3138 L. A. K. STAVELEY

ACKNOWLEDGMENT

Helpful discussions with Mr. M. ]. Hiza are grate­fully acknowledged.

* Permanent address: The Inorganic Chemistry Laboratory, South Parks Road, Oxford, England.

THE JOURNAL OF CHEMICAL PHYSICS

1 H. C. Longuct-Higgins and B. Widom, Mol. Phys. 8, 549 (1964).

2 N. S. Snider and T. M. Herrington, J. Chern. Phvs. 47,2248 (1967). - ,

3 F. E. Karasz and G. D. Halsey, Jr., J. Chern. Phys. 29, 173 (1958). '

4 w. B. Streett, J. Chern. Phys. 42,500, (1965).

VOLUME 53, "UMBER 8 15 OCTOBER 19iO

Effect of Vibrational Anharmonicity on the Isotopic Self-Exchange Equilibria HzX +DzX =2HDX

MAX W OLFSBERG

Chemistry Departments, University oj Calijornia,* Irvine, Calijornia 92664 and Brookhaven National Laboratory,t Upton, New York 11973

AND

AUDREY A. MAssAt

Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973

AKD

]. W. PYPER

Chemistry Department, Lawrence Radiation Laboratory, t University oj Calijornia, Li!"ermore, Calijornia 94550

(Received 3 April 1970)

The anharmonicity correction to the zero-point energy change in the triatomic dihydride isotopic self­exchange equilibria, H0C+D2X=2HDX, is discussed. It is shown that the inclusion of the usually ignored term Go in the expression for the anharmonic correction to the zero-point energy (Go+t:ZXij) causes the an­harmonicity correction to the zero-point energy change to be quite small for the cases considered (X is 0, S, Se), but when this term is ignored the correction is found to be nonnegligible. Equilibrium constants are calculated, and it is found that the agreement between theory and experiment is considerably im­proved by including the Go factor.

INTRODUCTION

The values of the equilibrium constant for the reactions

H 2X(g) + D2X(g) = 2HDX(g) (1)

have recently been determined mass spectrometrically where X is 0,1·2 S,3 or Se.~ The value of K(O) (where X is 0) had been of considerable practical interest in connection with isotope effects observed in mixed H 20-DzO solvents. The observed K(O) value at 25°C, ""'3.76, disagreed considerably with previous theoretical estimates, ,,-,3.4.4 It has been shown recently5-7 that this disagreement came about largely because of the usual neglect of the Go term in the expression for the energy of the nuclear motions of the molecules; Go appears as the term in n+t raised to the zeroth power in the energy expression for the ground rotational state when this energy is expressed in a power series in powers of n+t. where n refers to vibrational quantum number. It has been indicated'··6 that the inclusion of Go, which may be regarded as an anharmonicity term, appears to lead to the result that the correctly calculated value of K(O) is very

close to that calculated if anharmonicity is not taken into account at all.

THE Go TERM

With the minimum of the potential-energy surface taken as the zero of energy, the energies of a nonlinear triatomic molecule in the ground rotational state can be expressed, to second-order approximation, in the form

+ I: x,An;+t) (llj+t) , (2) i;;j

so that the quantum-mechanical zero-point energy is given by

Ev(O,O,O)/hc=Go+t I: Wi+t I: Xu. (3) i5:.i

The wi's, sometimes called zero-order frequencies, are the harmonic frequencies corresponding to the three normal coordinates Qi of the nonrotating molecule, obtained if all terms in the vibrational potential energy higher than quadratic are dropped. The xii's are

EXCHANGE EQUILIBRIA H 2 X+D 2 X=2HDX 3139

the usual8 anharmonicitv coefficients. Expression (3) differs from the usual ~xpression for the zero-point energy by the inclusion of Go. Go can be written ~s a sum of an anharmonic vibration term and of a rotatlOn­vibration interaction term,

Go= GOA,+GOrv. (4)

GOA' arises, just as Go does in a diatomic molecule/ from consideration of the nonrotating anharmomc molecule. Thus if the harmonic molecule is considered to be the unperturbed situation, then a perturbation term

H(!) / he= kSS8qs3, (5)

with q. a dimensionless normal coordinateS [Qs= (h2/A.) 1I4q., As = 4n-2e2ws

2], yields, in the second-order perturbation development, the following contribution to the energy of the state in which ns is the quantum number corresponding to qs:

I:!.E _ 2 ((ns! q.a! ns+1)2 + (ns ! qs3! ns+3)2 - - heksss 3h ~ -~~ - AA

+ (ns ! qs3! ns-3)2 + (ns ! qs3! ns-1)2). (6) 3hew. hews

to the classical vibrational kinetic energy (if vibra­tional angular momentum is ignored) is

rather than

tjL1I4 L Pk jL-1I2 Pk jL1I4

k

t L pZk, k

(9)

(10)

where Ph is the quantum-mechanical operator for the momentum conjugate to the normal coordinate Qk and jL-1 refers to a matrix, defined in standard textsS on vibration-rotation energies of molecules in terms of effective moments and products of inertia. A formula has been given6 which expresses GOrv in terms of the (principal) moments of inertia, the variations of the effective moments and products of inertia with extension of normal coordinates, and the Coriolis constants. The variations of the effective moments and products of inertia and the Coriolis constants can be expressed in terms of the transforma­tion coefficients which relate the Cartesian displace­ment coordinates to the normal coordinates. By manipulations, the details of which will not be pre­sented here, GOrv can then be expressed in terms of the (principal) moments of inertia,

The meaning of the matrix elements appearing in GOrv= (-h/167re) [l/Ix/el+ 1/ I y)el+3/ Izz(el], (11) Eq. (6) is the usual one. Evaluation of the matrix elements leads to where Izz(el=Ixx(el+ly)e).

