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 -')