Normal Coordinate Analysis, Isotope Shift due to O¹⁶/O¹⁸ Substitution and Mean Amplitudes of Vibration Of MoO₃

Indian J. Phye. 52B, 193-200 (1978)

Normal coordinate analysis, isotope shift due to

substitution and mean amplitudes o f vibration o f MoOa

T . S. Ra w AT*, L. Di x i t* * , B. B. Ra i z a d a*** a n d A. K . Ka t ia b

Chemical Physics Research Crony, 1). R. S. College, Dehra Dm«-248(K)I {Received 4 JWne 1977)

A normal uoordinatt^ analysis o f Udolybdcnuni trioxido has hmn carried out using rect^nt vibrational data and Muller's i.-m atrix formalism. This is a first systematic study, wlu^re v o have inacie calculations for point group at^giUHl to M

0

O

3

instead o f till believed D.^ symmetry. Following sfecond order pf'rturbation theory o f Muller, the com puted isotope shifts for substitution at terminal atom were found to be Avi(Ai) 42-2 (uii ^ Av,^{A^) - 12*0 cm ” *, --4 8 -7 cm * and Av^{E') --- 15 b (mi * wl\i(»h agrws very Avell with the observed ones, i.e. 43-0 cm *, Av

2

(^ i)

— 11*5 cm *, Av

3

(.^') -- 41*9 cm~* and Av^{K') -- 14-b cm *. Moan amplitudes o f vibration for bondcHl [Mo—

0

] and non-bondod |

0

...

0

] have b(x>n compiite<l and compartHi witli the M(iO„'*' systcmis: (w

3

to b and n' — 2 to b). Results are critically disi,uss(Id in th(^ light o f molecular constants o f W O „ (n -

1

to (>) to look for ch.aract(eristic nature o f vibrational constants.

1. In t r o d u c t io n

Much interest is attached to the structural and sxiecdroscopic problems o f tiaimi- tion metal chalcogon comi>ounds (Schmidt & Muller 1974) particularly to tlie oxides o f tungsten and molybdenum, Inwauso f)f their sc^voral important indus­

trial applications at relatively high temperatures. A complete aspect o f harmonic force-fields and mean amplitudes was studied by Cyvin & Hargittai (1974) in the case o f tungsten-oxides. However, no (equivalent study exists for m olyb­

denum oxides because o f complexities involved in their spectra and non availa­

bility o f reliable sptxjtral data. Recently Hewott et al (1975) have studied absorption spectra o f molybdenum-oxide mokxsules and have deduced M

0

O

3

to possess a pyramidal (C ^) structure. Thus earlier assumtption o f a planar M

0

O

3

is in contrast with the current study b y Hewett et al and warrants fiirther investigation o f molecular vibration. In present work, some speotioscopic com putations for simple molybdenum oxides, are reported. Harmonic force-fields, mean amplitudes o f vibration are discussed in particular, where substitution O*® and O** are available.

* To whom all the correspondence should be addressed.

♦* Present address : Indian Institute o f Petroleum, Dehradun.

To whom request for reprint be made.

B -9

193

2. Vibrational and Structural Data

Nagarajan (1966) us(-.(l vibrational data o f Weltner & M cLood (1965) in his study o f normal coordinate analysis o f MoOg, W O

3

and UO

3

. The vibrational frequencies used were either unobserv(»d or were estimated for a M

0

O

3

mole- C

5

ule. The recent iissignment o f MoOj, by Hewett et al (1975) was based on the appearance o f two absorption bands at 976 cm~^ and 922-2 cm~^, which led them to assign this molecule a non-planar symmetry rather a configuration.

