chapter 6 mean amplitudes of vibration as a 1'001,...
TRANSCRIPT
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 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)