I:!.E/he= (ks.//24w.) (-90nl-90ns-33)

= -lS/4(k •• //ws)(ns+t)2-7 /16(kss//ws). (7)

Thus the perturbation term (5) by itself contributes (-lS/4)(ks.//ws) to X.S and (-7/16)(k ss//ws) to Go.

Expressions have been given8 which permit the evaluation of the anharmonic Xij coefficients in terms of the cubic and quartic force constants (of the potential expanded in dimensionless normal coordinates). The normal harmonic frequencies also appear in the Xij

expressions, as well as the Coriolis constants and the largest principal moment of inertia (from consideration of the vibrational angular momentum). The cor­responding expression for GOA' of the nonlinear tri­atomic molecule in terms of these force constants is6 •7 •9

(8)

The term GOrv arises from two causes. One of these is the internal angular momentum of the molecule. The other cause is the fact that the term in the quantum-mechanical Hamiltonian which corresponds

FERMI-DENNISON RESONANCE

If there exists a Fermi-Dennison resonance, WI;::::;2w2, a formula of the type of Eq. (2) does not suffice for many of the energy levels of the molecule in its ground rotational state. Under these circumstances,1O de­generate perturbation theory must first be used in order to obtain the effect of the term k122qlq22 on the positions of many energy levels, e.g., nl = 1, n2= 0, n3=0, and nl=O, 111=2, ns=O. When such resonance occurs, the theoretical expressions for X22, X12, and Go must be modified in order to take out those terms involving kl222 which are now handled by degenerate perturbation theory (i.e., terms with energy de­nominators WI- 2W2 which arise from interaction between almost degenerate states nl = a± 1, n2= b, 1ls=e and nl=a, 1l2=b±2, 1ls=e). Thus the following replacements are made:

replaced by

1 8W22_3W12 kl222

4 4W22_WI2 WI

1 SWI + 8wz kl222

- 8 wl+2wz WI ;

31-l() \\' 0 L F SHE R G, ~I ASS .\, A '\ [) I' Y I' E R

replaced by

in Cn:

replaced by

-i2[k1ZN (wl+2wz)].

The ground state is not affected by the Fermi-Dennison resonance, and one can easily verify that the contribu­tion of the terms involving k1222 to the zero-point energy expression (3) is the same before and after the above replacements.

Benedict et alY (BGP) noted that the Fermi­Dennison resonance in HDO affected the origins of all but three of their observed upper states. In order to describe the positions of the states involved in the Fermi-Dennison resonance they introduced the inter­action constant b which is proportional to k122, and they handled the interaction between these states by the methods of degenerate perturbation theory. Their values of XZ2 and X12 calculated from experiment there­fore correspond to the theoretical ones discussed above in the case of Fermi-Dennison resonance. Papousek and PllvaI2 (PP) have carried out a theoretical calculation of the Xu values for H 20, D 20, and HDO which will be further mentioned in the next section. These workers tried to find a force field for water which tends to reproduce the experimentally observed Xij values for the three isotopic species. The Fermi­Dennison resonance in HDO is not explicitly referred to by them. However, their theoretically calculated XZ2 and X12 values do apparently take into account the resonance.13 Then the calculated Go value to be combined with the Papousek-Pllva Xij values in order to evaluate the anharmonicity correction to the zero-point energy of HDO should also take into account the resonance. This procedure was followed by one of us6 in a previous evaluation of the HDO zero-point energy correction. Hulston,7 in a similar calculation, combined the experimental Xij values of Benedict et al.n with Go calculated from Papousek and Pllva's MUBFF force field without considering the Fermi-Dennison resonance in HDO; he therefore obtained a somewhat different result.

CALCULATION OF THE ANHARMONICITY CORRECTION TO THE ZERO-POINT

ENERGY CHANGE

The calculation of the effect of the anharmonicity correction on the zero-point energy change of the equilibrium (1) requires the calculation of the an­harmonic contribution to the zero-point energy,

AZPE=Go+i LXii, i5:i

(12)

for the three isotopic molecules involved in the equilib­rium. Even when experimental data on Xij values arc available, t is probably preferable to use theo-

retically calculated Xij values in order to obtain the X'J contribution to the zero-point energy. The isotope­illdependent Illolecular force field used in this calcula­tion should ideally be chosen to give the best possible agreement with the available experimental data on the isotopic molecules. Since Go cannot be obtained spectroscopically, at least directly, the program that has been followed here is the theoretical calculation of both the contribution of Go and of the xu's to the zero-point energy change.

