Calculation of inertia defect

Calculation of inertia defect

JOURNAL OF MOLECULAR SPECTROSCOPY 8, 9-21 (1962) Calculation Part II. Nonlinear TAKESHI Department of Chemistry, OKA Faculty of inertia Defect ...

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JOURNAL OF MOLECULAR SPECTROSCOPY 8, 9-21 (1962)

Calculation Part II. Nonlinear TAKESHI Department

of Chemistry,

OKA

Faculty

of inertia

Defect

Symmetric XV, Molecules AND \7~~~~~

of Science,

MORINO

The Ukversity

of Tokyo,

Tokyo,

Japan

The general formula for the inertia defect previously reported by the authors was applied to the case of nonlinear symmetric X17, molecules. The potential constants adopted in the calculation were those determined from the vibrational frequencies and additional data which were either vibrational frequencies of the isotopic molecule or the cent,rifugal distortion constants. In case where the additional data are not available, potential constants were determined with the assumption f,, = 0. The calculated values of inertia defect agreed well with t,he experimental values for most molecules. The agreement was satisfactory also for the inertia defects of the HZO, HZS, and H2Se molecules in their vibrationally excited states. 1NTROI)UCTION

The general formula for the inertia defect was derived by the present authors in a previous paper (1). In the present article, the formula is applied to the cases of Cz, triatomic molecules. In these molecules there are three normal modes of vibration: two of them belong to A1 species and t*he other one to B, species. Symmetry considerations for C?, molecules will be found elsewhere (2). In the case of Cz,, triatomic molecules, it is seen t,hat only two Coriolis coupling constants, lja’ and <:n’, are nonvanishing. The out,-of-plane vibration is absent for triatomic molecules. Thus the expression for the inertia defect of CaUtriatomic molecules is writ,ten as follows. A = A,il, + Aoent+ Aelec7

10

OBA AX11 MORISO

whcr~ A\.il, is the part, of inertia defect drpendrnt on tht vibrational state ald A,.,.,,,and A,,,,., arc corrwtions due t,o the rffwt of wntrifngal distortion :LII~ clwtrolrir interaction, rcspectivcly. ATi,, agrees with the wpression of inertia defwt given 1)~’Ikuling and Ikminon for t,he Hz0 molecule (~3). In order to calrttlatr the inertia dcfwt, in amu ?I’?, the \+hrationul frt~c~unwirs as in rm-‘, wilt rifltgal fi4~,,h,,t,and rotationnl distortion cwistant constants L1, I<, and C’ ill NC ‘SN, mortwiits of inertia I,, ill ntnn X2, and 11.T’)I: = 181.!)01 and ~1 JI = I IXM shot&~ ht> suhst,itnt,rd in the expression. In addition, when the general wlatjions ( 19 ), (20), and (23 ) of Tkf. 1 :Irf’ :lpplicd to V2,, triatomk n~olt~cules, thr following relations arc swn to c&t,

In order to e\-aluutjr t)hr inertia dcfwt for C’?,.triatomii molrculr:: S IT2 , that vil)ration:~l frtquencies w1 , wa , and wi , thr geometrical paramctrrs r and TV( Fig. I 1, thr potential vonstnnt F,, , md the diago11:lI c>lrments of thr y-trnsor y,Irj , Qt,h, and y,., are necessary. When t’hcsr are knowl~, the C’oriolis coupling cwwtants :irc c*alcul:~trd from the following rrlntjions dctriwd lqr T\Ir~al alld 7’010 (_i 1.

INERTIA

DEFECT

TABLE STRUCTURES

Molecule

r(;\)

Hz0 H,S H2Se X0,

0.9572 1.328 1.460 1.197 1.278 1.432 1.49 1.3896 1.70 2.00

03 SO? ClO? F10 Cl20 Cl,S

11

I

OF Cz, TRIATOMIC

MOLECCLES

28 104” 92” 91” 134” 116” 119” 118” 104” 110” 103”

TABLE

Method 32’ 12’ 0’ 15’ 45’ 32 30’ 10’ 48’

