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Parametric Study of Functions of Molecular Vibrations IV. Coriolis Coupling Constants for the Molecules XY2 and X3 (Symmetry C2V)

A . A L I X a n d L . B E R N A R D

Laborato i re de Recherches Opt iques Facul te des Sciences, R e i m s (France)

(Z. Naturforsdi. 27 a, 593-^597 [1972]; received 27 October 1971)

I n this work , t h e authors give the parametr ic f o r m o f the Zeta constants f o r the case o f the molecules X Y 2 a n d X 3 ( symmetr ic species C2v), as a funct ion o f the mathemat i ca l parameter a wh i ch generates all the solutions, a n d o f the molecular coupl ing coef f i c ient b e t w e e n t w o s y m m e t r y coordinates {A\) and ^ ( ^ I ) .

Introduction

As is the case for various functions in the theory of molecular vibrations such as the force constant matrix F 1.2,3,^ the energy distributions MW 4>5, the compliance matrix (-?1-1) and mean square am-plitudes of vibration Ztj 6>7, the Coriolis coupling constants Cij can be studied using the parametric form in the general case of second order equations. The expressions we obtain are, then, dependent on the type of symmetry of the molecule which is studied.

As far as this work is concerned, the parametric study of the vibrations of molecules X Y 2 or X 3 of point group C2v leads to results which are very simple.

I. Parametric Study of the Coriolis Constants ty

M E A L and P O L O 8 ' 9 have defined the matrix Zeta of the Coriolis coupling constants

£a = £-1 C*L~ "

in which L is the well known matrix of the eigen-vectors of the matrix product (GF). Ca is a character-istic matrix of the studied molecule which has a certain analogy from the point of view of the de-finition and the transformation properties, with the matrix G which defines the inverse kinetic energy of the molecule. These matrices are easily obtained by the "vector method" J0-11

with Gtj = 2 f a («<„• Sja), A

CIJ X SH) G<X • a

R e p r i n t requests t o A . ALIX, Laborato i re de R e c h e r -ches Opt iques B P 347, 51-REIMS, France .

The matrix £ being invariant under scaling is then studied in the normed system SN of the symmetry coordinates by using the definition of the molecular coupling coefficient tpy between two symmetry co-ordinates St and Sj which are defined by the relations 6>12,

LN = aL, of = (ÖÜ)-1/2 • Thus we get: CNa = a" 1 C a a~x,

£iVa - LN-l QNo. jr/ft-1 = £ - 1 L-lt _ f a

By using the parametric form of the matrix ZA related to the solution LN° corresponding to a completely characteristic lower vibrational mode, which is identical to the model of "progressive rigidity of molecules" 13 and to the approximation14

Li] = 0 for i > j15, we get LN = LN°R with: RR* = I .

We have then f « = f t f « = Rt{£N')*R.

1. Case of non-linear molecules XY2 (group C2v)

This is the very well known case of the water molecule model with

S1(A1) = (liyS){R1 + R2), S2(A1) = L-OL, Ss(B1) = (1I]/2)(R1-R2).

The different matrices GN, CNv and Cy take the form

1 —sin2y»i2 0 ( 3 ^ = '—sin 2 y>i2 1 0

0 0 1 G12 with sin 2 y>i2 = - = —G 12N,

0 0 sin2y>i2 CNy = 0 0 - 1

— sin 2^x2 1 0 0 0 f13 0 0 123 .

— £13 — £23 0

The matrix 2>v of order 3, taking into account the expression for the sub-matrix of order 2 ( L ^ ^ i ) ) , is written as

= LN°(Ai) R(x),

1 0 0' LN = — s i n 2^12 c o s 2 ^ i 2 0

1 0 1 cos a

= — s i n (2y>i2 — a) 0

cos a — sin a sin a

0

0 cos a 0

0 — sin a

cos(2y»i2 — a) 0

Thus we get the final expressions for the constants C?3 a n d £23 a s

Cvl3 = sin (2 ipi2 - a) = - |/G22£i2 = - Lf2 , C23 = — c o s ( 2 V12 — a) = — ]/G22 L22 = — L%2.

