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welcome back to the lectures on chemistry
and introduction to molecular spectroscopy
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in this short lecture we shall ah examine
the microwave spectra of poly atomic molecules
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again subject to the rigid molecular approximation
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spectra of symmetric top molecules are the
first region in which we start getting more
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and more details from microwave spectra regarding
the structures and so on but you be surprised
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to see some patterns of for symmetric top
being very similar r identical to the pattern
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of the spectra of diatomic molecules
now please recall that we always take the
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rotational kinetic energy and the rotational
kinetic energy when the moments of inertia
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are all identical is given by the angular
momentum squared divided by the moment of
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inertia however if a molecule is such that
its moment of inertia above three mutually
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orthogonal axis are not the same then we have
to worry about those axis systems and write
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to the moment of inertia as i x i y and i
z
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let see some examples n the case of water
molecule we first try atomic molecule with
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the two hydrogen atoms and the oxygen atom
here the center of mass is somewhere here
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its on the axis which bisects the two hydrogen
that is the hydrogen o hydrogen bond therefore
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this is ah half of the bond angle and if this
is a center of mass this axis which is also
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the molecular symmetry axis is usefully axis
ok then what you have is a y axis which is
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perpendicular to that ok and in x axis which
is actually perpendicular to the plane of
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the screen that you have so which we normally
write as like that so if these are the three
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axis x y and z and if the ah moments of inertia
are calculated about these three different
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axis you would see that the ix is not the
same as iy and is also different from iz
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such molecules in which all the three moments
of inertia these are principle moments of
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inertia we have chosen the axis system such
that we worry only about these three ah moments
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and these principle moments of inertia which
are different lead to characterizing these
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molecule as an asymmetric top
and other example of asymmetric top in fact
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most molecules are asymmetric top other example
is again ozone in which you have all the three
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oxygen atoms now of which the one oxygen atom
is connected to the other two in a different
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way this is also i mean its identical in symmetry
to the water molecule and you have sulfur
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dioxide is an example if you think about this
nitrogen dioxide nitric ah there are many
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many examples of molecules which are asymmetric
tops
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now it is possible that in molecules two of
the moments of inertia may be equal to one
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another but different from the third and that
happens in molecules such as methyl chloride
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or chloroform for example any molecules which
has a c three symmetry c three meaning a threefold
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axis about which the molecule has symmetry
any such molecule will have two of the moments
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of inertia in this case of course the c lets
take ch three cl so we have c cl bond axis
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and then we have the three hydrogen atoms
which form an equilateral triangle plane i
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mean if you connect that in the plane of the
three atoms be an equilateral triangle and
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such molecules have the moment of inertia
about the z axis iz is different from the
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moment of inertia about the x axis which is
equal to the moment of inertia about the y
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axis hm and there are two categories here
that the different moment of inertia from
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the other two which are equal may be less
than the other two or greater than the other
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two if it is less than the other two we call
that as a prolate symmetric term if iz is
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less than ix which is equal to iy it is called
prolate symmetric term typically such symmetric
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tops are written or depicted using an ellipsoid
of this form the other in which the different
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moment of inertia is actually greater than
the the two equal moments of inertia this
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is called oblate symmetric term the oblate
symmetric term is usually written or depicted
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like an ellipsoid of this is of course an
ellipse into dimension but the whole thing
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is an ellipsoid the three moments of inertia
actually become ah the three axes system for
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drawing the ellipsoid and the axes are the
one of the axes and the other two axes are
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like that perpendicular
if all the three moments of inertia are equal
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such a molecule is called as spherical top
now methane for example or ah the carbon tetra
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chloride sorry or carbon tetra chloride these
are examples of ah symmetry of spherical tops
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that is its easy to see that if you draw a
cube approximately and envision the tetrahedral
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symmetry using the cube ok the tetrahedral
symmetry essentially means that the cube has
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opposite corners of the face occupied by the
four atoms so here is the hydrogen atoms so
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if you want this as the carbon then the face
diagonal to that is this so if you draw the
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bond like that and the bond here and the other
two face diagonals are these so these are
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the face diagonals in this face these are
the face diagonals in this face and you can
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see that they come out and if you connect
all four of them you see that the tetrahedron
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is actually half the symmetry of a cube or
an octahedron and in this structure it does
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not matter to you whether you are looking
from the top face or whether you are looking
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from the side face or this side etcetera because
the molecule is identical whichever way you
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look at it and therefore if you draw the three
mutually perpendicular axis as the ah axis
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for the principle moments of inertia it is
seen that he atoms are atom at the same distance
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away from each of these axis and so its immediately
clear that the three moments of inertia should
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be the same and like ways for a octahedron
molecule perfectly octahedral molecule also
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has all the three moments of inertia ah equal
ah identical if all the six atoms are the
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same ok
so these are three different cases that you
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can think about and the linear molecule is
a special type where two moments of inertia
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are equal and unequal to the third but unfortunately
the third is zero its a point mass approximation
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therefore its not a spherical top a is not
a symmetric top he symmetric top has both
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the moments of inertia non zero reasonably
