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welcome back to the lectures we move on to
the last section of this namely the review
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of ah elementary vibrational spectroscopy
carried out over several lecture modules in
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this course the important starting point for
us was the simple harmonic oscillator model
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and i recall that i had a fairly elaborate
session on the quantum mechanics of the harmonic
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oscillator model i gave you the energy level
expressions as well as the wave functions
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for the harmonic oscillator now recall the
harmonic oscillator hamiltonian is the kinetic
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energy two mu d square by dx square and this
being p square operator by two mu plus half
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m omega square x square and this is nothing
but half k x square where k is the force constant
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and omega is the ok i am using mu so let us
use mu here and omega is the angular frequency
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of the harmonic oscillator of the oscillator
therefore and its given by ah two pi mu thats
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equal to omega he relation between k and angular
frequency obvious from here namely k is equal
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to mu omega square therefore omega is k by
mu and the frequency nu is one by thats a
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square root here one by two pi square root
of k by mu
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so given the harmonic oscillator frequency
from the experiment and knowing what the molecule
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is therefore calculate we are able to calculate
the mu t is possible for us to calculate the
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force constant such exercises were given to
you for different molecules simple diatomic
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molecules the gave you the what is call the
fundamental frequency this is call the fundamental
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harmonic oscillator frequency and the fundamental
frequency is related to the force constant
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that it is this proportional to the square
root of the force constant it is proportional
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to the inverse of the ah reduced mass therefore
if we have molecules such as hd and h two
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and we know that the force constant between
the two hydrogens and the hydrogen and deuterium
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are approximately the same then the reduced
mass for this two species are different produced
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masses are different and therefore the harmonic
oscillator frequencies will also be different
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in fact u of isotope one divided by nu of
isotope two is in very straight forward to
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show that it is a reduced mass of the isotope
two divided by the reduced mass of isotope
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one such exercises were also given to you
to give an idea that how by a very simple
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mass shift it is possible for the harmonic
oscillator to have the slightly different
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frequency and that is particularly important
for isotopic species where the chemical nature
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the force constant which determines the chemical
association between the two atoms doesnt change
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by much therefore keeping the force constant
constant it is possible for us to calculate
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the shift in the harmonic oscillator frequencies
and vice versa i think such calculations were
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given
now the one important difference for harmonic
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oscillator from the microwave ah spectroscopy
in terms of the absorption of radiation infrared
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radiation in this case s the fact that the
dipole movement mu must change during the
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vibration ok therefore molecules which have
permanent dipole movements such as hcl co
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then no h ah br i mean all those molecules
during vibration they already have a zero
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non zero dipole movement and during vibration
the dipole movements also change t is the
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change in dipole movement during vibration
which leads to vibrational energy absorption
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and the vibrational states are all given by
the expression as the eigenvalue for this
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hamiltonian h psi n is equal to en psi n and
en is h bar omega n plus half psi n this omega
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is the same as the the angular frequency therefore
you see that the harmonic oscillator has introduced
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a new quantum number n whose values starts
from zero one two etcetera and even when n
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is zero the harmonic oscillator still has
what is call the zero point energy namely
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half h bar omega and thats important thats
a quantum mechanical ah outcome the application
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of quantum mechanics leads to this kind of
prediction that there is a zero point energy
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for the molecule even at zero kelvin the harmonic
oscillator energy levels are all equidistant
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because the lowest energy level when n equal
to zero is half h bar omega n equal to one
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is three half h bar omega and n equal to two
is five half h bar omega and so on
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therefore n th harmonic oscillator approximation
there is no vibrational spectrum for diatomic
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molecules there is a single line because the
transitions between n and n plus one are the
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only allowed positions leading to one line
in the frequency this to this or this to this
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or this to that and so on the transition between
vibrational levels which differ by more than
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one s not possible within the harmonic oscillator
frame work and one has to introduce what is
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known as the molecular anharmonicity the vibrational
motion is no longer harmonic that is the vibrational
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motion is no longer half k x square but there
will be other terms like x cube and x to the
