Molecular force field and structure of water: Recent microwave results

JOURNAL

OF MOLECULAR

SPECTROSCOPY

Molecular

53,62-76

(1974,)

Force Field and Structure of Water: Recent Microwave Results ROBERT

L. COOK

Department of Physics, Mississippi State University, Mississippi State, Mississippi 39762 AND FRANK Department

C. DE LUCIA AND PAUL HELMINGER’

of Physics,

Duke University, D&am,

North Carolina 27706

Recently, microwave studies of the rotational spectra of water and its various isotopic species have been reported. These studies provide rotational constants and among others the quartic distortion constants, which depend on the quadratic part of the vibrational potential function. These data are collected and discussed, and the molecular force field and structure of water is considered in light of this recent microwave data. The quartic distortion data gives force constants which are very reasonable considering the difficulties in the distortion analysis of these light molecules, where as many as 22 parameters are being evaluated to fit the observed spectrum. The infrared and microwave data are combined within the theoretical framework of the small oscillations model and the results compare favorably with the true harmonic force field. The infrared and microwave valence bond force constants of Hz0 are (mdyn/_&): jr = 7.746, js = 0.700, j,, = - 0.093, fre = 0.3’79. The results further confirm the usefulness of rotation-vibration data in the determination of force constants, and show that even for water with extremely large anharmonicity effects, a very representative force field can be obtained by combining ground state infrared and microwave data. Various molecular structures have been evaluated, and the average structures in the ground vibrational state for HzO, D20 and TtO are found to be: (0) (r) O-H = 0.9724 ii HOH = 104.50° DOD = 104.35” O-D = 0.9687 ii TOT = 104.26’ O-T = 0.9671 ii A one-dimensional approximation to the anharmonlcity effects is applied to determine the equilibrium molecular structure of Hz0 from the average structure data. The result is as follows: ra = 0.9587 d and Be = 103.9’. I. INTRODUCTION The important and familiar water molecule has been one of the most thoroughly studied molecules in the infrared region. Many vibration-rotation bands and numerous

* Present address: Department of Physics, University of South Alabama, Mobile, Alabama 36688. 62 Copyright

Q 1974 by Academic

All rights of reproduction

Press. Inc.

in any form reserved.

FORCE FIELD AND STRUCTURE

OF WATER

63

pure rotational transitions have been measured with varying accuracy and resolution over the past decades. Its equilibrium structure and internuclear potential energy function have been evaluated to various degrees of accuracy from analysis of this spectral data. Recently, extensive microwave studies of the rotational spectra of several isotopic species of water have been reported. Using techniques (I) which have enabled transitions in the submillimeter wave region to be measured, we have studied the spectra of both H20 (Z-6) and H&S (7-9). Other measurements on Hz0 at lower frequencies have been made by use of “conventional” techniques [see e.g. (IO-16)]. The characterization of the spectra of these light asymmetric tops requires a detailed analysis of their centrifugal distortion effects, and to achieve a satisfactory description of the spectra of these molecules, numerous distortion constants have to be incorporated into the analysis. In previous communications we have described the computations and the methods of analysis (2, 7,8) based on the reduced Hamiltonian of Watson (17-19). The microwave studies furnish us with accurate rotational constants and among others the quartic distortion constants, which depend on the quadratic part of the potential function. It would thus seem appropriate at this time to collect and briefly review the recent microwave data, and to investigate its relation to the molecular force field and structure of water. Water is one of the few polyatomic molecules for which anharmonicity corrections are known, so that, the observed vibrational frequencies may be reduced to harmonic frequencies which in turn yield the harmonic force field. The usual situation is to have only observed infrared (ir) data and possibly microwave (mw) data, both of which are contaminated with zero-point vibrational effects. This infrared and microwave data may be combined within the theoretical framework of the small oscillations model and, in the present case, the results can be compared directly with the harmonic force field. Thereby, one may discern how representative the ir and mw force field is of the true force field. II. SUMMARY

