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so we continue with the rotational spectra
microwave of polyatomic molecules and ah i
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recall that in the case of polyatomic systems
we were only looking at the spectra of symmetric
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tops
and ah you remember that the symmetric top
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is one of the special ah types of molecules
in which the movements of inertia have certain
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relationships we for a general polyatomic
molecule the three movements of inertia about
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the mutually three mutually perpendicular
orthogonal axis all passing through the centre
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of mass of the molecule or in general different
however the special case when the principle
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the so called the principle movements of inertia
when they are equal in all directions i x
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x i y y i z z this was called the spherical
top and some examples are given cubic molecules
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and perfectly tetrahedron molecules are the
best examples for spherical top
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then we had the special case namely i x x
equal to i y y greater than i z z that is
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not equal to i z z and the other case the
i x x is equal to i y y less than i z z then
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this was called prolate and at the i z z is
greater than the other two equal movements
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00:02:06,180 --> 00:02:16,150
of inertia it was called oblate symmetric
top both are symmetric tops the spectrum of
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00:02:16,150 --> 00:02:21,380
symmetric top and the energy levels of the
symmetric top are easy to calculate because
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of the special fact that the hamiltonian for
the symmetric top has a specific as a special
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from please remember for a molecule which
had all the three movement inertia being different
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the it is written as j x square by two i x
x plus j y square by two i y y and j z square
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by two i z z and this i did not solve in this
course because its a its called an a symmetric
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00:02:53,829 --> 00:02:59,739
top the three cases i x x not equal to i y
y
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00:02:59,739 --> 00:03:09,389
not equal to i z z is called a symmetric form
and some examples were given in the lecture
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00:03:09,389 --> 00:03:13,799
and you are suppose to look at the structure
of the molecule and any molecule which has
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a threefold axis of symmetric will most likely
be a symmetric top and ah and molecules which
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do not have a threefold axis of symmetric
which have only twofold axis of symmetric
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or by they are a symmetric top so there are
some special rules i think i mentioned that
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in the class in the course this was not solved
but what was solve was the special case j
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x square plus j y square by two i x x then
i x x is equal to i y y and j z square by
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00:03:48,760 --> 00:04:03,230
two i z z you recall that i wrote this hamiltonian
as a j yeah i believe it is b j into j j square
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00:04:03,230 --> 00:04:17,650
j z square let me right b j square plus a
minus b j z square ok this was the ah hamiltonian
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00:04:17,650 --> 00:04:24,960
form and for a prolate this is a b is are
the inverse movements of inertia one by two
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00:04:24,960 --> 00:04:32,880
one by i x x was related to b and one by i
z z was related to a inverse movements and
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00:04:32,880 --> 00:04:41,660
this when a is greater than b which means
i z z is less than i x x it was called the
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00:04:41,660 --> 00:04:46,531
prolate symmetric top and a is less than b
it was called oblate symmetric top and you
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00:04:46,531 --> 00:04:53,570
had the energy level diagrams written for
j square and j z square operator
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00:04:53,570 --> 00:05:00,670
you may recall that angular momentum quantum
mechanics particularly for the molecular system
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00:05:00,670 --> 00:05:06,280
in does not the components do not commute
with each other and for molecules spectroscopy
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00:05:06,280 --> 00:05:14,230
we use the commutation relation minus i h
bar j z because the axis x y z are fixed in
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00:05:14,230 --> 00:05:21,700
the molecule and this essentially means the
other components as well j y j z where minus
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00:05:21,700 --> 00:05:32,910
i h bar j x and j z j x were minus i h bar
j y what it means is that simultaneously you
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00:05:32,910 --> 00:05:37,630
cannot measure the x component of the angular
momentum and the y component of the angular
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00:05:37,630 --> 00:05:43,790
momentum for the same system but you can always
measure the absolute square of the angular
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00:05:43,790 --> 00:05:49,770
momentum and one of the three components because
in the lecture i remember telling you that
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00:05:49,770 --> 00:05:58,620
j square command j x is equal to zero or j
y is equal to zero or j z that is j square
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00:05:58,620 --> 00:06:02,710
commutes with all the three components but
these do not commute with each other therefore
