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The structure of the H3O+ (hydronium) ion
Volume 21, number 2
-15 August 1973
CHEMICAL PHYSICS LETTERS
THE STRUCTURE
OF THE H,O+ (HYDRONIUM)
ION
Peter A. KOLLMAN Department
ofPharmaceutical
Chemisfiy, School of Pharnucy, University of Ca!ifomia. San Francisco, California 94122, USA
and
Charles F. BENDER Lawrence Liverlnore Laboratory, Livermore, California
94550,
USA
Received 25 April 1973
Near Hartree-Fock level ab initio molecular orbital ticulations on HsO+ fsnd a minimum energy structure with tJ(HOH) = 112.5” and r(O-H) = 0.963 BLand an inversion barrier of 1.9 k&/mole. By comparing these results to calculations on NH3 and HaO, where precise experimental geometries are known, we estimate the “true” geometry of isolated HsO* to have a structure with e(HOH) = 110-l 12”, r(O-H) = 0.97-0.98,4 and an inversion barrier of 2-3 k&/mole. Our prediction for the proton affiiity of water is = 170 kul/mole, which is somewhat snelter than the currently accepted value.
1. Introduction In view of the importance of the hydronium ion in acidic water solutions, it has received considerable experimental and theoretical attention. De Paz et al. [I] and Kebarle et al. [2] have studied protonated water clusters in the gas phase and a recent paper by Lundgren and Williams [3] discusses X-ray and neutron diffraction, NMR and IR studies on H,O+ in some molecules in the solid state. Zundel has described some spectral studies of acid hydrates in liquids and films [4]. Theoretical studies of H,O+ have included ab initio (nonempirical) molecular orbital calculations using a one-center basis on the oxygen [S], a floating
spherical gaussian basis [6] and ST0 [7] and gaussian [8-131 atomic centered bases. The most recent and accurate studies in this area have been t!!ose of Moe+ kowitz and Harrison @UI) [8] , Hopkinson et al. (H) [9] , KoUman and Allen (KA) [I Cj, Newton and Ehrenson (NE) [I I], Kraemers and Diercksen (KD) [12], and Almldf and Wahlgren (_4W) [ 131. In cases where the geometry was optimized, those studies which em-’ pioyed only s and p functions on oxygen and only s
functions on H all found a planar H@’ ion to be the minimum energy form [8,10,1 I] whereas those that included p functions on hydrogen fmd a slightly nonplanar (e(HOH) 120’) structure (KA did not fuiIy optimize their structure since they were mainly interested in the hydrogen bonding of the HsOi ion, but it appears from their quoted energy values for the HsO+ ion with p functions on hydrogens that the minimum energy would occur between 8 = II 5 and 120 degrees. The same geometry search without the p’s clearly showed a minimum energy at d = 12O’j.
AW found a non-planar minimum energy at 0 = 116.6” with an inversion barrier (~inv,) of 0.3 k&/mole. The calculation of KD was the most accurate, but no geometry optimization was carried out, since.these authors were most interested in the properties of the H50; ion. None of the above studies give us definitive evidence on the geometry of H,Ot; this can be most clearly seen from the theoretical work by Rauk et al. [ 141 and Stevens [ 151 on NH, and Kari and Csizmadia [16] on CHT. These groups note the importarrce of d polarization functions in stabilizing the minimum energy pyramidal (C,,) relative to the planar (D3h) 271.
Volume 21, number 2
CHEMICAL
PHYSICS JXTTERS
t5 August 1973
Table 1 Hz0
(experiment) ref. [ 191
r(o-m
(A)
efHOH1 Et c-w-
(degrees) -
(A)
GWXH) (degees) aEhvet (kQ~mole) Et (au)
f04.52°
76.05 103 (-76.06283)
HxO+bj
f(X-HI
0.957
0.941 (0.941) 1’06.1 (106.6)
0.963 (0.959) 112.5’ 1.9 - 16.33260
NH3 (theory)
CH;
NH3 (expt.) ref. [2]
ref. [14] 1.000 107.2 5.1 - 56.22131
ref. [ 161 1.105 105.2 7.0 - 39.50820
1.0116 106.7 5.8
a) Ref. [IS]; same basiz set as here (largest basis set results in parentheses). b) This work; bond length in parentheses is minimum energy bond length for 6 = 120”.
