Reprinted from:

THE JOURNAL OF CHEMICAL PHYSICS VOLUME 59, NUMBER 5 1 SEPTEMBER 1973 moment of water from Stark measurements of H20, HDO, and D20

Shepard A. Clough Air Force Cambridge Research Laboratories (AFSC) Bedford, Massachusetts 01 730

Yardle!! BeeFeand Gerald P. Klein National Bureau of Standards, Boulder, Colorado 80302

Laurence S. Rothman ir Force Cambridge Research Laboratories (AFSC) Bedford, Massachusetts 01 730 (Received 7 May 1973) equilibrium dipole moment of the water molecule has been determined from Stark effect easurements on two H,O, one D,O, and six HDO rotational transitions. The variation of the dipole moment projection operator with rotational state is taken into account and expressions are given for this operator evaluated in the ground vibrational states of the three isotopes. The value obtained for the equilibrium dipole moment is lopx\= 1.8473 & 0.0010 D. The effective dipole moments in the principal axis energy representation are lpb(HOH)I = 1.8546 + 0.0006 D, /pb(DOD] = 1.8558 * 0.0021 D and lpb(DOH)I = 1.7318 0.0009 D, Ipa(DOH)I = 0.6567 & O.ooo4 D.

INTRODUCTION where the superscript on M denotes the order of magnitude in an appropriate expansion parameter The dipole moment of the water molecule has X, is the direction cosine between the a-molec- been determined by several authors using methods ular-fixed axis and the space-fixed axis, q is a di- including Stark effect and bulk dielectric measure- mensionless normal coordinate, and the quantities ments. '-' This paper describes the determination in parentheses are electric constants of the mole- of the equilibrium dipole moment for the water cule and for convenience may be given in Debye molecule using Stark effect measurements on two units (lo-'* esu - cm). The equilibrium dipole mo- H,O transitions, six HDO transitions, and one D20 ment ('pa), within the restrictions of the Born- transition. In a subsequent article the determina- Oppenheimer approximation, is independent of tion of the dipole moment will be described by Dyke isotopic substitution for a given molecule. The and Muenter7 using a molecular beam method on geometric representation that we will use in the three H,O and three D,O rotational levels. present study is that shown in Fig. 1. It is to be noted that the permanent dipole lies on the x axis and is assumed to point in the negative x direction, The ability to obtain a single value of the equilib- and consequently OpX will be designated as a nega- rium dipole moment to a precision consistent with tive quantity. The phases of the dipole moment the experimental limits of the Stark measurements derivatives are chosen to be consistent with this is due to the inclusion of electrical distortion in convention. the dipole moment function, the availability of im- proved eigenvalues and eigenvectors for the mole- It is next necessary to obtain the dipole moment cules considered, improved methods of treating the function to second order in the representation in data, and significantly more accurate Stark effect which the vibration-rotation Hamiltonian is vibra- measurements. tionally diagonal to second order. It is most con- venient to use the contact transformation method to THEORY obtain this result. If S' is the transformation func- The Hamiltonian for the Stark effect is given by tion which diagonalizes the Hamiltonian to second H=- *I= -M6, where p is the dipole moment order, then matrix elements of the dipole moment operator, 8 is the , and M is the pro- function in this representation (the t representation) jection of the dipole moment on the space-fixed are given as (field) axis. M is expanded through the second order as follows: OM!, = OM^^,, (4) 1Mtif- -1M fj, (5) a 2M!,=2Mi, +{'Him, '%}Reim +{'Mim, 'Hm,IRejm +['Hi,, OMIRE~~ (6) where we follow the notation of Rothman and

