JOURXAL
OF MOLECTJLAR SPECTROSCOPY 16, 5167
Calculation
(1965)
of Inertia
Part IV. EthyleneType Kozo .!Iepartment
KUCHITSU,TAKESHI of Chemistr:y,
OKA,*
Defect Molecules
ANT) ~ONEZO
l’acdty of Science, Tokyo, Japan
~IORINO
The University
of Tokyo,
The inertiadefects of some ethylenetype molecules. C’,S, . (S = H, 11, F, Cl, and Br), are calculated on the hasis of the general formulation developed in the first part of this series. The L’ mat,rires, the Coriolis coupling constants, and the contributions t,o the inertia defect from individual vibrational modes are evaluated by using the force ronstants and the fundamental frequencies taken from the literat,urr. For C&H, and CrI>, , they are also calculated hy using the approximate normal frequencies which are estimated from an empirical observation that the anharmonic corrections for a number of simple hydrides and deuterides are systematic. The results are compared with those derived from the fundamental frequencies in order to est.imate the uncertainty of the calculated values. The inertia defects for the ground vibrational state, A,, , thus obtained are: 0.050 (&HA, 0.070 (C21)4), 0.15?(CzF4), 0.25n (C&I,), and 0.11s(CeBr4) in amu A* units. They are found to be nearly equal to those of the corresponding S&O molecules; the A,)values of C?F, and C&l, are also found to be close to the observed values of the corresponding vinvlidene and c,is C.‘HL.\z molecules. These facts are explained hy c*onsidering the effects of inplane and outofplane vibrations of a planar molecule on A,, ; it is demonstrated in general t,hat the Coriolis int)eractions hetween the outofplane and inplane vibrations make definitely negative contributions t,o An , which nearly compensate the positive contrihut,ions from t,he interactions among the inplane vihrations.
111 a series of our preceding papers (IS) t, a general fornlula for calculation of the inertia defect has been derived (I) and applied to a number of nonlinear symnletric triatomic molecules (II) and planar fouratomic molecules (III ). The calculation has here been extended to some ethylenetype molecules, i.e., (‘zH4 , (‘?D1 , CzF4 , C2C14, and C2Rr4 ,
* Present address: Division of Pure Physics, Sational Research Council, Ott,awa 2, Canada. t Recently, Herschbach and Laurie published a paper on this subject which forms Part III of their excellent series of papers on the influence of vibrations on molecular st,ructure determination [U. R. Herschhach and 1.. W. Laurie, b. (‘he,,z. Phys. 40, 3142 (1964)]. Their results are essentially the same as ours (1S). 51
52
KUCHITSU,
OKA,
AND
MOWN0
In contrast to the cases studied in Parts II and III, little experimental information regarding the iuertia defect of the C?X, molecules is available at present, except that Dowling and Stoicheff (4) obtained, by rotational Raman spectroscopy, the inertia defects of CgHz and C&D, in the ground vibrational state as 0.025 f 0.020 and 0.134 f 0.0.54 arnu A’, respectively. They considered that the small positive values indicated the planar structure of these molecules. It seems important in this regard to calculate the inertia defect of these molecules (on the assumption of the planar structure ), since the result may serve as a useful guide for further improvement of the rotational analysis, which will result in a more accurate experimental inertia defect. The planarity will then be verified, as discussed in Paper III, provided the experimental and calculated values agree with each other. In addition, the present calculation for C&X4 will be one of the steps toward the understanding of the inertia defect of part’ially substituted ethylenes, C2H2X2 and C,HgXY for example, for which reliable experimental values have been obtained in recent years by microwave spectrosCOPY* As denionstrated in the previous papers, the parameters of primary importance for calculating the inertia defect are the force constants of the molecule, by the use of which the vibrational frequencies and the Coriolis coupling constants can be calculated. Icor the C,X, molecules with which we are concerned, all fundamental frequencies have been obtained by spectroscopic nieasurenlents, and plausible sets of the force constants which reproduce the frequencies fairly satisfactorily have also been reported. Even though the normal frequencies and the corresponding exact quadratic potential constants are not known for any of these molecules, the inertia defect sufficiently correct for our practical purpose, for the ground vibrational state at least, may be obtained by using the fundamental frequencies and the approximate force constants derived from them, as
b
a
Fro. 1. Cartesian molecular plane.
