Atomic structure

• The atom is mostly empty space.
• Two regions.
• Nucleus- protons and neutrons.
• Electron cloud- region where you might
find an electron.
Atoms
Subatomic particles
Electron
Proton
Neutron
Name Symbol Charge
Relative
mass
Actual
mass (g)
e-
p+
n0
-1
+1
0
1/1840
1
1
9.11 x 10-28
1.67 x 10-24
1.67 x 10-24

Bohr’s Model
Nucleus
Electron
Orbit
Energy Levels
Increasingenergy
Nucleus
First
Second
Third
Fourth
Fifth
Further away from the nucleus means more energy.
There is no “in between” energy

•Few difficulty Bohr model had
•The spectra of larger atoms. At best, it can make
some approximate predictions about the emission
spectra for atoms with a single outer-shell electron
•The relative intensities of spectral lines
•The existence of fine and hyperfine structure in
spectral lines.
•The Zeeman effect - changes in spectral lines due to
external magnetic fields
Bohr model: A semiclassical model

The Quantum Mechanical Model
• Energy is quantized. It comes in chunks.
• Quanta - the amount of energy needed to move
from one energy level to another.
• Quantum leap in energy.
• Schrödinger derived an equation that described
the energy and position of the electrons in an
atom
• Treated electrons as waves

Photoelectric Effect
Emission of electrons from metals when exposed to
(ultraviolet) radiation.

Observations
1. No emission of electrons if the frequency of
radiation is below a threshold value characteristic
of the metal, however high the intensity of the
light.
2. Kinetic energy of emitted electrons varies linearly
with the frequency, and is independent of light
intensity.
3. For frequencies above the threshold value,
emission of electrons is instantaneous, no matter
how low the intensity of the light.

Explanation (EINSTEIN 1905)
1. Light of frequency  may be considered as a
collection of particles, called photons, each of
energy h.
2. If the minimum energy required to remove an
electron from the metal surface is  (work
function), then if h < , no emission of electrons
occurs.
3. Threshold frequency 0 given by  = h0
4. For > 0, the kinetic energy of the emitted
electron Ek = h   = h(  0).

Line Spectrum of Hydrogen atom
when subjected to higher temperature or a electric discharge emit
electromagnetic radiation

Atomic Orbitals
• Principal Quantum Number (n) = the
energy level of the electron.
• Within each energy level the complex math
of Schrödinger's equation describes several
shapes.
• These are called atomic orbitals
• Regions where there is a high probability of
finding an electron.

• 1 s orbital for every energy level
• Spherical
shaped
• Each s orbital can hold 2 electrons
• Called the 1s, 2s, 3s, etc.. orbitals.
S orbitals

P orbitals
• Start at the second energy level
• 3 different directions
• 3 different shapes (dumbell)
• Each can hold 2 electrons

D orbitals
• Start at the third energy level
• 5 different shapes
• Each can hold 2 electrons

F orbitals
• Start at the fourth energy level
• Have seven different shapes; 2 electrons per shape

Schrödinger's equation, what is it ?
Newton’s law allows you to describe motion of mechanical systems and
mathematically predict the outcome of the system.
In quantum mechanics, the analogue of Newton's law is Schrödinger's
equation for a quantum system (usually atoms, molecules, and subatomic
particles whether free, bound, or localized).
It is a wave equation in terms of the wavefunction which predicts analytically
and precisely the probability of events or outcome.
The Schrodinger equation gives the quantized energies of the system and
gives the form of the wavefunction so that other properties may be
calculated.

The wave equation developed by Erwin Schrodinger in
1926
(one-dimensional form)
About the Wavefunction
The wavefunction is assumed to be a single-valued function of position and
time, which is sufficient to get a value of probability of finding the particle at a
particular position and time.
The wavefunction is a complex function, since it is its product with its
complex conjugate which specifies the real physical probability of finding the
particle in a particular state.