We notice, in agreement with H E R S C H B A C H and L A U R I E 1 6 that the constant is always negative.

Re?narJcs

The solution of the molecular potential function based on the hypothesis of complete uncoupling of the vibrations corresponds to the case of the "splitting" of vibration9-16

with = sin 2 yj12 , C°3 = — c o s 2 V12

= ( £ - i ) H C « (L-I<)h

in which (£ _ 1 )h is the part of the matrix L~l cor-responding to the high frequency vibrations and we deduce that

\py

Different authors14 '17»18 have shown that this approximation is suited very well to the molecules XYn when the coupling between $1 and $2 is not too important (| sin 2 121 < 0,3 with our definition6

of the coupling between S1 and S2). The influence of the value of the masses mx and

my on the values of the Coriolis constants has been studied recently 14>18 '19.

Fig. 1. Internal coordinates and the coordinates axes for X Y 2 ( C 2 v ) t y p e molecules.

We can find again one of the limits of £13 by taking into consideration the variation of the molecular coupling coefficient sin 2 xpi2

sin2 •on Q Sm <P .,1 -"y sin2 2 w 12 = v 7 o — i — — with — - = 1 + 2q + g 2 s i n 2 CP MX

my fx Hy = Q

Let sin2 2 ipi2 (o —> co) = 1

s i n 2 y i 2 = ± l , y i 2 = ± w / 4 .

This solution corresponds to the maximum cou-pling between the symmetry coordinates S1 and S2 • The application of the energy criterium 5 defines the only solution that is physically acceptable and is given by a = 0. Thus

(Ci3)e_>oo = ( & U 0 0 = sin 2 12^00 = + 1 .

2. Case of linear molecules XY2 (group D^)

These molecules behave in the same way as the diatomic molecules: the molecular potential model for these diatomic-like molecules is given by the solution corresponding to completely characteristic vibrations.

Let yu = 0 , »,7 = 1 , 2 , 3 .

We get the classical result18

C23 = - 1 , Ci3 = 0 .

«VÖ^

Fig. 3. Vibrat ions o f bent X Y 2 t y p e molecules.

(Q,)

Fig . 2. Mix ing o f the s y m m e t r y coodinates in normal coordinates (QI and Q2) f o r N = 2 cases.

II. Molecular Potential F and Coriolis Constants Ca

The force constant matrix F, the parametric ex-pressions for which have been given1, is entirely defined by knowledge of the parameter a and can be related to Cij as

F(Ai) = Lfä '22(a) A* R(- a) L^), F(Bi) = ABl (G-i)Bl

with t a n _ tan 2^12 + (I13/I23)

— 1 — tan 2^12 (£13/123) ' Besides, it is possible to find again the Coriolis

constants, except for the sign, by developing the relation of M E A L and P O L O 9

which gives

T2 — sis —

L*{FCv)L*-1 = A&

2 _ M J u f l f l ^ h i ] 13 — Ai - Aa

and q 3 + C|3= 1-

Consequently, by using the parametric expression of Fn, we get

Fn={G-!)n (

which leads to

Xi — 2.2 . Ai — A2 0 — 2 + 2 - c o s 2 (2^12 — a) j

t 2 — <>13 — 1 — cos 2 (2 Y»i2 — a) = sin2 (2 y)i2 — a) .

III. Energy Distributions and Normal Coordinates

1. Energy distributions M\k)

The energy contributions of the symmetry co-ordinates Si to the normal modes of vibration Qt are defined by 2 0

M\k) = Z L t k L j x F i j K 1 = = I & I & - 1 • j

Factorization of the matrices G, F, L, A . . . leads to

Mf 1 - Mf 0

M = 1 - Mf Mf 0

0 0 1

M(I) = 1 + tan 2yi2 tan « { = 1 2 i 1 + tan2 a ' ' ' "

