I%ENI)IN(; ENERGY h l l N l h l l Z A T l O N CRITERION APPI.IED T O XY,

P\ It,\I\lIDAI. SYSI'EMS FOR PREDICWNG I'IIEIR (;EORlEl'RY'

~ ~ ~

Abstract

1 'She nc\\ criteriori developed in the previous chapter to estimate tllc

/ geornetry of s~rr~ple molecules like ,\1Y2 bent symmetric system is now used to

predict the gcti~r~clry oTS); pyramidal systems .A study of tllc variation or the

various conllibuhoris to the vibrational potential energy wit11 interbond angle in

XYl pyram~dal s\ s tcrn cot~firms the ohservatlon prev~ously made fbr X'hbcnt

sy~nmctric syslern, that tllc actual equilibrium configuration lies in the prcmiscs

of il~initnu~n I,;,, . i ; . ' I 11c criterion is used lo predict the interbond angle in .\'I>

pyramidal systeti t

$3.1 Irttrotluctio~l

The elucidation of the structure of the XY, type molecules are of great

interest in the study of molecular dynamics . The vibrational potential energy of

thcsc molcculcs arc dctcrrnined ,usirig a mathematical formalism which has hccn

developed rccentlv(47 I . I h e new criterion applied to the S Y 2 bent syrnrnctric

systems sho\v 11i;it the hending energy exhibits a minimurn value which

comespoiids to tile ecluilibrium geometry of the molecule . The various

coi~tributions to the F'otential energy V versus inter bond atlglc in XY2 bcnt

syminetric systerns seem to suggest that the actual equilibrium conliguration

corresporids to ri~inirnum for V m,, . The extension of this analysis to XY,

pyramidal systerris is discussed in this chapter. llere the potential energy

contributions conle from pure stretch, various interactions and the bending

enerby.

$3.2 Symmetr? consideration

Group theoretical analysis show the following symmetries in the case of

X& pyramidal <!stems ..l'hc synmetry elements are [4]

I . C '- 3 lbld rotati011 axis passing through the X atom lying at the apex

and peq~endicul:~~ to the plane containing he Y-Y-Yatoms

2 3 r i ; :,I plane of reflection passing through the X atorn and having

one Y atoln I ticrc arc tliree such planes

3 b,-ldcnt~tv operation

h4olecules with this type of symmetry are said to belong to the ( ' 3 , ,

point group I he vibrational representation is

It has rwo / I l species of vibration and a doubly degenerate 1; species of

vibration. 7 tie symmetry co ordinates for the two species are

E species

S;,, 6 "1'2 6 . 1 -&= )

,Sd,, 6i2r(26a.32-~Sa~3i-8a,J

,Y > 2 - I 2/6r2 - 61.~)

, ~ 1 2 s , r / 5 uil -6u13 ( 3.7)

53.3 hlatliertlatical forn~alisn~

'I'lle p~tential energy function in inten~al valence co ordinates is given as

.]'he polcnt~al energy expression for the XY3 pyramidal system is

The average rwtential enerbT is given by

( 2 V ) r. z fq (6 r ,6 r , )

'/

Also the rneall square amplitude a, -(fir, 65)

Thus , the average potential energy is

(2V) 2/,n;~-7/Uu~~6/~u~~6fU~Uu~~2fr,~,u~Cijru~Gu~ (311)

where or (17 I -)

, - r2 (6 cx, ')

The rnenrl square amplitudes are evaluated at absolute zero

The nberagc potential energy in terms of the internal symmetry co

The force field elements P q and Z, elements in symmetry co ord~nates are

related to the ~ritemal valence co ord~nates f, and cr,, respect~vely as

I- - 2 ~ r o ! 4,;

I ' -, :~, rIr - CT, a

1 hus tlie aicragc beriding energy is represented as

Sin~ilar expressions for tlie other contributions to potential energy naamcly

, I/,-pure stretch energy, I,',., the stretch -stretch interaction energy, I/; -tlw

bending interactiori, /:,and V, ,I' the stretch -bent interaction terms can also he

written. 'lhew help in the evaluation of the various contributions to the potential

energy. A coniputer prograrn has been developed for the purpose .The method of

evaluatiori is silnilar to tlie one fbllowed for theXY2 bent synrnetric system . The