and internal
coordinates
of C,?i,
The c axis is perpendicular
to the
INERTIA
23
UEFECT
discussed in Paper III. It seems worth while to estimate the uncertainty in this approximation for such basic nlolecules as C’gHt and C,D, ; they are suitable for this examination, because the effect, of anharmonic vibrations appears nlost conspicuously in hydrides and deuterides. While a theoretical prediction of the rionnal frequencies of these molecules by calculating the anharmonic corrections (/S) is not practicable at present, it, may not be impossible to make a reasonable guess by utilizing the knowledge on analogous molecules. Comparison of t,he inertia defect thus obtained with that based on the observed fundanlcntal frcquwcies mill show to what cstent the uncertainty from t,his origin is involved in the tatter. CC)KIOI,IS COUPLIN(;
COIWTANTS
I’or the CzX4 molecule, norlvanishing elements of the C’oriolis coupling COILstauts (twentythree ill total 1 and sum rules among them have been given by Meal and 1’010 (6) on the basis of their extensive considerations 011 molecular qmnwtry. In the coordinate systenl given in l:ig. 1 and in Herzberg’s not,atioIl ( 7 J of the vibrations,
;1,,,( VI , V” , V:I), &,C ~5,
v6
) , B?,, i ~9 , VIII1, and B:,,, ( vll , vl? ) for inplane vi
hratiow., and illi,(
Bl,,( Y;), and Bz,,i vxi for out,ofplane vihratiotis,
clj
thcb sun1 rules among the constants arc shown by then1 to be cr:aA,? + cr:PA,2 = (<&Y ({$,’
+ ({$P,‘,” + ([:$”
cc::; j2 + (&,”
+ ($s’)’
.t (j&))” = (j:ag),‘?+ ({:;‘o,?
= ({:;‘,‘+ = ($6))?
({:;‘o)” = ($;:,‘$ + ({$+’
= 1,
($l\L)? = 1,
+ (jj;/ )?
= (.$1’1)2 + (j:&,’
= (<&)’
+ (ri$,)’
j + ( r:&
I (r:&)
= 1,
(2,
( r::i! ) ( r:r;s’1 + (ii:; ) ( r&i ) + ( si:;’ ) ( rar’; ) = ( rCA 1i.G
= 0.
On the other hand, the following relations are shown to exist arllotlg the rotatiotlvibration constants of this n~olecule by applying the ccluatiotls listed in Paper I. $“I + ($“’ = n,t”“, is = 1, 2, :3>. (a;aa,)‘? + (a;““‘)’ + (a;““’ ? I = II,,) (aa = 0. (s = 4, 5, . . 12; (Y = a, b, L’) as (&hl )? +
A::’
(&h)
2 ) = ilanlbb:ICc ,
[email protected], for all others (LY# 8). a, ==0
+ A,:p”’ = A::”
= 1
for inplane modes Cs = 1, 2, 3, 5, 6, 9, 10, 11, and 12 ),
(:j )
54
KUCHITSU,
OKA,
AND
MORINO
and
[email protected]“) 236 = A;5”’ = I , for outofplane
A’““’ = () SS
modes (s = 4, 7, and 8)
From Eqs. (2) and (3), and from Eq. (19) of Paper I, the following additional relations among the Coriolis constants may be derived (r:PA)” = (rb%))“,
(s::A)” = (s?A)‘,
(ri:,‘)”
= (S20)“,
(~%l)2 = (rm”,
cs&V
= (&3,
The Coriolis constants of Meal and Polo,
(4)
(~&Q2 = (&ill)‘.
can be calculated by the wellknown matrix equation 3” = Ll&i,
where
(5) and The symmetry coordinates and the corresponding elements of the svectors are given in Tables I and II, where the symbols s, c, and p denote sin .$;a( XCX), cos >&x(XCX), and rccx/Rcc , respectively. The symmetry coordinates of the inplane modes are identical with those listed in Refs. 8 and 9 with trivial changes TABLE THE
SYMMETRY
I
COORDINATES~
Q R, r, OL,+: see Fig. 1; 6, angle between the two CX:! planes caused by rotation about a common axis which includes the C atoms; 7, angle between the CC bond and the GXZ plane.
INERTIA
DEFECT
TABLE
II
THE SVECTORS FOR THE SYMMETRY COORDINATEP Cl x CX 
&sx 0 sY
(2P + C)Y
 l/d22 l/42(1
+ 2PC)Z sY CY
&z 11s = sin (a/2), c = cos (a/2), p =rcx/Rcc ; 11, = (!i)(cz + sy), u2 = (;g(c.z L’, = (1$i)(s.z + cy), ~‘2= (i$)(sy  cy); x, y and z are the unit vectors directing
the coordinate

s/J),
tomxrti
axes a, b, and c, respectively. TABLE
III
NONVANISHINGELEMENTS OF THE C MATRIX P’(4, 11) = s/Lx (‘“(4, 12) = &px P(5, (“‘(6, P(7, (“‘(7, (“‘(1, (‘b(2,
8) 8) 9) 10) 8) 8)
= = = = = =
s/v%2(1 + 2wk + /aI 1/&[2(2p + c!(l + 2w)pc + q.~.rl s/Z/Z(2rc + fix) c/v5(2rc + /JX) &l + Bp(‘)/.~c c/&[2(1 + 2p(.)c(c + /~.ul
(‘h(3, 8) =
[email protected](1 Ph(i, 9) = c/Lx (‘“(4, 10) = 8/*x
+ 2pc)~c + PXI
C”(1, 5) cq2, 5) C”(3, 5) P(1, 6) (‘“(2, 6) P(3, 6) P(9, 11) crC(lO, 11) C”(9, 12) CC(10, 12)
= aspc = 2sQlc = = = = = = = =
v%wJc + /ax) 2(2p + c)/.K 2C(2P + c)crc + P_X 2&s(2p + c)/.W 2st;uc (2l”$c + /.&X) 2/3(2s$c + /Lxx) 2&wpc
(Th(7, 11) = c/d2cac + PX) (‘h(7, 12) = \/5,S(2MC + /Jr)
in the notation, given
by Arnett
while those of the outofplane aud Crawford
( IO). The
modes are taken
Gmatrix
nlodes are listed in Refs. 8 and 9.’ For the outofplane G44 = !Jx , The nonvanishing
G,T =
PC +
elements
h.~x,
and
of the C lnatrix
elements
equal to those for the inplant
modes, the elemeuts
Gsa = (1 + 2pcJ2pc + lip,. are shown in Table
are (6)
III.
