WEBVTT

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hello and welcome to the twenty fifth class
of this course so in the last class we learnt

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about the the relation between group theory
and quanta mechanics and there we saw that

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the brief functions which are i can functions
of any given operators for example hamiltonian

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operator which is an energy operator can
act as the basis of more ire illusive the

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representation for any particular molecule
belong into a particular point grip

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also we learnt about the direct products
of the introduced will be representation which

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will be extremely useful in in the coming
weeks and we also learnt how this direct

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product are used they can be used so we
took an example to show that direct product

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can be used to find out the elements and also
to find out the spectral intensity most

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precisely to probability of transition
whether its allowed or disallowed and if it

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is allow then in which particular polarization
it will be allowed now one thing we

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didnt talk about in the last class that
not only the wave function which is an

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i can function of an operator can found the
basis for illusive representation but also

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the linear combination of the wave function
can also act as the basis for the same

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ire illusive representation which is a
you know quiet understood because an

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if i have an i can functions i i for an operator
each the hamiltonian then i can express i

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i as a linear combination of various emphasize
so therefore its no wondered that linear com

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proper linear combination of the again functions
can also act as a basis of the ire illusive

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representation and this property is
of at most important when we we

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try to find out which kind you knows
atomic orbitals will combined to give molecules

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orbitals or which will contribute to
the habituates orbitals also to find out

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like which particular warn vectors will
combine to keep an internal motion like so

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in general you know the chemists they try
to use the symmetry restrictions and to

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you know facilitate themselves to understand
chemical bonding and molecular dynamics for

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example as i said constructing habit orbitals
or molecular orbitals or finding proper orbitals

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sets under say align feed or also analyzing
vibrations of molecules

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now in ordered to do that that is developing
such understanding based on symmetric restrictions

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well phases common problem the problem
is that the you know one is to take one

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or more sets of orthonormal functions as required
by a both group theory and quanta mechanics

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and here this orthonormal functions
in case of this you know this chemistry

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problems this functions are taken to be
the atomic orbitals ok or in case of motions

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are there other incular motions they are intertie
coordinates of a molecules and to in order

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to make proper combination which will
keep the in the resultant function orthonormal

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and that will act as a basis of ire illusive
representation one needs to be quiet

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you know careful in his formulation of the
problem and you know as we learnt in the last

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class you know this kind of you know
the relation between the quanta mechanical

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observables the wave functions
rather not the and the you know symmetrical

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strength this really is important for
solving this problem ok

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so in case of you know when when
one tries to find out which atomic orbitals

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will form the hybrid molecular orbitals
or in general molecular orbitals one

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needs to take care of the symmetry constraints
and then ultimately the linear combination

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that one gets is known as symmetry adapted
linear combination or in sort s a l c so

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when s a l c so how one can find out that
which orbitals will be able to combine in

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a linear fashion and keeping the you know
symmetric constraints to give a proper

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you know basis for a particular ire illusive
representation of particular point curve

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to which our constraints molecules belongs
to now you see why the symmetric constraints

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needed because for example you take an any
molecules if you for example ammonia so in

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the case of ammonia what we have is
this bond that we write

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now bond means what this is the you know combination
of the atomic orbitals so hydrogen has one

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is orbitals and nitrogen has one is two is
two p orbitals so which orbitals will combine

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to form this successful bond is not only
governed by the energetics but also governed

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by the symmetric so until unless the orbitals
of this atoms have the same symmetric they

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cannot combine so there is a symmetry restriction
and therefore the you know the combination

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that we form to in the facilitated the
bond formation will be symmetry adapted

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linear combination combination of atomic orbitals
ok so how do you do that so there is a fundamental

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tool which is inversely accepted and this
particular tool is known as projection operator

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so now we will learnt about this particular
projection operator which will help us in

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finding on the s a l cs ok for any given molecule
so even to you know understand the

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position of at as functionally is we better
learn how to get this position of it refers

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and then will go for the actual application
of this projection operator by showing some