Hence with tana = tan(2y>i2 — (2^12 — a)) j^(i) __ 1 + tan (2 Vi2 — a) tg 2^12

1 -f tan2(2yi2 — a) 1 — tan 2 y>i2(li3/l23)

1 + (I13/I23)2

<N

<N

bo"

8 S

a-<N

<N a

+ + +

—1 O M 00 SO CO OO O IO t> SO C<J O H C<) SO OS ohp5 d oo d ao h as +++

0 0 0 I I

©(M^wiMia as as 00 co so ö e<i d 06 h r-rtfflHH^M

d i s o oo«Maeq(N a t> n \a cq co © so H H so H © © d d + + + I I I I

O OS CO ^ w o o o w t -o o o) oooacooo-H OS OS OS CO »O OS SO l> as d d d co os d h oo OS O5 OS as as os os ao oo

w O O O O O O O " © CO <M h n H h t ^ N co so ( o o t ^ o n ® HH{lj HIoOOHfli + + + I I I M I

(M sc m os 10 so os os 00 os os os os Os os o d d I I I

0 0 0 CO <N CO

o d d I I I

Tt< os ao <n (MffliQ®T(iO 00 os 10 so H os 00 10 >0 IO H 00 os os os os os 00 0 0 0 0 0 0 * I I

5 so 00 10 so c q o h H so m 0 0 0 0 0 0

0 0 0 0 0 0 CO © to © h o H CO <M SO o 00®OCC0O H co so so 10 o 0 0 0 0 0 0

lOOONCH 00 <M SO 00 Tf 00 h o H r> IOQOOOHIO h (M <N so TjH 0 0 0 0 0 0

.8

0?00O o g o f o . 0 C1 CI CI o M ZL -1 WSW 3 q ü m ü ^ O

From the vibrational energy distributions point of view, we find that the exact coupling of the vibrations is entirely defined by the constants (see Table 1).

2. Form of the normal vibrations (symmetry species A\)

a) in the space of s y m m e t r y c o o r d i n a t e s : S

The use of the molecular coupling coefficients ipu and of the normed system of the symmetry co-ordinates makes it possible to define the normal coordinates Qk in the space SN 6>21 as

C O S (Si ,Qk) = ^ = Li = c o s (Sf, Qk).

Then we get

tan (Si, Qt) = LZ/LZ , tan (S2, Q2) = C13/C23

and (Äi,S2 ) = 7r/2 + 2v>i2.

b) in the space o f the in terna l c o o r d i n a t e s R

The relation between the internal symmetry co-ordinates S and the internal valence coordinates R can be written as

S=lTR or R=UtS

so that the form of the normal coordinates Qk is

S — L Q, R — IQ yielding Q = l~iR = (2,-1 [/*) R.

The deviations of the normal coordinates Q1 and Q2 from the symmetry coordinates S1 and S2 are defined, by the angles (pi and (p2 as

cot cpi (ot) = j/G/2 G2n (tan 2^12 + 1/tana),

tan9?2(a) = }/G\2G\X (tan2xpi2 — tana).

The solution of non-coupling based on the con-sideration of energy distributions is limited by

tan cpi (0) = 0 , <pi(0) = 0 or Qi = L^ Si,

+ /n\ T / Ö22 s i n 2 V12 - 12 tan992(0) = ] / G n = Y2GÜ, •

Thus, the bigger the molecular coupling coeffi-cient ^12 5 the more the normal vibration v2(Q2)

1 A. A L I X and L . BERNARD, C. R. Acad. Sei. Paris B 2 7 0 , 66 [1970].

2 M. PFEIFFER, J . Mol. Spectrosc. 3 1 , 181 [ 1 9 6 9 ] . 3 F. TOROK and P . P U L A Y , J. Mol. Structure 3, 1, 2 8 3

[1969]. 4 A . A L I X , These 3° cycle Reims 1970. 5 A . A L I X and L . BERNARD, C. R. Acad. Sei. Paris B 270 ,

151 [1970].

will be a mixture of the coordinates Si and S2, although these coordinates contribute respectively to Q1 and Q2 only from the energy point of view (see Table 1).