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power four and other types of terms and these
basically distort the vibrational motion from
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harmonic motion to un harmonic motion we looked
at the vibrational wave functions also we
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wrote them as psi n a constant root alpha
x as a normalization constant which depends
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on the quantum number n and exponential minus
alpha x square by two and the hermite polynomial
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with the argument root alpha x where alpha
is square root of k mu by h bar square
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so k and mu are the parameters of the harmonic
oscillator therefore alpha is determined by
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the nature of the oscillator this is the planks
constant and it is the same alpha that goes
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here and alpha has the dimensions of one by
length square x is the displacement from equilibrium
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therefore this is length square alpha has
one by l square has a dimension so root alpha
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x is dimensionless and this hermite polynomials
where given to you odd and even depending
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on whether n is odd or n is even this formulas
are given to you in the lecture notes and
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therefore you might look at the wave functions
because the transition movement the transition
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movement i between transition n and n prime
the initial state lets write it as n double
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prime to the n prime is given by the absolute
square of the transition movement integral
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namely psi n prime of x let us see psi n prime
of x here psi n prime of x mu psi n double
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prime of x dx between minus infinity to plus
infinity d absolute square and mu being proportional
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as a displacement the rate of change of mu
mu has to change during the vibration and
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that extent is given by the displacement itself
the displacement connects a wave function
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n prime and n double prime to give you non
zero values for this integral only if n prime
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is equal to n double prime plus or minus one
for all the other values the dipole movement
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does not lead to connecting a y wave function
n to a wave function n double prime which
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is far away it is required the gap that is
required to be just one because of the fact
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that the dipole movement that we have is proportional
to the displacement and this is within the
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harmonic model therefore the selection rule
for harmonic oscillator is delta v is equal
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to plus or minus one this is not to be forgotten
so this is the harmonic model that we have
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and then there was the slight improvement
from the harmonic model by considering anharmonicity
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in the form of a morse oscillator for the
morse oscillator we wrote down the morse potential
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proposed by philip morse first [vocalized
noise] namely the displacement the potential
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has the function of the displacement is a
constant times his functional form one minus
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e to the minus alpha x whole square i would
write this as x minus x e whole square where
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v at to the equilibrium distance xe is zero
you can see that right away when x is equal
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to xe this term is the exponent is zero therefore
the exponential of zero is one one minus one
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zero therefore the vf so this is called the
minimum in the potential the morse oscillator
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model allows us to consider molecular dissociation
as a result of vibration because if you plot
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the morse oscillator potential as a function
of the potential energy as a function of the
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displacement coordinate this is xe and this
is x you would see that the potential is minimum
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and then it raises until it reaches a and
also it raises to a very steep value for x
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less than xe because when x is less than xe
this is negative and there is no minus sign
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therefore the exponential alpha becomes positive
keeps increasing as this becomes more negative
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therefore you see that as x becomes more and
more ah i mean less and less than the equilibrium
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distance when the atoms compressed together
very closely you see that the potential raises
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very steeply that is given and you can also
see the value of d as nothing but when x minus
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xe is very very large or x goes to infinity
ideally as a mathematical formula when x goes
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to infinity this whole exponential goes to
zero therefore what you have is one whole
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square and so you have d and d is essentially
the asymptotic value and this d with respect
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to this value is called the dissociation energy
because at that extension of x the molecule
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the atoms do not come back therefore they
permanently separated from each other and
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this is called the dissociation energy
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the morse oscillator also led to a very unique
form of the energy expressions you might recall
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the energy level associated with the morse
oscillator were given by h bar omega e times
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v plus half minus ah let be yeah i think i
used only ev by hc to keep in mind that this
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is only a frequency factor the centimeter
inverse or a wave number and then we have
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minus omega exe v plus half square i think
this is the expression i probably use in the
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lecture notes and you can see that the vibrational
quantum number v is zero one two etcetera
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immediately what you see is that e naught
is no longer half omega e but it is half omega
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e minus one by four omega e x e because of