Kef. 5

I. 11. I. R. M. w. M. W. M. w. M. w. I. II. M. w. 15. D. I’. D.

6

7 8 9 10 11

12 15

14

II

VIBRATIONAL FREQUEXCIES 011’ THE CzU TRIATOXIC MOLEWLES IN cm-1 Molecule

wi

Hz0 D,O

3832, Ii 2763.80*

HZ8 D,S

2721.92 1906 (1952.98) 2344.50 (2357.11) 1686.70 (1677.76) 1357.8 1110 1151.2 962.8 929 688 514

H?Se D& NO2 0, so, CIO? F,O Cl& Cl,S

1648.47 1206.39 (1205.93) 1214.51 898 (872.17) 1034.21 i103s.00) 741.42 (738.73 756.8 705 517.8 155.1 161 (320) 208

1

w

Wror wg

Ref.

3942.53 2888.78

e e

5 5

2733.36 1940 (1963.88) 2357.80 (2371.09) 1697.36 (1687.97) 1665.5 1043 13Gl 1128.2 826 969 535

e 0

6 15

0

16’

0

16

0 0 0 0
I7 18 19 it $0 $1 2

* This value was corrected by a private communication from Prof. E. K. I’lyler. The values in the parentheses are the values calculated using the adopted potential constants. \vhere

(‘-27~cw)‘). F and G arp mat,ricw with respect used in the calculat,ion of the normal vibration.

X, =

ordinates

the G matris

and is easily calculated

parameters are known. The molecules for which

t’o the symmetry coG-’ is the inverse of

when the atomic masses and the geometrical

geometrical

parameters

and all of t’he vibrational

frequencies are available, are considered in the present paper. They include &O, &S, &Se, n-0, , (JI , SO2 , C102 , 1c20, Cl&, C&3, and their isotopic speries

OKA ANI)

12

hTENTIA1,

.\l&CLllC! II,0 11,s II.&~ so: 0: so: (:I( Jz I’,0 Cl,0 (‘I+

C~KSTANT~

TABLE:

III

0~

(I?,,

.fd imtl)

J’r imd/A) 5.454 4.2s4 3.277 10 ,927 5.701 10.006 7.16 :s.os 4 xs 2.s’L

MORISO

TRE

TRIATOMK MOLECULES

./‘H (mtl A)

0 “‘74 -i 0.087 0” 0 .4tji 0.4w3 0 .‘271

0. tini

0.27

1.40

-0.20 0.76 -0.11

o.i513 1),6732 1 .ti11\1

2.09x3 1.

J‘rr (md/.\) -0.100 -0.011 -0 .(I19 ‘7.038 1.523

WBi

2.10 1 .x7 1 ‘77 __

0.024

w 0” w

0”

p‘i, (mtllh1 s.353 4.273 3.258 12.97 7.324 to .03 T.16 3.08 4.88

f f f i i dc f i f

0.02 0.05 0.0 0.1 0.1 0. I 1 1 1

2.32 It I

3 .\ssllrI~c~cl vnlllc~s. with CYzPsymmetry. The geomet rknl parameters :md vibrational frcclrwwirs rwd arc listed in Tahks I and 11. The ~~a1uw shown in thp parent,hwts in ‘I’ahk II arc thcb c&wlat~ed values ohtaiwd hy using the adopted potential (wI1stant~ list4 in Table III. I’OTESTIAL III ordcl~ t’o

F,, is nccwsary. pl’c’sd

(X )NSTA?;TS

calculate the Coriolis coupling voustant’s the potrntial constant The G and F matriws for the C’?,,triatomic molec*nlrs arc’ cc-

:1x

0 ‘, n11d

1’

=

where Jl,~ denotes the mass of the central atom and II, The potential funct’ion of t,he general form

2V = .frl(Ar,)’

is that of the cqd atom.