In a prior evaluation of the equilibrium (1) with X oxygen, one of us,6 as already mentioned, used Xi/S theoretically calculated by Papousek and Pllvalz

(PP) and combined these with Go's calculated from Eqs. (8) and (11), using the normal-coordinate cubic and quartic force constants evaluated theoretically by PP. In this article, in order to ensure isotopic consistency, the data taken from other sources consist of isotope-independent geometries and force fields in valence coordinates; the Xi/S and Go's for the isotopic modifications of a given molecular species are evaluated consistently from this data.

The harmonic frequencies of the molecules were found from the harmonic (quadratic) part of the potential by solving the standard GF matrix probleml4 in internal coordinates, and the linear transformation between Cartesian displacement coordinates and normal coordinates was obtained. Modifications of programs which had been kindly furnished to us by J. H. Schachtschneider were used for this purpose and also to calculate the relevant equilibrium constants in the harmonic approximation. These programs were, of course, also employed to evaluate necessary Coriolis constants. The anharmonic force field in internal displacement coordinates (which are nonlinearly related to Cartesian displacement coordinates and, hence, to normal coordinates) was transformed to a force field in terms of Cartesian displacement coordinates by a method which is essentially that of Pariseau, Suzuki, and OverendY' With knowledge of the linear transformation between Cartesian displacement co­ordinates and normal coordinates, the necessary cubic and quartic force constants in dimensionless normal coordinates are then obtained easily.

Kuchitsu and Morinol6 (KM) used the experimental harmonic vibrational frequencies of H20 and D 20 to deduce the harmonic force constants in internal valence coordinates. They then employed the observed rotation-vibration interaction constants (ex) and the observed anharmonicity coefficients (Xi;) to deduce the cubic and quartic force constants in the expansion of the potential in valence coordinates. The latter procedure was carried through independently for H20 and for D 20. Within the framework of the Born­Oppenheimer approximation, the two anharmonic force fields so found should be the same. In actuality the constants obtained for the two molecules are not exactly the same but the values are quite similar.

EXCHAKGE EQUILIBRIA H,X+D,X=2HDX 31-+ t

TABLE I. Anharmonic parameters calculated with KM force field.'

H20 1)20 HDO

Calculated Observedb Calculated Observedh Calculated Observedb

.1.'11 -42.6 -42.6 :r!:~ -17.0 -15.9 .\'1:1 -165.9 -165.il .r:!~ -18.il -16.1l .Y2::1 -20.1 -20.3 :rail -47.7 -47.6 GOav 19.3 (;Orv -17.7 AZPE' -76.4

a In em-I. /) These are the observed BGP val ue:-; ,

In the calculations reported in this article, an isotope­independent force field is, of course, used in any given calculation.

Table I lists the x;/s calculated with the KM force field for H20 and compares the values found with the experimental ones of Benedict et alY (BGP). A comparison of the values shows that the agreement between calculation and observation for the three molecules is somewhat better over-all than that noted by pp12 with their best force field (MUBFF). However, there is no reason to believe that the force field used here gives the closest possible agreement between calculation and observation for the three molecules. It is also not really certain that the observed values deduced by BGP from their experiments are the "best" ones since a somewhat different set has been deduced by KhachkuruzovY The anharmonicity contribution to the negative of the zero-point energy change of the equilibrium will be designated as Ll(AZPE),

- Ll (AZPE) = 2AZPEHDO - AZPEII20 - AZPED2().

From the values in the last row of Table I it is seen that Ll(AZPE) =0.1 em-I. This very small number would lead to a correction of the equilibrium constant of the order of 0.05% at room temperature.

-22.1 ·-22.6 -43.3 -4,.4 -6.1 -7.6 -21.6 -IlJ)

-1l6.2 -il7.2 -15.3 -13.1 -10.7 -9.2 -9.6 -11.il -10.1 -10.6 -26.6 -20.1 -27.1 -26.2 -82.4 -82.9

9.2 4.1 -9.3 -13.0

-40.7 -51l.6

Table II lists values of Ll(ixij) , Ll(AZPE), and Ll (tw) for water obtained with the use of the force field above and several other force fields. Here

tw here denotes n:::iWi and iXij denotes n:::i,;;jXij. It is obvious that

Ll(tw) is the negative of the zero-point energy change for equilibrium (1) in the harmonic approximation and Ll(tw)+Ll(AZPE) is the corrected negative of this zero-point energy change.

The second entry in Table II refers to calculations carried out with the quartic and cubic force field deduced by KMI6 to fit the data for D20( the quadratic force field is unchanged from the previous calculation). The next entry in Table II gives results obtained with the KM aa- a4 force field. This force field consists of the same harmonic force field as that previously discussed. The anharmonic correction consists purely of cubic and quartic "diagonal" stretching force constants, irrr and irrrr, which are defined in terms of aa and a4. The Xij values calculated here for H20 and D20 agree with those calculated with the same force

TABLE II. Ll (IXii) , Ll (AZPE) , and Ll ( !w) values. a

System Force field Ll( lX'i) Ll(AZPE) Ll(!w) ---- ------------ - . --~- ------- ------_ ...

