Chapter 6

MEAN AMPLITUDES OF VIBRATION AS A 1'001, FOR

STRIJCTIJRAI, ANALYSIS OF SIMPLE MOLECULES

Abstract

The vibrational amplitiude of the bonded X-Y atom pairs in the case of

XY2 bent symmetric systems :, XY2 linear symmetricand pyramidal XY3 systems

have been estimated in the present work. This vibrational amplitude is then

used to analyse the geometry of the molecule.

56.1 Introduction

It is well known that vibrational amplitudes of an atom pair in a

molecule are indicators of the elecbon overlap between atoms and hence the

energy of the system during their vibrations. It has been already established that

the bending energy will 1~ a minimum during their vibrations at their actual

molecular geometry. 'rh'us ,a variation of geometry searching for possible

minima of mean amplitude of vibration is found to yield interesting results. The

experimental data on mean amplitudes of vibration from electron dimaction

studies can provide a helping hand in this context.

s6.2 Mean Amplitudes (of Vibration

The rriean amplitutles of vibration [, for any atom pair i j of a molecule

is defined as 1 lo]

I ,, ~ ! ( ( r , - 7 ~ I)*) (6.1)

r, and r , ' refer to the instantaneous and equilibrium inter nuclear distance

bclwccn atoms I and,/ respectively. Cyvin has developed a detailed formalism

for the evaluation of thefse quantities from a knowledge of vibrational

frequencies and geometry of the molecule. Cyvin's formalism mvolvcs the use

of symmetq co-ordinates and employs the basic equation for the symmetrised

mean square alnpl~tude matrix Zas

Here , 1, refers to the normal coordinate transformation matrix and A is

a diagonal matrix related to the vibrational frequencies vi obtained from

spectroscopic data as

The I. matrix is of basic importance in the present context and it is

related to the symmetrised intramolecular force tield Fthrough the relation

A being the diagonal matrix. with the elements Ai - 4 r r 2 0 i 2 c ~ . The L matix

obeys a further baqic relation I, I, ' (; where (; is the well known inverse

kinetic energy matrix calculable merely from the geometry and atomic masses

of the molecule [6] 'l'he mean amplitudes I , are dxectly related to the elements

of C matrix and can be calc~ulated

56.3 Mathenlatical Formalism

In the present approach we assibm a certain geometry for the molecule

and calculate the inverse kinetic enerby matrix G using Wilson's recipe.[6] The

parameter fonnalisrn described in chapter 1 enables us to evaluate a lower

triangular matrix I,,, satisfy:ing Wilson condition [6] Lo Lo ' (; so that the

actual normal co-ordinate transformation matrix L becomes

where the general fbrm of orthogonal matrix (' has been described already

Equation 1 1 Y in Chapter I

Using the principle of invariance of the force field under isotopic

substitution of the atoms along with the parametric approach the 1,' and Z

elements can be calculateti .This fwther requires a solution of the quadratic

equation,

p, q, and r are given by the Equat~ons [2.34-371. The mean amplitude I, can be

now he evaluated from the Z elements . The procedure is continued by

uniformly altering the geometry of the molecule . A minimum for the mean

amplitude would imply minimum encrby for vibration and should naturally

correspond to tl~c actual equilibrium geometry ofthe molecule.

$6.4 Application to Bent Symmetric XY2 System.

As alreadq rnent~oned in chapter 2 , XY2 bent symmetric system belongs

to C po~nt group and have vibrational representation

The A , species present!; a vibrational problem of order 2 and hence

would need vibrational liequencies after isotopic substitution of atoms for

unambiguous fising o C I;, I. and Z matrices. The (: mam'x will be 2 x 2 in

structure in the (inn .as in equation [1.19] . The H2species contains only one

element for I,;/, and C.and hence i t i s uniquely solvable . This requires the

evaluation of (; matrices as given in the basic equations [2.24- - 2.271.

The relation between the .C elements and the bonded mean amplitudes of

vibration is given by

I , , , (22 , ) (A,) srn2{u 2) 1 2 ' 2 2 1 2 ( ~ ~ . \ ~ n . ( a ) + ~ 2 2 ( ~ ~ cos2(a2)) ' ' (6 9 )

Thus the mean amplitudes of vibration corresponding to the bonded atom

pair S - Y and non bonded atom pair Y... Yare evaluated for various values of

interbond angle. The interbond angle versus I,,, and I , ,, curves can now be

plotted

56.5 Application to X Y , linear Symmetric System

As already explained in chapter 4 the X Y2 linear symmetric system is

assibmed an interbond angle ,so that it can be treated as a bent symmetric

structure (C 71 ) With the same vibrational representation a$ for the XY bent

symmetric structure .the A , and the B2 species are assigned the vibrational

frequencies ,given in the Table IV .l.Proceeding in a similar manner as

explained above C elements are first evaluated and hence the bonded mean

amplitudes of vibration I,, and the non-bonded value Iy , are evaluated using

the equations6 8 arid 6.9.lhe ('02 and CS2 molecules are analysed in this

system.

1'0 begin with , an arbitrary value is assumed for the interbond angle of

the molecules ihe I .., and the I, ,,terms are evaluated . The interbond angle is

now systemat~cally varied and the bonded and the non-bonded mean

amplitudes are evaluated li'he variation of the I,, and the I,. , with interbond

angle arc now prcsentetl in ;I gaph.

$6.6 Application to ,YY, Pyramidal systems.

As alrcadv mentioned in chapter 3 pyramidal systern belongs to C.?,

pint group and have vibrational representation

The A, species presents a vit~rational problem of order 2 and hence would need

the frequencies after substituting the atoms with their isotopes. The 1- I, and Z

matrices can then be evaluated using the frequencies. As explained in chapter 2

the C malrix will be 2 x :! in structure in the form as given in equation 1.19 The

E species is of' order 2 and i n a similar way I T , L and Z are evaluated from the

(:matrix. The hasic equations for the (; elements necessary for the evaluation

of the I. matrix are given i m equations 3.26- 3.31. The relation between the

mean amplitude I , , and t a r e given below.