structure. An exhaustive s,p basis cn N and s basis on H predicts a far too low inversion barrier and a too large B(HNH). The nearly quantitative prediction of the correct barrier in NH, with the inclusion of polarization functions is due 1141 to the fact that alI 5 of the d orbit& are symmetry allowed to mix into the molecular orbitals in the C,, structure, but two (d,, and dYi) cannot contribute to the occupied MO’s in rhe planar (Djnf structure. Thus, one needs near Hartree-Fock Ievel wavefunctions to determine the minimum energy structure and the inversion barrier in H Oc, NH3 and CHF uld such a calculation on HsO j. is the main purpose of this study. In addition, there has been scme question-whether the “true” proton affinity of H20 is nearer 7.9 [I] or 7.2 eV [17;1and we hope to shed some Light on this question with our results.
ergies for the geometries of H,O* examined in this study are in table 2.
3. Results and discussion The inw-polated
minimum
bond length to be about 0.01-0.02 the “true”
The basis set used in these ab initio LCAO MO SW calcu!ations js the (9s5pZd/4slp) primitive gaussian basis employed by Dunning et al. [ 181 in ,?.heirstudy of HzO. This basis set, which was contracted [ 181 to (4s3p2d/%Ip), predicts virtually the same geometry as the most exhaustive basis set employed by Dunning et al. [18] and these geometrical parameters are within 2’ and Q.02 A of the experimental values. A summary of relevant c~culations on HzO, NH,, H30+ and CHT is presented in table 1. The total en272
i% shorter
minimum energy bond distance.
of
than There
is
Table 2 Potential surface of H30’ ~ ~-f (au)
2. Computational details
energy structure
H,O’ is$O-H) = 0.963 a and B(HOH) = 112.5”. In view of the accuracy of the NH3 and Hz0 calculations, we estimate our calculated angle to be l-2” larger than the “true” minfmum energy angle and the
1.90 1.85 1.80 1.90 1.90 1.8s 1.80 1.90 1.82 1.82 1.82 1.84 1.80 1.82
E (au) 114 114 114 108 120 120 120 111 114 120 112 112 112 110
-76.327023 -76.331268 -76.331776 -76.326615 -76.323464 -76.328321 -76.329472 -76.327337 -76.332065 -76.329500 -76.332559 -76.331766 -76.331681 -76.331588
Volume 21, number 2
CHEMICAL. PHYSICS LETTERS
no data on the completely isolated H30f ion, but Lundgren and Williams [3] find an average HOH angle of 110.4’ in their neutron diffraction studies of ptoluenesulphonic acid monohydrate. The average O-H distance they fmd is 1 .Ol 8,, longer than our estimated 0.97-0.98 A for the isolated H,O+ because of hydrogen bonding to neighboring oxygens (R(O-0) = 2.53 A). In view of the near Hartree-Fock results of Rauk et al. [14] we estimate an experimental inversion barrier of 2-3 kcal/mole for H-,0+ (we calculate a barrier of 1.9 kcal/mole), which can be compared with a barrier of 5.8 kcal/mole for NH3 and an estimated barrier of 8 kcallmole for CH: (assuming the barrier of ref. [ 161 to be underestimated by about the same as that of Rauk et al. [14]). Since negative ions are treated with less accuracy than neutral molecules or positive ions by MO theory, we expect that the CHT predictions may be further from the “true” geometry. However the qualitative trend of increasing barrier and decreasing HXH angle if one goes from H30+ to CHT is clearly evident and one would expect H,Fzt to be planar. Using Dunning’s calculations on Hz0 [ 181 and the zero point energy estimate of Newton and Ehrenson [I l] (AI&.,JD,O+-D20) = 0.27 ev), we predict a deuteron affinity for D,O of = 170 kcal/mole, and a proton affinity for Hz0 (assuming a bEZp.e. = 0.38 ev) of = 167.5 kcal/mole, significantly less than the proton affinity found by DePaz et al. [I] (!$O+) and close to the proton affinity (166 kcaI/moIe) found by Munson 1171. Estimates of the correlation energy of H30+ and H,O indicate that the latter may have a greeter correlation energy and thus, the experimental proton affinity should be !ess than that predicted here *. It is of interest that Schaad [21] has estimated a correlation contribution of 1 to 3 kcal/mole to the proton affinity of Hz. In light of these calculations, we have some evidence that the proton affinity of water is likely to be near 170 kcal/mole, but the uncertainty in the correlation energy estimate fur Hz0 and H30f and the fact that our total energy for Hz0 is still about 0.015 au from the estimated HartreeFock Limit [ 181 preclude a definitive statement_ Clear-
15 August 1973
ly, the experimental and theoretical evidence needs to be examined more precisely on this point. In addition to calculating the geometry ofH,O’: Almlb’f and Wahlgren (AW) [ 131 examined the effect of neighboring electrostatic fields (as found in certain crystals) on the structure of the zi,Oc ion. Although their results were qualitatively reasonable (nearby
negative charges increase the O-H distance and.the HOH angle decreases found are due to the tendencies to make 0-H...X angles as near linear as possibiej, some of them may need to be reexamined in view of the results reported here. The minimum energy geometry, B(HOH) = 112.5’, and inversion barrier (bending potential curve) Found here are quite different from that found by AW. From the calculations presented here, it appears that a near tetrahedral HsO+ ion (O(HOH) Z= 110-l 12’) is the optimum for the isolated H30+ ion rather than the 0(HOH) = 116.6” calculated by AW and the shapes of H30f ions in
crystalhne environments shouId be interpreted in this light. In the crystals examined by AW, the H30C HOH angles are found to be between 105 and f LO”.