Water KM(H2O) -19.1 0.1 -17.6 Water KM(D2O) -18.2 0.4 -17.6 Water KM(a:;-a4) -25.0 0.4 -17.6 Water PP -23.S 0.0 -17.6 Hydrogen sulfide KM(H2S) -15.8(-15.6) 0.2 -14.1 Hydrogen sulfide KM(a3- a4) -19.4(-19.0) 0.3 -14.1 Hydrogen selinide KM(a3- a4) -14.0(-13.5) 0.3 -12.7 Hydrogen selinide KM(H2Se) -28. 7( -28.6) 0.1 -12.7

11 In em-I.

3142 \\ () L F S B ERG, :\1"\ S S "\, "\ '\ j) P \' I' E R

TABLE ITI. Xi/S" for H,O calculated with Papousek-Pliva force field in valence coordinates (MUBFF).

Present calculation Papousek-Pliva

XIJ -52.4 -52.3

Xl:.:! -17.3 -IS.O

Xl3 -174.9 -174.6

X22 -13.2 -14.4

X2:l -23.2 -22.9

.\';13 -46.7 -45.S

It In em-I.

fields by KM to better than 1.5%. The last calculation for water in Table II was carried out with the an­harmonic force field MUBFF in valence coordinates given by Papousek and PHval2 in their Table VII together with their harmonic force constants and water geometry. It must be mentioned that, while the agree­ment of the x;/s calculated here with those calculated by PP from the same force field is good, it is not perfect (see Table III).

Some conclusions can be readily drawn from the water calculations in Table II. The anharmonicity correction to the zero-point energy change of the equilibrium is of the order of 20 cm-1 if the Go term is neglected, [-~ (tx,j)]. Such a correction would have a rather large effect on the equilibrium constant calculated in the room-temperature region and is of the same magnitude as the zero-point energy change calculated in the harmonic approximation, [ - ~ (tw) ]. The correctly calculated anharmonicitv correction, including Go, 'gives rise to an anharmonicity correction less than 0.5 cm-1 in magnitude. Such a correction affects the theoretically calculated equilibrium constant by less than 0.3% in the room-temperature region. Further calculations of ~(AZPE) have been carried out with arbitrarilv chosen cubic and quartic force fields. While no co~pleteness can be claimed for these calculations, it appears that physically reasonable force fields lead to very small ~(AZPE) values. If cubic and quartic force constants are chosen to be much larger of magnitude than thought physically reasonable, large ~(AZPE) values can result. This last state­ment is made to demonstrate that the small deviations of ~(AZPE) from zero recorded in Table II for various force fields appear to be real and do not just represent roundoff errors in the calculations.

Calculations have also been carried out for the hydrogen sulfide and hydrogen selenide systems. The geometries and quadratic force fields given by Kuchitsu and Morinol6 were employed. The first hydrogen sulfide cubic and quartic force field used in Table II is the one deduced by KM for H 2S while the second one is their a3- a4 force field, which was deduced in a similar manner to the corresponding water force

field. For hydrogen selenide, the first calculatioll was carried out with the a,-aj force field, while the second was carried out with the cubic force field deduced by KM for H 2Se from experimental data. Because of lack of data, KM were not able to deduce the quartic constants; these were therefore set equal to zero in the latter calculation. Table II reports two sets of ~(tXij) values for hydrogen sulfide and hydrogen selenide. The first one is that calculated with the Fermi-Dennison resonance assumption in HDX while the second one (in parentheses) corresponds to no Fermi-Dennison resonance. As pointed out before, the presence or absence of Fermi-Dennison resonance does not affect ~(AZPE) since the effect on ~(tXij) will be exactly cancelled by the corresponding effect on ~(Go).

The hydrogen sulfide and hydrogen selenide calcula­tions lead to a generalization of the conclusion drawn from the water calculations. The correctly calculated anharmonicity correction to the zero-point energy change in the equilibrium (1) is quite small when the Go term is taken into account in the calculation.

THE EQUILIBRIUM CONSTANTS

Harmonic Approximation

The equilibrium constants for the equilibrium (1) were first calculated in the harmonic approximation at a number of temperatures in terms of ratios of isotopic partition function ratios,

K = QHDX/QI!,X . QD2X /QHDX

These calculations were carried out with classical rotational and translational partition functions. ~ 0

rotation-vibration interaction was assumed, and the moments of inertia were calculated in the rigid-rotor approximation. For discussion of such calculations the reader is referred to the literature. 18 The calculations were carried out with the modified Schachtschneider programs.19 For water either Kuchitsu-Morinol6 force constants and geometry or the corresponding quantities given by Papousek and Pllval2 were used. To the accuracy reported here the results were identical. For H 2S and H2Se, data given by KM16 were employed. The KM force constants are listed in Table IV. In

TABLE IV. KM harmonic force constants."

