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SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
UNIT 1
BONDS TO BANDS
Introduction to Quantum Chemistry – Motion of a Quantum Mechanical Particle in One
Dimension (Time- Independent) – Schrödinger Wave Equation for Hydrogen Atom (No
Derivation) – Physical Meaning of Wave Function - Angular and Radial Wave Functions and
Probability Densities – Quantum Numbers – Principal, Azimuthal, Spin and Magnetic
Quantum Numbers – Wave Functions and Orbital Shapes - s, p, d, f - LCAO-MO of H 2
Theory of Solids: Conductors, Semi-Conductors and Superconductors – Role of As and Ga
Doping on Band Structures.
INTORDUCTION TO QUANTUM CHEMISTRY
1. Compare Classical Mechanics with Quantum Mechanics
S. No. Classical Mechanics Quantum Mechanics
- It deals with macroscopic particles. It deals with microscopic particles.
- It is based upon Newton’s laws of
motion.
It takes into account Heisenberg’s
uncertainty principle and de Broglie
concept of dual nature of matter.
- It is based on Maxwell’s
electromagnetic wave theory according
to which any amount of energy may be
emitted or absorbed continuously.
It is based on Planck’s quantum
theory according to which only
discrete values of energy are
emitted or absorbed.
2. Tell de Broglie’s hypothesis.
According to de-Broglie, the microscopic particle like electron has dual character. It
can behave as a wave and as a particle.
3. Define Heisenberg’s uncertainty principle.
For any microscopic particles like electron that has a dual character of wave and particle,
“ it is not possible to determine the position and momentum simultaneously”.
4. Write time independent Schrodinger wave equation.
𝝏
𝟐
𝜳
𝝏𝒙
𝟐
−𝟖𝝅
𝟐
[𝑬−𝑽]
𝒉
𝟐
5. Write Schrodinger wave equation in Hamiltonian form.
𝑯
̂
𝜳 = 𝑬𝜳
𝒘𝒉𝒆𝒓𝒆, 𝑯
̂ = [
−𝜵
𝟐
𝒉
𝟐
𝟖𝝅
𝟐
𝒎
- 𝑽] 𝒊𝒔 𝒌𝒏𝒐𝒘𝒏 𝒂𝒔 𝑯𝒂𝒎𝒊𝒍𝒕𝒐𝒏𝒊𝒂𝒏 𝒐𝒑𝒆𝒓𝒂𝒕𝒐𝒓.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
6. Define the term wave function.
In quantum mechanics, there is no distinction between the particle and wave, and hence
the microscopic system (like electron, atom, or molecule) is described by a mathematical
function known as wave function. It is represented by the symbol, Ψ (psi).
7. What are the conditions for acceptable wave function?
✓ Ψ must be a single valued function.
✓ Ψ must be continuous with respect to the change of variables.
✓ Ψ must have finite value.
✓ Ψ must be normalized.
APPLICATION OF SCHRODINGER WAVE EQUATION FOR A PARTICLE IN ONE
DIMENSIONAL BOX
1. Formulate the Schrodinger wave equation for a particle in one dimensional
box and determine the wave function and energy of the particle.
The Concept of 1 - D Box
Let us consider a particle moving along x direction in a confined space between the limits
x = 0 and x = L. As the particle is moving along the x - direction, its potential energy, V =
0 inside the box while the potential energy at the boundaries and outside the box is, V =
∞. The probability of finding the particle inside the box i.e. 𝛹 ≠ 0 while the probability
of finding particle outside the box is zero.
x = 0 x = L
V =
V=
x - axis
V =
oo
oo = 0
= 0
i.e. The boundary conditions may be expressed in mathematical form as:
(i) when
𝑽 = ∞
𝒙 ≤ 𝟎
𝒙 ≥ 𝑳
} → 𝑽 = ∞ 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒔 𝒕𝒉𝒂𝒕, 𝒏𝒐 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆 𝒆𝒙𝒊𝒕𝒔 𝒐𝒖𝒕𝒔𝒊𝒅𝒆 𝒕𝒉𝒆 𝒃𝒐𝒙 𝒂𝒏𝒅 𝒉𝒆𝒏𝒄𝒆 𝜳 = 𝟎.