References
111M. de Paz, l.J,Leventhal
and L. Friedman, I. Chem. Phy‘s. 51 (1969) 3748; hi. de Paz, A.G. Guidoni and L. Fried&n. I. Chen.
Phys. 52 (1970) 637.
PI P. Kebarle, S.K. SearIles, A. Zolla, J. Scarborough and
M. Arshadi, J. Am. Chem. Sot. 89 (1967) 6393. 1.M. Williams, 1. Chem. Phys. 58 (1973) 788. G. Zundel, Hydration and intermolecular in:eraction (Amdemic Press, New York, 1970). D.M. Bishop, J.Chem. Phys. 43 (1965) 4453; D.N. Tripathi, D. Tiwari and DX. Rai, Indian J. Pure Appl. Phys. 7 (1969) 707. A.A. Frost, J. Phys Chem. Ithaca 72 (1968) 1289. R. Grahn, Arkiv Fysik 19 (1961) 1417. J.W. Moskowitz and NC. Harrison, J. Chem. Phys. 43 (1965) 3550. A.C. Hopkinson, N.K. Holbrook, K. Yates and LG.
131 J.O. Lundgren and 141 ISI
[61 171 IS! [91
Csizmadia, J.. Chem. Phys 49 (1968)
3596.
DOI P.A. Kollrnan and L.C. Alien, J. Am. Chem. SOC. 92 (1970) 6101.
1111M.D. Newton and S. E!uenson, J. Ant. Chem. Sot. 93 (1971) 4971.
* New-ton and Ehrenson [ll] have noted that the correlation cantiibution to the energy of HsO’ may be as much as - 1 eV less than that of HsO, based on the work of ref. [20].
1121W_P. Kraemer and G.H.F. Ditrcksen, Chum. Phys. Letters 5 (1970) 463. [I31 J. Ab-nlaf and U. Wahlgren, (1973) 161.
Thearet. ti_
Acta 28
‘273
15 Augsut 1973
CHEMlCAL PHYSICS LETTERS
Volume 21, number 2
[141 A. Rauk,L.C. AllenandE. Clementi, J. Chem. Phys. 52 (1970) 4133. [is] R. Stevens, J. Chcm. Phys. 55 (1971) 1725. [16] R. Kari and I.G. Csizmadia, 3. Chem. Phys. 50 (1969) 1443. [I?] M.S.B. Munson, J. Am. Chem. Sot. 67 (1965) 2332; ?if.k Haney and J.L. FrankLin, J. Chem. Phys. 50 (1969) 2028. [ 181 T.H..Dunning Jr., R.M. Pitzer and S. Aung, J. Chem. Phys. 57 (1972) 5044.
[19] W.S. Benedict,,M. Gailar and E.K. Pyler, J. Chem. Phys. 44 (1965) 1139. [20! V. McKay, J. Chem. Phys. 42 (1965) 2232; C.D. Ritchie and H. King, J. Chem. Phys. 47 (1967) 56C. [21] L. Schaad, private communication. [22] W.X. Benedict and E.K. Pyler, Can. J. Phys. 35 (1957) 1235.
:’
274 ,’
,.:.’ ,,_:,” ,.,
.’
.I
j
,. .’