WAVEFUNCTION () (PSI)
Classically, the state of a system is described by its position
and momentum
In Quantum theory, the state of a system is described by its
wavefunction

1. A wavefunction is a mathematical function (like sinx, ex).
Like any mathematical function it can have large value at
some place, small in other and zero elsewhere.
It can be real or complex
2. A wavefunction contains all information about the system
3. The wavefunction is a function of Cartesian coordinate
and time. ie.  (x, y, z, t)
4. If the wavefunction is large at a point in space, the particle
has a large probability at that point
5. The more rapidly a wavefunction changes from place to
place, higher the K.E. of the particle it describes

A wavefunction describes the state of a system
How?
The state of a system is described by some measurable
quantities such as mass, volume, momentum, position,
Energy etc. These quantities are called observables
How to determine the observables from wavefunction ()
By performing a set of well defined mathematically
operations on . These mathematical operations are
called operators

The state of a system (particle) is completely specified by its
wavefunction (x,y,z,t), which is a probability amplitude and
has the significance that
2 dV
(more generally 2dV since  may be complex)
represents the probability that the particle is located in the
infinitesimal element of volume dV about the given point, at
time t.
2Born interpretation

NORMALIZATION
As per Born interpretation, the probability of existence of
the particle in the entire space should be 1.
In mathematical term the wave function has to be normalized
For one dimension
1
1
2
22





dxN
or
dxN


Where N is the normalization constant
  2
1
2
1


dx
N


In 1913 Niels Bohr came up with a new atomic model in which
electrons are restricted to certain energy levels.
Schrödinger applied his equation to the hydrogen atom and found
that his solutions exactly reproduced the energy levels stipulated by
Bohr. The result was amazing and one of the first major achievement
of Schrödinger's equation and earned him the 1933 Nobel Prize in
physics.
Newton’s Law: Conservation of Energy ( Harmonic Oscillator example))
Time independent Schrodinger Equation

Quantum conservation of Energy Schrodinger Equation
H = E
In a wave equation, physical variables takes the form of “operators” (H),
Hamiltonian operator
In three dimensions,
ħ2/2m (2 ψ/x2 + 2 ψ/y2 + 2ψ/z2) + U(x,y,z)ψ(x,y,z) = Eψ(x,y,z)
Time dependent Schrodinger Equation

25
Eigen value equations
(operator) (function)= (constant factor) (function)
   axax
eae
dx
d

)4(16)4(2
2
xSinxSin
dx
d


26
•Let us test the Schrödinger equation for some simple physical
systems! Some model systems at first – easy to test ! Hopefully,
some quantum properties of matter will also be understood!
•If we can understand these ‘simple systems’ without doing any
approximation and can derive ‘exact solutions’, then we shall
proceed towards our major target – atoms and molecules
•Why we do not go right now to solve the Schrödinger equation
for atoms and molecules? Because the mathematical solution for
atoms and molecules is complicated.
How to proceed further?

27
Translation: Particle in a box
Consider a particle in one dimension confined to a length L by
infinite potential barriers at x = 0 and x = L (infinitely deep
potential well)
V = 0
inside box
V = 
V =  V = 
V = 0
inside box

28
For x < 0 and x > L,  = 0 since probability of finding the particle in
these regions is zero.
For 0  x  L, the Schrödinger equation is
(ħ2/2m) d2(x)/dx2 = E(x)
The General solution to this equation can be written as
(x)=N sin kx + M cos kx where k=(2mE)1/2/ħ

29
Boundary conditions
1. For x < 0 and x > L,  = 0
2. For x = 0,  = 0
3. For x = L,  = 0
Let us apply the boundary conditions to the solutions
1.  = 0 at x = 0
 M= 0
After applying 1st boundary condition, (x)=N sin kx
2.  = 0 at x = L
(L) =N sin kL=0
 kL = n, where n = 1, 2, 3, …….
(x)=N sin kx + M cos kx

30
Since k= (2mE)1/2/ħ
n/L = (2mE)1/2/ħ
Rearranging this equation
1,2,3...nwhere
8 2
22

mL
hn
E
1. Energy is discrete and quantized
2. The ground state energy is not 0, but h2/8mL2, the zero
point energy. This is a consequence of the uncertainty
principle. This is the minimum irremovable energy.
2
2
1
8mL
h
E 
2
2
2
8
4
mL
h
E 
2
2
3
8
9
mL
h
E 
2
2
4
8
16
mL
h
E 