OF THE MICROWAVE

DATA

A summary of the rotational constants and quartic distortion coefficients for water and the various isotopes we have studied are given in Table I. The statistical uncertainties for 9.5% confidence are also included. A few general statements about the distortion coefficients can be made. A look at the definition of these constants indicates that for

64

COOK, DE LUCIA AND HELMINGER

planar molecules AJ is expected to be positive, as found. On the other hand, 6.r should be positive for a prolate top, but negative for an oblate top. Furthermore, the sum AJK + AK is positive for a prolate top and negative for an oblate top. All the molecules in Table I are prolate tops, however, the above statements have been found to be true for the oblate tops H& and DzS. It is apparent from Table I that 8~ is consistently larger than 6~. The distortion coefficients listed in Table I are linear combinations of the more fundamental r-distortion constants. An alternate and for the present purposes a more convenient set of constants may be evaluated directly from the spectral constants of Table I. These are ranala, Tbbbb, rceec and the linear combinations 71

=

i-2 =

Tbbcc

ATbbec

+

Taacc

+

+

h-arm

T’aabb,

i-

CT’aabb,

(1)

(2)

with 7’ aabb = Taabb + 2Tabab. Explicit relations between the two sets of constants for planar or nonplanar molecules have been given elsewhere (8). The alternate constants which were not given previously for all the molecules of Table I are listed in Table II. The errors quoted are for 9.5% confidence. The results for DzO have been taken from the work of Bellet and Steenbeckeliers (II). It may be noted that the above definitions given for 71 and 72 are appropriate for a planar molecule, since we have taken racac = Tbcbc = 0. Furthermore, T2 has units of (MHz)~ as compared to the others which are in units of MHz. The value tabmated in Table II has been normalized by dividing 72 by S = A + B + C as has been suggested by Kirchhoff (20). be extracted from Tl and The distortion COMtalltS Taebb, T’aacc, Tbbcc and Tabab cannot 72 without employing the planar relations. Because of these three relations there are only four independent distortion constants for a planar molecule. For discussion of the force field, it is convenient to choose the four independent constants Taaaa, Tbbbb, rceec and Tabat, (21). For the symmetric isotopes, the first three constants depend on the force tonstants associated with the symmetric stretch and the bending motion (A 1vibrations), while Tabab is dependent on the antisymmetric stretching motion (BI vibration). AlTbbbb and Tcccc come directly from the analysis of the rotational spectrum, though Taaaa, ‘Tab&,,as noted before, can be obtained only with the aid of the planar relations. There are in fact various ways of calculating ‘T&b from the constants 71 and 72. Unfortunately, because of vibrational effects the results of the various calculations wiIl not be the same, and hence, it is difficult to obtain a reliable value of Tabab. The results obtained from three different procedures of calculation are also given in Table II. The three planar relations allow the constants Taabb, Tbbcc and Taacc to be calculated from the constants Taaaa, Tbbbb and Tcccc (8). In one case, the three calculated T’S are employed in conjunction with the observed 71 and Eq. (1) to evaluate Tabab. This yields the first entry given for 7&,& in Table II. The second entry is obtained as above except that the observed 72 and Eq. (2) are employed. The last entry is obtained by calculating Tbbcc from a planar relation, and employing this constant, 71 and 72 are solved simultaneously for 7’aabb and T,acc. The Tabab is then extracted from T’,&b by calculating IT&b from one of the remaining planar relations. In the planar relations and in subsequent calculations, unless otherwise stated, the effective rotational constants have been employed. It may be observed that the results obtained for Tabab from ~1 gives a value which is intermediate

FORCE FIELD AND STRUCTURE

OF WATER

.T&le11. quareicDiarortionConsranLs of water ___-_L I?20

cMw=

--B__

~ D20'

wp

T20

65

ED0

-p-ml

HTO

-p T

-3351.9i1.6

anaa

Tbbbb TCCCC '1

I 'abab

-966.5?1.5c

-32.161~O.C1?