+ (Ar,)“} + 2jr7(ArI)(Ar,) (0 )

+ .fdAd

+ 2.fv.e(A.r, + Arz)(AO)

INERTIA

13

DEFECT

is used. Since only three vibrational frequencies are available for the four unknown constants in the F matrix, t,he potential constants can not) be determined from the vibrational frequencies of one isotopic species alone. For such molecules as X02 , 0, , and SO2 , t]he pot#ential constants have been det,ermined by using either the vibrational frequencies of the isotopically substituted molecules or the centrifugal distortion constant,s (17, 93, 24). For H20 and HsSe, accurate vibrational frequencies of the deut’erated species are available (5, 16), and the pot’ent,ial constants were det>ermined from them. In the case of HzO, the reported vihrat,ional frequrncies of the isot’opic molecules .at*isfy well the product] rule

and the potential

constants

were determined from WI, wz , and w3 of Hz0 and

I -1

0

FIG. 2. F,, versus F,? for the H$

-

molecule

12

14

OKA

AND

MORINO

0.000 * 0 .004 f 0.000 I 0.001 3x 0 .OOll j, 0.001 1 0.241 31 0.400 f 0.101 It 0.08 i 0.75 + 0.41 zt 0.44 *

0.003 0

.oo:s

0.015 0.015 0 OOr: O.OOi 0.009 0.018 0.012 0.17 0.19 0.18 0.28

INERTIA

DEFECT

TABLE

V

VALUES OF THE II-TENSORS USED

Molecule H20 HsS D,S

SO2 oi

%a 0.585 0.355 * 0.194 * -0.606 f -2.960 f

0.742 0.195 f 0.086 f -0.123 f -0.227 f

0.008 0.01 0.008 0.008

8

Al

Hz016 H?O’* D,016 D,0’8 HzS32 H,P” D&P [email protected]” HsSe DrSe x 140” pljl& 03 0180? s3*02 wo, CPO2 C13’0a F?O Cl10 c13’2o ClW C13’,S

9cc 0.008 0.01 0.007 0.005

0.0000 0.0030 0.0023 0.0016 0 0000 0 .OOoo 0.0074 0.0058 0.0012 0.0065 0.0714 0.0703 -0.3611 -0.2871 0.0415 0.0392 0.042 0.040 -0.411 0.162 0.171 1.506 1.579

0.666 0.205 f 0.008

0.124 f -0.074 -0.086

f f

0.01 0.005 0.005

VI

OF Av,i,FOR c22,TRIATOMIC

&it> = c Molecule

IN THE CALCULATION

S=

TABLE CaLCUL.4TED VALUES

1.5

MOLECULES

IN anlu A2

&(v, + fz’) 47 0.0992 0.0995 0.1349 0.1361 0.1384 0.1385 0.1924 0 1928 0.1607 0.2256 0.1705 0.1786 0.2111 0.2419 0.2739 0.2797 0.325 0.331 0.106 0,279 0.274 0.427 0.422

A, -0.0073 -0.0099 -0.0118 -0.0113 -0.0121 -0.0121 -0.0241 -0.0226 -0.0147 -0.0253 -0.0547 -0.0566 0.3714 0.2747 -0.0402 -0.0398 -0.048 -0.04i 0.567 -0.068 -0.070 -1.360 -1.424

normal vibrations are nearly along the bond directions. For heavier molecules, bending vihrat,ion w’, mixes with the totally symmetric strekhing vibration W, and { becomes

larger. ELECTRONIC

CORRECTION

The contribution of t,he bonding electrons on the inertia defect is by no means very small. In t,he case of the ozone molecule, it amounts t,o about, ten percent

16

OKA AND MORIX0

of

the t,otal inertia defect. The values of the g-tensor were determined by Hurrus (5%) for H& D& SO? , and O3 . They are not available for other molerules. ;\Ioreover the signs of t#he components of g-t,ensor have not been determined except for the water molecule (26). Thus t’he magnit’ude and the sign of the gt,ensor were est,imated in the following manner. All t,he components of the g-tensor were assumed to he positive in the cases of H&3 and D,S by analogy with the water molecule. The c*omponents of the y-tensor were all assumed to be negative in the cases of (Ia and SO2 , btcause in these molecules bonding A electrons and lone-pair electjrons on the oxygen atoms protrttde out, of t,he molecular frame, t#hus producing a negative magnetic moment xhkh is larger in magnitude than the positive magnttir moment produced l)y thr tutclear charges and spherically closed shells. The A,,,, were assumed for the molecules for which the Zeeman effecst has not heen measured. For NO, and CIO, which have double bond structure, Arlrc WTC assumed to be equal t,o that of SO, . For other single-bonded molecules, the effec~t

Molecule &ih.” ~_ ~~___ H#” H,OlX I)tO’S D,O’K kI.LW HP D.&J’ I).&’ H!& &Se S’dO? s 150: ( ).I O’W? SW? d”“O$ CIW.~ C13’O.v I’J) (‘I :q ) C’1~7J, C13~,S C13~,S

* Ass~mhd

0.0460 0.0463 0.0627 0.0632 0.0631 0.0632 0.0879 0 .0880 0.0736 0.1031 0.0936 0.0961 0.1107 0.1148 0.1376 0.1395 0.160 0.162 0.131 0.187 0.187 0.286 0.2X9 values.