Molecule /, ,> (fa .fa

\-Vater 8.454 -0.101 0.228 0.761 Hydrogen sulfide 4.284 -0.012 0.101 0.429 Hydrogen selenide 3.493 -0.020 0.055 0.327

a In mil1idyn(>~/angstrol11

EXCHANGE EQUILIBRIA H 2 X+D 2 X=2HDX 3143

the absence of quantum effects, the calculated value of the equilibrium constant will be given by the ratio of symmetry number ratios. This classical value of the equilibrium constant is four. 20 As was first pointed out by Bigeleisen,20 who considered this equilibrium constant in terms of ordered quantum corrections, the classical value is the correct one even in the first quantum approximation and deviations from the value four come about only because of contributions from the higher-order quantum corrections. The calculated isotope effect will therefore be presented in the form 4(X) = Y. The isotope effect can be expressed in the form19

K=4(MMI) (EXC) (ZPE),

where MMI is the factor which results from the translational and rotational partition functions, EXC is the factor which results from vibrational excitation, and ZPE is the factor exp[hcActw)/kTJ which results from the zero-point-energy change of the reaction. The results of these calculations are indicated in Table V. It is seen that the calculated equilibrium constants tend towards the classical value four with increasing temperature. The isotope effects also tend towards four in the series 0, S, Se. This phenomenon mainly reflects the fact that the deviation of ZPE from unity becomes progressively smaller since the magnitude of A (!w) is decreasing.

When one considers quantum-mechanical corrections to the classical value of these equilibrium constants in terms of the Bernoulli expansion,20 one finds that the first-order correction term [in (h/kT)2-the first quantum correctionJ vanishes, as pointed out above, and that the first contribution arises from the second-order correction term [in (h/kT)4]. This second-order term depends on the sums of the fourth powers of the normal frequencies of the individual molecules. These sums can be expressed in terms of the force constants and atomic masses, and it is fairly straightforward to show the following in the second­order approximation: (1) The deviation of the equilib­rium constant from the classical value contains terms

TABLE V. Calculations of K in the harmonic approximation.

System T(OK) MMI EXC ZPE K --------- ---------

H2O 273 1.050 0.999 0.911 4(0.956) =3.82 298 1.050 0.999 0.919 4(0.963) =3.85 348 1.050 0.997 0.930 4(0.974) =3.90

H 2S 273 1.056 0.996 0.928 4(0.976) =3.90 298 1.056 0.994 0.934 4(0.981) =3.92 348 1.056 0.991 0.943 4(0.987) =3.95

H2Se 273 1.058 0.993 0.935 4(0.983) =3.93 298 1.058 0.991 0.940 4(0.986) =3.94 348 1.058 0.987 0.949 4(0.991) =3.96

TABLE VI. Calculations of K at 273°K with zero bending force constants (fa =jra=O.OOO).

System MMI EXC ZPE K

H2O 1.050 0.952 1.000 4(1.000) =4.00 H2S 1.056 0.947 1.000 4(1.000) =4.00 H 2Se 1.058 0.945 1.000 4(1.000) =4.00

which depend on the magnitude of the bending force constant f,,; (2) the deviation is independent of the stretching force constantfr; (3) the deviation contains terms which depend onf"r andfr' but these are expected to be smaller than those of (1); (4) the deviation is negative; (5) the deviation is independent of the mass of the X atom per se and of the bond distance and the bond angle. While the Bernoulli expansion does not con­verge at room temperature and while even the finite ex­pansions based on the orthogonal polynomial methods of Bigeleisen and co-workers20 do not provide a good approximation to the exact equilibrium constants here in second-order approximation, the calculations which follow show that the differences in the equi­librium constants for the H20, H2S, and H2Se systems in Table V are in accord with the above predictions. Such predictive power of the Bernoulli expansion has been noticed previously, and detailed study of the orthogonal polynomial methods, which do converge fairly well at room temperature when third-order terms are included, must lead to conclusions in agree­ment with the above predictions for the parameters used in the present study. It should be noted that these orthogonal polynomial methods lead to series formally very similar to the one of the Bernoulli expansion except that modulating coefficients are introduced for the terms in various powers of (h/kT).

Calculations were carried out for the water system with the KM force field and geometry for the ·water molecule but with the oxygen mass replaced first by a sulfur mass and subsequently by a selenium mass; the equilibrium constants obtained at 298°K were 4(0.962) and 4(0,960), respectively. Thus the differ­ences in the X masses per se do not account for the changes in equilibrium constants in Table V.

Calculations have been carried out for the three systems withf" andfr" in the KM force fields set equal to zero so that the "bending" frequency of each molecule becomes a zero frequency which does not contribute to the equilibrium constant.18 The equi­librium constants now have values very close to the classical value already at room temperature. The deviations of ZPE from unity disappear to the accuracy given in Table VI [as anticipated from previous calculations on methane,21 A(!w) is "-'OJ, and the EXC factors cancel the MMI factors as shown in Table VI. In such a calculation care must be taken in calculating the EXC factor since the "bending"

W 0 L F SHE R G, :VI;\ S SA, i\ N l) l' Y PER

TABLE VII. Calculations for the water system at 298°K in the harmonic approximation.