I'he nor1 1)onded mean an~plitude I, , is given by

The mean amplitudes of vibration for the X Y 3 pyramidal system for

both bonded and non- bonded atom pairs are evaluated using the above

equations . The molecules .NH 3 , t'H 3, A8H3, SbH3 are subjected for analysis.

$6.7 Results and 1)iscussions

'The molecules ( ' 1 2 0 , CIIU2, NO2, SO2, H20, H3Y are

subjected to analysis based on the approach outlined above The vibrational

frequencies employed in the present analysis are given in Table 11.1. The

calculated values of the mean amplitudes of vibration I,, and the non-bonded

value I , , for various values of the inter bond angles are presented in Table

Vl.1. The variation of the mean amplitudes of vibration in these molecules with

interbond angle are shown in the Figs 6.a to 6.e.The experimental values of

mean amplitudes of vibration reported from Electron dif ict ion studies

wherever available ,are also marked in the figures

It is seen that the bonded mean amplitudes of vibration of these

molecules vary with interbond angle .The curves show that I,,, passes through a

well defined minimurn for all the eases studied . Interestingly enough thc

interbond angles obtained from the minimum of I,, compare well with those

reported fro111 experimental methods and are given in Table V1.2. The I ,. , values obtained from these calculations are also included in the table along

with the values reported earlier The mean amplitudes of vibration for the non-

bonded atorri pair however remains a constant for the molecule.

'The molecules ('02and CS2 are subjected to analysis using the

above method . A mlnim~um for the mean amplitude would imply minimum

energy for vibration and should naturally correspond to the actual equilibrium

geometry of the molecule . The vibrational frequencies used for the analysis are

given in Table IV.1. The rnean amplitude of vibrations for the bonded and the

non-bonded atom pairs are now evaluated using the equations already

rnentiorled in 44 of this chapter. The calculated values of mean amplitudes of

vibrations for various values of interbond angle are presented in the table V1.3.

The variation of the rnean amplitude for both the molecules with interbond

angle arc shown in Figure 6 f . Both these curves in figure show a minimum

value for I,. , which corresponds to the actual geometry of the molecule. Thus,

the minimum vibrational a:mplitude would imply the minimum for the energy

of the molecule. 'lhe interbond angles obtained from minimum I,, compare

well nit11 the reported value from experiment. The non-bonded mean

amplitude of vibration 1 ,. , obtained from calculations are also included in the

table.

For the .YY~, pyramidal system the variations in mean

amplitudes of vibration with inter bond angles show a similar trend as that for

the other two types analyzed. The variations of I,, with interbond angle gives a

minimum value corresponding to the actual geomeby ofthe system. The values

of i,, are presented in the 'Table.VI.4. along with the experimental value. The

bonded mean amplitude I , ~ , is plotted against the interbond angle for all the

molecules .They pass through a well defined minimum for all the case studied.

The vibrational amplitude corresponding to non-bonded atom pairs I ,. . .

remains the snrnc for each molccule The inter bond angle obtained from

minimum I,~,, compare well with those reported from experimental methods

The figures arc presented for all the molecules studied in Fig6 g. 'The curves

show a well defined minimum for all the molecules and the interbond angle

corresponds to actual geometry.

.It should however tx recognised that the use of mean amplitudes of

vibration is not a very sens~tive method in the structural analysis of simple

molecules , hut the procedure introduces a new constraint namely ,

minimisation of amplitudes to fix the geometry.

TABLE V1 . I Meat1 umplrtudes for xy2 hen! -.synlmefric molecules

-~ -- -..I,, I 3.64 k 0.4

REF: 72

lr ,, for ( '0~ 10")MI

4.967

TABLE VI 4 lrrrer horzd u ,~glefrom I,, rnmrrnurn XY2 bent .syn:nte/rrc

Molecule ---

Inler bond angle $

Fig 6 a shows the variation of bonded mean amplitude with

inter bond angle for C 1 0 2 .

X-axis 'Inter bond angle a (in degrees)

Y-axis IMean Amplitude I ,,(lo "nm)

Fig 6 b shows the variation of bonded mean amplitude with

inter bond angle for NO2.

X-axis Inter bond angle a (in degrees)

Y-axis Mean Amplitude I,(lO"nm)

I-~g 6.c shows the variat~on of bonded mean amplitude with

inter bond angle for SO 2 and ( 'LO2.

X-axis Inter bond angle a (in degrees)

Y-axis Mean Amplitude 1,,(10-~nm)

Fig 6.d shows the variation of bonded mean amplitude with

~ r ~ t e r bond angle for (Ti)( ... )and (:/~Y.(-x-x)

X-axis Inter bond angle a (in degrees)

Y-axis Mean Amplitude l,,(lOJnm)

Ftg 6.e shows the variation of bonded mean amplitude with

inter bond angle for Ha.

X-axis Inter bond angle a (in degrees)

Y-axis Mean Amplitude l.,(lO~'nm)

Fig 6 f shows the variation of bonded mean amplitude with

inter bond angle for CS2 and C02.

X-axis Inter bond angle a (in degrees)

Y-axis Mean Amplitude I ,,(I0 "nm)

Fig 6.g shows the variation of bonded mean amplitude with

inter bond an;gle for SbHl(...) ASH,( + + + )

PH, (...)and NH,(l~r~rl)

X-axis Inter bond angle a (in degrees)

Y-axis Mean Amplitude 1 , ( I 0 "nm)