* The symmetry coordinates and the G matrix elements are also consistent, with the corm responding expressions given by rZrnett and Crawford (10).
56
HUCHITSU,
OKA,
NUMERICAL
AND
MORINO
COMPUTATIONS
F MATRICESANDVIBRATIONALFREQUENCIES The elements of the F matrix are first taken from the UreyBradley force coustants listed in the papers by Scherer aud Overend ( CzH4 , C2D4 , and C2Br4) ( 11) , and by Mann, Fauo, Meal, and Shiruanouchi (CzF4and C&l,) ( Ii?). These constants were derived by fitting the calculated frequencies to the observed fuudarnentals by means of a leastsquares procedure. The frequencies calculated from these F matrices are all in agreenieut with those reported by the above authors. Approximate correction for the anharrnouicity to obtain the normal frequencies is made, as discussed by Arnett and Crawford (lo), by estilnatiug an effective parameter .L‘,in the expressiou taken after the case of a diatomic rnolecule, wa = we  2zeue , where wgarid we denote the fundamental aud the uornlal frequencies, respectively. Siuce the CH( CD) bondstretching vibrations are well localized, it seems plausible to assume that the xe value is nearly equal to that of the CH( CD) diatornic molecule, 0.0225 ( 0.0165) ( 13), because the anharnionicity of vibratiou of a diatomic molecule is in many cases showu to be trausferable to that of the correspouding baudstretching vibratious in a polyatomic molecule (5). If this is true, the above .c, must also be a good guess for other polyatomic hydrides and deuterides. This proposition is giver1 support in Table IV where the effective parameters .re are sunmarized for molecules for which reliable uorrnal frequencies have beeu deterruiued (14~23). Accordingly, the corresponding value giveu by Arnett and Crawford derived by the use of an isotope relation, so N O.OM? for C&H, , seems to be au overestimate by a factor of two. TABLE I\. EFFECTIVE PARAMETERS OF ANH.LRMONICIT~ ze FOR POLY.~T:JMIC MOLE~VLEV Hz0 Yl
“2 V3 Y4
0.026 0.016 0.024
KS
0.020 0.013 0.019
NH3
0.005 0.010
0.019 
0.019
15
16
ND:3
0.017 0.011 0.017
&
14
a Italicized
numbers
Yt YS
0.018
0.086 0.012
14
VI
CzHs
0.024

SO
HCN
0.043 0.019

1%
‘35
0.021
0.019

0.020
co2
(0.007) 0.004 0.010

0.010
17
18
0.013 19
CDI
DCX
cse
X20
0.015 0.030 0.017
0.026 0.009
0.007 0.009
0.011 0.002
0.009 O.OOfi
0.010 0.009
0.016
0.016
0.008
0.013
0.007
0.014 16
0.015 17
18
21
22
25
correspond
to the CH(CD)
stretching

vibrations.
20 ocs

INERTIA
I)EFEC:T
TABLE
\.
\.IBR.ITION.\I. I?REQIIENCIES OF CsH4 AND
CZDC (IN CM') C2D.l
C& calc(lP (L‘1
UJ., w:, WI w.I wti wi WY
W!,
3150 1652 1384 1052 3198 1’271 974 91%. 6 3240 835.5
w, ,( WI1
3127
WI.'
1493
calc(II)h 3004 1649 1333 1O“T Y 3100 1228 949 043 3112 814 3023 1447
ohs 3019.3 1623.3 1342.4 1027 3075 1236 949.2 943 3105.5 810.3 2989.5 1443.5
calc(I)c’
calc(II)*
2324 1543 1005.i
2205 1506 995 726 2312 999 720 780 2314 586 2195 1OMi
i14
2387 1034 737 798 2421 598 2267 1102
‘1 Approximate normal frequencies est.imated in t,he present study. ‘I (‘alrulated hy using the force ronstants given in Ref. 11. The frequencies s:Pnti:tlly with those given in the reference. c Observed fundamental frequencies listed in Ref. 24.
ohs 2851 1515 9X1 7% 2304 1OOli 720 i80 2345 (5x4) 2200.2 1077 .!)