1 ,.
2
‘.
-15 August 1973
CHEMICAL PHYSICS LETTERS
THE STRUCTURE
OF THE H,O+ (HYDRONIUM)
ION
Peter A. KOLLMAN Department
ofPharmaceutical
Chemisfiy, School of Pharnucy, University of Ca!ifomia. San Francisco, California 94122, USA
and
Charles F. BENDER Lawrence Liverlnore Laboratory, Livermore, California
94550,
USA
Received 25 April 1973
Near Hartree-Fock level ab initio molecular orbital ticulations on HsO+ fsnd a minimum energy structure with tJ(HOH) = 112.5” and r(O-H) = 0.963 BLand an inversion barrier of 1.9 k&/mole. By comparing these results to calculations on NH3 and HaO, where precise experimental geometries are known, we estimate the “true” geometry of isolated HsO* to have a structure with e(HOH) = 110-l 12”, r(O-H) = 0.97-0.98,4 and an inversion barrier of 2-3 k&/mole. Our prediction for the proton affiiity of water is = 170 kul/mole, which is somewhat snelter than the currently accepted value.
1. Introduction In view of the importance of the hydronium ion in acidic water solutions, it has received considerable experimental and theoretical attention. De Paz et al. [I] and Kebarle et al. [2] have studied protonated water clusters in the gas phase and a recent paper by Lundgren and Williams [3] discusses X-ray and neutron diffraction, NMR and IR studies on H,O+ in some molecules in the solid state. Zundel has described some spectral studies of acid hydrates in liquids and films [4]. Theoretical studies of H,O+ have included ab initio (nonempirical) molecular orbital calculations using a one-center basis on the oxygen [S], a floating
spherical gaussian basis [6] and ST0 [7] and gaussian [8-131 atomic centered bases. The most recent and accurate studies in this area have been t!!ose of Moe+ kowitz and Harrison @UI) [8] , Hopkinson et al. (H) [9] , KoUman and Allen (KA) [I Cj, Newton and Ehrenson (NE) [I I], Kraemers and Diercksen (KD) [12], and Almldf and Wahlgren (_4W) [ 131. In cases where the geometry was optimized, those studies which em-’ pioyed only s and p functions on oxygen and only s
functions on H all found a planar H@’ ion to be the minimum energy form [8,10,1 I] whereas those that included p functions on hydrogen fmd a slightly nonplanar (e(HOH) 120’) structure (KA did not fuiIy optimize their structure since they were mainly interested in the hydrogen bonding of the HsOi ion, but it appears from their quoted energy values for the HsO+ ion with p functions on hydrogens that the minimum energy would occur between 8 = II 5 and 120 degrees. The same geometry search without the p’s clearly showed a minimum energy at d = 12O’j.
AW found a non-planar minimum energy at 0 = 116.6” with an inversion barrier (~inv,) of 0.3 k&/mole. The calculation of KD was the most accurate, but no geometry optimization was carried out, since.these authors were most interested in the properties of the H50; ion. None of the above studies give us definitive evidence on the geometry of H,Ot; this can be most clearly seen from the theoretical work by Rauk et al. [ 141 and Stevens [ 151 on NH, and Kari and Csizmadia [16] on CHT. These groups note the importarrce of d polarization functions in stabilizing the minimum energy pyramidal (C,,) relative to the planar (D3h) 271.
Volume 21, number 2
CHEMICAL
PHYSICS JXTTERS
t5 August 1973
Table 1 Hz0
(experiment) ref. [ 191
r(o-m
(A)
efHOH1 Et c-w-
(degrees) -
(A)
GWXH) (degees) aEhvet (kQ~mole) Et (au)
f04.52°
76.05 103 (-76.06283)
HxO+bj
f(X-HI
0.957
0.941 (0.941) 1’06.1 (106.6)
0.963 (0.959) 112.5’ 1.9 - 16.33260
NH3 (theory)
CH;
NH3 (expt.) ref. [2]
ref. [14] 1.000 107.2 5.1 - 56.22131
ref. [ 161 1.105 105.2 7.0 - 39.50820
1.0116 106.7 5.8
a) Ref. [IS]; same basiz set as here (largest basis set results in parentheses). b) This work; bond length in parentheses is minimum energy bond length for 6 = 120”.