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00:06:02,710 --> 00:06:09,810
you have this and one of these and the one
of these components was chosen always as j
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00:06:09,810 --> 00:06:18,730
z and j z on the ideal functions psi j was
ah shown in hydrogen atom earlier and in many
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00:06:18,730 --> 00:06:33,420
other cases to be your h bar m or k i believe
use the index k psi j k and they use k as
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00:06:33,420 --> 00:06:41,790
the quantum number and the k is the index
which is essentially is a quantum number which
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00:06:41,790 --> 00:06:50,580
to the ah projection of the angular momentum
on to a chosen molecular axis is an axis the
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00:06:50,580 --> 00:06:59,210
k has possible values from minus j minus j
plus one to j minus one two j namely two j
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00:06:59,210 --> 00:07:06,110
plus one values
the eigenvalues for the symmetric top where
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00:07:06,110 --> 00:07:18,110
or ah j z square acting on the function therefore
this will give you k square instead of k therefore
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00:07:18,110 --> 00:07:25,750
the energy for the symmetric top is given
by two quantum numbers j and k and that's
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00:07:25,750 --> 00:07:35,370
b j into j plus one plus a minus b into k
square and you can see immediately that k
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00:07:35,370 --> 00:07:42,970
is equal to zero is a unique energy eigenvalues
its the generate one one energy state and
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00:07:42,970 --> 00:07:50,720
that's state is called j zero when all the
other states plus minus one plus minus two
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00:07:50,720 --> 00:07:56,110
plus minus three all the way up to plus minus
k each one of them they have a slightly different
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00:07:56,110 --> 00:08:00,950
energy because of k square but both of them
will be the same plus k and the minus k will
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00:08:00,950 --> 00:08:12,770
be the same therefore for all case not equal
to zero the energy levels are doubly degenerate
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00:08:12,770 --> 00:08:19,550
and we also looked at to the prolate and the
oblate symmetric top and here also it is important
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00:08:19,550 --> 00:08:23,800
to note on the microwaves spectrum is possible
only if the molecules has a permanent dipole
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00:08:23,800 --> 00:08:30,560
moment but it is possible that the molecule
has a dipole moment in an arbitrary axis not
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00:08:30,560 --> 00:08:36,510
along axis or along the x axis or y axis that
you choose therefore the angular momentum
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00:08:36,510 --> 00:08:41,750
the dipole moment we have more than one component
in the respective directions leading to the
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00:08:41,750 --> 00:08:50,699
fact that there are different selection rules
but all of them will involve delta j recall
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00:08:50,699 --> 00:08:56,790
to plus minus one and depending on the special
molecular cases you may have the selection
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00:08:56,790 --> 00:09:04,089
rules delta j is equal to plus minus one and
delta k is equal to zero or plus minus one
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00:09:04,089 --> 00:09:09,730
that depends on the molecular nature we did
not elaborate this but we the i recall that
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00:09:09,730 --> 00:09:16,129
this was the stated selection rule for the
that the polyatomic molecular case we looked
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00:09:16,129 --> 00:09:20,829
at some of the polyatomic systems namely what
are symmetric tops and what are a symmetric
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00:09:20,829 --> 00:09:26,459
tops some examples are given and some of the
energy levels were given and basic calculations
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00:09:26,459 --> 00:09:32,629
regarding the dipole moment metrics element
that also given with that i think we sort
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00:09:32,629 --> 00:09:40,180
of completed the basic structure on microwave
spectroscopy we did not move onto the ah non
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00:09:40,180 --> 00:09:46,680
reject molecular system and also the a symmetric
top molecular system because they are slightly
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00:09:46,680 --> 00:09:53,800
more complex and usually one studies such
system in the next level of the introductory
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00:09:53,800 --> 00:09:56,490
course to spectroscopy therefore i stopped
there
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00:09:56,490 --> 00:10:02,249
now in the next part of this review overview
there is only one more part of this i will
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00:10:02,249 --> 00:10:07,459
talk a little bit about to the ah review of
the vibrational spectroscopy that we looked
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00:10:07,459 --> 00:10:12,620
at with the more important formulas that you
need to remember and you also have to keep
80
00:10:12,620 --> 00:10:20,949
in mind in taking this course further so we
will move onto the vibrational ah spectroscopy
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00:10:20,949 --> 00:10:26,110
of what is called the harmonic oscillator
model spectroscopy for the diatomic molecule
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00:10:26,110 --> 00:10:31,410
and then we will also look at the summary
of the normal modes that we study along with
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00:10:31,410 --> 00:10:38,589
this one extension model for the non un harmonic
model namely the morse oscillator that will
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00:10:38,589 --> 00:10:42,740
be the summary of the remaining part of this
overview of the whole course
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00:10:42,740 --> 00:10:43,519
thank you