wgr~
es
1'~ the (‘H( CID) bending and the CrC stretching vibrations, silnilar tentative estimates of the .rL , 0.016 (0.013) and 0.009, respectively, may be nlad(J fro111 the observation that they are also nearly constant for the molecules listed it1 Table IV. The latter guess is made by analogy with the corresponding .I’, for (‘02 ( CS~ , X20, and OCS. This is slightly larger than the .I’, of the C2 n~olrcul~~ t1.j). 0.0072; it is also shown t,hat by the Jlorse approximation with arl estilllat,cd valw of n, 2.3 A‘, and with a force constant k of 11 nld 4, .I’, = hcwa’ 21; _ 0.008. Approximate normal frequencies derived in this way from the obscrvcd futldamental frequencies (24) are further adjusted slightly (less than 0.4%;. ) in order t’hat the product rules Inay be satisfied; the results are given in Table \:. ‘lk elenlwts of the F nlat’ris are determined uniquely from those frequencies except for the elements of the ;l,, mode, where an additional relation between the offdiagonal elements, Ci) F’,,a(41,, 1 = Fll
is assunwd.’ The F elenwnts list,ed in Table VI as (I) are in close correspondwlw, except for F,.? and F9.1,1 ,with t,he values (II 1 which are derived from the fwdalIwrrta1 frecluencies. a This common approximation (postulated in the case of the simple UreyBradley force field, for example) is equivalent to assuming in the internalcoordinate system that 2J,,. = of the potential terms which are .Tro’ + jlB” , where .frn’ , f,+’ , and ,f,+” are the coefficients proportional to &~:&xs~ , AraAqbJ , and Ar4A+s , respectively (see Fig. 1).
TABLE
VI
ELEMENTS OF THE F MATRIX OF ETHYLENE (IN MD/A)
A,,
Al, B1*
1, 1, 1, 2, 2, 3, 4, 5, 5, 6,
1 2 3 2 3 3 4 5 6 6
o The elements
CdL(I)”
Cd%(II)*
9.519 0.427 0.320 5.619 0.05Oc 0.416 0.657 5.465 0.180 0.569
8.527 0.702 0.233 5.140 0.103 0.426 0.626 5.140 0.178 0.531
which correspond
& &l &
&I
to the approximate
7, 7 8, 8 9, 9 9, 10 10, 10 11, 11 11, 12 12, 12
C&W
COHN*
0.965 0.706 5.498 0.279 0.417 5.528 0.050 0.397
0.919 0.673 5.140 0.178 0.383 5.140 0.103 0.377
normal frequencies
given in Table
V. b Calculated by using the UreyBradley c Assumed to be equal to F(ll, 12).
force
TABLE
constants
listed
in Ref.
11.
VII
ELEMENTS OF THE L1 MATRIS (IN AMU)
1, 1 1, 2 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 6, 7,
3 1 2 3 1 2 3 4 5 6 5 6 7
8, 9, 9, 10, 10, 11, 11, 12, 12,
8 9 10 9 10 11 12 11 12
C&L(I)a
CdL(IIY’
CJWP
0.0805 0.9718 0.0175 1.8111 0.2353 0.2708 1.9311 0.0912 0.5122 0.8578 0.9491 0.0050 0.2055 0.7729 0.6830 0.5889 0.9424 0.0561 0.1193 0.9828 0.9784 0.0227 0.1082 0.5480
0.0059 0.9752 0.0351 1.4615 0.2198 0.3528 2.2090 0.0944 0.4586 0.8578 0.9496 0.0072 0.2028 0.7729 0.6830 0.5889 0.9493 0.0303 0.0361 0.9839 0.9753 0.0368 0.1334 0.5472
0.4397 1.2536 0.0360 2.2939 0.6641 0.1250 1.6022 0.0468 0.8091 1.2126 1.2573 0.0291 0.5070 0.9482 0.9029 0.7131 1.2605 0.0798 0.2378 1.3681 1.3457 0.0459 0.2590 0.7392
a Calculated * Calculated c Ref. .%.
