Virial Theorem ~ ~ c o ~of ~Molecular ~ o Force ~ i Fields ~ ~ ~ n t.ively. Two features are apparent. First, neither plot protiuces a linear function (because s is not constant) and secondly, the initial dopes in Figure 5 do not increase proprtionally with m. In Figure 7 , P/h, is plotted against l / m for exchange reactions with benzene (multiple) and ammonia (stepwise). Both sets of reridts give linear plots and show the validity of eq 4.15 for the stepwise and multiple exchange SiyC2t ems. T h e present procedure thus demonstrates that for an isotope exchange process possessing either stepwise or symmetrical multiple character, the transform E,(t), which is defined in eq 3.2, greatly simplifies the treatment of the mass spectral h e t i c data. ,4cknowledgmeni's. 'The authors thank the Australian Research Grants Committee and the Australian Institute of Nuclear Science and Engineering for the support of this research.
~ ~ ~ ~ I~ ~ n d i x h tabulation of K,(i) is provided for the special initial
Virial Theorem
989
conditions: undeuterated organic substrate and a fully deuterated source of deuterium (i.+?., 5 = 1, r = 0). '6he table is for N = 10. Tables for N = 1. to 9 are readily obtained as subtables of this Table II. In general for N = y; y = 1 to 9, subt.ract ( I O - y ) from all values of i at the top of Table I1 and disregard all columns for which i becomes negative. Then use only the first y rows of Table 11 to complete the tabulation for Ai = y. For example, if N = 6, subtract 4 from the values of i and disregard the first four columns. Use only the first six rows to complete the table. References and Notes ( I ) J. Wei and C. D. Prater, Advan. Catai.. 13,203 (1962) (2) P. 1. Corio, J. Phys. Chem.. 74, 3853 (1970), and references therein. (3) H . Bolder, G. Dailinga, and H. Kloosterziel, $J. Caiai.. 3, 312 (1964). and references therein. (4) R. J. Hodges and J. L. Garnett, J. Phys. Chem. 72, 1673 (1968) (5) G . Szego, "Orthogonal Polynomials," American Matherriatical Society, New York, N. Y., 1959, pp 35-37. (6) P. L. Corio, lnt. J. Quantum Chem.. 6, 289 (1972). ( 7 ) C. Kemball, Proc. Roy. Soc.. Ser. A. 214,413 (1952). ( 8 ) R. J . Mikovsky and ,I. Wei, Chem. Eng. Sci.. 1 8 , 253 (1963). (9) C.Kemball, Trans. FaradaySoc.. 50, 1344 (1954). (10) J . L. Garnett and J . C . West, unpublished data.
ecomposition of Molecular Force Fields
Gary Simons" and Jay
L. Novick
Department o i Chemistry. W,chita State University. Wichita. Kansas 67208
(Received December 77 7973)
A procedure for decomposing a molecular force field into its (1)electronic kinetic and (2) electronic and nuclear potential components is developed. By applying this procedure to the triatomic species X20, HCN, CS2, CQ2, S02, 0 3 , SeO2, ClO2, BO2-, H2S, H20, H2Se, ClCN, OCS, GeF2, NCO-, OFz, and NO2, a set of structural rules regarding the signs of various T and V components of the Born-Oppenheimer potential is obtained. The signs of the equilibrium values of aT/aO, d2Vla02, a2VjdRa0, a2V/8R1dR2, a3VlaR2ao, d3T/aRa02, d3V/aRldR2a0, aTIdR, a2VlaR2, and a3T/aR3 are shown to be generally negative, while the signs of the respective derivatives of the other components are positive. A discussion of the physical significance of the rules is given, and the decomposition procedure is used to assess the virtues and defects of three recently proposed model functions.