structure. An exhaustive s,p basis cn N and s basis on H predicts a far too low inversion barrier and a too large B(HNH). The nearly quantitative prediction of the correct barrier in NH, with the inclusion of polarization functions is due 1141 to the fact that alI 5 of the d orbit& are symmetry allowed to mix into the molecular orbitals in the C,, structure, but two (d,, and dYi) cannot contribute to the occupied MO’s in rhe planar (Djnf structure. Thus, one needs near Hartree-Fock Ievel wavefunctions to determine the minimum energy structure and the inversion barrier in H Oc, NH3 and CHF uld such a calculation on HsO j. is the main purpose of this study. In addition, there has been scme question-whether the “true” proton affinity of H20 is nearer 7.9 [I] or 7.2 eV [17;1and we hope to shed some Light on this question with our results.
ergies for the geometries of H,O* examined in this study are in table 2.
3. Results and discussion The inw-polated
minimum
bond length to be about 0.01-0.02 the “true”
The basis set used in these ab initio LCAO MO SW calcu!ations js the (9s5pZd/4slp) primitive gaussian basis employed by Dunning et al. [ 181 in ,?.heirstudy of HzO. This basis set, which was contracted [ 181 to (4s3p2d/%Ip), predicts virtually the same geometry as the most exhaustive basis set employed by Dunning et al. [18] and these geometrical parameters are within 2’ and Q.02 A of the experimental values. A summary of relevant c~culations on HzO, NH,, H30+ and CHT is presented in table 1. The total en272
i% shorter
minimum energy bond distance.
of
than There
is
Table 2 Potential surface of H30’ ~ ~-f (au)
2. Computational details
energy structure
H,O’ is$O-H) = 0.963 a and B(HOH) = 112.5”. In view of the accuracy of the NH3 and Hz0 calculations, we estimate our calculated angle to be l-2” larger than the “true” minfmum energy angle and the
1.90 1.85 1.80 1.90 1.90 1.8s 1.80 1.90 1.82 1.82 1.82 1.84 1.80 1.82
E (au) 114 114 114 108 120 120 120 111 114 120 112 112 112 110
-76.327023 -76.331268 -76.331776 -76.326615 -76.323464 -76.328321 -76.329472 -76.327337 -76.332065 -76.329500 -76.332559 -76.331766 -76.331681 -76.331588
Volume 21, number 2
CHEMICAL. PHYSICS LETTERS
no data on the completely isolated H30f ion, but Lundgren and Williams [3] find an average HOH angle of 110.4’ in their neutron diffraction studies of ptoluenesulphonic acid monohydrate. The average O-H distance they fmd is 1 .Ol 8,, longer than our estimated 0.97-0.98 A for the isolated H,O+ because of hydrogen bonding to neighboring oxygens (R(O-0) = 2.53 A). In view of the near Hartree-Fock results of Rauk et al. [14] we estimate an experimental inversion barrier of 2-3 kcal/mole for H-,0+ (we calculate a barrier of 1.9 kcal/mole), which can be compared with a barrier of 5.8 kcal/mole for NH3 and an estimated barrier of 8 kcallmole for CH: (assuming the barrier of ref. [ 161 to be underestimated by about the same as that of Rauk et al. [14]). Since negative ions are treated with less accuracy than neutral molecules or positive ions by MO theory, we expect that the CHT predictions may be further from the “true” geometry. However the qualitative trend of increasing barrier and decreasing HXH angle if one goes from H30+ to CHT is clearly evident and one would expect H,Fzt to be planar. Using Dunning’s calculations on Hz0 [ 181 and the zero point energy estimate of Newton and Ehrenson [I l] (AI&.,JD,O+-D20) = 0.27 ev), we predict a deuteron affinity for D,O of = 170 kcal/mole, and a proton affinity for Hz0 (assuming a bEZp.e. = 0.38 ev) of = 167.5 kcal/mole, significantly less than the proton affinity found by DePaz et al. [I] (!$O+) and close to the proton affinity (166 kcaI/moIe) found by Munson 1171. Estimates of the correlation energy of H30+ and H,O indicate that the latter may have a greeter correlation energy and thus, the experimental proton affinity should be !ess than that predicted here *. It is of interest that Schaad [21] has estimated a correlation contribution of 1 to 3 kcal/mole to the proton affinity of Hz. In light of these calculations, we have some evidence that the proton affinity of water is likely to be near 170 kcal/mole, but the uncertainty in the correlation energy estimate fur Hz0 and H30f and the fact that our total energy for Hz0 is still about 0.015 au from the estimated HartreeFock Limit [ 181 preclude a definitive statement_ Clear-
15 August 1973
ly, the experimental and theoretical evidence needs to be examined more precisely on this point. In addition to calculating the geometry ofH,O’: Almlb’f and Wahlgren (AW) [ 131 examined the effect of neighboring electrostatic fields (as found in certain crystals) on the structure of the zi,Oc ion. Although their results were qualitatively reasonable (nearby
negative charges increase the O-H distance and.the HOH angle decreases found are due to the tendencies to make 0-H...X angles as near linear as possibiej, some of them may need to be reexamined in view of the results reported here. The minimum energy geometry, B(HOH) = 112.5’, and inversion barrier (bending potential curve) Found here are quite different from that found by AW. From the calculations presented here, it appears that a near tetrahedral HsO+ ion (O(HOH) Z= 110-l 12’) is the optimum for the isolated H30+ ion rather than the 0(HOH) = 116.6” calculated by AW and the shapes of H30f ions in
crystalhne environments shouId be interpreted in this light. In the crystals examined by AW, the H30C HOH angles are found to be between 105 and f LO”.