by using the approximate by using the fundamental
0.3325 1.2668 0.0657 2.1862 0.6391 0.1804 1.7698 0.0402 0.7967 1.2126 1.2590 0.0258 0.5027 0.9483 0.9029 0.7131 1.2760 0.0352 0.1312 1.3700 1.3384 0.0654 0.2946 0.7377
2.286 0.260 0.013 2.305 4.344 0.142 3.803 0.244 2.513 3.656 2.028 0.183 3.620 1.597 1.646 0.873 2.505 0.180 1.603 3.832 2.665 0.319 2.760 1.687
normal frequencies. frequencies. 58
2.457 0.031 0.016 2.545 5.899 0.469 5.381 0.813 3.406 4.995 1.516 0.343 5.559 1.638 1.746 0.763 2.716 0.179 2.599 5.141 2.788 0.430 4.323 2.127
2.437 0.036 0.004 3.425 8.788 0.943 8.260 1.635 5.074 7.498 1.501 0.332 8.567 2.171 1.821 0.722 2.842 0.111 4.421 7.606 2.944 0.481 7.014 2.996
INERTIA
DEFECT
TABLE sQG.kRED
CORIOLIS
YIII
COTPIJNG COPSSTANTS, ({!z’)2,
OF
CZSP. h, ’
a, s,
s’
C&L(I)
C,H&I)
CPDI(I)
CzDd(II)
C?E’,
crctr
a, 4. a, 5, a, 7. II. 1. I), 2, c. 1, c, 2, c, 1, c, 2, (‘, 9,
11 8 9 8 8 5 5 (i fi 11
O.(i596 0.7039
0 .6350 0.7013 0.7325 0.3398 0.6144 0.0157 0.6803 0.7651 0.0290 0.0110
0.6098 0.7441; 0.8290 0.4628 0.3117 0.0748 0.2800 0.7729 0.0405 0.0604
0.5537 0.7417 0.7614 0.4733 0.3603 0.0854 0 .3438 0.73138 0.0352 0.0x0
0.197 0.924 0.879 0.948 0.005 0.788 0.001 0.190 0.150 0.4il
0.105 0.985 0.925 0.972 0.004 0.930 0.000 0.055 0.159 0. (i74
0.8003 0.3277 0.5405 0.0111 0.5199 0.7916 0.0573 0.0277
0.051 0.993 O.Sfiti 0.989 0.001 0.970 0.000 0.024 0.171 0. xm
,I Calculated by using the I,’ elements given in Table 1.11. * Othrr nonvanishing constants, <::?z ,
C!,4LWLA4TION
OF THE
Com0~1s
CIO~STANTS
The nornlalized elements of the L’ matrix given in Table VII are ohtaiwd by t’he standard method (25). For czH4 and &D, , most of the elements ohtained fro111 the approximate nornlal frequencies do not differ appreciably fro111 t#he corresponding elements of the fundamental frequencies, as cxpccted fro111 ?I the relation imposed on the L’ matrix, LP’GL = E. For C&r , C&l4 and CJzBr4 , the force constants in terms of the modified UrqBradley field, which lead to vibrational frequencies in closer agreement with thcb observed fundamentals, have recently beeu deterrniued by Shimanouchi and Hiraishi (26 1. Accordingly, calculations for these molecules are made by using tjhe L’ matrices based on both the old (11, 12) and the new (26) sets of the force constants. Since, however, the results are essentially similar, only those bawd on the latter will be listed in the following tables. Sonvanishing Coriolis constants arc calculated by using Eq. (5). Irldepetldwlt elelllel A” are listed in Table VIII. INERTIA
According
to Eq. (27) of Paper
I)EFECT
I, the inertia
defect A is given by
3 Rigorously speaking, S:i’, rig’, <:6’, and {hi’ are not independent relation
in Eq. (2).
because
of the last
60
KUCHITSU, A = c
A,(v, +
?hj) + Am
OKA, AND MORINO
+ Aelm 7
(8) !
’
where s, s’ = 1, 2, . . 12, and 6,t is a Bronecker’s 6 which shows that the last term exists only when s refers to one of the outofplane vibrations: t = 4,7, and 8. ” For the ground vibrational state, the vibrational contribution Ao,vibis given by Ao.vii, = p;c
A, .
(9)
Although the first term of A, in Eq. (8) has a dit?ereuce in the frequencies, as’  & , this “resonance” disappears from the expression of A0 , by partial cancellation of the contributions from A, and A,, . The centrifugal contribution, Acent, is shown by Eq. (27) of Paper I to be Are,rt = A2B’(2/AY + 2/B” + IS/C’,&,
(10)
where f56 = (a:“b’)L/Wsz+ (as(ab))2/w6?7 (a61 = a5
4srcx(cL&1
+ .sLgt),
arid Cab)
a6
= 4srcx(cL;;
+ sL&p.
Hence the term ACentcan be calculated by using the masses, structure, and the L’ matrix elements. amu The contribution from the 7 electrons, Aeiec, is assumed to be 0.0052 A’ after the argument given iu Paper III. The individual contributions and the total inertia defects for the ground vibrational state are given in Table IX.4 The inertia defect for any excited vibrational state (except in the case of accidental degeneracy) may easily be calculated by addiug v,A, to A, . 4As stated in a preceding section, computations based on the force constants for CZF, , C&l, , and &Bra taken from Kefs. 11 and 18 give A, values similar to the corresponding values listed in Table IS. The following diRerent from the tabulated values:
CzF4 : A9(0.377),
A, terms,
are found to be appreciably
An(O.469);
CLX,
: Aa(0.226),
a~(0.664),
C?Brd
: A*(0.033),
A3(1.845), Ae(0.595j
The resulting inertia defects &Bra , respectively.
however,
A. are 0.15~
A.9(0.474),
An(0.660),
Alz(O.435);
(in amu AZ).
