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MOLECULAR DYNAMICS OF KGTHYL + O N I U M IONS AND W N I U M BORANES
7.1 INTRODUCTION
This chapter deals with eevec compounds of the type
AX3YZ3 whole molecular conmtantl have been evaluated
theoretically using general quadratic valance force field.
lhe four isotopic forms (cH~NH~', C H ~ N D ~ ' C D ~ N H ~ ' and
CD~ND;) of methyl amnonlum ion and amnonla-boranas (BH3-NH3,
BD3-ND3 and BH3-ND3) are the compounds under consideration.
The fundamental assignments were made by Waldron ( 1 )
who recorded the lnfrared spectra of CH3NH3*c1- and
C H ~ N D ~ * C I - between 294 K and 77 K . The infrared spectra
of crystrlllne films of the 4-methylamnonlum halides were
measured at temperatures varying from -190' to 22.t by
Cabana and Sandorfy (2). They made lnvestlgations on the
effect of the crystalline environment upon the spectra and
also on the nature of hydrogen bonding In those molecules.
from the spectra of the methylamnonium chloride, Bromlde
and iodide taken at room temperature, they confirmed the
three-fold r y m t r y of methylamnonlum lon which probably
rotates around C-h axis. Later on, Theoret and Sandorfy
( 3 ) made ~ t u d i e s on a set of compounds of type CD~ND~'X-,
CH~ND~'X- and C D ~ N H ~ * X - (X-nalides) by recording lnfrared
and far-infrared spectra. Assignment8 of the infrared
active fundamentalr were made for each of there ions and
potential conrtantr were evaluated by Oxton et a1 (4).
Some limited information concerning the vibrational spectra
of armonia-borane in solution and one of its derivatives,
trlmethylamlne-borane, was reported by Taylor ( 5 ) using
infrared ar well as Raman spectra. Smith et a1 ( 6 )
recorded the infrared spectra of amnonia-borane BH3-NH3
and two of its deuterated irotopic specie#, BD3-ND3 and
BH3-ND3 lsolated ln argon matrix at liquid hydroeen
temperature. Uslng spectral frequencies, potentlal
constants were also calculated for BH3-NH3 and BD3-ND3 by
Smith et a1 (6). Based on the orthonormal set of symnetry
coordinates suggested by Oxton et a1 ( 4 ) , a normal
coordinate analysis has been performed for methylamnonlum
ions and the irotoplc species of amnonla-borane with the ald
of the most general quadratic valence force field and the
results are reported I n the present chapter.
7 .2 THEORETICAL CONSIDERATIONS
The compounds, m e t h y l a w o n l u m ion and awonia-borane
fall under CgV point ~ r o u p having 12 fundamental vrbrationr
dlotrlbuted ln the following manner1
Out of these three species, a 2 is infrared inactive and e is
the doubly degenerate species. Flgure ( 7.1) depicts the
structure and nomenclature of the parameters of this
molecular type.
The orthonormalised set of symnetry coordinates are
given below:
a 1 Species
s l = ( 1 I 4 3 ) ( A r , + A r 2 + A r 3
, 2 5 ( 1 l f i ) ( A d , + A d 2 + A d 3 )
S j = A R
s 4 : ~ [ r ( h a + Aa2 + A a 3 ) - ~ f i ( A B , + A B 2 + A B 3 ) 1
S 5 = Q[d ( A b 1 + A b 2 C h b 3 ) - P&R ( b y 1 + b y 2 + b y 3 ) ]
S B b = (11 f i ) ( A d 3 - A d 2 )
S,, = I ( A - h a 3 )
S l o b = (dl fi) ( A6 - AO 2)
s l l b = (JiTAlfl) ( A? - A@ 3 )
s,,, * ( c ~ l f l ) ( Ayg - by2)
where P = - # c o s g / C O B a12 = - 6 coa y / c o o 612
Ad, Ar and AR are the c h a n g e s in Y - 2 , A - X and A - Y bond
distances r c s p e c t ~ v e l y . A D , AB , A6 and by a r e the changes h A A A
in X A X , X A Y , Z Y 2 a n d 2 Y A interbond angles
respectlvely.
T h e relatlonshap between the s y m e t r i s e d potential
constants and valence potentla1 constants are obtained from
the most general quadratic potentlal energy f unct lons
(conslderlng all rnteractlon). lhey are,
where Y = -PO. fR, fr and f d are the stretching potential
constants whlle f a , f , f and i art the bendlng Y
potentlal constants.
