½ m (vx2 + vy2 + vz2)
+Ze
• PE = -e /r, (r = distance between the
2
electron and the nucleus.)
• H = {(-h2/8π 2m) ∇ 2 –e2/r}
∀ ∇ 2 Ψ + (8π 2 m/h2)(E+e2/r) Ψ = 0
• If the effective nuclear charge is Ze
• H = {(-h2/8π 2m )∇ 2 –Ze2/r}
25. 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
26. e (x,y,z)
H2 +
ra
{(-h /8π m)∇ + V} Ψ = E Ψ
2 2 2
rb
A Rab B
• PE = V = -e2/ra – e2/rb + e2/Rab
• H = (-h2/8π 2m)∇ 2 + ( – e2/ra - e2/rb + e2/Rab)
• The Wave equation is
∀ ∇ 2 Ψ + (8π 2 m/h2) (E+ e2/ra + e2/rb – e2/Rab) Ψ = 0
Born-Oppenheimer approximation
28. e1 (x1, y1, z1)
He Atom
r12
r1
e2 (x2, y2, z2) r2 Nucleus (+2e)
{(-h2/8π 2m)∇ 2 + V} Ψ = E Ψ
• V = -2e2/r1 – 2e2/r2 + e2/r12
• H = (-h2/8π 2m) (∇ 12 + ∇ 22) + V
• The Wave equation is
• (∇ 12 + ∇ 22 )Ψ + (8π 2 m/h2)(E-V) Ψ = 0
29. e1 (x1, y1, z1) r12 e2 (x2, y2, z2)
ra2
H2 ra1 rb2
rb1
A Rab B
• PE = V = ?
• H = (-h2/8π 2m)(∇ 12 + ∇ 22) + V
• The Wave equation is
• (∇ 12 + ∇ 22 )Ψ + (8π 2 m/h2)(E-V) Ψ = 0
30. V = -e2/4πε 0[1/ra1+1/rb1 + 1/ra2 +1/rb2 -1/r12 -1/Rab]
attractive potential energy Electron-electron repulsion
Internuclear repulsion
31. Particle in a box
An electron moving along x-axis in a field V(x)
a
V=0
x =0 x =a
32. d2 Ψ /dx2 + 8π 2 m/h2 (E-V) Ψ = 0 a
Assume V=0 between x=0 &
x=a V=0
d2Ψ/dx2 Ψ [8π 2at x = 20 Ψ a 0
Also + = 0 mE/h ] & =
x =0 x =a
d2Ψ/dx2 + k2Ψ = 0 where k2 = 8π 2mE/h2
Solution is: Ψ = C cos kx + D sin kx
• Applying Boundary conditions:
∀ Ψ = 0 at x = 0 ⇒ C = 0
∴ Ψ = D sin kx
33.
34.
35. ∀ Ψ = D sin kx a
• Applying Boundary Condition:
∀ Ψ = 0 at x = a, ∴ D sin ka = 0 V=0
• sin ka = 0 or ka = nπ,
x =0 x =a
• k = nπ/a
• n = 0, 1, 2, 3, 4 . . .
∀ Ψ n = D sin (nπ/a)x
• k2 = 8π 2m/h2[E] or E = k2h2/ 8π 2m
• E = n2 h2/ 8ma2 k2= n2 π 2/a2
• n = 0 not acceptable: Ψ n = 0 at all x
• Lowest kinetic Energy = E = h2/8ma2
36. An Electron in One Dimensional Box
a
V=∝ V=∝ ∀ Ψ n = D sin (nπ/a)x
• En = n2 h2/ 8ma2
• n = 1, 2, 3, . . .
• E = h2/8ma2 , n=1
• E = 4h2/8ma2 , n=2
• E = 9h2/8ma2 , n=3
x=0 x=a Energy is quantized
37. Characteristics of Wave Function:
What Prof. Born Said
• Heisenberg’s 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, ψ 2 or ψ ψ* may be related
to density or appropriately the probability of
finding the electron in the space.
38. The wave function Ψ is the probability amplitude
2
ψ = ψ *ψ
Probability density
39.
40. 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.
41. Characteristics of Wave Function:
What Prof. Born Said
• Let ρ (x, y, z) be the probability function,
∀ ∫ρ dτ = 1
Let Ψ (x, y, z) be the solution of the wave equation
for the wave function of an electron. Then we may
anticipate that
ρ (x, y, z) ∝ Ψ 2 (x, y, z)
• choosing a constant in such a way that ∝ is
converted to =
∀ ρ (x, y, z) = Ψ 2 (x, y, z)
∴ ∫Ψ 2 dτ = 1
The total probability of finding the particle is 1. Forcing this condition on
the wave function is called normalization.