2254 DIPOLE MOMENT OF WATER 2255

TABLE I. Dipole derivative constants for the water lator representation and cij l/('Hi, - (for i = j, molecule given in Debye units (lo-" esu cm). Eij GO). For the numerical constants needed in HOH DOH' DOD* the calculations, force constants were derived fi b a b a b a from the work of Benedict et al. Unfortunately, '(.3&/8q1) - 0.0216 0 0.0050 -0.058 -0,024 0 unambiguous values for '(8'pa/8q? ) which contrib- ~(W,J.3q2) 0.161 0 0.120 0.107 0,137 0 ute directly to the coefficient of aq in 'M, I(apm/aq3) o 0.0950 -0.04oi 0.0611 o 0.0868 - %onstants for DOH and DOD are calculated from ob- 5 served values for HOH. See Ref. 10. d'" are not available for water and this Contribution has necessarily been neglected. Using the con- Clough8 in which { , }, and [ , IR indicate rotational stants of Table I, the dipole moment projection anticommutators and commutators, respectively. operators for the ground vibrational states of the '€Iij is the matrix element of the vibration-rotation three isotopic species as a function of the direction Hamiltonian in the principal axis, harmonic oscil- cosine and angular momentum operators are

HOH: OM = 'M=O,

'M = - 0.0067 @b + 1.06 x {@b, P,"} - 3.34 X lo-' {e,, Pz} -4.0w1cr4{ib,, ~2,}-3.34~10-~{@,,(P,P,+P,P,)} + 6.69x {a,, (Pap, + P,P, )}; DOH: OM=Op, cos$,@, + '1.1, sin$,@, , 'M= 0, 'M= -0.0088@,+1.78~10~5{@b,Pi}-9.13x10-5{@b, P:} -9. 53x10'5{@,, P:}+5. 85X10-5{@,, (P,P,+P,P,)} - l.44XlV5{@, (P,P,+P,P,)}+3. 66X10-4{@c,(P,P,+P,P,)} +O. 0067@,+6.70x10'5{@,, P~}+2.34x10-4{@,,Pz) - 5.21 X {a,, P,"} + 1.18X {a,, (Pap,+ P,P, )}; DOD OM=OpX@,, 'M=O,

2 M=-0. 0052@b+5.33x10'5{@6,P~}+8.4OX10-'{@,, P:} - 2.25~ {@,, P:} - 2.21~ (P,P,+P,~,)} + 4.07X {a,, (Pap,+ P,P,)}. (9)

Using these expressions and the rotational constants in a principal axis representation for HOH, DOH, and DOD" obtainedfrom an analysis using all avail- where ,,ya is the coefficient of M" and includes the able microwave dataand some infrareddata, line higher order contributions to the dipole moment strengths were calculatedin the energy representa- and a! runs over the directions in the principal axis tion. These line strengths were then used to obtain the molecular coordinate system. For HOH and DOD Stark shifts as a function of M2, b", and Op:, where there is a contribution to 2*4E only when a! = b and M is the projection of the total angular momentum for DOH when a! =a, b. It is useful to write this along the space-fixed axis. Using the usual second expression in terms of the rotation angle to prin- order perturbation treatment, l2 we obtain cipal axes as follows 2256 CLOUGH, BEERS, KLEIN, AND ROTHMAN

the high correlation among the constants. Conse- 2c4E= '1; E2{ ,ya sin2@,+ zyasinz@,M2 quently, we have used calculated values for these +Oyb c0s2@O+2yb CosZ'&M2 } (''1 coefficients. The most convenient expression for or in a more useful form as this calculation has been given by Kirchhoff, l3 as

