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Energy levels of atoms and
molecules
Dr. Yin LI
Department of Biophysics , Medical
School, PTE
[email protected]
Outline
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Interaction of light with matter
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Atomic Energy levels
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Energy levels of hydrogen atom
Bohr model
Molecular Energy Levels
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Light properties
Polarization of light wave
Interaction of light with matter
From atoms to a molecule (Electronic structure of hydrogen gas molecule (H2))
Electronic structure of other molecules (Types of chemical bonds)
Electronic Spectroscopy (UV-vis Absorption & fluorescence spectroscopy, Jablonski diagram)
Reference book:
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Physical Chemistry, Authored by Peter Atkins & Julio De Paula, Oxford University Press
Light Properties: Wave-particle duality
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Light is electromagnetic radiation within
a certain portion of the electromagnetic
spectrum.
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Properties of light:
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Light travels in straight lines;
Light speed in vacuum: ∼ 3 × 108 m/s;
Light can be reflected;
normal
1 2
𝜃1 = 𝜃2
Horizontal plane
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Light can refract.
sin 𝜃1
sin 𝜃2
𝑛
= 𝑛2 (Snell's law)
1
𝑛 denotes the refraction index of medium.
Light Properties: Wave-particle duality
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Wave Properties of light
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Interference : two or more waves with the same frequency combine to reinforce or cancel each
other, the amplitude of the resulting wave being equal to the sum of the amplitudes of the
combining waves.
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Diffraction: when waves encounter a small obstacle or opening, they tend to bend around that
barrier or pass through it and become spread out.
Light Properties: Wave-particle duality
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Einstein's photoelectric effect
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Many metals emit electrons when light shines upon them. The emission of photoelectrons has
nothing to do with intensity of light (amplitude of electromagnetic wave) and irradiation time. As
long as the incoming light exceeds a threshold of frequency, electron will be discharged from the
surface of metal. Light behaves differently from ordinary waves.
ℎ∙𝑣 ≥𝑊
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h is Planck’s constant, W denotes the work function of metal, which means the
minimum energy for photoelectrons to escape
from the surface of metal.
Energy of an ordinary wave:
𝐸 ∝ A2 and 𝐸 ∝ 𝑡
But, energy of light:
𝐸 ∝𝑓
The photoelectric effect elucidates particle properties of light. “Particle” here
does NOT mean some staffs like beans or bullets. It represents packets of energy
that can not be further divided.
The particle of light is called “photon” which has a mass of 0 and energy of ℎ ∙ 𝑓.
Properties of Photons
(a) Velocity
The velocity (ms-1) of any wave is given by the frequency, n (or number of waves per second,
s-1) multiplied by the wavelength, l in metres. The velocity of light in vacuum is a fundamental
constant, given the special symbol, c.
c = n.l
c takes the value 3 x 108 ms-1, which is pretty fast.
The velocity of light (v) in any medium, of refractive index, n, is slowed compared to vacuum to
a velocity
v = c / n.
The value of n is always greater than or equal to unity.
(b) Energy
Energy is often a property we associate with a particle (kinetic, potential, etc.). The energy
associated with a single photon was deduced by Planck, one of the founders of the quantum
theory as
E = hn
in which h is another fundamental constant, the Planck constant, which has the value
h = 6.63 x 10-34 Js
so E is in Joules.
Clearly the energy of a photon depends on frequency (or wavelength) and this leads to the idea
of the electromagnetic spectrum
Polarization of light waves
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Light is an electromagnetic wave, with Electric and Magnetic fields, that are in
phase and orthogonal.
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Un-polarized light: electric field vectors vibrate in all orientations that are
perpendicular with respect to the direction of propagation. For example: sun light.
If the electric field vectors are restricted to some orientations by filtration of the
beam with specialized materials, then light is polarized.
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The Interaction of Light and Matter
Spectroscopy is the study of matter (for chemists matter = molecules) by their interaction with
light. What is the mechanism of this interaction?
Recall the simples picture of an atom, as a positive nucleus surrounded by a negatively
charged ‘electron cloud’.
Imagine this atom placed in a static electric field. Electrons will be attracted to the positive
pole, the nucleus to the negative. This is polarization, and results in a dipole moment in
the atom. This is refereed to as an induced dipole, as it is only present when the field is
applied. The size of mind depends on the atom’s polarizability.