Calculation Force field"

I KM 4(0.963) =3.85 II f, = f,' = 0.000 4(0.966) =3.86 III fa =0.429 4(0.981) =3.92 IV fa=0.327 4(0.986) =3.94 V I" = f,a=O.OOO 4(0.964) =3.86 VI Ia = 1. 522 4(0.932) =3.73 VII Ir= 16.908 4(0.963) =3.85 VIII /,,= -0.202 4(0.963) =3.85 IX f,a=0.456 4(0.963) =3.85 X Ia=0.3805 4(0.983) =3.93 XI f,=4.227 4(0.963) =3.85 XII f,' = -0.0505 4(0.963) =3.85 XIII f,a=0.114 4(0.964) =3.86

" Calculations carried out for the water system with the KM force field except for the force constant (s) indicated. The units are millidynes/ angstrom. ]n calculation I I no force COll-"tants were changed.

frequency of each molecule is equal to zero. The isotopic ratios of these zero frequencies must be evaluated through use of the Teller-Redlich product rule (in practice this difficulty is averted by using the Bigeleisen-Mayer18 formulation of the isotope effect equation, which introduces the Teller-Redlich product rule naturally).

In Table VII, equilibrium constants are given for the water system obtained with harmonic force fields which differ from the KM force field as indicated. Calculation I is identical to the water calculation at 298°K in Table V. In calculation II the "stretching" frequencies of the water molecule have been forced to zero, and only the bending frequency contributes to the equi­librium constant. K is almost the same as in 1. In calculations III and IV, the water bending force constant has been replaced by the sulfur and selenium bending force constant, respectively. The K values are very close to those reported for H 2S and H2Se in Table V. Calculation V shows that the equilibrium constant is little affected by the interaction force constants. Calculations VI-XIII demonstrate, by the carrying out of calculations with fairly large changes in the force constants from their natural values, that the equilibrium constants reflect largely the value of the bending force constant f". It is thus clear that the differences among the three systems in Table V depend largely on the differences in the bending force constants.

Correction Terms

The equilibrium constants calculated above must now be corrected for various omitted effects. Chief among these is the correction of the ZPE term for anharmonicity by multiplying the previously cal­culated ZPE term by exp [hc<l(AZPE)/kT].

Other corrections to the equilibrium constants, which

have been considered by Friedman and Haar 22.2:l

for water and hydrogen s~lfide systems have also been considered here.

(1) Correction factor to the rotational partition function for rotation- vibration coupling. The correc­tion of the rotational constants to the zero-point vibra­tional state, with use of the experimentally determined a values of Benedict e/ al.,ll leads to a cor;'ection factor 1.001 for K of the water system. This correction factor has also been calculated with the use of the a's theoretically calculated by Papousek and PlivaY With their force field yIU13FF a factor of 1.007 is obtained while with SUBFF 1.009 is obtained. On the other hand, if the experimental ground vibrational state rotational constants of BGPll are used in calculat­ing K, as they were in a previous report,6 then a correction factor to K of 0.999 is obtained. For the hydrogen sulfide system use of the 0,. values quoted by Haar et al.23 leads to a correction factor 0.987. The correction factor to K from changes of rotational constants in higher vibrational states is calculated to be negligible for both the water and the hydrogen sulfide systems. Since there seems to be considerable undertain t y in the correction factor from correcting the rotational constants to the ground vibrational state, it has been decided to ignore this correction although with use of presently accepted methods this correction factor may differ from unity by as much as 1%.

(2) Correction factor for nonclassical rotation. With the numbers given by Friedman et al.,22.23 the correction factor to K is found to be 0.998 at 273°K for both the water and hydrogen sulfide systems (and it approaches unity with increasing temperature). This factor is sufficiently close to unity to be ignored for the present purposes.

(3) Correction factor to the vibrational excitation term, due to anharmonicity. If one follows the approach of Friedman and Haar,22 one has to correct the vibra­tional excitation factor in the partition function, 1- exp( -hcw/kT) , by replacing the harmonic frequency w by the corresponding fundamental frequency. With use of the fundamentals calculated here, correction factors found over the room-temperature range are about 0.9999 for the water system and the hydrogen sulfide system, and even closer to unity for the hydrogen selenide system. The additional correction factors arising from the terms designated fij by Friedman and Haar are quite negligible in the room-temperature range. Correction factors to vibrational excitation from anharmonicity have consequently also been ignored for present purposes.

(4) Correction factor arising from centrifugal stretch. V.'ith lise of factors listed by Friedman and Haar it is found that this factor also ~an be ignored here. '

Thus, the only correction to the equilibrium constants listed in Table V which will be considered to be of

EXCHAKGE EQCILIBRIA H,X+D,X=2HDX 314.1

TABLE VIII. Comparison of calculated and experimental equilibrium constants.