-36.326?O.C14

-28.69'0.25

-27.2fL.5

-7.87'0.10C

-3.7"4f0.008

-14.173*0.0?6

-9.535to.C1?

-5.266fC.014

240.511.5

233.3t6.3

68.3611.69'

38.403?0.039

27.711.6

-124.dd

-134.Bd

-85.1e

-107.0e -183.7f

36%

H' C'

835895.3 435156.2 278279.2 835833.1 435094.0 278372.5

825422.3 435136.6 277089.3 825354.9 435069.2 277190.4

4.143~0.009

7.3C

-266.88t0.16 -52.141~0.035

-256.64'0.10 -44.C:*~o.322

-0.37to.07 -3.446fO.Ol.5

-15.gd

-191.0d

-148.G

-39.hd

-i7.4e.g

-13.4e

-:77.6e

-141.be

-35.gE

-45.4f'g

-20.4f

-222.9f

-172.Of

-47.3f

-6.0

-2.2

-13.4

-7.4

-3.5

-2.1

-0.8

-4.4

-2.1

-!_?

20%

312

21%

2;:

-33.4Q

27% Ground

A'

-5*0.02t0.07

-72.52740.026

-9.8

c

-1300.04f0.25

-29.461'0.008

-27.8

B

-1688.49i0.18

-67.12+0.25=

! -194.1*

A

-504.977f0.042

-271.1f1.5

31.6620.37

?eb 2

-3307'11

-272.06f0.25

state

RotaCional

canstonts

of liater (MHZ)

462293.0

338815.8

701941.8

677852.1

4LO180.0

217083.4

145639.7

272047.5

198?64.0

17206P.V

145301.3

100280.2

192110.1

150492.8

11915L.O

462276.3

338808.0

701846.3

677777.7

410160.3

272751.9

1VXC89.7

171050.2

192253.3

150604.4

llR183.6

217966.7 145726.3

145631.9 100291.9

between the other two values. The spread in values is large, for HJ60 the highest and lowest value differ by over 100 MHz. Because of the large range in ~~~~~~this constant is not considered further in discussions of the force field. For a nonvibrating planar molecule, a simple equation relating 7ccceto 71 and 72 may be derived (18,ZO). In Table II is tabulated the difference between the observed T,,,~ and the value calculated from the planarity relation. For the vibrationless state, Arccee would vanish. The effects of vibration are readily apparent from the rather large nonzero values obtained in Table II. Furthermore, there is a systematic decrease in this quantity with increasing mass or effectively decreasing amplitude of vibration. The values of

COOK, DE LUCIA AND HELMINGER

66

where Nielsen’s

B = @ + 16R,j(A

-

C)/(B

- C),

(4)

C = e -

-

B)/(B

- C),

(5)

(23) Rg constant

16R&l

is given by

Rs = The constant

(4&r + r&/32.

7bbce may be obtained from the planar relation

BY? Tbbec

=

~aaaa

__

-

__

2 This constant

in conjunction

tional constants Of more interest

~CCCO

B4

with the AJ from Table

+7.

1

(7)

I gives R6 and leads to the rota-

A, B, C given in Table II. Usually two iterations are required to stabilize are the rotational

ground-vibrational-state

moments

are particularly

A’, B’, C’ which are simply related to the

of inertia,

viz,

h

= -

IbO = __

8??B”

8lr2A’ ’ These are obtained by correcting

constants

8?r2C’

A’ = A +

Tabab/2

B’ = B +

T&b/2

(10)

h<~b&/k

(11)

,

=

c

the evaluation

This leads to uncertainties

which are orders of magnitude

h I,o=-.

A, B, C for the effects of T&b. In particular,

c

As pointed out previously,

large for these light molecules.

constants

h I,0

rotational

Tbbbb +-

A4

the values obtained since the corrections

be obtained.