Tiio planar model should give rise to a single allowed high frequtmcy absori>tion v^(E') (in IR ) wlu^re pyramidal structure permits the absorption o f the tw^o higher frequencies and V:i(E')\ stret(*hing vibration in I.R . as observed b y Hewett et ah Their o})Servations are in agreenu^nt with the intensity ratio o f corresponding symmetric and anti-symmetric modes due to Schmidt (1973), wlio has given the relation betwecm pyramidal angle /? and the ratio o f intensities o f symmetric and anti-symnu^tric models, tliat is :

194 Rawat, D ixit, Raizada and Katiar

^8ym

The observation o f Hewett ef al (1975) very well fits in the relation o f Schmidt (1973). Soim^ preliminary calculations o f force-constants with the assumption fa " ^h04 fr have also been givcm b y Hewett et al (1975) but no exact description o f symmetry force-constants; ^

12

( ^

1

) and with. Hence our interest is to throw light upon the complete description o f symmetry fonjc constants and valence force constants alongwdth often omitted interaction force constants /r« a n d /r'«. This will enables us to show two things viz., (i) w’l)ether the assump­

tion /a = 0-04 f r giv(\s equivalent results with X ,, =.

0

approximation o f Muller and (ii) whether substitution could bo used to d(>duoe exact force fields and if not, what is the c^ffect o f this substitution on mean amplitudes o f vibrations.

The spectral data foi* MoOg^® and MoOs^“ under the vibrational representation r = 2 A i+ 2 E ' lias been collected in table

1

alongwitli the pyramidal angle

= 61-5° and [M o-0] distance to be J

-8

A.U. as suggestenl b y Hewett et al (1975).

Table 1. Vibrational frequencies*** (cm~^) of MoOg^® and MoOg^^ molecules.

Molecule v ii A i )

»»M

o03

^® 976 260-9 922-2 275-0

®»Mo03i»

933 249-4 8S-03 260*4

* See Hewett et cd (1975) (Mo-0) bond distance = l«8 A and pyramidal angle fi =5= 61*5®)

Normal coordinate analysis o f M

⁰

O

³

3 . Pb o c e d u b a l De t a il s

195

W ilson ’s matrix method (1955) was uswl to ...ury ,.«t the nonual co ­ ordinate analysis. The kinetic energy matrix was com pu k d using W ilson’s

Stt

Vec tor method, '(’lie elements o f the JT-matrix relatcMl to the various valence force constants in each symmetry sixties o f a J K , mok.cular model can

1

m» written a« ;

l

2

frr'

■f'l2 --- »-(2/r.4/,.')

= '^(fa-l-

2

/a ,) J'xs -- fr frr, E

44 r H f a - U -

whore fr twid /« are tile bond streteliiiig an<l angh* bending forc^e <onstants nvs- pcctively, w h ile/rr a n fl/«* are tl\eir imitnal interaolions, /,-a. <hnioteH the ink^r- aetion between and A a having a eoinmon iiond; and ,/’V« denotoH inter­

action botwetm Ar and Aa having no connnon liond; A n^preseiits changes in bond lengths or bond angles.

The elements o f symmetrized mean square amplitiahvs ((^yvin 1968) were obtained from C yvin’s seluiar equation ' -A<A’ (

0

where (t ’ represents the inverse o f the kinetic energj' matrix, E is tJie unitary matrix and

■■ « . t , c ‘

h, k and (■ have th<>ir usual moaning. Bastsl on Oyvin's principle, Sundaram (1961) has oxtondod tho method for the evaluation o f mean amplitude quantities for X Y ^ pyramidal molotmles. Tho same procedure has be«m us*fd. Tn a regular pyram idal X Y ^ type nmlecule, there are two types ot distances, viz., bonded X - Y and non bonded Y ...Y . Hero we report the corresponding mean amplitudes o f vibration for bonded U (X—Y) and non bonded IJ {Y ...Y ) distances coit*»s- imnding to each pair o f atoms alongwith that arising due to angle bending o f tw o adjacent bonds. Other mean amplitude quantities, arising due to the inter­

actions o f bond stretch and angle bending o f various coordinates, have not been rejiorted.