(ii) when
𝑽 = 𝟎
𝟎 < 𝒙 < 𝑳
} → 𝑽 = 𝟎 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒔 𝒕𝒉𝒂𝒕 𝒕𝒉𝒆 𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆 𝒆𝒙𝒊𝒕𝒔 𝒊𝒏𝒔𝒊𝒅𝒆 𝒕𝒉𝒆 𝒃𝒐𝒙 𝒂𝒏𝒅 𝒉𝒆𝒏𝒄𝒆 𝜳 ≠ 𝟎.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
Determination of Energy of a Moving Particle in 1 - D Box
Squaring the equation (5) on both the sides, we have:
2
2
2
2
From equation (3), we have:
2
2
2
Comparing equation (3) and equation (7), we have:
2
2
2
2
2
2
2
2
Therefore, the energy of particle depends upon ‘ n ’ value, so we have:
𝒏
𝟐
𝟐
𝟐
Where n is the principal quantum number and can have the values 1, 2, 3, ….; h is the
Planck’s constant; m is the mass of particle (electron) and L is the length of box (space
between two quantum numbers).
When n = 1, then 𝐸
1
ℎ
2
8 𝑚𝐿
2
; this is the minimum energy possessed by the particle.
When n = 2, then 𝐸
2
4ℎ
2
8 𝑚𝐿
2
; energy level of particle increases by four times.
When n = 3, then 𝐸
3
9ℎ
2
8 𝑚𝐿
2
When n = 4, then 𝐸
4
16ℎ
2
8 𝑚𝐿
2
and so on ….
As the principle quantum number increases, the spacing between two energy level
increases. Further, the spacing between two energy level decreases when length of one-
dimensional box increases.
Zero Point Energy: It the minimum amount of energy possessed by a particle in one
dimensional box, and it is given by:
𝟏
𝟐
𝟐
Where h is Planck’s constant (6.626 × 10
‒
kg m
2
s
‒
and m is mass of electron (9.1 ×
‒
kg).
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
x = 0 x = L
n = 1
n = 2
n = 3
E
3
=
E
2
=
E
1
=
9h
2
8mL
2
4h
2
h
2
E
n
e
r
g
y
8mL
2
8mL
2
2. Determine the value of “A” in the wave function of particle in one
dimensional box by normalization method.
The wave function Ψ can have a positive real value and a negative imaginary value and
hence it has no significance.
The wave function for a particle in one dimensional box is given by:
On squaring both the sides, we get Ψ
2
, which will have always a positive value.
Integrating Ψ
2
with the limits 0 → L can be used for finding the particle in one
dimensional box, which is equal to unity (100%).
2
2
2
𝐿
0
𝐿
0
2
[
] 𝑑𝑥 = 1
𝐿
0
2
[∫ 𝑑𝑥
𝐿
0
𝐿
0
] = 1
2
[𝐿] = 1 (𝑜𝑟) 𝐴
2
Therefore, the wave function for particle in one dimensional box is given by:
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
2. Explain the physical significance of Ψ and Ψ
2
The Ψ is known as wave function while Ψ
2
is known as probability density function of
the particle in 1-D box.
The Ψ can have a positive value and a negative value while Ψ
2
always will have a positive
value.
The Ψ
2
can be used: (i) to find out the probability density of particle in 1-D box and (ii)
to calculate the number of nodes of a system in 1-D box.
3. What is probability density?
Probability density (function) refers to the probability of finding particle like electron in
a given region of space.
APPLICATION OF SWE FOR HYDROGEN ATOM
1. Outline the time-independent Schrodinger equation of hydrogen atom
using φ and θ****.
The hydrogen atom is an example of two particle system (electron and proton).
The proton has a mass of M with a charge of + Ze and the electron has a mass of m with
a charge of – e.
Since the proton has heavier mass than the electron, the electron revolves around the
nucleus with its reduced mass along the distance r.