K (11:0\1<11

x l' anharmonicity) "

OXY'gen 273 4(0.864) =3.46 29R 4(0.878) =3.51

348 4(0.900) =3.60

Sulfur 273 4(0.898) =3.59 298 4(0.909) =3.64 348 4(0.925) =3.70

Selenium 273 4(0.913) =3.65 298 4(0.922) =3.69 348 4(0.935)=3.74

a It is apparent from Table II that these values. unlike those in the next column. are fairly sensitive to force field. The difference between the

consequence is the anharmonic correction to the zero­point energy. This correction will be carried out: (1) without consideration of Go by using .1(tXiJ and (2) with consideration of Go by using .1 (AZPE) . For each system the first entry (not in parentheses) for that system in Table II is used. While the use of .1(tXij) gives rise to a large decrease in the previously calculated ("harmonic") equilibrium constants, Table VIII shows that the correctly calculated anharmonicity correction is negligible. Table VIII also lists for com­parison the values obtained experimentally in the gas phase by mass spectrometry for these equilibrium constants.24

The agreement between theory and experiment is considerably better when the anharmonicity correction to the zero-point energy is correctly calculated. The agreement between theory and experiment for the hydrogen sulfide and the hydrogen selenide systems appears satisfactory when one considers the experi­mental limits of error and the previously mentioned uncertainty of up to 1 % in the theoretical contribution of the molecular rotations to the equilibrium constants. It may, however, be noteworthy that for these two systems as well as for the water system, the "best" calculated values appear to be uniformly high with respect to the "best" experimental ones. For the water system, the disagreement is sufficiently large to lead one to question its origin.

Pyper and Newbury3 have mentioned the possibility of errors in the experimental determinations, and the reader should make reference to that discussion. Among the factors which lead to uncertainties in the theoretical values, the following ones should be mentioned.

(1) There are possible errors in the contribution of the molecular rotations and of the rotation-vibration

---- -" ------ - --------

K (including Go)

(" best" theoretiotl value) Kpxptl

4(0.957) =3.83 3.74±0.021 4(0.963)=3.85 3.76±0.021

3.74±0.072 4(0.974) =3.90 3.80±0.041

4(0.977) =3.91 4(0.982) =3.93 3. 88±0. 03" at 297°K 4(0.988) =3.95

4(0.985) =3.94 4(0.987) =3.95 3. 92±O. (W at 29SoK 4(0.992) =3.97

values below for oxygen and the value about 3.4 mentioned at the be­ginning of this report should not be clisturbing.

interaction. Some of the uncertainty has to do with the effect of vibrations on the rotational constants as discussed previously. In addition, it appears worthwhile to reinvestigate the accuracy of the approximations made in evaluating isotopic rotational partition function ratios.

(2) The adequacy of the anharmonic theory being used, which is accurate to second OJ der of approxima­tion, needs further investigation.

(3) Since the theoretical values calculated are essentially the harmonic values, one might question the accuracy of the harmonic force constants. It is seen from calculations VI-XIII in Table VII, where individual force constants of the water molecule were varied, that the only such force field variation which affects the equilibrium in the room-temperature region is the variation of the bending force constant. However, an increase in the bending force constant by almost a factor of 2 would be needed in order to bring the calculation into agreement with experiment. The increase of the bending force constant by a factor of 2 raises the calculated "bending" frequency in H20 from 1649 to 2342 cm-I, while the value deduced by Benedict et alY from the spectrum is 1648 cm- I .

While it has not been attempted here to carry out concurrent variations of force constants, it appears fairly obvious that a major part of the disagreement between theory and experiment for the water equi­librium cannot be blamed on uncertainties in the harmonic force field.

(4) It is possible that the discrepancy between theory and experiment reflects a failure of the Born­Oppenheimer approximation. The theoretical investiga­tion of such possibility would be quite difficult. It would be expected that a "small" failure of the Born­Oppenheimer approximation would tend to cancel in the evaluation of a self-exchange equilibrium and,

W () L F S B ERG, ,\1;\ S SA, .\ ~ j) P Y l' E R

therefore, this possibility as a source of discrepancy is discounted for the present.

CONCLUSION

It has been shown that the equilibrium constant for reaction (1) where X is 0, S, Se is subject to a fairly large correction which arises from the vibrational anharmonicity contribution to the zero-point energy change when previously standard formulations for this contribution are employed. The inclusion of the Go factor, Eq. (3), in the calculation does however tend to eliminate this correction for these isotopic self­exchange equilibria. The latter conclusion does not ap­pear very sensitive to the details of the anharmonic force field as long as this force field is reasonable. The pres­ently calculated values of the equilibrium constants agree fairly satisfactorily with the values observed in mass-spectrometric experiments. There is a slight dis­crepancy between theory and experiment which deserves further study.

ACKNOWLEDGMENTS

One of us (M. W.) wishes to acknowledge that his interest was first drawn to the anharmonicity problem by Dr. Lewis Friedman and Dr. V. J. Shiner, Jr., who convinced him that the difference between their experimental results and usual theory was real. He also wishes to acknowledge continuing discussions on self-exchange equilibria with Dr. Jacob Bigeleisen and Dr. R. E. Weston, Jr.

* Present address. Work at University of California, supported in part under U.S. Atomic Energy Commission Contract No. AT (04-3) -34, Project Agreement No. 188.

t Work carried out under the auspices of the U.S. Atomic Energy Commission.

t Present address: Chemistry Department, Columbia Univer­sity, New York, N.Y., 10027.