F or a unique solution o f tho » —

2

secular determinant in tho vibrational eigenvalue problem, an additional constraint, apart from vibrational frequencies, is needed. In the present case the same situation exists since both the species A i and E ’ are o f second order. Several investigations have proposed different constraints to overcome this difficulty but the i-m a tr ix approximation o f Muller (1968) has been foim d to bo moat satisfactory in calculating a reasonable set o f force constants as well as mean amplitudes, where the coupling o f masses is

196 Bawat, D ixit, Raizada and Katiar

small. B rb liy, foi (

2

x

2

) secular equations considering i y =

0

; j > i , the olements o f the and S-matrice.s can be obtained b y tb.e relation ;

F

^{— _L A a « i2 *}

Gn Gii6jat\G\

(7^2*^2

y

lii det|G^|

1

^ 2 2 __

(^U’^2 iht\a\

1 1 J

S x i - 1

1

1

1 }►

S.22 _ A 2 -d o t 1

(r

₁

i

1 (a22 1

J 'i^22

(

1

⁾

(

2

⁾

Wilson, Decius & Cross (1955) have developed a direct relation between isotope shift and iz-niatrix elements using perturbation theory. Accordingly

S (i(, ‘ )ifc<(-^o

w ... (3)

where At and A®* are the eigen values for the isotopic and the TU)rmal molecule respectively, Z<o'' stands for the corresponding normal coordinate tiansforma- tion coefficients and AC is the change in the

6

’ -matrix, which can be dctermincsl using the relation in matrix form ;

a

= C«+AC. ... (4)

For Second order eigen vahio problems the eej. (3) can bo WTitten as AA*

A«*

(I/O- M fc ,® A C „+ 2 (X ,-i)* iA C „+ (V ')* s ‘-‘AGi«!- (5) It is known that in cases whore ■» —

2

and where mass coupling is small (m , > my), Lii =

0

Ls a good approximation. Thus eq. (3) can bo further simpli­

fied under this approximation to yield AA^/Ai® and hence AAi

Ai®

AG O®,

11 11

A ^

V <uiTe|-

... («)

... (7)

Using the above formulation the isotope shifts for the (**MoOs“ -**MoOa“ ) systems have been examined. The above eq. (7) gives satisfactory results in the

Normal coordinate analysis of M oq ^ 197

isotopioaUy labelled trioxidea o f halogen (Sanyal & Dixit 1974) and tetralu^lral hydrides o f AI, B and Ga. (Rawat ei al 1976).

4. Results and Discussion

The G V F F constants calculatisl with the help o f X-niatrix approximation are presented in Table 2. .Symmetrized foree-oonstants, (Table 2) which an.

Table. 2

A. General valence force-field constants (mdyn/A) o f the MoOa'* and their coin-

M olecu le/ / ; ion

parison with M0O42 w d MoOg® ions.

frr frr

MoOe**"

8

.

10

s -

0:121

r)-94 0-5r) 3-899 0*288

0*01(>7 jffjt 0*03.7

A 0*380 0-35

A. Kofereiirt^

0*0958 ProHont study

0*02

(r/)

('>) {a) Basile L. J., Ferraro J. R., Labonville T. & Wall M. ('. Coord. (1973) Chem. Iter,

11. 21-69.

(5) Ahmad P., Dixit L. & Sanyal N. K. 1974 Indian ./. Pure Appl. 12, 489-94, B . Syiurnotrifciod: foroo-(^onstants of (m y d o /A )

Molecule Method F n (^ i)

M0O3IO 0 7*465 0*0543 0 189 8 429 0 0223 0*470

obtained, setting ^ 0; represent least kinoruatical coupling i.e., niylm^ ratio is small and m ay be verified by P E D metluKl or any other approximation method.

H owever, it was noticed in our earlier studies (Sanyal & Dixit 1974) and (Rawat et al 1976) that where method is applicable, L^i — 0 or P E D metluKls give com parable results. W e have collected some main force constants that is those o f Mo-*0 stretchings, bond-bond interactions and l>ending force (jonstants o f MoOn” ■ systems in Table 2. It is clear from table 2 that as the number o f oxygen atoms exceeds, GVFF stretching force constant fr decreases with simul­

taneous changes in bond-bond interaction force constant /ff. The bending force constant / « also follows the same pattern. This clearly shows that the reltative stability o f these oxides is governed mainly b y the valence statics o f molybdenum and number o f oxygen atoms involved in oxide formation. The force constants in magnitude show that this stability is directed towards systems having minimum non«bonded oxygen distances. Comparing the force constants o f W O j with M oO ,

viz. /r(W-O)