(6)

-

(9)

of Taba. is not unique and various values may

in the derived ground-state

larger than the experimental

rotational

uncertainties.

constants The A’, B’

of H20, for example, both exhibit a spread of over 50 MHz because

of the different *a!,& values. The effect of this on the structure determination, however, is not very important as compared to the well-known effects of vibration. This is discussed further in Section IV. The rotational calculated intermediate

from 71, as described previously, between

those obtained

constants

obtained using the r&b

are given in Table

value

II. These constants

using the other possible values of

T,,bab

are

given in

Table II. Parenthetically, it may be observed that for nonplanar molecules these distortion corrections to the rotational constants would have to be made by calculating R6, etc., from a known force field. III. MOLECULAR FORCE FIELD CALCULATIONS

The problem of determining molecular force fields has been a subject of keen interest for a long time, and the theoretical evaluation of the force constants has also been of continued interest (24). A convenient and general approach in the evaluation of the quadratic potential constants is an iterative least-squares analysis. In such an analysis, various experimental data dependent on the force constants may be incorporated, such

FORCE FIELD AND STRUCTURE

OF WATER

67

as vibrational frequencies, centrifugal distortion constants, Coriolis coupling constants, inertial defects, and vibrational mean square amplitudes. The importance of supplementing the vibrational frequency data with vibration-rotation data has been pointed out by Duncan and Mills (25). In the present case, considerable data on the vibrational frequencies and centrifugal distortion constants are available, and various calculations and comparisons can be made. For the various analyses described here, 16-digit floating point arithmetic has been employed. The statistical uncertainties in the force constants have also been evaluated. In general, requiring the data to be fit within the small oscillations approximation will introduce the major limitation on the results. Since the model errors will usually be dominant, the statistical errors obtained must be viewed with some caution. In addition, the observables have been analyzed with various weighting. In these calculations, the weight matrix is assumed

.-I. Infrared Vibrational Frequency Data

Let us first review the results for the quadratic force constants in the most general valence bond potential function when vibrational frequency data is employed. The vibrational frequency data utilized in all the calculations of this paper are summarized in the footnotes of Table III. The frequency VZ(WZ)of HDO is taken from Gailar and Dickey (26), while ~1 of Da0 has been corrected as noted by Shimanouchi and Suzuki (27). ,411other frequencies are from Benedict, Gailar and Plyler (28). The vibrational problem is linearized by means of a Taylor series expansion

= Wk” +

where Xk = (~s”c~/#)o~~, with X Avogadro’s constant, c the speed of light, and wk a fundamental vibrational frequency in cm-’ units. The Lii are elements of the L-matrix, where Xk = Ci,j LikLjkFij (29). The atomic masses and fundamental constants have been taken from Gordy and Cook (30). F rom an approximate set of force constants F”, the quantities who and Lij” are calculated. Calculation procedures have been described by various authors [see e.g. (31-33)]. Equation (12) is then used as the basis for an iterative least-squares analysis in which the corrections, 6F;j, to the constants, F$, are calculated. Using the harmonic vibrational frequencies for HzO, D20 and HDO, the results of Table III are obtained. The structural parameters r = 0.957 A, 0 = 104.5’ have been used in the calculations. The force constants of Table III represent the true harmonic force field. The constants tabulated here, and in other tables, are symmetrized force

68

COOK, DE LUCIA AND HELMINGER

8.355

8.357

0.761

0.763

0.345

0.367

8.555

8.555

0.6

0.7

Observed 7.310t.217

7.396

7.586

0.797f.073

0.771

0.722

-0.517

-0.183

7.838

7.853

10.5

13.1

-0.63br.279

7.826t.049 9.4

=Tbeuncertainties represent bHarmonic 3832.2, far ~~0 =Observed 3656.7, for ~~0 d1n unit*

constants

consistent

SZ = 66, S3 = (6~1 -

quoted

with

the internal

CO”stants

because of the small deviations

cm-l

cm-’

coordinates:

Note, the statistical

are well determined,

weighting schemes have little effect on the constants,

of all the frequencies

model and the experimental

for the observed

uncertain-

and that the interaction

which is due to the accuracy

data. Similar calculations,

with ours, have been reported by Nibler and Pimentel

The results

S1 = (6rr + 6r2)/ti,

F 22 and FE have been reduced to

bond distance.