, 0.258 , and 0.455 (amu AZ) for C&F, , C&21,, and
INERTIA TABLE CALCTL.ITED
AI,., /I, 1 SrIll &.I,., s<, ,,’
C&a (IN AMI .42)
C,D,(II,”
CzFS
GCl,”
C .JB r,”
0.0500 0.0628 3.81WcI 0.2265  0 OWli 3.1511c 0.0768 0.252G 0.0276 0.4138 0.02M 0.2M.ul
O.O5(i 0.023 0.520 2 .338c 0.176 0.043 0.255 0.287 0.157 2.22(iC 0.213 0.220
0.015 0.091 0.838 1.5106 0.258 0.521 0.555 0.235 0.273 1.3’32, 0.467 0.322
 0 .0:3:3  0 .19Q
0.0757 0.0007
0.1% 0.0003 0.005 0.152 0.010
0.257 0.0002 0.005 0.25’2 0.020
0 $20 0.0004
0. ona 0.0991 0.07K 0.1296 0.0221 0.3050 0.0135 0.1979 0.0151 0.313G 0.0107 0. 19lifi
0.0763 0.0‘491 0.6204 0.2581 0.0913 0.2421 0.1110  0.2586 0.0318  0 ,394:~ 0.0320 0. “$25
0.0532 0.0008 0.0052 0.0488 0.005
0.0553 0.0009 0.0052 0.0510 0.005
0.0731 0.0007 0.0052 0.0086 0.005
2 3 t 5 (i 7 x !I 10 11 1”
OF
&D,(I)”
0.0109 0. OiU 0.0108 0.1587 0.0249 0.2607 0.0518 0.1978 0.0222 0.29l9 0.0191 0.1850
x=1
IS
IXERTI.\ DEFECT
caL(
1,
61
DEFECT
0.0052 0.0713 0.005
1.527 2.lxL” 0.279 0.251 0.33li 0.278 0.Yll 2.217’ O.(iO(i  0 3w
0.005 0.115 0.0m
11Calculated by using the approximate normal frequencies. I, Calculated by using the fundamental frequencies. r 1Tnreliahle because the difierence between the frequencies is accidentally small. This. however. does not affect A(, , sinre in that caase the resonance terms cancel each othrr. rf Estimatted uncertainty. TABLE COMP.~RISON
H
s = S2C=CS2
(cslc) D
s,c‘=o (ObS)C S&=CH2 (ohs) cisHSC=C.YH (ok)
S
OF THE INERTIA DEFEI.T
0.050 0.057t
D
(IN AMI; .42)
Cl
_.I’.
0.070 0.0777
0.152 0.1565 0.152’id 0.1688”
0.252 0.251 0.23e 0.2290
Br 0.115 (0.3250)
values refer to the most abundant isotopic species. ‘? Present study. r Taken from Paper III. A calculated value is listed for Br&O. .( Ref. 29. ci Ref. 27. c Ref. 28. y Ref. SO. ‘1 Observed
EFFECTS OF INPLANE ANI) OUTOFPLANE (‘ouiparisou
\71BIZATTONSON A,,
of the preseut results with the inertia defects of analogous X.
It is interestiug
~uol~~
cules is given
in Table
to note that the inertia defects
the &Xtype uioleculw (X
molecules are close to those of the corresponding X&0type = H, n, Y, (‘1, and Br), H2CCX2type molecules (X = F and
of
62
KUCHITSU,
OKA,
AND
MORINO
Cl), and cisHXCCXHtype molecules (X = F and Cl), although the moments of inertia of C2X4 are certainly larger than those of the latter molecules. In Papers II and III of this series, it was shown that up to fouratomic molecules the heavier the molecule, the larger the inertia defect. On the other hand, there are reliable experimental results which show that the inertia defects of larger molecules are nearly equal to zero, or even negative (31) . There must therefore be a turning point beyond which the inertia defect no longer increases with the increase in the mass and size of the molecule. Ethylene derivatives seem to be the molecules which are nearly on this turning point. The reason why the A, of C&Xl is close to that of X&O may be understood by the following general consideration of the effect of vibrations on the inertia defect of a planar molecule. According to Eqs. (8) and (9), the vibrational contribution to the inertia defect for the ground vibrational state, A~,~ib, is given by k + $

w s (11)
Symmetry consideration shows that a set of two inplane vibrations, say n and n’, produces a vibrational angular momentum along the outofplane axis (the caxis). Since in this case the coefficient of the squared Coriolis constant in Eq. (11) is positive, interaction between two inplane vibrations always increases the value of A,. Consequently, if the subtotal of Ao,“ib which consists of the interactions between an inplane vibration n and other inplane vibrations 7~’ (71’ < n) is provisionally designated as D, , it follows that
(12) On the other hand, a set of an outofplane vibration and an inplane vibration produces a vibrational angular momentum only along the inplane axes, while a set of two outofplane vibrations makes no angular momentum. Hence a sum of the terms of Eq. (11) which are related to an outofplane vibration t, which we will represent by D, , may be written as follows:
where the summation goes over all inplane vibrations n. By using the sum rules which are generally shown by the equations given in Paper I to exist among the Coriolis constants of a planar molecule,
INEliTIA
I)EFECT
64
KUCHITSTJ,
OKA, AND MORINO
Eq. (13) is simplified as
&& Wn
G24’)”+
(ri?)'1
+
n
< 0.