2+4E= 'pf &' {( ,yb + zyb 144'') 4Ei=CGin /E'Him1HmnGim l2 2Ei /'HimEim/', n m I- m + sin2+, [(,ya - ,yb) + ( '7, - ZYb) I (12) (14) 9 where 'Eiis the result obtained from second order since perturbation treatment, Opb = OpXcos+, 'pa= OpL,sin+,, 'Ei =EI 'HimI2cim, (15) and m sin2@,+cos2~, = 1 . and the 'Himare the matrix elements in the energy This result for the Stark energy shift of a rotation- representation for the perturbation H = - IJ. -8 . a1 includes the effect of electrical and The other effect that must be considered is the mechanical distortion through fourth order. energy shift due to the of the mole- There are two other fourth order effects which cule. This effect has the functional dependence must be considered in the analysis of the Stark ef- 4~(p~1)=(or(poi)+zr(poi)~z)&2 fect for a molecule such as water. One is the con- sideration of higher order terms in the perturba- and may be evaluated from the expression tion treatment of the Stark calculations. It can be 4~(p~1)= - + 6 Z (pmm*m*m). (17) shown that the contribution from a fourth order 01 treatment will have the form Diagonal elements of the polarizability have 4E= OpZ E4(,A +,AMz + 4AM4). (13) been calculated for HOH by Liebmann and Mosko- witz14 and we have adopted the following polariza- Although in principle it is possible to obtain the bility constants: pa,= 1.651, pbb= 1.452, and p,, coefficients from an analysis of the Stark shifts, = 1.226 cm3). With these values the maximum this has not been possible from our results due to correction to the observed frequency shifts is 4 Wz for the HOH transitions, 0.4 kHz for the DOD transition, and ignoring the off -diagonal element of the polarizability (pab)2 kHz for the DOH transi- i'i tions. For the precision of our experimental data, these corrections are not significant. ANALYSIS AND RESULTS The experimental details of the measurements have been discussed at length by Beers and Klein" for the data taken by them. The Stark shifts for // the 22 GHz HOH line (61,6- 52,3)were kindly pro- vided by Kirchhoff. '' The values for the electric fields for the latter data were readjusted using the constants obtained by Muenter17 for the Stark shifts on the OCS molecule to be consistent with the field determination for the other transitions. In all FIG. 1. Coordinate systems for the water molecule. cases calculated fourth order corrections and ex- The coordinates of the center of mass with respect to the perimental corrections were applied to the data origin at the oxygen are given as xo and zo in ang- stroms. The primed coordinate system is the principle before the least square analysis. axis system and the rotation angle from the unprimed SYS- The analysis of the microwave Stark effect data tern is @. The system is used where a-Lcz', bcx', has been carried out using fie method of least and c-y'. The phase for a positive change in the normal coordinates is chosen as 41: DO stretch, OH stretch; q2: squares. Each observed Stark shift has been ap- increase in the band angle; q3: DP contraction, OH propriately weighted based on the error of the stretch. Data are R = 0. 9573412 A; f3 = 52.28026", and particular measurements and provides an input data point to the least squares analysis. The data for Xn Zn 6 the several rotational transitions, two for HOH, six HOH - 0.0655487 0 0 for DOH, and one for DOD, have been treated in DOH - 0.0930725 - 0.0400709 21.05735" several ways. In Method I, the analysis has been DOD - 0.1178299 0 0. made as a function of 8' and M2G2 to obtain quanti- DIPOLE MOMENT OF WATER 2257 ties orand 2r in the expression for the Stark shift,

v = vo+,r&2+,rM2&2, (18) where

Or = OF:( + sin'$,( OYa - OYb)) and Mti ll

Values for Op2 can be obtained from calculated t-v) et- g$ % Z2t-2 values of oya, 2ya, and sin@,. This approach gives 00 0000N0 0211 00 ONOOOO 0 the best fit to the observed data and provides a 00 000000 0 measure of the experimental precision of the data dd dddddd d since the rms error depends only on the assumed functional dependence. On the other hand, since there may be high correlation between orand 2r, this method does not provide reliable values for om o*u3moow 0 mm mwwmmw rl rlln NmrlrlOm m 'p. The transition frequencies (listed to the near- ~m WLOW~-*WN PYO mm~~oo est megahertz) and the results of Method I including 00 rlrloo*o d values of and are tabulated in Table 11. The dd drl*dddd d ,r I1 II I values given here vary slightly from those pre- viously reported by Beers and Klein. These dif - ferences, most of which are within expected error limits, are due to a revision of some of the modu- lation errors and to adjustments in the weighting of rl N some of the observations. rlrlo **t-m 00 04 NOrlONrl 0 -k, 00 ONOOOO 0 00 000000 0 For Method 11 the data from each transition are I1 .. 00 dddddd d utilized to obtain a value for 'p2 (also Op2 sin2@,, if f desired) using an expression for the shifts obtained P from Eq. (12). In a third approach, MethodIII, the % 2 data for each isotopic species are analyzed using 3 Eq. (12) to obtain a value of Op2 for the three iso- b topes. The final method, Method IV, combines w- all the data together to obtain a single value for P 5 (and again, if desired, Ob2 sin2@,). When the s data are combined, weights have been assigned to M the measurements for each rotational transition .3m which are inversely proportional to the rms error Y 2 obtained for that transition from Method 11. The 4 results from Methods 11, 111, and IV are given in Table 111. The methods described for analyzing the DOH data allow for the determination of the angle through which the equilibrium coordinate system must be rotated to obtain principal axes. The 3A change in the angle @, as determined from the data is of the order of magnitude of the error and the 4 improvement in the fit is not significant. Conse- ri w quently, for the calculations on DOH the angle de- J. a rived from the analysis of the structure has been c assumed, $,=21.05735". The dipole moment constants given in Table 111 W 3 are for the equilibrium dipole moment. The ef- zm 0 00 fective dipole moment for the ground vibrational + ma 8a states of the three isotopes may be obtained by 5 adding the second order coefficients of @a from 2258 CLOUGH, BEERS, KLEIN, IND ROTHMAN

