NORMAL COORDINATE NORMAL COORDINATE ANALYSIS OF XYANALYSIS OF XY22 BENT BENT

MOLECULE – MOLECULE – PART 1PART 1

Dr.D.UTHRADr.D.UTHRA

Head, Dept.of PhysicsHead, Dept.of Physics

DG Vaishnav College, Chennai-106DG Vaishnav College, Chennai-106

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

material for Postgraduate Physics students pursuing material for Postgraduate Physics students pursuing their their

programme under Indian Universities, especially programme under Indian Universities, especially University of University of

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

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

the purpose is achieved. the purpose is achieved. -D.Uthra-D.Uthra

I acknowledge my sincere gratitude to my teacherI acknowledge my sincere gratitude to my teacher

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

so much dedication and patience & for inspiring meso much dedication and patience & for inspiring me

and many of my friends to pursue research. and many of my friends to pursue research.

My acknowledgement to all my students whoMy acknowledgement to all my students who

inspired me to design this presentation.inspired me to design this presentation.

- D.Uthra- D.Uthra

Steps…Steps…

► Assign internal coordinates of the moleculeAssign internal coordinates of the molecule► Assign unit vectors and find their components Assign unit vectors and find their components

along the three cartesian coordinatesalong the three cartesian coordinates► Obtain the orthonormalised SALCs Obtain the orthonormalised SALCs ► Use the SALCs and obtainUse the SALCs and obtain

U - matrixU - matrix S - matrixS - matrix B - matrixB - matrix G - matrixG - matrix

► Apply Wilson’s FG matrix methodApply Wilson’s FG matrix method

The matricesThe matrices

U - matrix has the form UU - matrix has the form Ujkjk

S - matrix has the form SS - matrix has the form Sktkt

B - matrix has the form ∑B - matrix has the form ∑kk U Ujkjk S Sktkt

j - order of the symmetry coordinatej - order of the symmetry coordinate

k- internal coordinatek- internal coordinate

t- atomt- atom

In this presentation…In this presentation…

Learn to form Learn to form

U-matrix and S-matrixU-matrix and S-matrix

for an XYfor an XY22 bent molecule bent molecule

For XYFor XY22 bent molecule bent moleculeOrthonormalised SALCs areOrthonormalised SALCs are

SS11=(1/=(1/√2)[√2)[dd11++ dd22]]SS2 2 = = ααSS33=(1/=(1/√2)[√2)[dd11-- dd22]]

Internal coordinates areInternal coordinates are1- d1- d11

2- d2- d22

3- 3- ααAtoms are assigned as Atoms are assigned as

1 - Y1 - Y11

2 - Y2 - Y22

3 - X3 - X

Note : Order of assigning atoms and internal coordinates is according to the user and it canNote : 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 vary between person to person. But remember to follow that order and must not change it through out the analysis.change it through out the analysis.

To assign unit vectors…To assign unit vectors…

►Unit vectors are assigned along every Unit vectors are assigned along every bond of the moleculebond of the molecule

►They have unit magnitudeThey have unit magnitude►Consider (by convention) they are Consider (by convention) they are

positive if they point towards the atom positive if they point towards the atom and negative if they point away from and negative if they point away from the atom under considerationthe atom under consideration

For XYFor XY22 bent molecule bent molecule►There are two unit vectors vThere are two unit vectors v11 and v and v22

along the two bonds dalong the two bonds d11 and d and d22 respectivelyrespectively

►They point towards the end atoms YThey point towards the end atoms Y1 1 andand

YY2 2 respectivelyrespectively

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

►Y- axis bisects the angle Y- axis bisects the angle αα between the between the bonds, so that bonds, so that αα/2=/2=θθNote : Recap your knowledge in trigonometry and then Note : Recap your knowledge in trigonometry and then only proceed!!!only proceed!!!