I. Introduction As witnessed by the number of recent review articles which deal with the ~ u b j e c t , l -the ~ determination of molecular force fields including cubic and often quartic force constants has become an area of considerable research activity. There are now in the literature cubic or quartic force fields of varying degrees of accuracy for nearly 20 triatomic specie^,^-^^ and several more ambitious studies have been reported .24-26 Given the difficulty and expense of determining accurate potential surfaces from ab-initio computations,27 these "experimental" force fields remain a primary source of information concerning the interactions between atoms in molecules. The determination and interpretation of these fields is a formidable task, however, which has Led several investigators to propose simple model potential fonrtions.~~x-36 These models provide ad-
ditional physical content for the empirically determined force constants, and may eventually lead to the construction of model potentials for molecules about which there is little or no experimental data. In section Il of this paper we demonstrate that by employing the polyatomic quantum-mechanical virial theorem in the form derived by Nelaiider,37 one may rigorously decompose a molecular force field into i t s kinetic and potential components (the components due (1) to the electronic kinetic energy and (2) the electron-nuclear, electron-electron, and nuclear-nuclear electrostatic interactions, respectively). Since no physical model, empirical rule, or approximate wave function is employed, the resulting kinetic ( T ) and potential (V) constants may be regarded as "experimental," subject only to the errors involved in the determination of the force constants. The Journal of Physical ChemiSiFy, Vo/. 78, No. 70, 7974
Gary Simons and Jay L. Novick
990
By inspecting the T and V fields of a number of triatomic molecules, we have discovered several relationships of apparently general validity, which describe the way T and V contribute to certain force constants; these are given in section 111. These rules may prove useful in assessing the reliability of experimental force fields or of computed potential surfaces, but more importantly, they allow for an increased understanding of the "whys" of bonding and structure; for example, we show that in essentially all cases the kinetic energy tends to open the bond angles of bent triatomics, while the potential tends to close the angles. Further, the experimental T and V fields may be employed to analyze physical models of force fields. In section IV we use this approach to assess the virtues and defects of three recently proposed model functions.
TABLE I: Linear Angle Derivatives for Bent Triatomicsa'b
11. Decomposition of Force Fields
TABLE 11: Quadratic Angle Derivatives for Linear Triatomics
ppenheimer potential or force field of a polyatomic molecule may be expressed as a sum of kinetic and potential components
Molecule
Molecule
NzO HCN
w(@ih
=
2W
+ Z,R,(aW/aR,)
V(Ri,R2,O) = 2W(R,, RZ, 8 )
CS2 CQz
BO2-a BO2 -b ClCN OCS NCO --a NCO -- b NCO -- c
(2)
where 1 runs over the set 1, with all members of {RJ1 except R, held constant during each differentiation. For a triatomic molecule, eq 2 may be written
+ R,dW/aR, + R2dWIaRz
(3a)
a
T(R,, Rz, 8 )
=
-W(R,, Rz, 0) - R,aW/aR, - RZdWIdR2
(3b) where R1, Rz, and 0 are the two bond lengths and the bond angle. By successively differentiating eq 3, one straightforwardly obtains a series of expressions relating various derivatives of T and V to derivatives of W.36If the derivatives are then evaluated a t the equilibrium geometry, the T and V fields may be expressed in terms of the molecular force field. Thus if the force field W is written as = 2 (a(i+i+h)W/aq,iaqz'aq,h/,q,"q,iq,h/ll j ! kl (4)
w 7
L Jf
h 5N
i,j, h
2 0
where the q ' s are the displacement coordinates R1 - RI,, Rz - Rae? and 0 - Be, and l e indicates the derivatives are evaluated a t the equilibrium geometry, then eq 3 and its derivatives allow one to determine the kinetic field T -- B (d'"+'+"T/ dql1dq,'dq,h(,q,'q,'q,"/iljl kl ( 5 ) > . t ifk 5N-1 L,J h 2 0
and the analogous potential field. Note that an N t h order force field determines N - I t h order T and Vfields. We have determined the T and V constants for a number of triatomic molecules. Unfortunately, the derivatives of W are not normally known to high accuracy, and errors in the force constant values are magnified in the determination of the T and V constants. Although experimental uncertainties are not necessarily closely related to statistiThe Journalof Physicai Chemistry, Voi. 78, No. IO, 1974
70 error
Ref
1.51 1.38 0.05 0.49 0.36 0.86 0.23 -.0 .OB 0.59 1.55
27 28 67
10 12 13 14 16 17 16 21
24
I 3
23
h9T/h82
hW/hB2
% error
Ref
3.10 0.61
-2.43 -0.35 -1.14 -1.26 3.09 -0.43 0.77 -0.81 2.07 -1.50 -1.57
11 18 2 1 8 10 20 39 10 5 7
5 6 8 9 15 15 18 19 22 22 22
1.71 2.04 -2.42 1.10 -0.42 1.46 - 1.33 2.23 2.30
In KC1 matrix. In KBr matrix. In KI matrix.
