Normal coordinate analysis of xy 2 bent molecule part 1
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NORMAL COORDINATE ANALYSIS OF XY 2 BENT MOLECULE – PART 1. Dr.D.UTHRA Head, Dept.of Physics DG Vaishnav College, Chennai-106. This presentation has been designed to serve as a self- study material for Postgraduate Physics students pursuing their

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NORMAL COORDINATE ANALYSIS OF XY 2 BENT MOLECULE – PART 1

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Normal coordinate analysis of xy 2 bent molecule part 1

NORMAL COORDINATE ANALYSIS OF XY2 BENT MOLECULE – PART 1

Dr.D.UTHRA

Head, Dept.of Physics

DG Vaishnav College, Chennai-106


Normal coordinate analysis of xy 2 bent molecule part 1

This presentation has been designed to serve as a self- study

material for Postgraduate Physics students pursuing their

programme under Indian Universities, especially University of

Madras and its affiliated colleges. If this aids the teachers too

who deal this subject, to make their lectures more interesting,

the purpose is achieved. -D.Uthra


Normal coordinate analysis of xy 2 bent molecule part 1

I acknowledge my sincere gratitude to my teacher

Dr.S.Gunasekaran, for teaching me group theory with

so much dedication and patience & for inspiring me

and many of my friends to pursue research.

My acknowledgement to all my students who

inspired me to design this presentation.

- D.Uthra


Steps

Steps…

  • Assign internal coordinates of the molecule

  • Assign unit vectors and find their components along the three cartesian coordinates

  • Obtain the orthonormalised SALCs

  • Use the SALCs and obtain

    • U - matrix

    • S - matrix

    • B - matrix

    • G - matrix

  • Apply Wilson’s FG matrix method


The matrices

The matrices

U - matrix has the form Ujk

S - matrix has the form Skt

B - matrix has the form ∑k Ujk Skt

j - order of the symmetry coordinate

k- internal coordinate

t- atom


In this presentation

In this presentation…

Learn to form

U-matrix and S-matrix

for an XY2 bent molecule


For xy 2 bent molecule

For XY2 bent molecule

Orthonormalised SALCs are

S1=(1/√2)[d1+d2]

S2 = α

S3=(1/√2)[d1-d2]

Internal coordinates are

1- d1

2- d2

3- α

Atoms are assigned as

1 - Y1

2 - Y2

3 - X

Note : Order of assigning atoms and internal coordinates is according to the user and it can

vary between person to person. But remember to follow that order and must not

change it through out the analysis.


To assign unit vectors

To assign unit vectors…

  • Unit vectors are assigned along every bond of the molecule

  • They have unit magnitude

  • Consider (by convention) they are positive if they point towards the atom and negative if they point away from the atom under consideration


For xy 2 bent molecule1

For XY2 bent molecule

  • There are two unit vectors v1 and v2 along the two bonds d1 and d2 respectively

  • They point towards the end atoms Y1 andY2 respectively

  • We assume that the molecule is lying in XY plane and Z axis is normal to plane containing the molecule

  • Y- axis bisects the angle α between the bonds, so that α/2=θ

    Note : Recap your knowledge in trigonometry and then only proceed!!!


Normal coordinate analysis of xy 2 bent molecule part 1

Z

This matrix has

  • 3 col equal to 3 cartesian coordinates

  • Rows equal to number of unit vectors

    Magnitude of v1 and v2 is one

Components of unit vectors

X

v1

v2

Y


U matrix

U-matrix

U-matrix is formed with the help of symmetry coordinates

This matrix has

  • Columns equal to number of internal coordinates

  • Rows equal to number of symmetry coordinates

  • Entry Ujk of U matrix implies coefficient of kth internal coordinate of jth symmetry coordinate of the molecule


U matrix for xy 2 bent molecule

U matrix for XY2 bent molecule

Number of Rows = SALCs

S1=(1/√2)[d1+d2]

S2 = α

S3=(1/√2)[d1-d2]

Number of columns =Internal coordinates

1- d1

2- d2

3- α


S matrix

S-matrix

S-matrix matrix has

  • Columns equal to number of atoms

  • Rows equal to number of internal coordinates

    Entry Skt of S matrix indicates the unit vector

    associated with the vibration involving

  • tthatom of the molecule and

  • kth internal coordinate of the molecule

    Use symmetry coordinates to form S-matrix


For xy 2 bent molecule2

For XY2 bent molecule

No. of columns = No. of Atoms =3

1 - Y1

2 - Y2

3 - X

No. of rows = No. of Internal coordinates =3

1- d1

2- d2

3- α


Normal coordinate analysis of xy 2 bent molecule part 1

  • Entry of Skt matrix indicates the vector that is involved the change in kth internal coordinate, corresponding to tth atom

  • Rules to form Skt matrix are clearly described by Wilson, Decius and Cross


Normal coordinate analysis of xy 2 bent molecule part 1

How to write Skt matrix entries for stretching

vibrations?