31
The energy difference between two successive energy levels
2
2
1
8
)12(
mL
hn
EE nn


Energy difference between successive states depends upon
1. Mass
2. Length of the box
Mass of the particle increases – classical limit
Size of the box increases – classical limit

32
Wave function of the particle in a box
L
xn
N
L
n
knkL
kxN





sin
since
sin



The wave function has to be normalized ie.
2
1
2
0
2 2
1sin 











 L
Ndx
L
xn
N
L

So the normalized wave function is
1.2,3,....nwheresin
2 2
1













L
xn
L
n



33
The wavelengths could be  = 2L, L, (2/3) L……
In general  = 2L/n with n = 1,2,3,……

34
Characteristics of the wavefunctions
1. Wavelength = 2L/n
2. There are n-1 nodes (interior points where the wave
function passes through zero) in the wavefunction n
3. The energy increases with increasing number of nodes.
The ground state has no nodes.
4. The ground state energy is not 0, but h2/8mL2, the zero
point energy. This is a consequence of the uncertainty
principle.

35
Applications of this model
1. Calculation of energy of -electrons of conjugated olefins
2. Electrons in nano materials
3. Electrons present in cavities or color centers
4. Translational motion of ideal gas molecules
H2C C
H
C
H
CH2

36
Hydrogen atom
Hydrogen has special significance
•No approximation is required in solution of Schroedinger equation
•Can get expression for energy levels
•Spectral frequencies can be deduced
Since MN>>Me, the nucleus can be considered to be at rest
For H atom the Schrödinger wave equation can be written as
Ĥ ψ = E ψ
[- (h2/2m){2x2 +2y2 +2z2} +V] =E

37
V= - q1q2/ r
The potential, V between two charges is best described
by a Coulomb term,
(ħ2/2m 2 +Ze2/r ) ψ = E ψ
It is convenient to describe the solutions to the Schrödinger
equation in spherical polar coordinates (r, ,) rather than
cartesian (x,y,z)









E
r
e
Sinr
Sin
Sinrr
r
rrme





























0
2
2
2
222
2
2
2
4
111
2
h
The Schrödinger equation in spherical polar coordinate is
This equation can be solved by separation of variable technique
ψ(r,,) = ()()
ψ(r,,) = R(r) ()
ψ(r,,) = R(r)() () Angular part
() Angular part
R(r) Radial part
Solution may be a product of three functions.

39
ψn,l,ml
(r,,) = Rn,l(r)Yl,ml
(,)
where Rn,l(r) is called the radial part of ψ, and
Yl,ml
(,) its angular part.
The wavefunction of the electron in the hydrogenic
atom is called an atomic orbital. An orbital is a one-
electron wavefunction.
Electron described by a particular wavefunction is
said to occupy that orbital.
Atomic orbitals specified by three quantum numbers
n, l, and ml.

Orbital angular momentum = [l(l+1)]ħ

Energy is –ve  stabilization effect
Higher the value of Z  more
stabilization
When n increases energy
increases
Principal quantum numbers, n: Energy Levels
The energy levels are
En = e4Z2/32
ħn2
= hcRZ2/n2 where
Where R = (e4/32
ħ)/hc

Solution : Some example
2
2
3
0
1
2







e
a
R(n,l) =
For n=1, l =0
2
1
4
1







Y l, ml (,)=
For l=0, ml =0
 Y is a constant and does not depend on  and 
 For a given radial distance, same value of probability is
observed at all directions from nucleus
 S-orbitals are spherically symmetrical

Solution :
R(n,l) =
For n=2, l =1 For l=1, ml =0 , +1, -1


 cos
4
3
),(
2
1
0,1 





Y
 The angular variation of wavefunction depend on cos .
 The probability density is proportional to cos2.
 The probability density has maximum value along an
arbitary axis (z-axis) on either side of the nucleus
( at = 0 and 180o)
4
2
3
0
1
64
1 








e
a



i
e






 sin
8
3 2
1

The shape of orbitals
s orbital
p orbitals
d orbitals
f orbital

Atomic structure

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