this term and e one is three halves omega
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e minus nine by four omega e xe because this
is three by two whole square and likewise
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if i write e two it is five by two omega e
minus twenty five by four omega e x e the
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picture that the morse oscillator immediately
gives raise to from this kind of energy level
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expressions is the following you by remember
that he energy levels become closure to each
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other [vocalized noise] so if i have to plot
the lowest energy as the half omega e minus
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one by four omega e x e the next energy level
is not omega e but there is also he difference
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between the two which is two m nine by four
minus one by four is eight by four there is
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a minus two omega e x e so ideally if you
have to write this as half omega e this as
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three by two omega e is no longer that this
is slightly lower than that half omega e by
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this factor one by four and this is even lower
therefore you see that the morse oscillator
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energy levels close in the next energy level
if you look at e two is closer to e one then
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e one is closer to e naught and like wise
you can see that so this is v equal to zero
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v equal to one v equal to two three and four
so what it means is that the gap between successive
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energy levels is no longer the same between
any given pair
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so you can see that this is not equal to that
this is not equal to that morse oscillator
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also allows in principally jump from zero
to two and therefore what you see in vibrational
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spectra is that a center line corresponding
to this which is of a very large intensity
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and then smaller lines which correspond to
a slightly lower intensity and every weak
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line which is approximately be twice so if
you write this as the new ve so this is the
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first line corresponding to the difference
between e naught and e one which is omega
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e minus two x e omega e
and therefore if we note two of these differences
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we have two unknowns mainly omega e and xe
so the differences we can calculate these
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unknowns therefore we can attain the anharmonicity
parameter for the morse oscillator using the
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morse model and the nice thing about the morse
model is that the molecule for very large
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values in principle can be associate the harmonic
model doesnt allow that please remember the
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harmonic model is essentially that no matter
how far away the atom is how high in the potential
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energy is the atom will always come back to
its equilibrium and move around the equilibrium
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this thus the prediction of the harmonic model
which doesnt work in reality of course and
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therefore what you see as the morse model
as a very nice way of accommodating the actual
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molecular behavior there are other models
there are other anharmonic models due to cubic
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and harmonicities quartic anharmonicities
etcetera and we will not in this particular
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course worry about those things and i did
not even introduce them those are important
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when you want to study the high resolution
infrared spectroscopy of gas waves molecules
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of very very very highly resolved spectra
then you need to worry about all those terms
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therefore this is a precursor to understanding
some of those things later
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now we will just very quickly move on to the
polyatomic vibrational motion and review what
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we have done but before we do that again please
remember that for vibrational spectroscopy
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of the harmonic oscillator level the selection
role is delta d is equal to plus minus one
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and it requires the dipole movement you change
during the vibration for the more oscillator
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there is no dipole movement selection rule
all energy levels can be connected by the
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dipole movement operator but the intensities
of vibrational levels decreases very sharply
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[vocalized noise] as you go from v is equal
to zero to v is equal to one to v is equal
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to two and so on therefore the lines becomes
less intense as you go further away but there
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are lines unlike that of a harmonic oscillator
now if you look at the poly atomic molecular
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systems for vibrational degrees of freedom
i told you immediately that and n atom molecule
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linear there are three n minus five normal
vibrational modes vibrational modes and you
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recall for linear molecule there are two rotational
modes and there are three translational modes
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leading to a total of three n [vocalized noise]
for non linear molecules there are three n
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minus six normalized normal vibrational modes
three rotational degrees of freedom and three
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translational degrees of freedom giving the
estimate total of three n degrees of freedom
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and we were only looking at harmonic oscillator
model for all the three n minus five or three
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n minus four six degrees of freedom and the
harmonic oscillator model is obviously called
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the normal mode model in this particular case
and the if i remove recall i gave you a large
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number of animations and the simple example
of a carbon dioxide or ozone or water these
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examples are given for specific reasons namely
in the case of ozone all the three atoms which