Eqs. (7)-(9) to the appropriate results for 'pa, c-t-**u)o m from Table III, Method 111. We obtain for ma00 N*ddNN 00.. 000000...... N0 HOH, /~b= px = - 1.8546 * 0.0006 D ; 00 000000 0

DOH, /~,b= - 1.7318 * 0.0009 D,

pa= - 0.6567* I). 0004 D,

px = - I p:+ pi )'"= - 1.8521 io. 0012 D, DOD, pb = pX= - 1.8558 i 0.0021 D . These values for DOH are applicable in the repre- sentation in which the rotational Hamiltonian is of the form H~ =AP;+ BP;+ CP:

+(terms of higher power in angular mmw~oawdN*d+Nrl momentum). . .. 0000000 PI0 00 d d d d d d 0- Line strengths calculated from Eqs. (7)-(9) and a rl corroborated by the Stark measurements enable 0 rlN 0 0 N0 improved atmospheric transmission calculations 0 0 0 in the infrared. A tabulation of HOH energy levels, d 0 d transition frequencies, and transition intensities using the dipole moment function reported here, m W will be published shortly. '* c- 0In 2 m Aside from errors in the Stark data, the princi- rl 3 rl I I I pal source of error in our analysis is in the higher order coefficients in the dipole moment expansion.

More accurate experimental values of the first mt-00 N**t-" N*rlONrl 2 derivative constants, particularly for DOH and 00.. 000000...... 0 DOD, would be highly desirable. An estimate of 00 000000 0 the magnitude of the contribution to 'M from ne- glected terms of the type '(8'1~ /e& can be made mln w va m rl orl NNmoam N from the observed intensities of the 2ul, 2v2, and 00 0d0000 0 2u, infrared bands of HOH. The strongest of these 000000 dd d d d d 0. d 0 is 2u, and if it is assumed that all the intensity of that band is attributable to the dipole moment ex- pansion, we obtain l2(a2pa,/aq~)I=O.0O66D. A ot- at- 0Ina **mm calculation of the second order contribution to the ...... m band intensity from mechanical anharmonicity rld rlrlrlddrl d 2v, II IIIIII I is of the order of 0.008 D. Consequently it is ex- pected that the correction to the dipole moment from this source is of the order of L- m*maNN * 00NL- d*dONd 00 000000...... 0d dd 000000 0 A more accurate determination of the dipole mo-

ment projection operator depends on the availabil- e -- 6 om d ity of these higher order constants from experi- ..^ --NN 9 ment or from ab initio calculations. Nln m 1, 1 1 In conclusion we should like to discuss a number --ma - ^.. * of points about which there appears to be confusion dd rl -- I in the literature. The projection of the molecular mw * dipole moment on a space-fixed axis 2 for an asymmetric rotor in a given vibration-rotation state designated by [(v), J, K,, K,, M] is indepen- dent of field (neglecting polarizability) and is char- E n 3 0 acterized by the good M, since m n DIPOLE MOMENT OF WATER 2259