The potential energy of electron moving at a distance of r due to electrostatic attraction
with proton is given as:
𝑍𝑒
2
( 4 𝜋𝜀
𝑜
)𝑟
𝑒
2
( 4 𝜋𝜀
𝑜
)𝑟
Where Z is the atomic number, e is the electronic charge in Coulomb, and 𝜀
𝑜
is the
absolute permittivity of the medium, vacuum, in this case (𝜀
𝑜
= 8.85×
‒
Now there are two types of motions are involved in hydrogen atom and they are:
i) Translation motion of electron
ii) Internal motion of electron and proton
The Schrodinger wave equation for these two types motions is given by:
𝑇
𝑇
𝑇
Where Ĥ is Hamiltonian operator; Ψ T
is complete (or total) wave function and E T
is the
total energy of hydrogen atom.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
We know that, 𝐻
= [
−𝛻
2
ℎ
2
8 𝜋
2
𝑚
+ 𝑉] 𝑤ℎ𝑒𝑟𝑒 𝛻
2
= [
𝜕
2
𝜕𝑥
2
𝜕
2
𝜕𝑦
2
𝜕
2
𝜕𝑧
2
]
2
2
[
2
2
2
2
2
2
] + 𝑉} … ( 2 )
When Ĥ is substituted in equation (1), we get:
2
2
[
2
2
2
2
2
2
] + 𝑉} 𝛹 = 𝐸𝛹
𝑒
2
( 4 𝜋𝜀 𝑜
) 𝑟
Therefore, {
𝟐
𝟐
[
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
] −
𝟐
𝒐
Conversion of Cartesian Coordinate into Spherical Polar Coordinate
In order to solve the equation (3) to express the cartesian coordinate (x, y, z) into
spherical polar coordinate (r, θ, φ) for hydrogen like atom, ( i.e .: for the spatial rotation
of an electron around the nucleus with the angular variables θ (Zenith angle) and φ
(Azimuthal angle) and radial distance r ), we have:
x
y
z
x
r
y
z
P'
P
Q
O
R
S
Replacing ( x, y, z) with ( r , θ , φ ) in equation (3), we get Schrodinger wave equation for
hydrogen atom as:
{
−𝒉
𝟐
𝟖𝝅
𝟐
𝒎
[
𝟏
𝒓
𝟐
𝝏
𝝏𝒓
(𝒓
𝟐
𝝏
𝝏𝒓
) +
𝟏
𝒓
𝟐
𝒔𝒊𝒏𝜽
𝝏
𝝏𝜽
(𝒔𝒊𝒏𝜽
𝝏
𝝏𝜽
) +
𝟏
𝒓
𝟐
𝒔𝒊𝒏
𝟐
𝜽
(
𝝏
𝟐
𝝏𝝋
𝟐
)] −
𝒁𝒆
𝟐
𝒓
} 𝜳 ( 𝒓,𝜽,𝝋
)
= 𝑬𝜳 ( 𝒓,𝜽,𝝋
)
… (𝟒)
Equation (4) gives the complete wave function for hydrogen like atom.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
2. Summarize the significance of four quantum numbers.
The location, the energy and the spin of an electron, the size of orbits, the shape and
orientation of orbitals in an atom can be described by quantum numbers.
They are: principal quantum number (n), azimuthal quantum number (l), magnetic
quantum number (m) and spin quantum number (s).
1. Principle Quantum Number (n)
The principal quantum number gives the address of an electron in a shell or orbit.
It can have the values of 1, 2, 3, 4 … and here, n = 1 denotes K-shell, n =2 denotes L-
shell, n = 3 denotes M-shell and so on …
It also gives the information about radial distance of electron revolving around the
nucleus, in turn used to determine the radius, velocity and the energy level of electron in
various orbits.
According to Bohr’s atomic theory, the radius of orbits, velocity and the energy of
electron in an atom can be calculated by using the following expressions:
𝑛
2
× 0. 53 × 10
− 10
𝑛
- 18 × 10
10
𝑛
𝑛
− 1313. 31
𝑛
2
As the principal quantum number increases, the energy of electron also increases.