References
111M. de Paz, l.J,Leventhal
and L. Friedman, I. Chem. Phy‘s. 51 (1969) 3748; hi. de Paz, A.G. Guidoni and L. Fried&n. I. Chen.
Phys. 52 (1970) 637.
PI P. Kebarle, S.K. SearIles, A. Zolla, J. Scarborough and
M. Arshadi, J. Am. Chem. Sot. 89 (1967) 6393. 1.M. Williams, 1. Chem. Phys. 58 (1973) 788. G. Zundel, Hydration and intermolecular in:eraction (Amdemic Press, New York, 1970). D.M. Bishop, J.Chem. Phys. 43 (1965) 4453; D.N. Tripathi, D. Tiwari and DX. Rai, Indian J. Pure Appl. Phys. 7 (1969) 707. A.A. Frost, J. Phys Chem. Ithaca 72 (1968) 1289. R. Grahn, Arkiv Fysik 19 (1961) 1417. J.W. Moskowitz and NC. Harrison, J. Chem. Phys. 43 (1965) 3550. A.C. Hopkinson, N.K. Holbrook, K. Yates and LG.
131 J.O. Lundgren and 141 ISI
[61 171 IS! [91
Csizmadia, J.. Chem. Phys 49 (1968)
3596.
DOI P.A. Kollrnan and L.C. Alien, J. Am. Chem. SOC. 92 (1970) 6101.
1111M.D. Newton and S. E!uenson, J. Ant. Chem. Sot. 93 (1971) 4971.
* New-ton and Ehrenson [ll] have noted that the correlation cantiibution to the energy of HsO’ may be as much as - 1 eV less than that of HsO, based on the work of ref. [20].
1121W_P. Kraemer and G.H.F. Ditrcksen, Chum. Phys. Letters 5 (1970) 463. [I31 J. Ab-nlaf and U. Wahlgren, (1973) 161.
Thearet. ti_
Acta 28
‘273
15 Augsut 1973
CHEMlCAL PHYSICS LETTERS
Volume 21, number 2
[141 A. Rauk,L.C. AllenandE. Clementi, J. Chem. Phys. 52 (1970) 4133. [is] R. Stevens, J. Chcm. Phys. 55 (1971) 1725. [16] R. Kari and I.G. Csizmadia, 3. Chem. Phys. 50 (1969) 1443. [I?] M.S.B. Munson, J. Am. Chem. Sot. 67 (1965) 2332; ?if.k Haney and J.L. FrankLin, J. Chem. Phys. 50 (1969) 2028. [ 181 T.H..Dunning Jr., R.M. Pitzer and S. Aung, J. Chem. Phys. 57 (1972) 5044.
[19] W.S. Benedict,,M. Gailar and E.K. Pyler, J. Chem. Phys. 44 (1965) 1139. [20! V. McKay, J. Chem. Phys. 42 (1965) 2232; C.D. Ritchie and H. King, J. Chem. Phys. 47 (1967) 56C. [21] L. Schaad, private communication. [22] W.X. Benedict and E.K. Pyler, Can. J. Phys. 35 (1957) 1235.
:’
274 ,’
,.:.’ ,,_:,” ,.,
.’
.I
j
,. .’
1 ,.
2
‘.