P, commutes with the Hamiltonian. For micro- 'M. W. P. Strandberg, J. Chem. Phys. 17, 901 (1949). wave Stark measurements, it is not the dipole mo- 2C. I. Beard and D. R. Bianco, J. Chem. Phys. 20, 1488 ment of the transition that is observed as is some- ( 1952). times implied, but rather the energy difference 'G. Birnbaum and S. K. Chattergie, J. Appl. Phys. 23, 220 (1952). between two eigenstates, each state being charac- 4M. Lichtenstein, V. E. Derr, and J. J. Gallagher, J. Mol. terized by its own expectation value of the dipole Spectrosc. 20, 391 (1966). moment projection operator. From results pub- 5Y.Beers, G. P. Klein, and S. A. Clough, Bull. Am. Phys. lished up to this point it is not possible to establish SOC. 16, 86 (1971). a difference between the effective dipole moment 'A. H. Brittain, A. P. Cox, G. Duxbury, J. G. Hersey, and R. G. Jones, Mol. Phys. 24, 843 (1972). for the three isotopes, this difference being cer- 7T.R. Dyke and J. S. Muenter, "Electric Dipole Moments of tainly less than 0.003 D. Finally we have observed low I states of H,O and D,O," J. Chem. Phys. (to be published). no anamalous behavior of the dipole moment for 'L. S. Rothman and S. A. Clough, J. Chem. Phys. 55, 504 DOH and the dipole moment projection function is (1971). evidently given adequately by Eq. (8). 9W. S. Benedict, N. Gailar, and E. K. Plyler, J. Chem. Phys. 24, 1139 (1956). See also K. Kuchitsu and Y. Morino, Bull. Chem. SOC.Jap. 38, 814 (1965) and K. Kuchitsu and L. S. Note added in p~oofiThe representation for Bartell, J. Chem. Phys. 36, 2460 (1962). which Eq.'s (7), (E), and (9) are applicable, is that 'OW. S. Benedict and S. A. Clough (private communication). designaied as the_@form, Reduction (a), [i. e., The method of transferring the dipole moment derivatives is for I+, ~,(bbcc)= ~,(bbcccc)= 6;,(ccbbbb)= 66(bbccaa) described by J. Overend, Infrared Spectroscopy and Molecular =0] by K. K. Yallabandi and P. M. Parker, J. Structure, edited by M. Davies (Elsevier, Amsterdam, 1963), p. 362ff. Chem. Phys. 49, 410 (1968). These equations in "W. S. Benedict, S. A. CLough, L. Frenkel, and T. Sullivan, J. other representations, including that currently Chem. Phys. 53, 2565 (1970). The rotational constants used referred-- to as the Watson form in which @ =I?, in this work for HOH and DOH, respectively, are similar to = H, = HI, = 0, are available from SAC. The oper- those used by F. C. DeLucia, P. Helminger, R. L. Cook, and ator S' used to obtain these equations includes the W. Gordy, Phys. Rev. A 5, 487 (1972) and F. C. DeLucia, R. L. Cook, P. Helminger, and W. Gordy, J. Chem. Phys. effect of first order distortion, Coriolis, and an- 55, 5334 (1971). harmonic (cubic) terms in the Hamiltonian. 12M. W. P. Strandberg, Microwave Physics (Methuen, London, 1954), p.45ff. For a general discussion of Stark Effect see J. E. Wollrab, Rotational Spectra and Molecular Structure (Academic, New York, 1967), p. 244ff. ACKNOWLEDGMENTS 13W. H. Kirchoff, I. Am. Chem. SOC.89, 1312 (1967). I4S. P. Liebmann and J. W. Moskowitz, J. Chem. Phys. 54, 3622 (1971). We are pleased to acknowledge the many coh- "Y. Beers and G. P. Klein, J. Res. Natl. Bur. Stand. (U.S.)A structive discussions we have had with F. X. 76a, 521 (1972). Kneizys in the course of this work. Also we are I6W. H. Kirchhoff (private communication). grateful to W. Kirchhoff for allowing us to use his "J. S. Muenter, J. Chem. Phys. 48, 4544 (1968). "S. A. Clough and W. S. Benedict, "Transition Frequencies, results on the 22 GHz H,O transition and to W. S. Transition Strengths, Energy Levels and Radiative Lifetimes Benedict who has collaborated with us on the in- for the Ground Vibrational State H2'"0,"AFCRL Report (in terpretation of the molecular data on water. preparation).