The maximum number of electrons that can be accommodated in each shell can be
calculated by using the expression 2 n
2
- n = 1 represents K-shell, which can accommodate, 2 n
2
2
= 2 electrons
- n = 2 represents L-shell, which can accommodate, 2 n
2
2
= 8 electrons
- n = 3 represents M-shell, which can accommodate, 2 n
2
2
= 18 electrons
- n = 3 represents N-shell, which can accommodate, 2 n
2
2
= 32 electrons
2. Azimuthal Quantum Number (l)
It gives the address of an electron in various sub-shells or sub-orbits.
It can have the values of 0, 1, 2, 3, 4 …, and here l = 0 denotes s - sub-shell, l = 1 denotes
p - sub-shell, l = 2 denotes d - sub-shell and so on …
The l value depends upon principal quantum number ( n ), and its value varies from 0 to
( n ‒1).
When n= 1, l can have the values: 0 to ( n ‒1) = 0; and l = 0 represents 1 s sub-shell and
the prefix 1 represents s sub-shell belongs to K shell.
When n= 2, l can have the values: 0 to ( 2 ‒1) = 0 to 1; i.e .: l = 0, 1; in which 0 represents
2 s sub-shell and 1 represents 2 p sub-shell.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
When n= 3, l can have the value of 0 to ( 3 ‒1) = 0 to 2; i.e .: l = 0, 1, 2; in which 0 represents
3 s sub-shell, 1 represents 3 p sub-shell, and 2 represents 3 d sub-shell.
The maximum number of electrons that can be accommodated in each sub-shell = 4( l ) +
l = 0 represents s - sub-shell, which can accommodate [4(0) +2] = 2 electrons.
l = 1 represents p - sub-shell, which can accommodate [4(1) +2] = 6 electrons.
l = 2 represents d - sub-shell, which can accommodate [4(2) +2] = 10 electrons.
l = 3 represents f - sub-shell, which can accommodate [4(3) +2] = 14 electrons.
3. Magnetic Quantum Number (m)
It gives the number of orbitals present in various sub-shells.
It also gives the information about the shape and orientation of orbitals in various sub-
shells.
The magnetic quantum number ( m) depends upon azimuthal quantum number ( l ) and,
the m value explains the possible orientations of orbitals and the m values vary from ‒ l
to + l through 0, which can be calculated using the expression = 2( l ) + 1.
For l = 0 (denotes s sub-shell), magnetic quantum number, m = 2( l ) + 1 = 2( 0 ) + 1 = 1
(represents s orbital)
Since, m = 1 depends only on radial wave function and, hence the s orbital exhibits
spherically symmetrical in shape and the probability of finding electron is equal in all the
directions.
For l = 1, (denotes p sub-shell), m = 2( 1 ) + 1 = 3, m = 3.
m = 3 represents three p - orbitals and hence they exhibit three possible orientations (‒ l to
- l through 0): ‒1, 0, +1 and are designated as p x
, p y
and p z
orbitals.
For l = 2, (denotes d sub-shell), m = 2( 2 ) + 1 = 5, m = 5
m = 5 represents five d - orbitals and hence they exhibit five possible orientations (‒ l to + l
through 0): ‒2, ‒1, 0, +1, +2.
4. Spin Quantum Number (s)
It gives the direction of spin of electron along its axes. If an electron has spin in clock-
wise direction, it is designated as s = + ½ (↑). If an electron has spin in clock-wise
direction, it is designated as s = ‒ ½ (↓).
The total spin of electron in an atom is given by ½ ( n ), where n is the number of unpaired
electrons.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
3. Shapes of d-orbitals
✓ The angular wave function for d - orbitals l = 2 and hence m has five values (orbitals)
( m =2 l +1) through ‒2, ‒1, 0, +1, +2 and are designated as d z
2
, d x
2
‒y
2
, d xy
, d yz
, d xz
✓ The d xy
, d yz
and d xz
orbitals are similar in shape and size consisting of four lobes,
arranged between x & y , y & z and x & z respectively.