(14)
It is thus evident that each contribution D, from the outofplane vibration tends to reduce the A, value, and the lower the frequency of the vibration, the larger the decrease in the inertia defect. Returning to our particular problem, it is demonstrated in Table XI that by going from X&O to X&CX2 , the decrease in the inertia defect of C,X, , caused by two extra outofplane vibrations, is sufficient for compensating the increase caused by the additional couplings among the inplane vibrations. For a more specific demonstration of this compensation, the Au,vih for &X4 and for X&O are expressed in terms of D, and Dt as follows.
where
D, = D+, + D,,=6 + Dw
(15)
On the other hand, AO,vib( X2CO)
where Y, T, and
=
DI'= D:=,+ DL
+ DL
D',in place of n, t, and D, respectively, refer to the
X&O
mole
INERTIA
cule for the sake of clarity. expression
Tahlc
It is therefore
as Dr if the following
XI
shows that
especially
there
correspoudence
aud deutcrides,
of the W’S and c’s of C?X, and X,c’O. tak(bs place among
the additional
way t’hat Dir + D,,,
show11 that &’
is also a numerical
for the hydrides
fi.,
I>EFECT
of the indices
correspondence
because
Furthermore,
contributions
the sanlc
is made,
of 11, with &‘,
of the effective
almost complete
sinlilarity cancellatio11
to the A,,,,il, of CnX,
0; this nlakes the iuert’ia defect

has exactly
in such a
of <‘&X4 approxinlatcly
c>clual t#o that of X&O.
As may he seen from Table by using approximate
IX,
normal
the iuertia defects of C2H, and C,D,
frequencies
are nearly
values calculated
by using the fundamental
argunleut
to that
Table
similar
IX
are estimated.
(.$) are therefore
freyuencies.
given in Papers The
different
II and III,
experimental
values
from the present
results
calcula,ted
equal to the correspondiug From this, and from thcl the uncertainties
given in
of Dowling
Stoicheff
and
by ulagnitude
just
heyoud
that of their quoted uncertainties. Ou the other
hand,
from the present
(52j
and l’lyler
it is now possible
A, and the rotational by infrared
to obtain
coustants
spectroscopy;
the rotational
constants
R, and Co determined
&I,,
by Alle11
they are found to be
&(C,H,)
= 1.938 f
0.009 en’
il,,(CzDd)
= 2.459
0.007
and
These
values,
however,
values deterlnined f
0.008
en‘.
A
are in significant
by Dowling further
f
discrepancy
and Stoicheff
spectroscopic
cn‘. with
(,/t ) : 1.828 f
iuvcstigation
the
0.009
corresponding cn’
011 this poiut
and 2.43~ seenls to be
necessary. An empirical nearly
rule discussed
equal to the geometrical
Y2(‘Or carmot
IT&TX2 perfectly
in Paper III,
be applied to the relation
, and H,CCH, in the
cases
that
the inertia
mea11 value of the inertia
, although
the
of H&YZE’C’l
among the inertia same
(from
rule
H&C1T2
defect
defects
of XYCO of X2C0,
defects
is shown
to
in and
of X2CC’X2 apply
and H&CC&)
,
almost
($3)
and
cis HP’CCHCI (from HFCCHE’ aud ClHCCHCl) (,zU+).If the rule is also valid for the HC,I\;s molecule (from H&C’S, and X&CS,) this nlolrcule is expected to have an inert’ia defect almost the same as those of H&CX, and X,CCXn the inertia defects of the latter n1olecaules are nearly eyual.h
, sinc*r
’ Since t.he present paper was submitted to the journal, the following report has come to the
authors
attention:
I. A. Mukhtarov, Optics anti Spect,roscopy 15, 303 (1963). He has
66
KUCHITSU,
OKil,
AND
MORINO
It seems of much importance to determine by spectroscopic studies the inertia defects of ethylene and of various halogenated ethylenes, C&X, , C2HX, , CzHzXz , and C2H3X, in the excited vibrational states as well as in the ground state. Such studies will provide further valuable iuforruation on the normal vibrations and on the quadratic potential constants of these molecules.