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are vibrating about their equilibrium positions
of the same mass therefore the vibrational
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displacements of each one of the atom about
its equilibrium is the same as the other in
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the case of water molecule these are lighter
atoms and this is oxygen is heavier atom
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therefore the vibrational amplitudes of the
lighter atoms are slightly more in the vibrational
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period that of the heavier atom is much less
and in the case of a linear molecule you realize
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that there are one there is one additional
vibrational degree of freedom and that led
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to what is called the double degenerate bending
mode for all for all linear molecules one
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bending mode is o c o bending in this plane
and the other bending mode is the carbon going
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behind the board and the oxygen coming in
front that is also a bending mode and both
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of them have the same energy and the bending
mode frequencies are lower than the stretching
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mode frequencies the stretching mode for carbon
dioxide where two of them is symmetric stretching
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and this is called the symmetric stretching
then there is this unsymmetric stretching
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or asymmetric stretching in which one of the
bonds is lengthened and the other is shorten
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this was very special as well as the bending
mode because both of these induce a dipole
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movement in the molecule during the vibration
therefore the dipole movement changes from
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zero to here zero here to a finite value and
then goes back to zero as the molecule completes
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one vibrational motion this change in the
dipole movement means that this degree of
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freedom and the bending degree of freedom
or both infrared active the symmetric switching
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move for carbon dioxide does not involve any
change in the dipole movement from its original
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value of zero because carbon is in the middle
of both these oxygens thus center of mass
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charge center as well as the mass center therefore
there is no dipole movement so when the molecule
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undergo symmetric stretch both the oxygens
go away from the carbon and the same extend
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and then come back to the a equilibrium position
on the same time therefore for symmetric stretch
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does not involve any change in dipole movement
it is not high or active and this is true
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for a large number of linear molecules and
molecules which have no dipole movements such
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as tetrahedral molecules c h four or the ah
planar triangular i mean with an atom in the
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middle of triangle center bh three kind of
molecule those are molecules which do not
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have any dipole movement for certain type
of vibrational motions but the fact that there
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are dipole movements introduced during other
is symmetric stretches means that those modes
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or infrared detectable and we see down in
the introduction spectroscopy quite a number
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of such examples were given to you and with
this i think we sort of came to an end in
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telling you that vibrational spectroscopy
are the level of which we are ah we have introduced
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to a sufficient to understand most importantly
the relation between the force constants the
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reduced mass and the vibrational frequencies
the last lecture involved raman spectroscopy
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which was slightly different from all the
other things are been did normal spectroscopy
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is the spectroscopy due to scattering of light
not due to absorption of light so when light
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falls on a sample which is something like
this and if you measure the light that is
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scattered in a perpendicular direction the
light that is scattered has frequencies which
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are modulated which are changed from the original
frequency of the light that ah excited irradiated
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the sample and these changes are on either
side of the frequency they can be a radiation
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with an increased frequency than the original
one and those are called the anti stoke lines
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and the radiation which has a frequency which
is lower than the frequency of the original
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one which is called the stokes radiation and
the radiation of exactly the same frequency
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as the incident radiation was called rayleigh
frequency and raman spectroscopy introduce
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the concept of polarizability as and the induced
dipole movement as the reason for the observation
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of raman spectrum the induced dipole movement
is proportional to the externally applied
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field or the electromagnetic field that falls
on the radiation that falls on the substance
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and therefore this is directly proportional
to the magnitude and the proportionality constant
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polarizability was introduced to you as a
tenser of rank two of course we did not make
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use of that property in the lecture the brief
lecture that we had we use the scalar property
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and for raman spectroscopy to be observed
it is important that molecules have one polarizability
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for microwave region for rotational lines
of raman spectrum to be observed static polarizability
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is required
therefore when hydrogen for example undergoes
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rotation in the presence of an electro external
electric field the stretching the the separation
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of the hydrogen atom leads to what is called
the polarization of the two hydrogen atom