Expressing the symnetry coordlnates i n terms of
cartesran displacement coordlnates, the s vectors are
derlved and these have been used form the B-rnatrlx glven In
lable 7.1.
7.2 .4 G-MATRIX
The G-matrlx elements have been obtained from the
relation,
G - B u i
where 6 I S the transpose of B matrlx and u is a dlagonal
matrix formed with the reciprocal masses of the constituent
atoms. The evaluated ti elements are glven as under:
t 1 Species
- 4 b c p i 2 ) + ( 2 b / & ) ( a c p - v ) ) - u A 06 b ( 3 v - 3 f p )
t i l 5 = u
2 3 = u y S j b
t i L 4 = 0
tiz5 = uz Q ( ( a 1 6 ) ( 2 t - b c ' p ) - ( b 1 6 ) ( v - a c l p )
- ( r l h ) ( p - u + b c 4 p / 2 ) + a ( & q + 6 b l 2
e Specie.
G 7 7 = UX + ( 3 1 2 ) a 2 uA = U, + ( 3 1 2 ) a 2 uY
G q 9 = y y 1 ( 3 1 2 ) u 2 + ( 3 1 4 ) b 2 + s 2 1 + bp2yA
G l O , 1 0 ' u, 1 ( 3 / 2 ) U 2 + 3 1 4 b 2 + a 2 ] + bp2 u y
G 1 l , l l = UX c2 + uA ( 3 e 2 / 2 ) + ~ ~ ( 3 ~ ~ 1 2 )
G 1 2 , 1 z ' u, c d 2 + UA ( 3 i 2 1 2 1 + Uy ( 3 e t 2 / 2 )
'78 = 0
C 7 9 = - y( ( a u 1 2 + 3 a b 1 4 + b s ) + v A ( 3 a p )
' 7 , 1 0 ' O
= - uA ( 3 a e I 2 )
G , , , , = uA ( 3 a g 1 1 2 1
C u 9 = 0
- U, ( a u l 2 + 3 a b 1 4 + b s ) + % ( 3 2 p ) ' 8 , 1 0 - '
C 8 , 1 1 = uy ( 3 a g 1 2 )
C 8 , 1 2 = - Uy ( 3 a e ' / 2 )
c 9 , 1 0 = O
C ~ , l l = - UX ( " b e 1 2 + 3 b 2 c / 4 - a c s ) - U A ( 3 p e )
G ~ , 12 = UA ( 3 ~ 0 ' )
5 0 , 1 1 ' % ( 3 ~ 8 )
G 1 0 , 1 2 ' ' uz ( b c 6 u / 2 + 3 b 2 c 8 1 + - , C I S ) - U y ( 3 p e ' )
G 1 1 , 1 2 = - ( 3 e g 1 / 2 ) - by ( 3 e 1 g / 2 )
a = s i n a = srn @ = sin y = oin 6
b = C o r a = coo @ = c o s y = C06 6
u A , % , 9 and bz are the reciprocal masses of the
respective atoms.
7.2 .5 SECULAR EQUATION
T h e secular equation / FG - & i = 0, h a s been solved
with the h e l p of krnetlc constants (k). The k ~ n e t l c constant
rnatrlx ir related to G as, (k) = ( G ) " . F r o m the k ~ n e t r c
energy expression, the relatlonshrp between the valence
kinetic constants and symnetrized klnetic constants have
been obtained.
k l l = k r + 2krr
k 2 2 = kd + 2kdd
k 3 3 = k~
k q 4 = 3 [ Q 2 ( i n + 2kaa ) + Y' ( k g + 2kss ) + 4QY k a 8 ]
k55 3 IQ' ( k , + 2 k d 6 ) + y 2 ( k y + 2 k y y ) + 4QY k y 6 I
k 1 2 = k r d
k I 3 = 4 kRr
14 = 6 ( 2 Q k r a + Ykr8 )
k 1 5 = Jf ( Q k 1 6 + Yk ) r Y
k 2 3 = JS k ~ d
k 2 4 =J? (Qk + Yk ) d a dB
k7 a krd k? 9 r c k7,10 5 - k r c s -k
k7,11 = rp k7,12 ' -frf k89 = 'fds.
k e . i o '-kd6 k s , l l ' k e , ~ 2 = k d ~
k9.1u ' -ka6 I< 9 - 1 1 = -k *8 k 9 , 1 2 "k4'
klo,ll = -ke6 k 1 0 * 1 2 f'5 kll.12 = - k p r = -k
where Y = -PQ.