The authors are indebted to Professor T. Shimanouchi of the University of Tokyo for informing them of the result of his normalcoordinate calculations before publication. RECEIVED
January 14,1964 REFERENCES
1. T. OK.~ AND Y. MORINO, J. Mol. Spectry. 6, 472 (1961) (Paper I). 2. T. OKA AND Y. MORINO, J. Mol. Spectry. 8, 9 (1962) (Paper II). 3. T. 0~~1 AND Y. MGRINO, J. Mol. Spectry. 11, 349 (1963) (Paper III). 4. J. M. DOWLING .~ND B. P. STOICHEFF,Can. J. Phys. 37, 703 (1959). 5. K. KUCHITSU,T. IIJIM~, .~NDY. MORINO, International Symposium on Molecular Structure and Spectroscopy, Tokyo, 1962; K. KUCHITSUAND Y. MORINO B,uZl. Chem. Sec. Japan (in press); D. R. HERSCHBACH AND v. W. LAURIE, J. Chem. Phys. 36, 458 (1961). 6. J. H. ME.%L AND S. R. POLO, J. Chem. Phys. 24, 1119, 1126 (1956). 7. G. HERZBERG,“Infrared and Raman Spectra of Polyatomic Molecules,” Van Nostrand, Princeton, New Jersey, 1945. 8. Y. MORINO, K. KUCHITSU, .IND T. SHIMINOUCHI,J. Chem. Phys. 20, 726 (1952). 9. D. E. MANN, T. SHIMBNOUCHI,J. H. MEAL, .~NDL. F.%N~, J. Chem. Phys. 27, 43 (1957). 10. R. L. ARNETT .~ND B. L. CRAWFORD,J. Chem. Phys. 18, 118 (1950). 11. J. R. SCHERERAND J. OVEREND, J. Chem. Phys. 33, 1681 (1960). 12. D. E. MINN, L. FANO, J. H. MEAL, .~NDT. SHIMANOUCHI,J. Chem. Phys. 27, 51 (1957). IS. G. HERZBERG, “Spectra of Diatomic Molecules.” \‘an Nostrand, Princeton, New Jersey, 1950. i4. W. S. BENEDICT,N. &IL.IR, AND E. K. PLYLER, J. Chem. Phys. 24, 1139 (1956). 16. H. C. ALLEN AND E. K. PLYLER, J. Chem. Phys. 26, 1132 (1956). 16. W. S. BENEDICTAND E. K. PLYLER, Can. J. Phys. 36, 1235 (1957). 17. L. H. JONES AND R. S. MCDOWELL, J. Mol. Spectry. 3, 632 (1959). 18. A. E. DOUGLAS AND D. SHARMA, J. Chem. Phys. 21, 448 (1953); E. K. PLYLER, E. D. TIDWELL, AND H. C. ALLEN, J. Chem. Phys. 26,302 (1956); D. H. RANK, G. SKORINKO, D. P. EASTMAN, AND T. A. WIGGINS, J. Opt. Sot. Am. 60, 421 (1960). 19. T. A. WIGGINS, E. K. PLYLER, AND E. D. TIDWELL, J. Opt. Sot. Am. 61,121Q (1961). 20. C. P. COURTOY, Can. J. Phys. 36, 608 (1957). 21. D. AG.~R, E. K. PLYLER, AND E. D. TIDWELL, J. Res. Null. Bu.r. Std. 66A, 259 (1962). 26. J. PLIVA, J. Mol. Spectry, 12, 360 (1964). 23. T. WENTINK, J. Chem. Phys. 30, 105 (1959).
‘
given the rotational constants of HFCCF, determined by microwave spectroscopy: A = 10665.5 MC, B = 3872.4 MC, and C = 2837.9 MC. The inertia defect of this molecule, calculated to be 0.190 amu AZ, is appreciably larger than those of H,CCF, (0.1527) and F&CFz (0.152), so that the meanvalue rule does not seem to hold in this case.
INERTIA
$4. B. L.
67
I)EFECT
CRAWFORD, J. E. L.~Nv.~sTER, .YNU R. C. INSKEEP. J. Che?)I. Phys.
J. C. 1)ECIUS. .\ND P. C. CROSS, “Molecular
$5. li:. B. WILSON, New York.
\ibrations.”
21, 678 (1953). McGrawHill,
1955;
R. C. &LIKE,
I. M.
MILLS,
W. B. PERSON, AND A. CRA~FORL), J. (‘henr. Ph,ys.
26, 1266
(1956). 26. T. SHI.MANOUCHIAND J. HIR~ISHI, Symposium 1963 (private ,?7. \. W.
$9. 31.
33. 33. 34.
on Molecular
Structure.
Sendai, October,
).
LaIRIE .YND I). T. PEN(‘E, J. C’he~z. Phys.
38, 2696 (1963). 34, 291 (1961); \.. W. L.YVRIE AND 1). T. PEN(‘E, J. Chem. Phys. 38, 2693 (1963). 8. HEKINO AND T. NISHIHAW.\, J. Phys. Sor. Japan 12, 13 (1957). W. H. FLYG.\RE :\ND J. A. HONE, J. C’hem. Phys. 36, J10 (1962). See, for example, T. &JIM.\, .I. Phys. Sot. Japan 15, 281 (1960); II. li. LIDE, JR., J. (‘hem. Phys. 37, 2074 (1962); I:. L. POYNTER, J. CherlL. Phys. 39, 1962 (1963). H. C. ALIEN *ND E. Ii. PLYI~ER, J. .4m. Chem. Sot. 80, 2673 (1958). .J. Ii. BR%G, T. C. MADISON, .YND A. H. SH.~RB:+VGH, Phys. Rev. 77, 143 (1950). .J. .4. HOWE, J. Chem. Phys. 34, 1247 (1961).
28. \.. W. L.lIRIE,
SO.
communication J. (‘hem.
Phys.