42. ∀ ∫Ψ 2 dτ = 1 Normalized wave function
• If Ψ is complex then replace Ψ 2 by ΨΨ *
• If the function is not normalized, it can be done
by multiplication of the wave function by a
constant N such that
• N2 ∫Ψ 2 dτ = 1
• N is termed as Normalization Constant
43. 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 (dΨ/dr) must be continuous
• Ψ Ψ*dτ is finite, so that Ψ can be normalized
• Stationary States
• E = Eigen Value ; Ψ is Eigen Function
44. 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?
45. Operators
“For every dynamical variables there is a corresponding operator”
Energy, momentum, angular
momentum and position coordinates
Operators Symbols for mathematical operation
46. Eigen values
The permissible values that a dynamical variable
may have are those given by
αφ = aφ
φ- eigen function of the operator α that
corresponds to the observable whose permissible
values are a
α -operator
φ - wave function
a - eigen value
47. αφ = aφ
If performing the operation on the wave function yields
original function multiplied by a constant, then φ is an eigen
function of the operator α
φ = e2x and the operator α = d/dx
Operating on the function with the operator
d φ /dx = 2e2x = constant.e2x
e2x is an eigen function of the operator α
48. 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
αφ = aφ
Multiply both side of the equation by φ *
φ *αφ = φ *aφ
To get the sum of the probability over all space
∫ φ *αφ dτ = ∫ φ *aφ dτ
a – constant, not affected by the order of operation
49. Removing ‘a’ from the integral and solving for ‘a’
a = ∫ φ *αφ dτ/ ∫ φ *φ dτ
α cannot be removed from the integral.
a = <φ α φ >/ <φ φ >
50. Variation Method: Quick way to get E
• HΨ = EΨ
∀ Ψ HΨ = Ψ EΨ = EΨ Ψ
• If Ψ is complex,
• E = ∫ Ψ *H Ψ dτ/ ∫ Ψ * Ψdτ
• E=<Ψ H Ψ> /<Ψ Ψ> ……
(4)
• Bra-Ket notation
51. What does E = <Ψ H Ψ> /<Ψ Ψ> tell us ?
• Given any Ψ, E can be calculated.
• If the wave function is not known, we can
begin by educated guess and use Variation
Theorem.
Ψ 1 ⇒ E1
Ψ 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.
52. ∀ Ψ 1 ⇒ E1
∀ Ψ 2⇒ E2
The Variation Theorem tells that
• E1 , E2> Eg, Eg true energy of the ground state
• IF, E1 > E2,
• Then E2 and Ψ 2 is better approximation to the energy
and corresponding wave function Ψ 2 to the true wave
function
53. 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.
54. 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.
56. MOT: We can write the following principles
Describe Each electron in a molecule by a
certain wave function Ψ - Molecular Orbital
(MO).
Each Ψ is defined by certain quantum numbers,
which govern its energy and its shape.
Each Ψ 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.
57. Simplest possible molecule:
H2+ : 2 nuclei and 1 electron.
• Let the two nuclei be labeled as A and B &
wave functions as Ψ A & Ψ B.
• Since the complete MO has characteristics
separately possessed by Ψ A and Ψ B,
∀ Ψ = CAΨ A + CBΨ B
• or Ψ = N(Ψ A +λ Ψ B)
∀ λ = CB/CA, and N - normalization constant
58. This method is known as Linear Combination
of Atomic Orbitals or LCAO
∀ Ψ A and Ψ B are same atomic orbitals except
for their different origin.
• By symmetry Ψ A and Ψ B must appear with
equal weight and we can therefore write
• λ 2 = 1, or λ = ±1
• Therefore, the two allowed MO’s are
∀ Ψ = Ψ A± Ψ B
59. For Ψ A+ Ψ B
we can now calculate the energy
• From Variation Theorem we can write the
energy function as
• E = <Ψ A+Ψ B H Ψ A+Ψ B> /<Ψ A+Ψ B Ψ A+Ψ B>
60. Looking at the numerator:
E = <Ψ A+Ψ B H Ψ A+Ψ B> /<Ψ A+Ψ B Ψ A+Ψ B>
∀ <Ψ A+Ψ B H Ψ A+Ψ B> = <Ψ A H Ψ A> +
• <Ψ B H Ψ B> +
• <Ψ A H Ψ B> +
• <Ψ B H Ψ A>
• = <Ψ A H Ψ A> + <Ψ B H Ψ B> +2<Ψ AH Ψ B>
61. = <Ψ A H Ψ A> + <Ψ B H Ψ B> + 2<Ψ AH Ψ B>
ground state energy of a hydrogen
atom. let us call this as EA
<Ψ A H Ψ B> = <Ψ B H Ψ A> = β
β = resonance integral
∴ Numerator = 2EA + 2 β
62. Physical Chemistry class test answer scripts will be shown to
the students on 3rd March (Tuesday) at 5:30 pm in
Room C-306: Sections 11 and 12
63. Looking at the denominator:
E = <Ψ A+Ψ B H Ψ A+Ψ B> /<Ψ A+Ψ B Ψ A+Ψ B>
• <Ψ A+Ψ B Ψ A+Ψ B> = <Ψ A Ψ A> +
• <Ψ B Ψ B> +
• <Ψ A Ψ B> +
• <Ψ B Ψ A>
• = <Ψ A Ψ A> + <Ψ B Ψ B> + 2<Ψ A Ψ B>
64. = <Ψ A Ψ A> + <Ψ B Ψ B> + 2<Ψ A Ψ B>
Ψ A and Ψ B are normalized,
so <Ψ A Ψ A> = <Ψ B Ψ B> =
1
<Ψ A Ψ B> = <Ψ B Ψ A> =
S, S = Overlap integral.