energy constants

force constant is positive. Different

agreement

force

the

symmetry

The force constants

mdyn/A units by using the appropriate

both the theoretical

for-

throughouC

one standard deviation. frequencies used in the analysis are (w1,w2,w ): 1648.5, 3942.5 cm-l for H 0; 2763.8, 1206.4, 2 2 88.8 and 2824.3, 1441.4, 3889. a cm-1 for H”O. frequencies used in the analyses are (u,,“~,v~): 1594.6, 3755.8 cm-l for H 0; 2671.5, 1178.3. 2788.1 and 272f.7, 1403.4, 3707. 3 cm-’ for HW. of ma- .

&2)/a.

ties indicate the potential

Freque”ciesC

vibrational

frequencies

of

in excellent

(34).

are also given in Table

III.

The structural parameters r = 0.956, 8 = 105.0” have been employed here, and in the remaining calculations utilizing the distortion constants. This type of calculation gives the effective force field for the ground state rather than for the vibrationless state. The deficiencies of the small oscillations model are apparent in the large deviations obtained, well beyond

the experimental

uncertainties.

Because

of these model errors, different

weightings have a significant effect on the constants obtained. In the calculations, three different weighting schemes have been employed. In analysis I all frequencies are weighted equally, ZQ~= 1. This by definition of the least squares criterion gives the best overall fit of the data. For analysis III, on the other hand, the weighting is taken as inversely proportional to the square of the frequency, W; - l/vi2. This effectively gives force constants which minimize the sum of squares of percentage deviations. The rationale behind such a weighting scheme is that the larger the frequency, the larger the associated anharmonicity correction, and such frequencies should not be required to fit as well as lower frequencies. A similar statement seems reasonable in regard to the distortion constants. As is apparent from Table III, Analysis III gives force constants

FORCE FIELD AND STRUCTURE

OF WATER

60

closest to the harmonic force constants, In Analysis II, where wi - l/vi, one minimizes C~AY,(A~JZJ;), and the effect is to give a force field in between that obtained from analysis I and III. In any case, the interaction constant Frz, is negative as compardr to the positive value obtained when harmonic frequency data is used and is rathee sensitive to the weighting chosen. R. Microwave Distortion Constant Data The distortion constants depend explicitly on the elements of the inverse constant matrix. In particular, under the assumption of small oscillations

force

(CT= a, b, or c) ~auua = Rm C J,,(i)J,,(‘)Fijel, (13) i,j where R, = - ._lo?/Rfe~, R = 10-22/2h, and A, = JI/SR~~~is the rotational constant in MHz, Fij-’ are the elements of the inverse force-constant matrix, and the Jam(;) = dI,,/dSi have been given by Kivelson and Wilson (35). Using atomic mass units, bond distances in angstroms, and the bond stretching force constant in dyne/cm, bond interaction force bending force constant in dyne-cm, and bond bending-stretching constant in dynes, the 7huaa are given in units of MHz. The three symmetry force COLdantS of Species A r for water can be evaluated from raaaa rbbbb and recee. x0 reliable information on Fa3 can be obtained from 7&b, because of the large ambiguity in this constant as mentioned previously. In the calculations utilizing the distortion constants, the effective rotational constants A’B’C’ (corrected for 7”bab as obtained from 71) have been employed throughout. The least-squares results of combining’the various isotopic data are summarized in Table IV along with the vibrational frequencies predicted from the derived force constants. Similar results are obtained using just the data of each