1 L. Friedman and V. J. Shiner, Jr., J. Chern. Phys. 44, 4639 (1966) .

2 J W. Pyper, R. S. Newbury, and G. W. Barton, Jr., J. Chern. Phys. 46,2253 (1967).

3 J. W. Pyper and R. S. Newbury, J. Chern. Phys. 52, 1966 (1970) .

4 J. W. Pyper and F. A. Long, J. Chern. Phys. 41,2213 (1964); R. E. Weston, Jr., ibid. 42, 2635 (1965).

5 M. Wolfsberg, Advan. Cbem. Ser. 89,185 (1969). 6 M. Wolfsberg, J. Chern. Phys. 50, 1484 (1969). 7 J. R. Hulston, J. Chern. Phys. 50, 1483 (1969). 8 H. H. Nielsen, Rev. Mod. Phys. 23, 90 (1951). Note that

Nielsen uses sum ~ij while in this paper sum ~iS;j is employed. 9 W. H. Shaffer and R. P. Schuman, J. Chern. Phys. 12, 504

(1944). 10 Reference 8, pp. 126-129. 11 W. S. Benedict, N. Gailar, and E. K. Plyler, J. Chern. Phys.

24, 1139 (1956). 12 D. Papousek and J. Pliva, Collection Czech. Chern. Commun.

29,1973 (1964).

1:: Papousek and [,liva used two different anharmonic force Jiclds designated as SUEFF and :VIUBFF. The latter was con­sidered by them to yield better agreement with experiment. Since k122 for HDO is comparatively small in SUEFF, the values of X22 and XI2 evaluated with this force field are not very sensitive to whether or not the Fermi-Dennison resonance is taken into account. However, for MUBFF (the force field which was used in the Go evaluations of Refs. 6 and 7), the calculated values of x" and Xl2 are quite sensitive to the Fermi-Dennison resonance. The calculated values of kijl, k ijlt , and Wi for HDO given in the paper by l'apousek and Pliva for MUBFF and SUBFF (together with the Coriolis constant and the moment of inertia needed to calculate .1'12) have been used to calculate the .\'22 and .1'12 values given in the table at the end of this footnote. This table is self­explanatory. The l'apousek and Pliva values are close to those which one obtains by taking resonance into account for MUBFF. It is, however, an open question whether l'apousek and Pliva obtained this result because of an error. The units below are\\'avl'­numbers.

-~--~- -----

X~t .lIZ : (.1',,+.1'1') ----- ------

MUBFF" -S.9 -16.3 -6.3 MUBFFb -4.6 -33.8 -9.6 MUBI''F" -8.1 -17.9 -6.5 MUBFI·'d -8.1 -17.9 -6.5 SUBFI-'a 4.0 -14.2 -2.6 SUBFFh 4.3 -15.1 -2.7 SUBFF" 4.3 -14.3 -2.5 SUBFFd 4.2 -1.1.1 -2.7

a Calculation taking Fermi-Dennboll resonance into accounL. b Calculation not taking Fermi-Dennison resonance into account. e Calculation not taking Fermi-Dennbon resonance into account but

arbitrarily making the 4wz 2 -W12 energy denominators 10 times a:-; large a:-, they actually are. It \vas found accidentally that this recipe gives good agreement with the MCBFF values reported by Pa}lou~ek and Pliva.

d Calculated valves reported by Papou~ek and Pliva.

14 E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1955).

15 M. A. Pariseau, 1. Suzuki, and J. Overend, J. Chern. Phys. 42,2335 (1965).

16 K. Kuchitsu and Y. Morino, Bull. Chern. Soc. Japan 38, 814 (1965) .

17 G. A. Khachkuruzov, Tr. Gos. Ins!. Prikl. Khim. 42, 109 (1959) .

IS J. Eigeleisen and M. G. Mayer, l Chern. I'hys. 15, 261 (1947) .

19 M. Wolfsberg and M. J. Stern, Pure Appl. Chern. 8, 225 (1964).

20 J. Bigeleisen, Proc. Intern. Symp. Isotope Separation, Amster­dam, 1957, 121 (1958); J. Bigeleisen and T. Ishida, l Chern. Phys. 48, 1311 (1968); T. Ishida, W. Spindel, and J. Bigeleisen, Advan. Chern. Ser. 89, 192 (1969).

21 l Bigeleisen, R. E. v"eston, Jr., and M. Wolfsberg, Z. Naturforsch. 18a, 210 (1963).

22 A. S. Friedman and L. Haar, l Chern Phys 22,2051 (1954). 23 L. Haar, J. C. Bradley, and A. S. Friedman, J. Res. Natl.

Bur. Std. 55, 285 (1955). 24 There have also been two recent determinations of the

equilibrium constant K(O) in the liquid phase by indirect methods. V. Gold [Trans. Faraday Soc. 64,2770 (1968)J reports a value 3.8 at 31°C while A. J. Kresge and Y. Chiang [J. Chern. Phys. 49,1439 (1968)J report a value 3.85±0.03 at 25°C. The liquid-phase-gas-phase correction to the equilibrium constant is negligible if the vapor pressures adhere to the relationship P HDo = (PD20PH20)'12, as closely as indicated by the work of L. Merlivat, R. Botter, and G. Nief, J. Chim. Phys. 60, 56 (1963),