= 9-826 m d yn /1 and

/r(MO-O)

= 8-108 m dyn/A,

it ia evident

198 Bawat, D ixit. Raizada and Katiar

in thoao iso*atructural molecules, most stable bond is formed in the case o f heavier central atom. This idea is also supported by the magnitudes o f vibrational frequencies : v^{Ai) o f W O

3

= 1045 cm “ ^ and v^{A^) o f M

0

O

3

- 976 cm~^ An exa­

mination o f table

2

reveals some remarks on the force fields o f Hewett et al (1975).

W hat we speculate is that probably Hewett et al have presumed / « = 0*04 fr in order to avoid difficulties associated in solving (

2

x

2

) eigen values o f A^ and E ' species. Their assumption is tentative on the ground that they have omitted to mention physical significance o f the propoae(l relation except that bending frequencies were unobserved and were omitted in calculation. The universality o f this relation in pyramidal molecules is still questionable. However, the results resemble with the magnitudes o f valence force constants obtained b y Muller's method (1968).

In general as shown in Table 3, the root mean square amplittide quantity due to non-bonded oxygen pair is greater as compared to tlio [M O-01 bondofl mean amplitude i.e.,

Table 3. Mean amplitude o f vibration (in A) for bonded (Mu-O) and uon-bondtHl ( 0 .. .0 ) distances o f MoOg^^/MoO^^** moJetmles and their comparison with Mo

0 4

*“ and MoO^**" ions.

Molecule/

ion

Distance

Mean Amplitude of Vibration T -- 0"K T = 298"K T r>00"K

Roforenco

«»Mo03"« Mo-^O Bonded

0-0361 00365 0-0386 Present study

0 .. .0 Non-bonded

00624 0-0706 00982

®®Mo03^8 Mo— 0 0-035J 00356 00378 I^resent study

Bonded 0 .. .0 Non-bonded

00607 00792 0 0979

MoO*^- Mo— 0 00379 0-0386 0-0414 (a)

Bonded (0.039)*

0 ...0 Non-bonded

00638 00745 0-0885

MoOe«“ Mo— 0

Bonded 00439 00461 00518 {f»)

0 ...0 0-0650 0-0571 0-0632

Non-bonded 0 ...0 Non-bonded Non-linear

0*0762 00868 0-1044

<a) Sharma D. K., Pandey A. N., Publish A. K. & Kai S, K. 1975 Z, Natufforsch, 20a, 1504-6.

(6) Ahmad Parvez, Dixit L. & Sanyal N. K. 1974 Indian J. Pure & Appl, Phye. 12, 489-94.

* Value in parenthesis is from Muller A. & Cyvin 8. J. 1968 J. Mol. Spectroscopy 26, 315.

Normal coordinate analysis o f iHfoOg

io-.-o 0-0705 A > \Mo-f, 0-0364 A.

199

The trend in quantity (erd)^ > (err)^ it? contrary to the eorreaponding force cons­

tants. It m ay bo mentioned here, tliat the cahailated mean amplitudes, in the proJient case, d o not depend upon bond-distnance but deptmd mainly on valence- bond angle, which has been estimated to be 99"’45' by using th(^ relationsiiip sin/? =

2

/

3

s in a

/2

where /? is the p>Tamidal angle which has }>een taken to be ()

1

*

5

° as suggested b y Howett et al. The bond angles in W O

3

and MoO., ar<»

nearly comparable and mass-coupling contribution in them is such tliat Uu' vstretching mean amplitud(^s are nearly ec|ual : /«> « 29K“K has beem reportcKi to bo CK)385 A b y Cyvin & HaJrgittai (1974). Extending this idea along the series o f available oxides o f Molylwhmum and (tomparing [MO-01 mean amplitudes, it may lie saf(4y inferrtid that in thest^ metal oxides, moan vibrational amplitudes are diaractteristic to some vxUmi. For example /mo « at 298^’K in and MoOe*^" is0*0d9 A, find 0-04t)1

A

nsspwttiwly (s(m» Table 3).