ah big enough for them to be observed in one
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way or the other but in the case of a linear
molecule it doesnt have there is only one
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moment of inertia two of the two components
are equal but the third is zero so there is
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a difference in the fundamental classification
of the molecular moments ok moments of inertia
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now what happens is the rotational kinetic
energy when there are three different moment
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of inertia is given by the angular momentum
components in those directions divided by
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the moments of inertia about that axis y y
square by two iy plus j z square by two iz
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we shall we shall now study the general ah
hamiltonian and energy levels of the asymmetric
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top if you write this as the operators then
its two ix plus j y square the operator for
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that two iy and the j z square by two iz we
shall not ah do that in this course now it
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requires slightly more detailed analysis
so what we will look at the case ah look at
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is the case of symmetric tops
for the symmetric top let us consider the
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particular case that ix is equal to iy and
iz is different from both of them therefore
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you have the hamiltonian rotational which
is only the kinetic energy component of the
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rotational ah motion is j x square plus j
y square divided by two ix because x and y
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are the same and then you have j z square
divided by two iz this is the symmetric top
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hamiltonian but the operators and one must
know that the angular momentum operators jx
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and jy do not commute in the sense of in in
in quantum mechanics and and in the these
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have a particular commutation relation minus
i h bar jz i use the minus sign because these
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axis the x and y and z are fixed in the molecule
and they are actually rotating and moving
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with the molecules why they are fixed on the
center mass of the molecule but they are rotating
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with the molecule so these are called body
fixed axis system and this is different from
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what you studied in the case of a hydrogen
atom where you would have studied that the
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angular momentum lx and ly be the components
of the angular momentum
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do not commute but give you something like
that and then correspondingly ly lz give you
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ih bar lx and lz lx give you the commutator
gives you i h bar ly and so on whereas for
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molecular spectroscopy the components that
we have here are n such that they have the
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minus sign they are called anomalous sorry
they are called anomalous commutation relation
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commutation relations
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and
what they have is j alpha j beta is i using
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some specific notation called a third rank
tension rotation epsilon alpha beta gamma
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its called j ok if you dont know about this
dont worry all it means is that jx plus comma
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jy the commutator is sorry that one should
be minus i h bar ok minus i h bar and this
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is j x comma j y minus h bar jz these are
operators and jy jz are operators with the
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minus h bar jx and the third one jz jx is
minus i h bar jy all of them commute with
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the square of the angular momentum operator
whether write jx or jy or jz all of them that
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commutator is zero
therefore what it means is that in quantum
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mechanics it is possible in our ah description
of the molecular rotations it is possible
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for us to find ah eigen values for operators
which commute with each other and here the
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operators that commute are square of the angular
momentum and one component of the angular
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momentum and usually if this component is
chosen as the z component or the component
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in the molecular symmetry axis which is the
highest symmetry axis for any molecule given
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that therefore when you write the hamiltonian
for this system the symmetric top particular
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when you write this as jx square plus j y
square by two ix and jz square by two iz you
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replace this combination by the total angular
momentum square minus the z square component
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and you write two ix plus j z square by two
iz this is the hamiltonian which contains
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the operators j square and j z q and they
of course commute to each other therefore
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it is possible for us to actually find the
solutions to the angular momentum problem
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by solving the equation h psi is equal to
e psi but there are two operators j and square
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on jz and both of these are found to satisfy
the property that jz acting on the molecular
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function gives you a quantum number k h bar
psi and j square on the quantum number on
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the wave function gives you h bar square j
into j plus one psi therefore you see that
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the wave function is simultaneously the eigen
functions of these two operators we write
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this by writing the values eigenvalues here
j and k j and k and j and k to denote that
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the difference such the different eigen functions
you have namely k going from minus j minus
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j plus one to plus j so there are two j plus
one eigen functions this was also mentioned
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in the last lecture then i talked about the
bracket notation jk these ate psi jk or this
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so there are two j plus one eigen function
and this is important because this differentiates
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the spectroscopy in principle of a symmetric
top from that of a linear diatomic molecule
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now these are the input for us to calculate
the hamiltonian the on psi jk and hamiltonian
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is of course h bar square by two what we have
here is h bar square by two ix and the operator
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j square gives you j into j plus one and the
operator here the j z square gives you h bar
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square k square k square and the other term
in the hamiltonian which is jz square by two
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iz gives you simply h bar square k square
by two iz all of this being be eigen value
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multiplying the eigen function psi jk this
is the quantum mechanical solution
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so the notation that spectroscopy use or ah
keep in mind remember b j into j plus one
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times hc as the energy of a di atomic molecule
and now the energy for a poly atomic molecule
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which is a symmetric top ha has that and some
additional terms let us call the quantity
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h by eight pi square izc which comes from
this term h bar square by two iz if you call
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this hc times a again a wave number then a
will be h by eight pi square iz c and so this
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is a this will be b therefore the energy level
if you look at it when you write h psi jk
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as the energy now which is dependent on the
two quantum numbers ejk psi jk and if this