∴ Denominator = 2(1 + S)
65. Summing Up . . .
E = <Ψ A+Ψ B H Ψ A+Ψ B> /<Ψ A+Ψ B
Ψ A+Ψ B>
Numerator = 2EA + 2 β
Denominator = 2(1 + S)
E+ = (EA + β)/ (1 + S)
Also E- = (EA - β)/ (1 – S)
E± = E A ± β
S is very small
∴ Neglect S
67. 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.
68. 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 (Hund’s rule)
Bond order = ½ (bonding electrons – antibonding
electrons)
69. 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
ψA ψB
ψ AB = N(cA ψ A + cBψ B) c – extent to which each AO
contributes to the MO
ψ 2AB = (cA2 ψ A2 + 2cAcB ψ A ψ B + cB2 ψ B 2)
Probability density Overlap integral
70. Constructive interference
. + .
+. +.
ψg bonding
cA = cB = 1
ψ g = N [ψ A + ψ B]
Amplitudes of wave
functions added
71. ψ 2AB = (cA2 ψ A2 + 2cAcB ψ A ψ B + cB2 ψ B 2)
density between atoms
electron density on original atoms,
72. 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
73. node
+. -. +. .-
cA = +1, cB = -1 ψu
antibonding
ψu = N [ψA - ψB]
Destructive interference
Nodal plane perpendicular to the
H-H bond axis (en density = 0)
Energy of the en in this orbital is
higher.
Ψ A-Ψ B
Amplitudes of wave
functions
subtracted.
74. The electron is excluded from internuclear region destabilizing
Antibonding
75. When 2 atomic orbitals combine there are 2
resultant orbitals.
orbitals.
Eg.
Eg. s orbitals
σ*s
1
E high energy antibonding orbital
1sb 1sa
σ1s
Molecular
orbitals
low energy bonding orbital
76. Molecular potential energy curve shows the variation
of the molecular energy with internuclear separation.
77. 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
78. H2
11.4 eV
LCAO of n A.O ⇒ n M.O.
109 nm
Location of
Bonding orbital
4.5 eV
79. The overlap integral
The extent to which two atomic orbitals on different atom
overlaps : the overlap integral
S = ∫ ψ A ψ B dτ
*
80. S > 0 Bonding S < 0 anti
Bond strength depends on the
S = 0 nonbonding
degree of overlap
81.
82.
83.
84. Homonuclear Diatomics
• MOs may be classified according to:
(i) Their symmetry around the molecular axis.
(ii) Their bonding and antibonding character.
∀ σ 1s< σ 1s*< σ 2s< σ 2s*< σ 2p< π y(2p) = π z(2p)
<π y*(2p) =π z*(2p)<σ 2p*.
88. First period diatomic molecules
H H2 H σ1s2
σ u*
Bond order: 1
Energy
1s 1s
σg
Bond order =
½ (bonding electrons – antibonding electrons)
89. Diatomic molecules: The bonding in He2
He He2 He
σ1s2, σ *1s2
σu*
Bond order: 0
Energy
1s 1s
σg
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.
90.
91. Second period diatomic molecules
Li Li2 Li σ1s2, σ *1s2, σ2s2
2σu* Bond order: 1
2s 2s
Energy
2σg
1σu*
1s 1s
1σg
92. Diatomic molecules: Homonuclear Molecules of the Second Period
Be Be2 Be
2σu*
σ1s2, σ *1s2, σ2s2,
2s 2s
σ *2s2
Energy
2σg
Bond order: 0
1σu*
1s 1s
1σg
102. Bond lengths in diatomic molecules
Filling bonding orbitals
Filling antibonding orbitals
103.
104. Summary
From a basis set of N atomic orbitals, N molecular orbitals are
constructed. In Period 2, N=8.
The eight orbitals can be classified by symmetry into two sets: 4 σ
and 4 π orbitals.
The four π orbitals from one doubly degenerate pair of bonding
orbitals and one doubly degenerate pair of antibonding orbitals.
The four σ orbitals span a range of energies, one being strongly
bonding and another strongly antibonding, with the remaining
two σ orbitals lying between these extremes.
To establish the actual location of the energy levels, it is necessary
to use absorption spectroscopy or photoelectron spectroscopy.
107. 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 φ A