1, matrix ant1 the 1,'and Z elements are evaluated which is made possible through

the solution ol.the quadratic equation in the parameter c. Also the c values arc

solved for both the~1,species and the /:'species. As in the,YY2 system here also the

suitable value oTc which is acceptable for the particular set up is used for

evaluation of the matrix elements

'The (; elerner~ts reilu~red for the evaluation of the I , matrix are given by thc

relat~ot~s ( 1 1

(; (,,f ) ( 4 ~ I . Y . ( a 2) -1) p 1. p ( 3.26)

; , 2 1 1 ) -2(4cos.2(a2)-I)p,r/ana (3 .27)

2 ; ) ( 4 -set. a)(Jp,si,z. a 1 ,uy) (3.28)

2 ' 1 . 2p,sm. a1 py ( 3.29)

2 ( I 2 a t u n a ( 3.30)

2 2 ( , I ) 2 I . a t u t ~ a + ( I t 'i (sec a)) A,,, (3.31)

where tx IS the Inter bond angle , p, and fi. are the reciprocal of the masses

of the X an0 the Y atoms respectively.

Fortunately there arc few S Y 3 pyramidal molecules in the literature [48 -51 ] for

which the inter bond angles are uniquely fixed and the frequencies before and

after isotoplc substitution of the atoms are exactly determined. Moreover these

happened to he hydrides for which the isotopic frequency shiAs are

con~paratively large. In the evaluation of the potential energy contributions the

vibratior~al frequencies are assumed for the molecules. These data are presented

in thc Tahle I l l . 1 . As explained in Chapter 2 the 1; and Z elements are to be

evaluated (br both the species. Proceeding in a similar manner the solution of a

cluadrat~c cq~~~l t lon so formed for each specles would make it psslblc to evaluate

1, and ' elcrnents I he parameter c is to be determined for both the spccles

To begin wit11 an arbitrary value is given for the interbond angle and the encrby

evaluated '1 lie variation in the poter~tial energy contribution with interbond angle

is studied ' I hc variations are represented in graphs.-the ene rg versus interbond

angle plots

53.4 llesults and tliscussions

'She variation of the vibrational potential energy contributions with inter

bond angles In ,YY3 pyramidal molecules are shown in the various plots. Fig 3a

shows the variation of the bending energy contribution with interbond angle for

ShHi molecule .'lhis shows a well defined minimum for the bending energy

contributioll This corresponds to the actual geometry. The experiniental valuc

li,r the inter borid ;uigle is 9 1 . 5 "The present study shows the valuc as 91" A

fairly good agreement Fig 3b shows the variation of the purc stretch

interaction enerby w~th interbond angle. This shows an extremum ,but sliglitly

away from tile actual geometry. Fig 3c represent the interaction energy oof the

various otlicr types namely stretch-stretch, bent-bent, and those between the

different spccics

I lle Fig 3d shows the nature of variation of I,' ,,,, with interhond

angle l i~r lllc iVI / , n~olcculc :l'his also gives a singlc minimum for the bcndilig

energy which corresponds to an interbond angle closely agreeing with actual

geometry o f the molecule. In Fig 3e the pure stretch enerby is plotted against the

interhond angle I'ig 3f shows the other contributions and their variations with

interbond angle. Fig 3g , 3h,3i are the similar plots for the AsH3 molecule. Fig

3j,3k,31 are the plots for the I ' t l , molecule. This gives a single minimum for the

bending enerby versus interbond angle graph. The interbond angle so obtained

92.5"agrees well with the experimental vaa1ue92~l'hus it is seen that all these

curves s h o ~ a general trend ,that is the minimisation of the bending energy

contrihutiorl .and extrernum value for the interaction enerby.