The isotope-effect on mean amplitudes, in tlu^ (tase of inorganic rnoleeuk^s, is known for few compounds only. However, the possibility of such obsc^rvatiiai has b(Mm rwently emphasized by Mulh^r & Mohan (1972) (excluding the case o f H /D isotope substitution). On this ground the obst^rvc^l isotopes c^fh^tt on vibrational freqiumcic^s o f MoO;^ lias Ixnm examim^l (see table 4) in the liglit of Mohan-Muller tlieory and lias Ikmmi found that Hiu. to tei-minal atom Table 4. Observed and calculat(xl freepumey slufts (in cm ^) d.ue to O^^O

isotopes foT MoOjj.

Molecnile pair

ooMoO^s

Avi(dd obs, cal.

42*2

A»^2(-4i) ot)S. Cttl.

11-5 120

A»»3(/^') ohs. • cal- 41*9 4K-7

obs. cal.

1 4 0 1 5 0

substitution in the caso o f pyramklal trioxid<«. like MoO, tb.,

in magnitude and valuos arc W o “ =" ^ f f \ ■ f “ • nectivelv at T = 298“K . Before utilising observed isotopt* effect m freqiiencies T m e a n amplitude calculation, we confirmed the obsc*rvation o f Hewett H al keeping in view the second order perturbation t h ^ r y p n .p o s ^ b y MuUer ^ (1972). The computed and observed vibrational frequencies for MoO, /MoO, (table

4

) show the basic soundness o f the assumption made m the theory ol MuUer, Schmidt & Mohan (1972). Tlie method works satisfactorily for isotope effect studies. N o experimental evidence in support o f the computed mean ampUtudes are available in the present precision o f electron^iffraetion meaeure- m e L , we hope that the data will be useful in future when such techniques will

200 Bawat^ D ixit, Baizada and Katiar

be developed. Concluding with this study, it is well to remark that Muller’s JD>matrix approximation is very good application to deal with molecular vibra­

tions in molybdenum trioxides.

Ac k n o w l e d g m e n t s

Authors are thankful to the Director, Indian Institute of Petroleum, Dehra Dun for constant encouragement. One of us (LD) records his sincere thanks to Prof. Nitish K . Sanyal, Department of Physics, University of Gorakhpur, for his keen interest in the problem.

Re f e r e n c e s

Cyvin H. J. 1968 Molecular vibrationtf and mean square amplitudes, Univorsitetsforloget Oslo and Elsevier, Amastordam (1968).

Cyvin S. J. & Hargittai I. 1974 Acta Chemica Academiae Scientiarum Hungaricmi Tomua 83, 324, 321.

Howett Jr. W. H., Newton J. H. & Wetner W. dr. 1975 J. Pliys. Chem. 79, 2640.

MuHer A. 1968 Z. Phya. Chem. (Leipzig) 288, 116; Z. Naturforach, 23A, 1029, 29.

Muller A. & Mohan N. 1972 J. Chem. Phya. 58, 2994.

Muller A., Schmidt K. H. & Mohan N. 1972 J. Chem. Phya. 57, 1752.

Nagaragan G. 1966 Indian J. Pure & Appl. Phya. 4, 158.

Kawat T. S., Dixit L. & Raizada B. B. 1976 Indian J. Pure & Appf. Phya. 14, 656.

Sanyal N. K. & Dixit L. 1974 Z. Naiurjorach 29A, 697.

Schmidt K. H. & Muller A. 1974 Vibrational spectra of transition metal vhalcogen compounds.

Coordination Chem. Rev. 14, 115-179.

Smit W. M. A. 1973 J. Mol. Structure 19, 789.

Sundaram S. 1961 J. Mol. Spectroae 7, 53.

Weltner Jr. W. J. R. & McLeod Jr. D. 1966 J. Mol. Spectroae. 17, 276,

Wilion E, B., Dccius J. C. & Cross P. C. 1956 Molecular Vihrationa, McGraw-Hill, New York.