'Ihe method of kinetic constants for evaluating the
force constants has been found to glve qulte similar results
in different molecular types (7-10). lhe determlnatlon of
symnetry force constants involved In the secular e q u a t ~ o n
from the vlbratlonal frequencies alone has been a
mathematically underdetermlned problem so far. Therefore
any genulne attempt to evaluate all the symnetty force
constants associated wlth a problem In the order of n 7 1
should lnvolve the ~ n c o r p o r a t i o n of at least n l ( n l - l ) i ~
additional data other than the nl trequencies. lhls methoc
seeks to relate the Off-diagonal elements to the dlagonal
elements of the F-matrix through the relatron
lhua the equations lnvolvlng A' and E species are solved
easily.
7.2.6 YEAN AMPLITUDES OF VIBRATION
lhe mean amplitude of vibration ( T ) at 298.16 K ha\-e
been evaluated from Cyvln's (11) relation,
= LA;
L and A have their usual meanings. From these syrrmetrized
mean Square amplitude constants ( L ) , the valence mean square
amplitude quantities (r) have been obtained. And, from
6 , the mean amplitudes of vibratzon (L) have been
evaluated.
7.2.7 CORIOLlS COUPLlNG CONSTANTS
Corlolis coupling constants occur as a result of the
C o r ~ o l l s forces whlch arise when a molecule is both rotating
and vlbratlng. The Coriolis forces dlrectad at right angles
to the axls of rotation are proportional to the masses of
the particles, thelr apparent velocities wlth rsspect
to a coordinate system rotetlng with the molecule and the
angular veloclty of thls rotating coordinate system wlth
respect to a flxed coordinate systen, ( 1 2 ) .
C o r i o l ~ s coupllng of vlbratlon-rotation effect can
a!!ect the sprctra of linear, symnetrlc and asymnetrlc top
o~olecules, although lts presence may be more pronounced In
sy rmetrlc top where first order effects are possible.
Coriolls c o u p l ~ n g constants ( J values) for the coupllng
between degenerate pairs may be obtalned experimentally by
rneasur~ng the separation betu,een the sublevels cf the
perpcndlcular bands In v i b r a t ~ o n - r o t a t i o n spectra of
rnoleculer. Meal and Polo ( 1 3 ) have developed vector method
f o r the calculation of these constants for both degenerate
and non-degenerate couplings.
According to Jahn's rule (14) two vibrational
atates can couple through a Coraolas interaction, i f the
dlrect product of symnetry species of the two vibrational
states contalns a rotational species. Hence, from the
character table, the Coriolis interaction allowed i n a
molecule may be determined.
The Corlolls coupllng constants have been
evaluated uslng the expression:
5%. (L-l) y-1
Ihe rows and columns of the ~ ~ m a t r l x are labeled according
to the numbering of the symnetry coordinates. In terms of
the H matrlx the C* elements are given by
c ~ = B I* B' + where lN = p I &
P
A knowledge o f even the order of magnitude of these
constants 1s often extremely useful for band assrgnments,
preliminary analysis of rotational structure and predlctlon
of the band shapes.
-1 4 ?'he C o r ~ o l l s coupllng constants $'( = L C L ) In
thls type of molecules arise due to the couplings,
( a 1 x e)' and (e x e ) '
o(, lhe Corlolis rnatrlx elements C , utilrsed ~n the evaluation
of $&are as follows:
( a 1 x e)' coupling:
c X d 8 = 0
c~~~ = u x Q [ ( v - a t p ) JljTi - Ji s (4 q )
+ ( h l 2 ) b - ( h 1 2 ) b c p
- ( a c p - v ) ( h b 1 2 f i ) 1
+ uAQ ( 3 v - 3 f p ) ( 2 6 1 f i ) p
7 . 2 . 8 CENTRIFUGAL DISTORTION CONSTANTS
h h e n t h e h i g h resolution i n f r a r e d a n d microwave
t e c h n i q u e r a r e u s e d t o s t u d y t h e p u r e rotational s p e c t r u m oi
a m o l e c u l e , ~t 1 s s e e n t h a t t h e m o l e c u l a r e n e r g y l e v e l s a r e
n o t predicted e x a c t l y b y r l g l d r o t o r t h e o r y b u t a r e
l n t i u e n c e d b y perturbations s u c h a s t h o s e resulting I r o m
v i b r a t i o n - r o t a t i o n interactions a n d centrifugal d l s t o r t l o n .