&.,:,I, 0 .ouOE 0.000x 0 .ooos 0.000x 0.0007 0.0007 0.0007 0.0007 0.000x 0.000s 0 .OOlO 0 .OOlO 0.0011 (~.OOll 0.0004 0.0004 0.001 0 .OOl 0.001 0 00 1 0.001 0.001 0.001

0 0000 0 0000 0 ooou

o.ouou 0 .ooOI 0 .OOOl 0 ooou 0 .oouo 0:’ 0:’

--O.O03i* -0.0037* -0.0104 -0.0104 -0.0037 -0.003i - 0 ,001:’ -0.001~~ 0:’ (y 0:s 03 0:’

0.0467 0.0470 0.0635 0.0640 0.0639 0.0640 0.0X86 0.0887 0.0744 0.10-12 0.0909 0.0934 0.1014 0.1055 0.1343 0.13a 0.157 0.159 0.132 0.1% 0.18s 0.2xi 0.289

f f. & f + f + zt f f f f * f f + f + zk zt f zt f

.OOOl .OOOl .0002 .0002 .0005 .0005 .OOlO .OOlO .0005 .OOlO ,ooso .oofio .OOlO .OOlO .OOlO .OOlO ,015 ,015 .015 ,020 .O%O .060 .060

0 .0486 0.0648 0 .0660

0.0595 0.1045

0.1017 0.1046 0.134R 0.1365 0.168 0.217 0.197

INERTIA TABLE

17

DEFECT VIII

INERTIA DEFECT FOR THE CPUTRIATOMICMOLECULESIN THE VIBRATIOKALLY EXCITED STATES (IN amu AZ) Molecule

Hz0 DzO HzS

A1

obs. talc. obs. talc. obs. talc.

0.003 0.000 O.OOi 0.002 -0.001 0.000 -0.005 0.001 -0.007 0.007

0.121 0.099 0.143 0.135 0.145 0.138 0.146 0.161 0.217 0.226

A.3

Ref.

-0.002 -0.007 -0.017 -0.012 -0.012 -0.012 -0.013 -0.015 -0.004 -0.025

s 5 6 16 16

of electrons on the inertia defect was neglected. The former assumption would justified, because the X02 and Cl02 molecules have odd electrons, and their electronically excited states are thought, to be closer to the ground state than in the case of the SO, molecule and hence electronic cont,ribubion may be larger. The adopt’ed values of the components of t’he y-tensor are listed in Table T’. The abnormally large g,, of the ozone molecule may be due to the mixing of the electronic excited states owing t’o the rot’ation of the molecule. It is easily shown for the water molecule that I,,g,, + Ibbgbbis nearly equal to I,,g,, . This shows that the bonding electrons rotating with the molecular frame are almost exactly confined to the molecular plane. It indicates also that if we adopt, the familiar model of the wat’er molecule in which the lone-pair electrons on t,he oxygen at,om protrude out of the plane of molecule, we must conclude that the lone-pair electrons rotate, not adhered to t’he molecular frame but with “slipping,” fixed in orientat’ion in space. not easily he

IKERTIA

DEFECT

The inertia defects for Czr,triatomic molecules were calculated from the above considerations. The calculated values are listed in Tables VI and VII. For almost, all molecules there exists a relation w3 > WI > w2, so that A, and A2 are positive and A3 is negative. For E‘ZOand 03 , WI > wa and A, is negat’ive and A, is positive. In any case, however, A,,)>,, is always positive for all the molecules as was mentioned before (1). Aeentis small and the order of magnitude is nearly equal for all molecules. For the sake of brevity of the table, the inertia defect, for S33 isotopic species was omitted. It can be obtained by t’aking the average of t,he values for S3’ and S34 isotopic species with a good approximation. Here we must consider t’he uncertainties of the calculated values. They arise from

a number

of sources

as discussed

below.