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about the electric field it introduces an
alpha this alpha is static and this static
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polarizability is responsible for the rotational
raman lines now for the presence of non zero
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alpha means that even homo nuclear diatomic
molecules n triple bond n oxygen all homo
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nuclear diatomic molecules and even molecules
which do not otherwise have a dipole movement
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and cannot be detected using microwave spectroscopy
can be actually detected using raman spectroscopy
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and the relation then the reason for that
is the presence of polarizability that is
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the extend that the molecule can be polarized
in the presence of the external field i also
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reminded you that the polarizability leads
to a selection rule being a second rank tenser
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even the static polarizability is like xx
or xy or xz two quantities two directional
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displacement and therefore the selection rule
for raman spectroscopy is either zero or plus
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minus two or micro wave rotational raman lines
in the case of microwave spectroscopy we were
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only looking at the dipole movement of the
molecule here the induced dipole movement
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being proportional to the polarizability it
is the polarizability which is responsible
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for the rotational lines and that being a
second rank tenser leads to a selection rule
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of plus minus two and therefore rotational
raman lines are always seen between j and
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j plus or minus two so if you start from zero
the first line is two and you remember this
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is energy is zero this is six b and therefore
the first line has an energy of ah radiation
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the radiation that is seen in the roman spectrum
has the characteristic frequency of six b
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the next one if you look at say one to three
this is two b and three is of course twelve
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b the difference between the two ways and
b then you have two to four b two to four
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which is this is j these are all j these are
all j and two to four is six b to twenty b
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and therefore you have the next line fourteen
b
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therefore you see successive raman lines come
from the six b they are of by four b to the
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next line ten b and from ten b the next line
is at fourteen b therefore the raman spectrum
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if you look at it the rayleigh line which
doesnt have any frequency shift new zero is
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now shifted by the anti stocks lines namely
new naught plus six b as the first gap and
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then it is new naught plus four b this is
ten b this is new naught plus six b and this
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is new naught plus ten b and the gap between
the two is four b this is six b and the same
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thing happens in the anti stokes region when
this is sorry this is the anti stoke region
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same thing happens in stokes region when essentially
the energy is absorbed by the molecules then
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that frequency is missing in the radiation
and you can see that happens here when energy
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is dumped by the molecules in to the radiation
from coming down in the energy that increases
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the radiation frequency on the side so this
is stokes and this is anti stokes this is
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rayleigh lines most important thing for raman
spectroscopy ah for us to remember was the
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fact that the polarizability is important
and in the case of rotational roman lines
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the static polarizability results in this
radiational in this kind of selection rule
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for vibrational spectroscopy it is again the
change in the induced dipole movement and
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correspondingly it is the change in polarizability
from the static polarizability during the
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vibration that was required and therefore
that need not change the selection rule for
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infrared raman active lines raman active vibrational
modes the selection rules or the same as the
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infrared active selection rules namely delta
b is equal to plus minus one therefore vibrationally
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the only difference between the raman spectroscopic
lines and that of the infrared lines is that
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molecules which have a center of symmetry
seem to choose mutually exclusive spectroscopies
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either this or that so molecules which have
a center of symmetry have vibrational modes
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which are either roman active or infrared
active but not both but molecules which do
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not have a center of symmetry and which do
have a permanent dipole movement or both raman
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active and also a ir active
thus for example for water molecules we have
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raman lines as well as infrared lines for
carbon dioxide we dont have the roman line
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for the symmetric stretch and the infrared
line is for the anti symmetric or the asymmetric
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stretch and the bending mode so these are
some of the elementary properties that come
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from the polarizability the nature of polarizability
and its change with respect to the vibrational
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mode from that happens in a molecule so with
all of these elementary ideas now been told
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to you and hoping that you have followed some
of these lectures i wish you all the best
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and i hope that you will continue this study
of spectroscopy in a more profitable way in
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the future in to other areas as well as in
to more advanced areas i wish you all the
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best
thank you