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illustrative examples so let us start
with

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lets assume that we have a orthonormal
set l i number of functions ok such

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as phi one phi two up to phi l i ok so we
have a i number of such functions and we also

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assume that this functions from the basis
of the i th ire illusive representation of

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a point group to which our concerned molecules
belongs to ok so if this forms a basis for

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and i ire illusive representation so were
put a superscript i ok that will tell you

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that to which ire illusive representation
this functions act as basis form and the order

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of the group that will are concerned lets
take as h as visual now if i consider any

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operator any symmetry operator as r and e
i take anyone of this functions and operate

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a symmetry operation on that function then
what should i get so if i take any imbricate

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function phi t phi t is one of this you know
l i number of this which formed of basis for

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i th ire illusive representation
so that i can write as right so i can write

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this one as i have taken this function
to be the basis for i th ire illusive representation

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and this gamma r having superscript i is a
representation for this particular symmetry

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operation r giving i th ire illusive representation
and s t you can understand is a particular

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element ok so this gamma r is a matrix representation
so s t gives me the particular element of

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that representation right so now what we have
to do in order to find out how to

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get the projection operator what i will do
i will multiply both the sides with another

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representation for j th i r say
so i will multiply both the sides by a

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quantity which is
which is gamma of the same operation r and

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belongs to j th ire illusive representation
and some element is s prime t prime ok and

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i we will take a complex conjugate of that
particular element and after that we will

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some it over all the symmetry operations r
so we have to do the same theme on the right

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hand side as well so in that case i have summation
over r summation over s phi i s gamma of r

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s t multiplied by gamma of r for j th ire
illusive representation having elements is

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prime t prime ok
so now one thing i notice here this phi

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i is an function so it is independent of the
symmetry operation right it it doent clash

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with the symmetry operation so i can you know
very easily separate this phi part and this

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representation part ok so what i can do i
can rewrite this one as summation over s phi

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s i and then summation over r and this part
that means gamma r for i th ire illusive representation

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and element s t multiplied by gamma of r for
j th ire illusive representation having element

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is prime t prime and take the complex conjugate
of this one ok so this we can very easily

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do right
now we have a situation where this right hand

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side tells me that i have you know i can
have such l i number of know such products

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ok where you know each one we be a function
phi s that belongs that forms basis of

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ire illusive representation and and that is
multiplied by a co efficient ok so this part

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is a co efficient which is multiplied this
phi s ok and i can get such l i number

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of products fine so now if i considered
this particular co efficient what does this

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co efficient tell so this co efficient themselves
r the summation of the products of two ire

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illusive representation over all the symmetry
operations thats what it tells me right

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now sets sum of the products of ire illusive
representation over all symmetry operations

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are governed by the great ortholonomatic theory
so what does the great ortholonomatic theory

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tells me about such an such a co efficient
ok so if i just take this one so it is sum

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over r gamma of r for i th ire illusive representation
and have been elements s and t multiplied

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by gamma of r for j th ire illusive representation
having elements s prime t prime is equal to

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if you remember the great ortholonomatic theory
that we discussed earlier so we have considered

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the order of the group to be h so here order
of the group and then i have two different

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ire illusive representation i th and j th
right so their dimensions

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lets assume l i and l j therefore this will
have l i and l j right and then i will have

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series of delta functions so here what we
have we have two ire illusive representation

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ok get theorem will you know guide you how
you can write the delta functions here so

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for two different ire illusive representation
i will have the delta i j and then i have

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elements as s t for the i th and s prime t
prime for the j th ire illusive representation

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therefore i will have two more delta functions
one will be delta s s prime and other one

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we will delta t t prime alright so now
what does that tell me it tells me that i

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can replace this one over here right
so this part will remain as it is and i can

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write here as h sorry summation over s
phi s for i th ire illusive representation

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and h by route over l i l j and then three
delta functions right

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and then whatever i have on the other side
that remains so here this right hand side

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immediately tells me that phi s will you know
this side actually will give non zero