IV. Mean Square Amplitudes of Vibration and Centrifugal Stretching Constants 6'7> 19

The parametric study 6>7 of the mean amplitudes of vibration Eij leads in that particular case to a study of a fundamental relation between the con-stants Z 2 2 and the Cij 1 9 which is

£22 = G22 2" 02 — c o s 2 ( 2 F 1 2 — <*))

= £22 t^is"12 ( 2 V12 ~ a ) + cos2 (2 tpi2 — a)] .

Hence we get

r 8 a = G22(<5i& + toä) wit11 Cf3 = 1 — C|3 , -£22 , (F_1) 22 . - 11

2 = ^ z l 2 = -Ö22 g2 =

13 <5l — 2 Cl — f2 Al — A2

These relations between Z22 and (F~1)22 are a simplified form of the ones we obtained in the general case of second order7.

It may also be possible to express the various elements of the matrices 0 and 0 in the matrix theory of the centrifugal constants22, which are, for instance, given by

£2 = ($22/022) - h = gl - [flllflg-1)!!] e t c 13 Ai — A2 a 1 — a2

These relations make it possible to test the results we get with the different experimental data for a given molecule (27, A, ...).

Conclusion

The parametric study of the Coriolis coupling constants leads to some expressions which are parti-cularly simple and which directly define the mole-cular potential, the energy distributions and the normal modes of vibration. The relations between the matrices F, E, 0 and 6 are obviously explicit.

6 A. A L I X and L . BERNARD, C. R . Acad. Sei. Paris B 2 7 2 , 5 2 8 [ 1 9 7 1 ] .

7 A . A L I X and L . BERNARD, part no. ILL, J . Chim. Phys. 68 , 1 1 - 1 2 , 1611 [ 1 9 7 1 ] .

8 J . H . MEAL and S. R , POLO, J . Chem. Phys. 24 , 1 1 1 9 [ 1 9 5 6 ] .

9 J . H . MEAL and S. R . POLO, J . Chem. Phys. 24 , 1 1 2 6 [ 1 9 5 6 ] .

1 0 B . N . CYVIN, S. J . CYVIN, a n d L . A . KRISTIANSEN, J . C h e m . P h y s . 39, 1967 [1963] .

1 1 E . C. WILSON, J . C. DECIUS, a n d P . C. CROSS, Molecu lar V i b r a t i o n s , M c GRAW-HILL, , L o n d o n 1955.

1 2 F . TOROK a n d P . PULAY, J . Mo l . S t ruc ture 3, 1 [1969] . 1 3 M . LARNAUDIE, These Par is 1952. 1 4 A .MÜLLER, Z . P h y s . C h e m . 238, 116 [1968] . 1 5 C. J . PEACOCK a n d A . MÜLLER, J . Mo l . S p e c t r y 26, 4 5 4

[1968] . 1 6 D . R . HERSCHBACH a n d V . N . LAURIE, J . C h e m . P h y s .

37, 1668 [1962 ] ; 40, 3142 [1964] .

17 D . E . FREEMAN, Z . N a t u r f o r s c h . 2 5 a , 217 [1970] . 1 8 A . MULLER, B . KREBS, a n d S. J . CYVIN, Mo l . P h y s . 14,

491 [1968] . 1 9 S. J . CYVIN, Molecu lar V i b r a t i o n s a n d M e a n S q u a r e

A m p l i t u d e s , Univers i te t for laget , Oslo 1968. 2 0 P . PULAY a n d F . TOROK, A c t a Ch im. H u n g . 44, 287

[1965] ; 47, 273 [1966] . 2 1 P . PULAY, A c t a Chim. A c a d . Sei. H u n g . 47, 373 [1968] . 2 2 S. J . CYVIN, B . N . CYVIN, a n d G . HAGEN, Z . N a t u r -

forsch. 23 a, 16 [1968] .