18

OBA

AND

NIORI~O

(i ) 7'hc uncertainties CL the pofPntiaZ cmutant.s. The assumption .frr = 0 1tsc4 in the detcwnination of the poknlial constants for some molecules is not always justified. By this assumpt,ion t’hc frcquewy of wzjmust he nearly rclual to that of W, . This is the case for almost all molecwles, esccpt for C&O, whew wl is much smaller than CQas is swn from Table II. Thus if the adopkd vihratiotlal assigtlmcnt is corrwt, the assumpt~ion ,fp,. = 0 is uot, suitable for this molccuir~. This is thv reason for t,he abnormally large value of fvB mentioned earlier. (kwr:ill? speaking, for C?,. triat,onG molwules, t)hr pokntial vountantjs can lx, csprwwtl with one cwmponent as a paramrtrr sinw WC have only thrre olwrvcd fwctucnc’iw for four tmknown wnstants in Ect. (6). (traphs showing the valtw of /I’,, as a fruwtion of Fr2 arc illust,rated in Pigs. 2---l. It is seen that thv straight linv Fn = Yaaintersects the ellipse in t,he upper part and middle part for (‘I( )! :u~tl Ii+ ), respwtively. It will hc understood from thaw graphs that E’ll for II,S at~d (‘lOc is rather stat,ionary when F1? is varied, while that of P’J) rhaugw rapidly. The unwrtninties for Fn quoted in Table II wrc wtimated from swh cwusid~~rat ions. Some mwrtaintics must be take11 into arcount ~WII for P’n of the molccwlw

-3

-2

-1

Fro. 3. F11 versus

0

1 Flz for the

2 ClO,

3 molecule

4 Fxz

IXERTIA

19

DEFECT

],I FIG. 4. P,, versus FL2 for the F&

molecule

which provide us additional dat,a t)o det’ermine the potential constants. The uncertainties in Fll are rat,her serious in determining A1 and A3 , because w1 is nearly equal t,o w3for almost, all molecules and the factor multiplied to (
OIiS

20

AXD

MORINO

( iii ) lrlaccwacy of the zibrational jrequrncies and structural pararrwtcrs. l“m (‘IO! , Cl&), and C&S, the vibrational frequcw+s and the structures havr trot awurately been &ablixhed. This is the reason for the relatively large unw~ taintics assumed for the inertia defect of these molecules. I iv) Lark of the e.rperimental ~a/ues of the g-tensor. This is seriolw wpwiall~ for (‘lOz alld X0, for the reason described in the previous paragraph. ( 1’) Neglect of the high,er-order ferns in the theory. The present trwtmc~lt ot thca vi~)ratiou-rotation interaction is based on the second-order pert twbat ioIl. Higher ord(>r t,erms will be of importance cxpwiallg for light molwulcs for whkh t hr, rot,ational energy is not very small compared with the vibrational t~~wrgy. Aloreo\-cr the ncnr dcgencrwy of wI and w:( will make this wntrihlltiou more swions.

COMPARISON

WITH

THE

ESPERIMENTAL

VALUES

‘lb observed lralues of the inert’ia dcfert in the ground vibrational stntc arc listed in the last c&mm of Table VII. It is seen that the agreement between the observed and calculated ~alucs is quite satisfactory except for the (xsw of H2Sc and F20. The exact agreement in the wses of SO2 and 0, furnishes an evidewe ill favor of the pokntial c’onstants determined for these molwl&~s from t.he wnt’rifugal distortion ronstants. I:or 1120, D,C), and H2S, the c&alatt~d \~~lucs are always smaller than t’he observed values. This disagreement wnnot be wmovctl by adjusting the potential constant s. It’ would he due to the neglwt of the higher order terms in the throrct~ical formulation if the esperimellt al \-slurs are corrert. For H28e t)herc is a collsiderablc disagreement between the c&&ted value and the one ohsrwed by the infrared spwtrum. However the I-ulur for IMc determined by the present aut,hors (27) from the analysis of thr mkrowavr spe&um of .Jache c,t a/. (7) agrees well with thr calrulatrd 1x1~s. The large disagreement in the case of E‘?O wnnot he rcmovrd by adjltstillg the pot,rntial constants. If the experimental value is wrrcrt,, it will provide ,ls :~n interesting problem. The inertia defects for the molecules in the vibrationally excit,ed statrs have IXWI observed for some molecules. The observed alld c*alcl~lated \~~lues fol A,, are compared in Table VIII. The observed \-alucs of A, were derived by the lea&squares procedure from the inrrt,ia defe& in various excited statw. It is swn that, the agreemrnt is satisfactory although it is llot as good as in th(, ,*a~ of &I .