Z

XThis matrix has This matrix has

► 3 col equal to 3 cartesian 3 col equal to 3 cartesian coordinatescoordinates

► Rows equal to number of unit Rows equal to number of unit vectorsvectors

Magnitude of vMagnitude of v11 and v and v22 is one is one

X X componencomponen

tt

YYcomponencomponen

tt

Z Z componencomponen

tt

VV11 -v-v11 sin sin θθ = =

- - vv11 s = s = -s-s-v-v11 cos cos θθ = =

= -= -vv11 c = c = --cc

00

VV22 vv22sin sin θθ = =

vv22 s = s = s s

-v-v22 cos cos θθ = =

- - vv22 c = c = -c-c00

Components of unit Components of unit vectorsvectors

Y

v1 v2

U-matrixU-matrix

U-matrix is formed with the help of U-matrix is formed with the help of symmetry coordinatessymmetry coordinates

This matrix has This matrix has ►Columns equal to number of internal Columns equal to number of internal

coordinates coordinates ►Rows equal to number of symmetry Rows equal to number of symmetry

coordinatescoordinates►Entry UEntry Ujkjk of U matrix implies coefficient of of U matrix implies coefficient of

kkthth internal coordinate of j internal coordinate of jthth symmetry symmetry coordinate of the moleculecoordinate of the molecule

U matrix for XYU matrix for XY22 bent bent moleculemolecule

Number of Rows = Number of Rows = SALCsSALCsSS11=(1/=(1/√2)[√2)[dd11++ dd22]]SS2 2 = = ααSS33=(1/=(1/√2)[√2)[dd11-- dd22]]

Number of columns Number of columns =Internal coordinates=Internal coordinates1- d1- d11

2- d2- d22

3- 3- αα

d1 d2 α

S1 1/1/√2√2 1/1/√2√2 00

S2 00 00 11

S3 1/1/√2√2 -1/-1/√2√2 00

S-matrixS-matrix

S-matrix matrix has S-matrix matrix has ► Columns equal to number of atomsColumns equal to number of atoms► Rows equal to number of internal coordinatesRows equal to number of internal coordinates

Entry SEntry Sktkt of S matrix indicates the unit vector of S matrix indicates the unit vector associated with the vibration involving associated with the vibration involving

►ttth th atom of the molecule andatom of the molecule and►kkthth internal coordinate of the molecule internal coordinate of the molecule

Use symmetry coordinates to form S-matrixUse symmetry coordinates to form S-matrix

For XYFor XY22 bent molecule bent molecule

No. of columns = No. of Atoms =3No. of columns = No. of Atoms =31 - Y1 - Y11

2 - Y2 - Y22

3 - X3 - XNo. of rows = No. of Internal coordinates =3No. of rows = No. of Internal coordinates =31- d1- d11

2- d2- d22

3- 3- αα

Y1 Y2 X

d1

d2

α

►Entry of SEntry of Sktkt matrix indicates the vector matrix indicates the vector that is involved the change in kthat is involved the change in kthth internal coordinate, corresponding to internal coordinate, corresponding to ttthth atom atom

►Rules to form SRules to form Sktkt matrix are clearly matrix are clearly described by Wilson, Decius and Crossdescribed by Wilson, Decius and Cross

How to write SHow to write Sktkt matrix entries for stretching matrix entries for stretching

vibrations?vibrations? It should be noted, during any stretching of any bond, It should be noted, during any stretching of any bond,

two atoms are involved. two atoms are involved. The atom towards which the unit vector representing The atom towards which the unit vector representing

the bond points at is called as end atom, while the the bond points at is called as end atom, while the other atom from which vector starts is called apex other atom from which vector starts is called apex atom.atom.

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

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

► For an XYFor an XY22 bent atom, bent atom, when dwhen d11 changes, changes,

atom Yatom Y1 1 is involved - vector is involved - vector vv11 is +ve as is +ve as vv11 points towards Ypoints towards Y11, ie Y, ie Y11 is end atom is end atom

atom Yatom Y2 2 is not involved – so is not involved – so nono vector is vector is involved w.r.t Yinvolved w.r.t Y22

atom X is involved –vector vatom X is involved –vector v11 is –ve as is –ve as vv1 1 points points away from X, ie X is apex atom away from X, ie X is apex atom

when dwhen d22 changes changes

atom Yatom Y1 1 is not involved – so is not involved – so nono vector is vector is involved w.r.t Yinvolved w.r.t Y1 1

atom Yatom Y2 2 is involved - vector is involved - vector vv22 is +ve as is +ve as vv22 points towards Ypoints towards Y22, ie Y, ie Y22 is end atom is end atom

atom X is involved –vector vatom X is involved –vector v22 is –ve as is –ve as vv2 2 points points away from X, ie X is apex atom away from X, ie X is apex atom