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value only when this is equals to s prime
right i am summing it over s so if i operate

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this operator on this particular function
phi t form i is the representation then i

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will have this right hand side having one
zero value only when s is equals to s prime

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so i will get phi s prime here with two more
delta function delta i j delta t t prime so

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which means that not only that it has to
be a particular function phi s prime in other

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two have non zero value for this part but
also the two ire illusive representation that

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we are considering here must be identical
meaning i equal to j and other element also

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should be the same that is t equals to t prime
so therefore if consider all this

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what do i have is following so what we
have here the left side sum over r i can just

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slightly rearrange because this two are not
interfering each other so i will write this

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term first so i have gamma of r for j th
ire illusive representation having elements

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s prime t prime multiplied by the operation
r and optimum phi t forming basis for

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i i th represent ire illusive representation
ok so this is equals to from here whatever

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i said if i can write that h by l j because
when l i i equals to j then only this right

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side i can write with survive so therefore
for i equals to j i have l i equals to l j

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so l j square and take a route i get l j ok
so this and i have phi now s prime because

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for s equals to s prime this will survive
and for at this is representation and if i

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still keep
the deltas took it the change relative skill

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i will have this part
now by rearranging i can have l i by h whatever

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we had phi of s prime i delta i j delta t
t all right so this this part of the left

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hand side is known as the projection operator
and this is abbreviated as p of s prime t

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prime ok for j th ire illusive representation
ok so this is equals to the projection operators

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and then you have this
right so this is my projection operator so

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now you can see here that this particular
operator when it is operated on any arbitrary

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function phi t it projects out another function
phi s prime ok

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so this projection will in a takes place only
when this you know function phi t contains

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or it it itself is phi t prime ok so
then it will give you know that you know survive

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ok because i have this delta function delta
t t prime moreover it has to you know phi

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you know belong to the same particular ire
illusive representation ok so then i can have

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this phi is prime being projected out from
an arbitrary function or the combination

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of the function which is given by phi t
phi t all right so ultimately by taking

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care of this delta function what i can right
is this making ok so i will rub this part

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so therefore what i can write is p of acting
on general function any arbitrary function

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t which will give me ultimately phi t prime
if i take care of this delta t t prime right

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so a sincerely it means that this phi t
must contain some component of phi t prime

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in that case i will have when phi t prime
and for any arbitrary any ire illusive

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representation see i can write this general
form of this ok now this is the most general

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form but i can have the special case so
the special case is as follows so if i take

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this

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so this is a particular special case that
we can considered so what does this special

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case tells me so it tells me that this
p j t t prime that is the projection operator

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which will project out phi t prime out
of an arbitrary function which is phi t that

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we started with late so by using l i such
projection operators based on l i diagonal

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matrixes ok
so that is element rather we need generate

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from some arbitrary function which is phi
t that we discussed as you know a state

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of functions which will form the basis of
the j th ire illusive representation ok so

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this is what we have as projection operated
now this projection operator will be used

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in specific cases when we consider in the
following class taking a particular example

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of an molecule their we will consider the
atomic orbitals there were functions and we

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will operate this projection operators here
so here like a we say that this is particular

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ire illusive representation
now what we can do we can you know take

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the real ire illusive representation so you
have to look at the correctable find out the

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you know ire illusive representation that
you want to work with so for example some

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ire illusive representation beet to you
so can have that projection operated written

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as p b to u ok so in the following class
we also see that there is there are two type

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of projection operator one is like complete
projection operator another is incomplete

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projection operator so most of our purpose
we are pretty good with the incomplete

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projection operator itself which deals with
not the matrix elements as such but only with

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the characters ok so we can use the characters
from character table belong to particular

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ire illusive representation to form the projection
operator and thereby you know we can work

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on the particular orbitals of in you know
atom within a molecule to project out which

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are the orbitals that can combine a linearly
to form the you know say orbit orbitals

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so in the following class we will learn
more about this and try to form different

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l i s a l c is from for different molecules
till then have a nice week

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thank you very much