'1 hus the new criterion applied to the XY, pyramidal system very well

predict the geometry of the molecule o r in other words the bending encrby

minimisation criterion can be used as a tool for the study of the geometry of

simple molecules. The data used for the evaluation are given in Table 111 1 . The

results are ~~rcserited in Tables 111.2 and 111-3.

Table 111 1 1)ulu iuhlc JorXY3 I'yrumrdal lype molec111e.r

~ ~

Molecule cm I

~

A l S=cs

~ --

NtJ3 3504 NDz 2496 ~-

1'11s - - ~ 2448

PD3 .. ~. 1763.8 Sbli3 --- 1986.0 SbD, 1412 AsH3 . . 2204.0 . -

A s h 1583.7 E species NI 13 ~ -~ 1592 ND3 ~ 2643.2 PHs 2457 1'11~ 1763.3 sb1r3 - 1976 SbD3 1403.5 ASH? .~ 2225 AS& .-- 1583.7

.I'ahlc l i l 2: Inter tior~d angle determined from Energy ccor~sideratioris

J

MOI,I:CIJL.IL,

Sbl I, SbD? -~ ~

Asfl? AsD,

- ~-

PH, PD? .. ~

NI I? ND3

MOLt:.CUI.I<

SbH,

SbD3 -~ ~

INII:I2 BOND ANGLE RASI111 ON

V, r i i i r ~

9 1.00"

92.0"

.~

92.50"

107.5"

EXF:I'I~IIIMI:N'I'AL,. VALIJI:

liEF

IN-lER BOND ANGLE BAS1111 ON

v ..+o

92'

EXEPEKIMENI'AI~. VALUE

REF

91.5"

92 "

93.50'

107"

ASH? I 88"

91.5"

92

51

5 1

5 1

51

51

.-

5 1

XY3.Pyramidal I'ype Molecule 4 z.

Figure 3.a shows the variation of Bending energy with

lnter bond angle for ShH3

X -axis lnter bond angle(a in degees)

Y-axis Bending energy (V,, in cm-I)

Figure 3.b shows the variation of Stretch energy with

Inter bond angle for ShU3

X -axis lnter bond angle(a in degrees)

Y-ax~s Stretch energy (V, in cm - ')

Figure 3 c shows the variation of interaction energy with

Inter bond angle for SbH,

X -axis lnter bond angle(a in degrees)

Y-axis Bending energy (V,,V, in cm-')

Figure 3 d shows the variation of Bending energy with

Inter bond angle for NH3

X -axis Inter bond angle(cc in degrees)

Y-axis Bending energy (V, in cm-')

Figure 3 e shows the variation of Stretch energy with

Inter bond angle for NH3

X -axis lnter bond angle(a in degrees)

Y-axis Stretch energy (V, in cm-I)

Figure 3 i'shows the variation of Interaction energy with

lnter bond angle for NH,

X -axis lnter bond angle(a in degrees)

Y-axls Interaction energy (V, & V, in c m ' )

Figure 3.g shows the variation of Bending energy with

Inter bond angle forAsH3

X -axis Inter bond mgle(u in degees)

Y-axis Bending energy (V, in cm -')

Figure 3 h shows the variation of Stretch energy with

lnter bond angle for AsH3

X -axis lnter bond angle(a in degrees)

Y-axis Stretch energy (V, in cm-')

Figure 3.i shows the variation of Interaction enera with

lnter bond angle for AsH3

X -axis lnter bond angle(a in degrees)

Y-axis lnteraction energy (V,, in cm -I)

Figure 3.3 shows the variation Bending enerky with

Inter bond angle for pH3

X -axis Inter bond angle(a in degrees)

Y-axis Bending (V,, in crn -I)

I-~gure 3 k shows the vanation of stretch energy with

Inter bond angle for PH3

X -axis Inter bond angle(a in degrees)

Y--axis stretch energy (V, in cm -')

Figure 3 1 shows the variation of Interaction energy with

lnter bond angle for pH3

X -axis lnter bond angleta in degrees)

Y-axis Interaction enerby (V, in cm -')