✓ The d x
2
‒y
2
orbital has also four lobes, arranged along x - axis and y - axis.
✓ The d z
2
orbital consisting of two lobes along z - axis and a ring of electron density
around the nucleus.
✓ Since, all the five-d orbital have same energy and are known as degenerate orbitals.
d
xy
Z
Y
X
d
yz
Z
Y
X
d
xz
Z
Y
X
d
Z
Y
X
x
2
-y
2 d
Z
Y
X
z
2
LCAO-MO TREATMENT OF HYDROGEN MOLECULE
**1. Define atomic orbital.
- Elaborate the LCAO-MO method in the formation of hydrogen molecule.**
Atomic orbital: The single electron wave function is known as atomic orbital.
According to linear combination of atomic orbitals (LCAO) theory, the atomic orbital of
one atom linearly combines with the atomic orbital of similar energy atom (overlap each
other) to give two types of molecular orbitals namely a bonding molecular orbital [BMO]
and an antibonding molecular orbital [ABMO].
Usually a wave function of a wave possesses a crust portion and a trough portion, and
are arbitrarily assigned the sign + and ‒ respectively.
Formation of bonding molecular orbital
If a crust of one wave overlaps with the crest of other wave, they interact to give a
constructive reinforcement wave.
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
Let us consider, there are two hydrogen atoms H A
and H B
in which each atom has one
electron in 1s atomic orbital in the ground state.
When the crust one hydrogen atom (+Ψ A
) overlaps with the crest of other hydrogen atom
B
), they interact to give a more stable, less energy bonding molecular orbital. The
combination of two atomic orbitals of hydrogen atoms is Ψ b
A
B
The formation of more stable bonding molecular orbital is due to the force of attraction
of high electron density with the nuclei.
Constructive wave
+
Crust
Trough
+ +
Formation of antibonding molecular orbital
If a crust of one wave overlaps with the trough of another wave, they interact to give a
destructive wave.
When the crust one hydrogen atom (+Ψ A
) overlaps with the trough of other hydrogen
atom (‒Ψ B
), they interact to give an unstable stable, higher energy antibonding molecular
orbital. The combination of two atomic orbitals of hydrogen atoms is Ψ b
A
B
The formation of unstable antibonding molecular orbital is due to the force of attraction
of less electron density with the nuclei.
Destructive wave
+
+
Crust
Trough
Determination of Energy of Molecular Orbitals for Hydrogen Molecule
According to wave mechanics, the atomic orbitals can be represented by wave functions.
The energy of molecular orbitals for hydrogen system can be determined by using the
Schrodinger wave equation, and it is given as:
Since the wave functions of two atomic orbitals are in involved in overlapping to form
molecular orbitals, the equation (1) is multiplied by another Ψ on both the sides, and we
obtain:
2
Integrating the equation (2) over the spatial configuration, we get:
2
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
The energy level diagram of hydrogen atom through LCAO is given in the following diagram.
Higher energy ABMO
1s
1
H
A
of atom
Lower energy BMO
E
a
=(Q )
1s
1
H
B
of atom
E
b
=(Q + )
E
N
E
R
G
Y
BAND THEORY OF SOLIDS
**1. Explain the band theory of solids.
- Explain conductors, semiconductors and insulators with a neat diagram.**
It is based on molecular orbital (MO) theory.
Overlapping of atomic orbitals gives rise to two types of bonding orbitals namely:
bonding molecular orbital (BMO) and antibonding molecular orbital (ABMO).
In order to understand band theory of solids, let us consider, the element sodium atom.
Sodium atom has an atomic number of 11 and it has the electronic configuration 1 s
2
2 s
2
3 p
6
3 s
1
When two sodium atoms are brought together, the valence electron of 3 s orbital starts
overlapping with the other 3 s orbital of sodium atom to form one bonding molecular
orbital (BMO) with two electrons and one antibonding molecular orbital (ABMO) with
no electrons.