70

COOK, DE LUCIA AND HELMINGER

isotope. It is encouraging that the distortion data gives vibrational frequencies which are very reasonable, considering the problems associated with such a calculation, and the difficulties in the distortion analysis of these light molecules, where as many as 22 parameters are being evaluated to fit the observed spectrum. The largest discrepancy is about 5%. In general, the results are similar to that obtained for heavier molecules. Furthermore, it is noted that the interaction force constant is positive as found for the harmonic force field. It is apparent from Table IV that the different weightings have a small effect. Also, that it is rather difficult to fit ~~~~~well. As we go from analysis I to III, the fit of ~~~~~gets worse. Interestingly enough, the predictions of the vibrational frequencies improve slightly as the larger distortion constants are weighted less. The statistical analysis indicates the data is not very sensitive to the stretching force constant Fn. C. Combined Infrared and Microwave Data

In many cases, the vibrational frequency data available for determining the quadratic force constants is not sufficient or satisfactory, and it is very helpful to supplement the data with other vibration-rotation data, such as centrifugal distortion constants. In the present case, although the potential constants have been evaluated from the infrared data, it is of interest to combine both the infrared and microwave data and thereby obtain a force field most consistent with both spectral regions. To combine both the ir and mw data, Eq. (12) is employed

in conjunction

with

= f”uam - C {R, 2 J,,‘k’J,,‘l’F~i-lFi~-l)~~F~i i - C { 2R, C J,,‘k’J,,“‘FbilFir’}~~F~~ i
where R, has been defined previously

where the derivatives

of the distortion

from Eq. (13). By taking we find

k.1

(14)

and (Y = a, b, or c. This follows from

constants

the partial

with respect to the Fkll

derivatives

aF-l -=-

of the matrix

F-1

aF,i

follow directly

equation

F-‘F = E,

06)

which gives aFhl-1 = _

dF%j

Fki-lFjt-‘.

(17)

FORCE

FIELD

AND

STRUCTURE

OF WATER

71

General expressions for the partial derivatives of various rotation-vibration constants have been given by Mills (36). Equations (12) and (14) thus provide the basis for a weighted least-squares analysis, It should be emphasized that both Eqs. (12) and (14) really apply to constants for the vibrationless state rather than the observed constants. Table V gives the results from combining ir and mw data of Hz0 and DgO. These force constants are most compatible with all the data. It is apparent from the Table that it is difficult to fit satisfactorily the vibrational and rotational data. This is really not surprising considering that the Y’Shave large anharmonicity effects and also that the I’S are contaminated with unknown, but significant, vibrational contributions. As we proceed from weighting I through TIT, the interaction constant decreases, the average deviation of the fit rises, and the higher vibrational frequencies and largest r’s fit more poorly. Unit weighing favors the vibrational frequencies, whereas weighting III, weights the distortion data fairly heavily. Some prejudice can be developed for weighting II, which gives a reasonable distribution of residuals and an acceptable interaction constant. Different weighting could be considered; however, any reasonable scheme would give results within those found here. A comparison of the best force fields obtained in various ways are summarized in Table VI. For the ir and mw entry, Fs3 comes from only the Y? data, because of the 1imitatiOnS in Cahhting an aCCUrate value of rabab as discussed in Section Il. Clearly-, one advantage of the ir and mw force field for the ground state is that it tits both spectral regions, and the interaction constant is positive and close to the value found for the harmonic force field. The valence bond force constants obtained from the symmetrized force constants are also compared in Table VI. Clearly, the ir and mw constants compare favorably with those of the harmonic force field. The present results indicate that if corrections for the effects of vibration are to be ignored, the “best” or most representative force field is obtained by combining both ir and mw data, even whenlthe vibrational frequency. data is rather extensive. Furthermore, even though the anharmonicity effects are particularly large, the vibrationrotation data when included leads to a meaningful potential function. These results

COOK, DE LUCIA AND HELMINGER

72

m/x) F~+&

8.355

7.396

7.481

7.653

0.761

0.771

0.675

O.700

F12hdlx)