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e s therefore given in terms of hc b j into
j plus one minus k square plus this term and
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then you have the other term which is given
as plus hc a k square on psi jk
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so all i have done is to associate the rotational
constant with the this term as the old ah
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b constant this one with the hc and the rotational
constant with the principle moment of inertia
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along these bond the symmetric axis as the
a constant so now we have two constants for
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the symmetric top because there are two moments
of inertia so if you write the energy divided
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by hc jk then this is b j into j plus one
plus a minus b times k square
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therefore you see that the energy levels are
no longer two j plus one fold degenerate as
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as they were in the case of a linear diatomic
linear or diatomic molecule the energy levels
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for a diatomic molecule where the same for
all values of k for any given j and we use
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this in the last lecture in determining the
intensities by saying its each ah energy ah
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corresponding to a given j is two j plus one
fold degenerate now in the case of symmetric
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top we see that that is no longer there the
symmetry is now produced and what you have
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is the energy level is now a function of the
square of k and it is of course a bj into
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j plus one with the additional term also look
at the magnitude of a and b if you look at
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it a is proportional to one by iz the principle
moment of inertia a along the symmetry axis
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and b is proportional to one by ix and this
ix and iy are equal iz is different
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therefore you see if iz is smaller than ix
and iy we said it was a prolate symmetric
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top therefore for a prolate symmetric top
a is therefore greater than b because the
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moment of inertia is smaller than the equal
moment of inertia the unequal one is smaller
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than the other two moments of inertia which
are equal therefore the rotational will have
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the inverse relation the rotational constant
a which corresponds to the inverse moment
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of inertia along the symmetry axis is actually
greater than that of the other moments of
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inertia namely the rotational constant b so
this is important
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now since for any given j we have k going
from minus j to plus j we can write this as
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zero plus minus one plus minus two to plus
minus j k therefore we can write the e j with
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k is equal to zero as one energy and that
will be simply bj into j plus one because
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the other term if you look at it bj into j
plus one plus a minus b times k square so
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lets keep that in a corner somewhere ejk is
bj into j plus one minus sorry plus a minus
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b k square lets keep that this is what we
are using to write down for ah exp write down
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the expression for different energy levels
and when k is zero you get only that and when
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k is plus or minus one you get b j into j
plus one plus a minus b and so on so let us
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right now the energy levels for j is equal
to one to start with this is non trivial case
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therefore e one zero is two be e one plus
minus one is a two b plus a minus b two b
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plus a minus b k square is one therefore that
is a plus b and please remember a is greater
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than b therefore a plus b is greater than
two b so if you write the energy level expression
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the j is equal to one k is equal to zero which
corresponds to two b will be slightly lower
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than the energy level for j is equal to one
and k is equal to one this will be a plus
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b the difference here is of course a minus
b which is positive ok
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therefore for a prolate symmetric top the
energy levels actually increase for a given
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j from k is equal to zero to k is equal to
plus minus one to plus minus two plus minus
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three and so on therefore if you write this
again now for j is equal to two and j is equal
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to three we will write this for two or simple
examples this will be six b when two zero
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two is the quantum number for ah j and zero
is the quantum number for k so this will be
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six b and two plus minus one will be remember
bj into j plus one plus a minus b this is
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six b plus a minus b so what you have is a
plus five b and for j is equal to two you
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have twelve and this is of course k square
therefore that is four a so what you will
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get is twelve b plus four times a minus b
which will give you four a plus eight b
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so for a symmetric top for any given j there
are j plus one energy levels ok
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the plus one is due to the fact that the k
is equal to zero say it is non degenerate
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and for all the other states j is equal to
two this is ah k is equal to plus minus two
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00:29:25,020 --> 00:29:30,840
plus minus two and that so maximum and for
j is equal to three of course there be four
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energy levels corresponding to three zero
three plus minus one three plus minus two
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and three plus minus three and so on this
is for prolate symmetric top for the oblate
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symmetric top the energy level expression
is still the same it is ejk which is b j into
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j plus one plus a minus b k square but for
oblate symmetric top a is less than b and
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therefore what you see is this term is a negative
contribution to the j k is equal to zero zero
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therefore ej zero will be the highest among
all the energy levels e j zero e j k where
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k goes from plus minus one plus minus two
to plus minus j among all the energy levels
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00:30:26,830 --> 00:30:32,370
highest one will be the one with k is equal
to zero because the other term is negative
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ah for any value of k therefore the energy
levels decrease precisely what you see here
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as in increasing progression of energy levels
you will see the reverse with the highest
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00:30:42,200 --> 00:30:47,760
level being the two zero and the next one
being two plus minus one and the next lower
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00:30:47,760 --> 00:30:53,380
one being to zero and so on so on the relations
are inversed ok this is about the symmetric
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top energy levels
so in summary what we have to remember is
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that ejk is equal to b j into j plus one plus
a minus b times k square and for any given
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quantum number k quantum number j for any
given quantum number j there are two j plus
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00:31:29,040 --> 00:31:41,980
one there are sorry j plus one levels j plus
one levels so we shall stop here and take
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a look at the some of the examples of the
symmetric top molecules ah and study the energy
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level and the transitions in the next half
of the lecture until then
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thank you very much