3s
1
Filled valence bond
Empty conduction bond
E
n
e
r
g
y
3s
1
BMO
ABMO
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
When four sodium atoms are brought together, the valence electron of 3 s orbital starts
overlapping to form two bonding molecular orbitals (BMO) with two electrons each and
two antibonding molecular orbitals (ABMO) with no electrons.
3s
1
Filled valence bond
Empty conduction bond
E
n
e
r
g
y
3s
1
BMO
ABMO
3s
1
3s
1
Na-atom 2 Na-atom 3
Na-atom 4 Na-atom 1
We know that one-gram equivalent of sodium will have 6.023× 10
23
atoms (i.e., n number
of Na-atoms).
When n number of sodium atoms overlap each other, they form N/2 number of BMO
with 2e
each and N/2 number of ABMO with no electron.
The n number of closely spaced filled valence bonds appears like a band known as filled
valence band, VB), while n number of closely spaced empty conduction bonds appears
like a band known as empty conduction band, CB.
The energy gap present between filled VB and empty CB is called as forbidden gap or
energy gap ( E g
Energy gap, E
g
3s
1
of n number
of atoms
Filled
valence band
Empty
conduction band
Energy
Classification of Solids on the Basis of Band Theory
Based on E g
values, the solids are classified into three types namely: insulator,
semiconductors and conductors.
(i) Insulators:
Solids in which the valence band and conduction band are largely separated [> 5 eV] in
such a way that, there is no electronic conduction takes place between them are known
as insulators (Examples: Rubbers, Mica, Quartz).
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
Role of Arsenic Doping (n-type Doping)
The process of adding impure metal atom directly into a pure semiconductor is known as
doping and the added impure metal atom is known as dopant.
The arsenic metal atom is an example of pentavalent impurity. When an arsenic atom is
added into a silicon crystal, it replaces one of the silicon atoms and thereby forming four
covalent bonds with nearby four silicon atoms.
Further, the fifth electron of arsenic atom remains totally free and can move
independently in the crystal lattice in turn can act as charge carrier when subjected to an
electric field and is responsible for maximum conductance of semiconductor.
Si Si
Si
Si
As
Si
Si Si
Si
Excess
Free
electron
Role of Gallium Doping (p-type Doping)
The gallium metal atom is an example of trivalent impurity.
When a gallium atom is added into a silicon crystal, it replaces one of the silicon atoms
and thereby forming three covalent bonds with nearby three silicon atoms. Since gallium
is an electron deficient atom, it creates a positive hole with respect to the fourth silicon
atom in the crystal lattice.
Further, this positive hole of gallium atom can move independently in the crystal lattice
in turn can act as charge carrier when subjected to an electric field and is responsible for
maximum conductance of semiconductor.
Si Si
Si
Si
Ga Si
Si Si
Si
Positive
hole
SCYA 1101 : Engineering Chemistry UNIT 1: Bonds to Bands
SUPERCONDUCTORS
**1. What are superconductors? Give examples.
- Define Meissner effect.
- Mention any two applications of superconductors.
- Mention some important properties of superconductors.**
Definition: Metals/ alloys, which conduct electricity with zero resistance when cooled
below to its critical (threshold) temperature are known as super conductors.
Examples:
S. No. Superconductor Critical Temperature
- He 4K
- Hg 4.2K
- Pb 7.2K
- Alloy of Pu, Ga and Co 18.5K
- Alloy of Cu, Ba, Ce, and Tl 125K
Meissner Effect: The expulsion of magnetic lines of forces during the change from
normal conducting state to superconducting state when cooled below to the critical
temperature is known as Meissner effect.
Properties:
✓ They possess greater resistivity than other elements at room temperature
✓ On adding impurity to super conducting element, the critical temperature can be
lowered
✓ In superconducting state, all electromagnetic effects disappear
✓ Superconductors can exhibit Meissner effect
Applications:
✓ Quantum computing
✓ Sensors
✓ Magnetic Resonance Imaging (MRI)
✓ Military and marine transportations