0.345

-0.517

0.443

0.536

0 F33(md/A)

8.555

7.838

F

11

Valence fr

Bond

Force fo

7.838 constants frr

W/X) fro

IRbe)

a.455

cl.761

-0.1OU

0.244

IR(lQ

7.617

0.771

-0.221

-0.366

thus further confirm the importance of vibration-rotation data in the determination of force constants (25). Although water is by far the worst case from the point of view of large anharmonicity effects, the results of combining ir and mw data are actually very satisfactory. Therefore, for other molecules techniques of this nature can be expected to give, in principle, a very representative force field. Moreover, it would seem to be advantageous to consider weighting the data. If all the data fits well, such weighting will have little effect. On the other hand, when sufficient data is available so that deficiencies of the model are apparent, e.g., in a poor fit of the data, the weighted calculations can be useful. Obviously, the weighting chosen here is rather arbitrary providing in effect arbitrary anharmonicity corrections. Nonetheless, the range in the constants obtained from different weightings might be used as a further guide to their reliability. IV. MOLECULAR STRUCTURE

For a planar bent XY2 molecule, the molecular structure may be obtained directly from any two of the three principal moments of inertia. The ground state rotational constants, A’, B’, C’, have been given in Table II. Because of the difficulties associated with correcting for the distortion contributions, the rotational constants are uncertain, and the bond distances derived from them will be correspondingly uncertain. However, this effect is very small. Using, for example, the three sets of A’, B’ for H20, the spread in the effective bond distances obtained is about 5 X 1OW A. The effect is in the 5th significant figure, and would ultimately limit the accuracy of the bond distance only if the vibrational corrections to the rotational constants were known with greater certainty. On the other hand, the effects of vibration produce a much more serious limitation on the structure. Because of the nonvanishing inertial defect, various rostructures may be obtained from different pairs of moments of inertia. Such effects are well known, however, these light molecules provide a very basic and interesting example of this as illustrated in Table VII for the various isotopic species. The effects of vibration are obviously rather large. Also, it will be observed that the range in the bond distance decreases as the mass of the vibrating atom increases. On the other hand,

FORCE FIELD AND STRUCTURE

OF WATER

73

it is not clear whether the tendency is for the bond distance to decrease or increase with isotopic substitution. The r,-structure has also been calculated for comparison. Here the coordinate of each atom of Hz0 is obtained by Kraitchman’s equations using the appropriate isotopic (HzO, H2018, HDO) we find r = 0.9585 A, 8 = 104.6” species. For the combination and for the combination (HzO, HpOl*, HTO) we obtain r = 0.9590 A, 19= 104.4”. In the above calculation, the u-coordinate of the oxygen atom is assumed to be zero bysymmetry. A value of 0.006 A is calculated for this coordinate by Kraitchman’s equation. For small molecules like HTO, the average structure, (r), can be uniquely determined, and has the advantage of being physically meaningful like the equilibrium structure. The average structure can be evaluating by correcting the effective moments of inertia (IGo, Jb”, 1,O) for effects which depend only on the harmonic part of the potential function. The elongation of the average bond distance over the equilibrium bond distance is due to the anharmonicity in the vibrational potential. The appropriate equations have been given by Laurie and Herschbach (37). In these expressions the quantity cos x and sin x are required. These were evaluated from the equations CDSx = (ra*/1c*);[13 sin x = -

(r**/Ic*)k[83

(16*/1,*)!{1~ -

(1a*/le*)i{23

(18) (19)

which can be obtained from the expressions given by Kirchhoff (20). The Coriolis coupling constants required in the above equations have been obtained from the matrix equation < = L-X&-’

(20)

where the C-matrix has been given by Meal and Polo (38). Using the ir and mw force field of Table VI, the average structures for the various isotopes have been evaluated and are shown in Table VII. As expected, because of the anharmonicity effects, the bond distance decreases as the amplitude of vibration decreases. The replacement of H by D leads to a shortening in the bond length of 0.004 A, while replacement of H by T gives a shortening of 0.005 A. The residual inertial defect for the average moments of inertia