Zum Rotations-Zeeman-Effekt in Dimethylketen D . S U T T E R , L . CHARPENTIER u n d H . DREIZLER

Abtei lung Chemische Physik im Institut für Physikalische Chemie der Universität Kiel

(Z. Naturforsch. 27 a, 597—600 [1972] ; eingegangen am 19. Dezember 1971)

MOLECULAR ZEEMAN EFFECT AND THE DETERMINATION OF THE MOLECULAR G VALUES, PARAMAGNETIC SUSCEPTIBILITIES, AND MOLECULAR QUADRUPOLE MOMENTS IN DIMETHYLKETENE

The rotational Zeeman-Effect in the microwave spectrum of dimethylketene was investigated at fieldstrengths c lose to 22 kG. Only Aj= 1 rotational transitions with Z l M = ± l selection rules did show appreciable splittings due to the magnetic field. From the splittings the diagonal elements of the molecular gr-tensor were determined to b e :

GAA = + 0 .020 (3 ) ; GBB= + 0 .0165 (8 ) ; GCC= + 0 . 0 1 2 6 ( 5 ) .

(Only the relative signs of the ^-values are obtained from the experiment) . The susceptibility an-isotropics were found to be close to zero.

Über das Mikrowellenspektrum von Dimethyl-keten, (CH3)2C = C = 0 , wurde mehrfach berich-tet 1. In der vorliegenden Arbeit werden die Ergeb-nisse einer Untersuchung des Rotations-Zeeman-Ef-fekts bei magnetischen Feldstärken um 22 kG vor-gelegt. Der verwendete Zeeman-Spektrograph ist in einer früheren Arbeit2 beschrieben worden.

Im Vergleich zu anderen Präparationen 3 hat sich folgende hinsichtlich der Reinheit des Endproduktes und der Ausbeute (60%) gut bewährt.

(CHS) 2C (COOH) 2 + HCEEC-0 - C2H5 O II

(CH3) 2 C O + CH3COOC2H5

II 0

o II

(CH,), C* ^ 0 —60-120 ° c _ ( C H . c = C = O + C02. 3 - 1 0 — 2 — 1 0 — 3 Torr 3 2

II o

Die Herstellung von Dimethylmalonsäure erfolgte nach 4.

Eine 60-proz. Lösung von Ethoxyacetylen (Fluka) wurde durch Destillation bei — 30 °C, 10~2 Torr, von einem Rückstand befreit. 2,2 ml der so gereinigten Ethoxyacetylen-Hexan-Mischung (11 mmol Ethoxyace-tylen) wurden auf —10 °C gekühlt und zu einer eben-so gekühlten Lösung von 0,6 g (4,5 mmol) Dimethyl-malonsäure in 4 ml getrockneten Äthyläther gegeben. Die Mischung wird 40 Minuten unter Rückfluß und Feuchtigkeitsabschluß (CaCl2) gekocht. Das Produkt wurde anschließend im Exsikkator abgekühlt.

Essigsäureäthylester, Äther und Hexan wurden in einem Stickstoff ström bei etwa 10~2 Torr entfernt, zu-nächst 4 h bei Zimmertemperatur, danach 12 h bei 30 °C und schließlich 24 h bei 40 °C. Es verbleibt das Dimethylmalonsäureanhydrid als weißes Pulver. Nach Anschluß eines mit flüssiger Luft gekühlten Auffang-gefäßes wurde unter Vakuum (10~2 —10~3 Torr) bei 60 °C bis 120 °C ansteigend die Pyrrolyse durchge-führt. Das Dimethylketen und Kohlendioxid sammelt sich als hellgelbe Substanz im Auffanggefäß. Bei etwa — 60 °C wird das C02 abgetrennt. Dimethylketen ist bei - 1 8 0 °C über Monate haltbar.

Viele Rotationsübergänge des Dimethylketens (DMKT) erscheinen im Magnetfeld lediglich etwas ver-breitert. Nur die Übergänge mit AJ=1, AM = ± 1 Auswahlregeln werden nennenswert aufgespalten. Sie erscheinen im Magnetfeld als annähernd sym-metrische Dubletts (vgl. Abb. 1). Die beiden Kom-