I n r e a l i t y t h e m s l e c u l e c a n n o t be r e g a r d e d a s r l g l d a n d b o n d
distances and bond angles will vary due to centrifugal force
produced by rotation. Such centrifugal distortions are
usually large for llght molecules because of their small
moments of inertla. In many cases, it can be treated as
perturbation of the rigid rotor Hamiltonian, the rnfluence
of centrifugal distortion 1s only a small fraction of the
rotational energy.
The general theory with respect to asynrnetrlc rotor
molecules has been formulated by Wllson ( i s ) , Wilson and
Howard ( t 6 ) and Nlelson ( 1 7 ) .
The dlstortlon constants are deflned as
-1 1 J J ' a p v 6 - - ( a l e l e y l e 6 , ' a , S y6+'i] ( 1 ' 2 4 )
where I:x, I* and are the prlnclpal moments of ~ n e r t i a Y Y
at equlllbrlum and J a 6 s represent the partial derivatives
at equilibrium of the instantaneous inertia tensor
components with respect to the symnetry coordinates, N 1s
the lnverse force constant matrix.
Cyvln (48 , I S ) has reformulated the theory definlng a
new quantity t a s a6rd
t = - 2 1 ~ le le le T where oa yy 66 a6y6
t = J'_ N J -
The centrifugal distortion constants are linear combinations
of the ~ k a n d are Civen below:
-(1132) [3lXxxx + 3 1 +Z(T + T ) ] t i 4 . YYYY XXYY XYXY
D~-(1'4) "zzzz - ('xxzz +ZTxzxz ) -(Tyyzt + L T y z y z ) lfi4 -DJ - DK - ( 1 1 4 ) r,,,, f~~
R 5 -(1132) [ T~~~~ - T YYYY -2( rx,,z + 2 ~ x z x z )
+ 2 ( lyyzz * z l y z y z ) l h 4
The centrifugal distortion constants have been
evaluated with the ald of Teg introduced by Cyvin et a1
(16 ) . The T matrix elements are glven ln Table 7.2.
7.3 RESULTS AND DISCUSSION
The vibrational frequencies and molecular parameters of
methylamnonium ions and isotopic compounds of amnonla-borane
(1-6) are presented in Table 7.3. U s ~ n g these
pararaters and observed frequencies as the baslc
quant 1 ties, a11 the molecular constants have been
evaluated and the results are reported from Table 7.4 to
7.7. The molecular k ~ n e t i c consants provide the additional
constrrlnts tor the solutlon of the secular equation. One
expects 36 force constants for this type of molecules from 5
frequencies in A 1 type and six lrequencies in B 1 type. In
this preoen: work 34 vital valence force constants evaluated
f o r these cases usrng kinetic constants are reported in
Table 7.4. The following observations are made from this
lable.
( 1 ) The major potentlal constants fH(A-Y), fr(Y-2.) and
fd(A-X) are in the expected range for all the cases
under conslderatlon here. lhey also agree well wlth
Oxton et a1 ( 4 ) values.
( 2 ) Considering the first two constants fR(A-Y) and
f d ( A - X ) , it is observed that they decrease as the
mass of 'A' atom decreases.
I I ( 3 ) lhe statement, slnce the lsotopic molecules have
the same electronic structure, the potentral function
under the influence of whlch the nuclel are movlng 1s
the same to a very high degree of approximation'' b y
Herzberg ( 2 0 ) is certainly reflected in the unique
sets of values obtained here for both the types of
compounds.
( 4 ) lhe interaction potential constants of a hrghly
delocrllsed bond molecules tends to take a positive
value while the locallred bond molecule takes up
a negative value.
lhe vlbratlonal maan rmplltudes of bonded and non
bonded atomlc dintrnces are grven in lable 7.5. Here it 1s
seen that lR(A-Y), the vlbratlonal mean amplitude oi C-N
bond 1s constant for a11 the four types of methylamnonium
I o n . The same c o n c l u s i o n h o l d s good f o r t h e o t h e r compound
t o o . The non-bonded v i b r t t a o n a l mean a m p l i t u d e s b e a r
h i g h e r v a l u e s t h a n t h e bonded v i b r a t i o n a l mean a m p l i t u d e s .