COOK, DE LUCIA AND HELMINGER

74

(la*, Ib*, I,*) are small, but depend slightly on which set of A’, B’, C’ are employed in the calculation. For the A’, B’, C’ obtained via ~1, the residual inertial defects are 0.0007, -0.0007 and -0.0014 amu AZ for HzO, D20 and TzO, respectively. If the harmonic force field is used to evaluate the average structures, the bond distances obtained for the various isotopes are about 0.0006 A less, and the residual inertial defects are much larger (c. a factor of 4). From the Coriolis coupling constants and the vibrational frequencies, the vibrational part of the inertial defect may be evaluated (39). In calculating the inertial defect, vibrational frequencies consistent with the force field have been used rather than the observed values. Using the calculated values for the symmetric isotopes, the inertial defects for the asymmetric isotopes have been evaluated from the emperical relation of Oka and Morino (40). The calculated and observed values are compared in Table VIII. The observed value quoted is the average of those obtained from the three different sets of A’, B’, C’ obtainable from the three rabab values quoted in Table II, and the uncertainty covers the range of values obtained. The agreement between the calculated and observed inertial defects is very good. To evaluate the equilibrium structure, the rotation-vibration constants are required to correct the various rotational constants, e.g., A, = A’ + C aia/2 I

(21)

etc. Sufficient information to evaluate the (Y’Shas only been measured in the infrared Gailar and Plyler (28) are assumed, the region. If the infrared results of Benedict, equilibrium structure that is calculated is ye = 0.9575 A and ee = 104.51”. The effect of different A’, B’, C’ do not affect the structure to the significant figures quoted. The residual inertial defect does, however, depend on which rotational constants (A’B’C’) are employed. The three values are A = - 0.0011 via ~1, A = - 0.0008 via Q and A = - 0.0015 amu A2 via (71, Q). The structure is in good agreement with the infrared value of re = 0.9572 f 0.0003 A and 8, = 104.52” * 0.05” (28). On the other hand, if the vibration-rotation constants of Hall and Dowling (41) are assumed, the equilibrium structure is re = 0.9594 A and ee = 104.20”. The residual inertial defects are slightly higher; A = - 0.0048 via rr, A = - 0.0046 via 72 and A = -0.0053 amu AZ via (71,72).

Table

“III.

Observed and Calculated Inertial Defect (mu 12,

AcObs.)

.I(lzalc.)

Llm.=

0.0492r0.0004

0.0486a

0.0006

0.0656vl.0002

0,0664=

-0.0008

0.0771~0.0002

0.0786a

-0.0015

0.0557~0.0004

0.0568'

-0.0011

0.0586'0.0004

0.061~3~

-0.0032

0.0707f0.0002

0.0722b

-0.0015

75

It is also possible to obtain the equilibrium structure from the average structure data under certain approximations. For a diatomic molecule, the average structure is related to the equilibrium structure as follows (42) (r) = re -

(3a,/8a)(h/2cao/$

(22)

where p is the reduced mass, and where a0 (cm-l) and al (dimensionless) are, respectively, the harmonic and cubic potential constants. From the average bond distance and the reduced mass of at least two isotopes, Eq. (22) can be used to calculate re. In the present case, if we assume the O-X bonds may be treated separately as onedimensional oscillators, we may apply the above relation. This approximation is not unreasonable (43) and has been used by Oka and Morino (44) to derive the equilibrium structure of HzSe. The equilibrium structure obtained for Hz0 making use of the data in Table VII is re = 0.9587 A f 0.0001 A, 8, = 103.89” =t 0.06”. In deriving B,, the same reduced mass dependence has been assumed for (0) as for (r). The quoted uncertainties only represent the spread of values obtained from different combinations of isotopes. Note this value for re lies between the values obtained from the infrared vibration-rotation constants. Finally, the various structures of water are compared in Table IX. The r,-structure is close to the r,-structure, while r. is less than ye. ACKNOWLEDGMENTS One of us (R.T,.C.) would like to thank the Biological and Physical Science Research Institute of Mississippi State University for support of this work. Work at Duke University was supported by NSF grant GP-34590. RECEIVED:

h’ovember

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