Since light is an oscillating electric field it is not surprising that it interacts with the atom (the
magnetic field is of much less significance in spectroscopy). Energy is transferred from the light
to the molecule, and in so doing an induced (oscillating) dipole is generated.
This process is referred to as absorption.
The reverse process is also possible. When the light field is off the induced dipole relaxes
back, and gives out energy as radiation.
This process is called emission.
Atomic Energy levels
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Hydrogen atom is the simplest model to study atomic energy levels.
Emission spectrum of atom hydrogen was detected by the following device.
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It was observed that the spectrum of atom hydrogen appears in a series of lines.
This observation manifests that the energy levels of hydrogen atom is discrete
(quantized).
continuous
discrete
Energy levels of hydrogen atom
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In 1885, a swiss mathematician-Balmer analyzed the hydrogen spectrum and
proposed an empirical formula for the visible spectral lines of hydrogen atom.
1
1
1 / l  R 2  2 
2 n 
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Where 𝑛 is an integer larger than 2, and 𝑅 is
a constant, 0.010972 nm−1 .
Wavelength (nm)
n
656.2
3
486.1
4
434.0
5
410.1
6
364.6
∞
Rydberg expanded Blamer’s formula to all series of hydrogen spectrum.
1
1
1 / l  RH  2  2 
 n1 n2 
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𝑅𝐻 is called Rydberg constant.
𝑅𝐻 ≈ 13.6 eV, or 1.097 × 105 cm−1
𝑛1
Name of Series
1
Lyman
2
Balmer
3
Paschen
4
Brackett
5
Pfund
Energy levels of hydrogen atom
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In 1913, Bohr developed a model for hydrogen atom.
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Electrons move around the nucleus with high velocity in some certain
allowed “stationary” orbits.
Each stationary orbit corresponds to a definite energy. The orbit that is
closer to nucleus has a less energy compared to the orbit that is far away
from nucleus.
Electron can jump between orbits by absorption or emission of radiation
whose energy is equal to the energy difference between two orbits.
Angular momenta for electrons are quantized.
Limitations of Bohr model
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According to classical mechanics, any charged particle that is moving is going to emit
electromagnetic radiation. Therefore, electron can’t circle around nucleus in “stationary” orbits. The
final result is that electron spirals into nucleus. Due to smooth change of electron orbits, it would
result in a smooth energy level instead of discrete energy levels.
Bohr model just can give a good description of hydrogen atom. It failed in explanation of the multielectron atoms.
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Uncertainty principle
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Heisenberg proposed that you can be certain about the position or momentum of a subatomic
particle but never both at the same time. This gave us a “electron cloud” model in which electrons
do not have precisely described orbits.
In classical mechanics, you can be certain about both
momentum and position of this moving car running on
road.
In quantum mechanics, you can only
know the probability that electron
would appear in a certain position.
Here comes quantum mechanics!
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To correctly describe the quantum states of a physical system, the classical wave
expression must be abandoned. In 1926, Schrödinger published his equation which
was later known as Schrödinger equation to describe the quantum states of a system.
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Time-independent Schrödinger equation
𝐸Ψ = 𝐻 Ψ
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Where, Ψ is wave function, 𝐸 is the eigenvalue of a quantum state, 𝐻 is an operator.
To calculate the whole energy of the system, we sum both kinetic energy operator and potential energy
operator 𝐻 = 𝐾 + 𝑉 , 𝐻 is called Hamiltonian.
The requirements for Ψ:
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Ψ must be continuous; and its derivative must also be continuous.
Ψ must be single-valued.
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Ψ 2 must be integrable, and
an electron should be 1.
+∞ 2
Ψ d𝜏
−∞
= 1meaning that in whole space the total probability of finding
Which function below is a qualified wavefunction?
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Figure of wavefunction for 1s electron in hydrogen atom plotted as a function of radial distance between
electron and nucleus .
∞
Ψ 𝑟
0
2
∙ 4π𝑟 2 d𝑟
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Electron orbitals in atom
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Electron configuration of atom
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Formation of H2 molecule
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When two hydrogen atoms approach each other, the electron of one atom begins to feel the
attraction from both nuclei of these two atom. Electron cloud of each atom gets distorted and the
electron density around each nucleus shifts toward the region between two atoms. As the distance
between two atoms decreases, one electron becomes be able to enter the orbital of another
electron. Finally, these two electrons are shared by both nuclei and become indistinguishable. At
this moment, the covalent bond between these two atoms is formed, H2 molecule is generated.