T a b l e 7 . 6 p r e s e n t s C o r a o l a s c o u p l i n g c o n s t a n t s .
A l l t h e c o n s t a n t s b e a r r e a s o n a b l e v a l u e s . r i s: 3:, and a r e z e r o f o r a l l t h e s e v e n c a s e s u n d e r
5 f 0 , W
c o n s i d e r r t i o n h e r e . l h e h i g h v a l u e s of 1 L.,il 13;,,a a
* I 1 L , I:, , 1 , , 5:. 1 n d i c a t e t h a t t h e
c o u p l i n g s be tween t h e v l b r a t l o n e c o n c e r n e d a r e f i a g n a f l c a n t .
The C o r i o l l s c o u p l a n g c o n s t a n t s obey t h e following sum
r u l e s .
a1 x l c o u p l i n g
e x e c o u p l i n g
u h e r e l A and I , , a r e t h e moment of l n e r t i a a b o u t t h e
symnet ry a x i s and t h e o t h e r a x e s r e s p e c t : v e l y .
The c e n : r l f u g a l d i s t o r t i o n c o n s t a n t s a r i s a n g due t o t h e
n o n - r i g i d i t y of t h e a t o m l c bonds a r e g l v e n an l ' a b l e 7 . 7 .
Among the three evaluated centrifugal distortion constants,
Dk is greater than the other two in all the caues.
The last mentioned three constant have been evaluated
for there two cares for the flrst time here. To the
author's knowledge, no experimental data on Corrol 1s
coupling constants and centrifugal distortion constants are
available for these compounds to compare wlth the present
results. It may be seen that a systematic set of molecular
constants have been obtalned by the method of klnetlc
constants whlch proves the validity of the method.
TABLE - 7.2 : Ts UATRIX FOR AX3YZ3 TYPE MOLECULES
S q 39 ( a c p - v ) ( 2 r b - R) 3Q ( a c p - v ) ( 2 r b - R)
+ 2 6 arQ (6 Q t 6 b12 -6 b c p l 2 ) + 2 a r Q ( 2 t - ( 3 / 2 ) b c p - p + u )
-3RQ ( v - f p ) -3R9 ( v - f p )
- ( , c 2 ) b c d p ) -3RQ ( v - f l p ) -3RQ ( V - f ' p ) -3RQ ( v - f l p )
S!?: $Z ad ( 2 q - ( A l 2 ) b -(6) ad ( 2 t + 2p + u )
S l I a -("312) a b c r (fi) a b c r
S I Z a -(=) a b c l d (m) a b c l d
S 7a -A a r b
' ~ a & adb
( 1 1 6) 1 2 ( R I Z - r b ) ( - t + 2p + u ) - 3pR + ( a r s - v a r ) ]
S I o a ( 1 1 ITb) [ 2 ( d b - RI2 ) ( - t + 2p + u ) - 3pR + ( v a d - a d s ) ]
S I l a ( c 2 ) bc ( R / Z - r b ) + (6) eR + (v'jTi) a 2 c r + ( v '%)~R
S l Z a ( J 3 / 2 ) b c l ( d b - RiZ) - (m) e b R -(*)a2 c l d - ( f i ) g l ~
A l l o t h e r terms a r e z e r o .
- T x ~ , m T y z , s
s~~ ( 5 2 ) a 2 r -6 a r b
SBb (m) a 2 d adb
s,, -(/El a r ( 1 + u + b I 2 ) - ( a 2 ) [ ( b - 1 ) ( r b - RIZ)
S l o b - ( m 2 ) ad ( 1 + u + b l 2 ) -(fi) [ ( I - b ) ( d b - R I 2 )
S l l b (JZ) a b r c (/3/2) I ( b c (HIZ) - r b ) + a 2 r c
+ ( R I 2 ) ( e + g ) J
TABLE 7.3: VIBRATIONAL FRERUENCIES (ern-' AND YOLECULAR
PARAMETERS OF SOME A X 3 Y Z 3 TYPE WLECULES (1 -6)
FREQUENCIES (cm-')
Spe- Desig cH3NHC3 c H 3 ~ D t 3 c D 3 ~ H t 3 C D ~ N D * ~ BH3NH3 BD3ND3 BH3ND3
cie8 nation
are tetrahedral
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