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 ) ]

FIG 7 . 1 STRUCTURE NOMENCLATURE AND MOLECULAR

PARAMETERS OF AXjYZj TYPE MOLECULE

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

TABLE - 7 . 4 1 POTENTIAL CONSTANTS OF SOYE l U 3 Y Z 3

TYPE MOLECULES (NC')

L a l u e s I n t h e p a r a n t h e s l s were taken from Oxton e t a l ( 4 )

TABLE - 7.6: CORIOLIS COUPLING CONSTANTS.

TABLE - 7.7: CENTRIFUGAL DISTORTION CONSTANTS (Wz)

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