Y1 Y2 X

d1 v1 0 -v1

d2 0 v2 -v2

α

►How to write SHow to write Sktkt matrix entries for matrix entries for bending vibrations?bending vibrations? In a bending, 3 atoms – two end atoms and one apex In a bending, 3 atoms – two end atoms and one apex

atom are involvedatom are involved Also, the angle between two bonds dAlso, the angle between two bonds d1 1 and dand d2 2 are are

involved (in this case, dinvolved (in this case, d11=d=d2 2 =d )=d ) Hence, two vectors are involvedHence, two vectors are involved For the end atom towards which vFor the end atom towards which v11 points (here, Y points (here, Y11), ),

use the expression use the expression (v(v11coscosαα – v – v22)/ (d)/ (d11dd22))½ ½ sinsinαα (here, (here, dd11=d=d2 2 =d and so=d and so, , (d(d11dd22))½ ½ = d.= d.

For the end atom towards which vFor the end atom towards which v22 points (here, Y points (here, Y22), ), use the expression use the expression (v(v22coscosαα – v – v11)/ (d)/ (d11dd22))½ ½ sinsinαα

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

-[(v-[(v11coscosαα – v – v22)+ (v)+ (v22coscosαα – v – v11)]/ (d)]/ (d11dd22))½ ½ sinsinαα

Y1 Y2 X

d1 v1 0 -v1

d2 0 v2 -v2

α

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

Now find the components of SNow find the components of Sktkt matrix matrix entries along the three cartesian entries along the three cartesian

coordinatescoordinates►Now your S matrix contains Now your S matrix contains

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

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

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

X X componencomponen

tt

YYcomponencomponen

tt

Z Z componencomponen

tt

VV11 -v-v11 sin sin θθ = =

- - vv11 s = s = -s-s-v-v11 cos cos θθ = =

= -= -vv11 c = c = --cc

00

VV22 vv22sin sin θθ = =

vv22 s = s = s s

-v-v22 cos cos θθ = =

- - vv22 c = c = -c-c00

How to proceed ?How to proceed ?► From the table containing entries of components of unit From the table containing entries of components of unit

vectors, note the components of vectors and in Svectors, note the components of vectors and in Sktkt matrix in matrix in respective positionsrespective positions

► X component of vX component of v11 is is –s–s (S (Sxx

dd11YY11 = -s, S = -s, SYY

dd11YY11 = -c, S = -c, Sxx

dd22YY11 = 0, S = 0, Sxx

dd22YY22 = =

s, Ss, SYY

dd22YY22 = -c ) and so on = -c ) and so on

► Now the entry corresponding to Now the entry corresponding to αα for atom Y for atom Y11

SSXXααYY

11 = = (v(v11coscosαα – v – v22) / d) / d sinsinαα

= (-s cos= (-s cosαα –s) /d sin –s) /d sinαα = -s(cos = -s(cosαα + 1) /d sin + 1) /d sinαα

= -s[2cos= -s[2cos22((αα/2) -1+1] /[2d sin(/2) -1+1] /[2d sin(αα/2) cos(/2) cos(αα/2)] /2)]

= -c /d [as cos(= -c /d [as cos(αα/2)=c and sin(/2)=c and sin(αα/2) =s ]/2) =s ] S SXXααYY

11 = = (v(v11coscosαα – v – v22) / d) / d sinsinαα

SSYYααYY

11 = (-s cos= (-s cosαα –s) /d sin –s) /d sinαα = -s(cos = -s(cosαα + 1) /d sin + 1) /d sinαα

= -s[2cos= -s[2cos22((αα/2) -1+1] /[2d sin(/2) -1+1] /[2d sin(αα/2) cos(/2) cos(αα/2)] /2)]

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

►Similarly, find x and y components of Similarly, find x and y components of SSktkt matrix for matrix for atom Yatom Y22

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

SSktkt matrix matrix

Y1 Y2 X

X Y Z X Y Z X Y Z

d1

-s -c 0 0 0 0 +s +c 0

d2

0 0 0 +s -c 0 -s +c 0

α-c/d s/d 0 c/d s/d 0 0 2s/d 0

In this presentation, you have learnt In this presentation, you have learnt to form to form

U matrix and S matrix for a bent XYU matrix and S matrix for a bent XY22

molecule. molecule.

C U in the next presentation to learn C U in the next presentation to learn toto

form B-matrixform B-matrix

-uthra mam-uthra mam