  • It should be noted, during any stretching of any bond, two atoms are involved.

  • The atom towards which the unit vector representing the bond points at is called as end atom, while the other atom from which vector starts is called apex atom.

  • When the symmetry coordinate represent stretching, then entry in Skt matrix for the atoms involved in that vibration is equal to the unit vector representing the bond that is involved in that stretching.

  • By convention, unit vector for the end atom (involved in stretching) in Skt matrix takes +sign and unit vector for the other atom involved takes –ve sign.


Normal coordinate analysis of xy 2 bent molecule part 1

  • For an XY2 bent atom,

    • when d1 changes,

      atom Y1 is involved - vector v1 is +ve as v1 points towards Y1, ie Y1 is end atom

      atom Y2 is not involved – so no vector is involved w.r.t Y2

      atom X is involved –vector v1 is –ve as v1 points away from X, ie X is apex atom

      • when d2 changes

        atom Y1 is not involved – so no vector is involved w.r.t Y1

        atom Y2 is involved - vector v2 is +ve as v2 points towards Y2, ie Y2 is end atom

        atom X is involved –vector v2 is –ve as v2 points away from X, ie X is apex atom


Normal coordinate analysis of xy 2 bent molecule part 1

  • How to write Skt matrix entries for bending vibrations?

    • In a bending, 3 atoms – two end atoms and one apex atom are involved

    • Also, the angle between two bonds d1 and d2 are involved (in this case, d1=d2 =d )

    • Hence, two vectors are involved

    • For the end atom towards which v1 points (here, Y1), use the expression (v1cosα – v2)/ (d1d2)½ sinα(here, d1=d2 =d and so, (d1d2)½ = d.

    • For the end atom towards which v2 points (here, Y2), use the expression (v2cosα – v1)/ (d1d2)½ sinα

    • For the apex atom (here, X), sum up the expression for end atoms and prefix it with –ve sign, ie.,

      -[(v1cosα – v2)+ (v2cosα – v1)]/ (d1d2)½ sinα


Now find the components of s kt matrix entries along the three cartesian coordinates

Now find the components of Skt matrix entries along the three cartesian coordinates

  • Now your S matrix contains

    • No.of rows = no.of internal coordinates, in this case, 3

    • No.of columns = 3x no.of atoms= 3x3=9, in this case

  • Use the table containing entries of components of unit vectors.


How to proceed

How to proceed ?

  • From the table containing entries of components of unit vectors, note the components of vectors and in Skt matrix in respective positions

  • X component of v1 is –s (Sxd1Y1 = -s, SYd1Y1 = -c, Sxd2Y1 = 0, Sxd2Y2 = s, SYd2Y2 = -c ) and so on

  • Now the entry corresponding to α for atom Y1

    • SXαY1= (v1cosα – v2) / dsinα

      = (-s cosα –s) /d sinα = -s(cosα + 1) /d sinα

      = -s[2cos2(α/2) -1+1] /[2d sin(α/2) cos(α/2)]

      = -c /d [as cos(α/2)=c and sin(α/2) =s ] SXαY1= (v1cosα – v2) / dsinα

    • SYαY1= (-s cosα –s) /d sinα = -s(cosα + 1) /d sinα

      = -s[2cos2(α/2) -1+1] /[2d sin(α/2) cos(α/2)]

      = -c /d [as cos(α/2)=c and sin(α/2) =s ]


Normal coordinate analysis of xy 2 bent molecule part 1

  • Similarly, find x and y components of Skt matrix for atom Y2

  • For atom X, sum up the entries of end atoms and prefix with –ve sign


S kt matrix

Skt matrix


Normal coordinate analysis of xy 2 bent molecule part 1

In this presentation, you have learnt to form

U matrix and S matrix for a bent XY2

molecule.

C U in the next presentation to learn to

form B-matrix

-uthramam


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