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H atom has only one electron in 1s orbital. The covalent bond formed from the combination of these
two 1s electrons is called 𝜎 bonding orbital whose energy is lower than the sum of two hydrogen
atoms. Simultaneously, there is also a 𝜎 ∗ antibonding orbital formed at a higher energy level. The
energy level of antibonding orbital is higher than bonding orbitals and it raises the energy of
molecule and destabilizes the molecule. Therefore, without any extra stimulation, electrons always
stay at the bonding orbital.
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The essential of molecular orbitals (MOs)
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Molecular orbitals (MOs) is the linear combination of atomic orbitals (LCAOs).
Take H2 molecule as an example.
Ψ𝐻2 = 𝑐1 ∙ 𝜓1𝐻 +𝑐2 ∙ 𝜓2𝐻
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Where 𝑐1 and 𝑐2 are coefficients, 𝜓1𝐻 and 𝜓2𝐻 are atomic orbitals of each hydrogen atom.
𝜎 bonds (head to head)
𝜋 bonds (back to back)
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Potential energy surface
The energy is plotted as a function of position of the atoms, which for H2 is just the internuclear separation, r.
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At infinite separation the energy is defined as zero.
As the atoms approach the AOs overlap to form MOs.
The bonding MO is more stable than the separated atoms.
The stabilization energy reaches a minimum, when the
favorable overlap is balanced by the repulsion between
the two nuclei- this is the bond length.
At smaller r the repulsion increases dramatically.
Anti-bonding
orbital
Bonding orbital
Electronic transition
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Analogous to electronic transition in atoms, in molecule electron is also able to jump to a higher
energy level after absorption of photons with a given wavelength which corresponds to energy
difference between two energy levels.
𝜋 → 𝜋 ∗ transition
𝜎 → 𝜎 ∗ transition
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Hybridization of s and p electrons in carbon atom
E
E
sp3 sp3 sp3 sp3
Sp3 hybridization
Sp3 hybridization
Formation of CH4
+
sp3 hybridized
carbon atom
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Hybridization of s and p electrons in carbon atom
E
2pz
E
sp2 sp2 sp2
Sp2 hybridization
Formation of CH2=CH2
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Formation of formaldehyde (CH2O)
sp2 hybridized carbon atom
2pz
E
sp2 sp2 sp2
sp2 hybridized oxygen atom
2pz
E
sp2 sp2 sp2
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Energy levels of formaldehyde (CH2O)
The order of energy required for transition :
S0 → S2
S0 → S1
300
Collecting together data on molecular electronic transitions we can say that
np* are weak
np* are of long wavelength (low energy)
ss* are strong, but of high energy
pp* are strong and at intermediate energies.
Such rules can be useful in assigning the UV vis spectra measured in the lab. In
addition we have to remember that long conjugation length decreases the energy
of pp* transitions. Some of these ideas are evident in the table below.
Chromophore
transition
lmax / nm
strong / weak
C—C, C—H
s  s*
~150
strong
O
n  s*
~185
weak
N
n  s*
~195
weak
S
n  s*
~195
weak
O
n  p*
~300
weak
O
n  s*
~190
weak
p  p*
~190
strong
p  p*
250
strong
C C
H2C
C
H
H
C
C
H
H
C
CH2
• Jablonski Energy Diagram
• Problems:
1. Work function of metal sodium is equal to 2.29 eV, please
calculate whether a radiation with a wavelength of 400 nm
can drive electrons to escape from the surface of metal
sodium. (1 eV ≈ 1.62 × 10−19 J).
2. Calculate the wavelength of the light emitted by hydrogen
from level 4 to level 3 using Rydberg equation. What is the
nature of emitted light?(Infrared, visible, ultraviolet or x-Ray)
3.
The oxygen atom in water molecule is sp3 hybridized. Please
draw the energy levels of this oxygen atom before and after
sp3 hybridization. Please also draw all the chemical bonds
within water molecule and mark their types (𝜎 or 𝜋 bond?).
Explain why water molecule has a permanent dipole.