One of the authors (T. 0.) expresses his thanks to Japan Society Science for the award of a post-doct,oral fellon-ship.

RPX~EIVE~: March

6, 1961

for the Pronlotion

of

INERTIA

DEFECT

21

REFERENCES 1. T. (>K.AAND Y. MORINO, J. ~lfol. Spectroscopy 6, 473 (1961). 2. T. oI(A AND Y. MORINO, J. Phys. Sot. Japan 16, 1235 (1961). 5. B. T. DARLING AND D. M. DENNISOX, Phys. Rezl. 67, 128 (1940). 4. J. H. MEAL AND S. It. POLO, J. Chem. Phys. 24, 1126 (1956). 5. W. S. BENEDICT,N. GAILAR, AND E. K. PLYLER, J. Chew&. Phys. 24, 1139 (1956). 6. H. C. ALLER.,JR. AND E. Ii. PLYLER, J. Chew&. Phys. 26, 1132 (1956). 7. A. W. JACHE, P. W. MOSER, AND W. GORDY, J. Fhem. Phys. 26, 209 (1956). 8. G. R. BIRD, J. Chem. Phys. 26, 1040 (1956). 9. R. H. HUGHES, J. Chew Ph,ys. 24, 131 (1956). 10. M. H. SIRVETZ, J. Chem. Phys. 19, 938 (1951). 11. A. H. NIELSEN AND P. J. H. WOI~TZ, J. (‘hem. Phys. 20, 1878 (1952). 12. A. W. JACHE, American Chemical 15ociet.vMeeting, Sept., 1960. 1s. J. I). DUNITZ AND K. HEDBERO, J. .-ln~. Chews. Sot. 72, 3108 (1950). 14. D. P. STEVENSONANDJ. Y. BEACH, J. .4m. Chem. Sot. 60, 2872 (1938). 15. C. R. BAILEY, J. W. THOMPSON,AND J. B. HAI,E, J. (‘hem. Phys. 4, 625 (1936). 16. E. D. PALIK, J. Ililol. Spectroscopy 3, 259 (1959). 17. E. T. ARAKAWA AND A. H. NIELREN, J. Mol. Spectroscopy 2,413 (1958). 28. M. K. WILSON AND R. M. BADGER, J. Chew&. Phys. 16, 741 (1948). 19. G. HERZBERG, “Molecul,zr Spectra and Molecular St,ructure,” Vol. II. Van Nostrand, New York, 1949. LO. H. J. BERNSTEINAND J. POWLING, J. Chena. Phys. 18, 685 (1950). 21. K. HEDBERG, J. Chem. Phys. 19, 509 (1951). 22. E. STAM~MREICH, R. FORNERIS,AND K. WONE,J. (‘hem. Phys. 23, 972 (1955). dS. L. PIERCE, J. Chewa. Phys. 24, 139 (1956). 24. D. KIVEI~SON,J. Chews. Phys. 22, 904 (1954). 25. C. A. BTRRUS. J. C’hem. Phys. 30, 976 (1959). 26. C. Ii. JEX, J. W. B. BARGHAUSEN,AXD R. W. STANLEY, Phys. Rev. 86, 717 (1952). 27. T. OKA AND Y. MORINO, J. Mol. Spectroscopy (to be published). 28. W. E. SMITH, dustdim J. Phys. 12, 109 (1959). 29. R. H. JACKSON,(private communication). 30. R. F. C~TRL,JR., et al., Phys. Reu. 121, 1119 (1961).