TABLE 111: Quadratic Bond-Angle Derivatives for B e n t Triatomics Molecule
Or
av/ae
a In this and following tables, T and V are in milliergs, R I and RS are in angstroms, and e is in radians. In this and following tables, % error is per cent uncertainty in T component derivative determined from statistical dispersions; see text.
18hI) = V(IR,\, fob)) 4- T((RiI,( O h } ) (1) where {R,) are a set of internuclear distances and { 0 k } a set of angles which together are sufficient to determine the molecular structure. Nelander3? has shown that these quantities are related by the polyatomic virial theorem
V
hT/DB
-1.51 -1.38 -0.05 -0.49 -0.36 -0.86 -0.23 0.06 -0.59 - 1.55
tPT/dRM
@V/.3Rd6
70error
Ref
4.76 5.32 -0.40 -15.19 1.oo 0.68 0 .I1 0.78 2.04 0.80
-4.24 -4.78 0.42 15.35 -0.86 -0.23 -0.03 -0.80 -1.83 -0.14
11 88 210
10 12 13 14 16 17 16 21 1 23
152
158
cal dispersions, we have treated the dispersions (when available) as errors, and carried them through the calculations. The final "uncertainties" are not quantitatively correct, but they do suggest which derivative values are meaningful and which are not. Some results are presented in Tables I-VII.
111. Structural Rules An inspection of Tables I-VI1 reveals a number of rules regarding the signs and magnitudes of various T and V derivatives. I . I n bent triatomics t h e kinetic component of the Born-Qppenheimer energy favors larger angles while the potential component favors smaller angles (dT/M < 0, a v i d 0 > 0). This rule is rather surprising, as it attributes the same type of energetics to diverse molecules such as SO2 and HzO; nevertheless, it is well supported by the data. Nine of the ten molecules listed in Table I obey the rule, and the uncertainties in the force constants of the
Virial Theorem
Decomoosition of
Molecular F o r c e Fields
TABLE VI: Cubic Bond-Angle2 Derivatives
uadratic Bond-Bond Derivatives _ ^ ~ I _ _ _
Molecule
dT/hR~bR2
bZV/bRibR2
% error
-0.26 8.68 I .58 3.02 5.28 3.44 2 .a8
1.29 - 2.65 -1.80 -2.38 -4.02 -3.32 -1.05 -0.98 -16.63 -3.72 -3.63 0.07 -1.54 -0.43 -3.10 7.12 0.14 -7.52 -6.96 -5.94 -1.28 0.87
318
1.01
1sI72
4 82 4.71
- c ,023 1 4. 0 .$I
3.49 -6 08 a 12 E, 77 8 17 7 07 2 .I1 I. .27 a
In Nl matrix.
In KCf matrix.
In KBr matrix.
48 9 9 10 171 188
714 74 208 5 238
140
Ref
5 20 6 8 9 10 12 13 14 15 15 16 17 16 18 19 21 22 22 22 1 23
902 0 3
ClQZ H2S
HzQ NO2
baV/bRW
9.54 17.06 170. 2.4 53. .69 0.10
- 12.72 -20.60 -161.7 --3.2 -52.67 -0.68
% error
74 63
Ref
10 12 14 16 17 23
N-N C-H C-N 6-S
S-0
0-0
Cl02 BO*B0z-b H2S H20 ClCN ClCN
61-0 B--0 53-0 H-S H--0 6-N C-CI
N2O
6-0 N-0 N-0
ocs ocs
NO2 NCO--a NCO -h NCO NCO-a NCO -b NCO-c
-5.96 2.37 1.06 -4.72 -4.24 -10.16 -9.40 35.48 -4.14
c-0
0 1
a
l _ l . l _ s _ l _ l
VT/bR*hB
cs2 coz soz
In KI matrix.
LE V: Cubic Bond2 Angle Derivatives for Bent Triatomic8 Molecule
NzO HCN HCN
-1.39
e--s c-0 6-0 C-0 e-N C-N 6-N
In KCl matrix.
4.28 -0.48 -1.72 3.99 3.02 7.86 5.34 -37.68 4.84 0.08
4.76 6.46 -4.67 -5.84 -3.61 -9.71 -6.28 -4.19 -11.24 -5.43 -5.56 -4.54 -4.14 -4.87
42 152 63 12 14 18 317
99 110 15 6 236 152 31
3.90
6 ‘43 .39
.52
a .’TI 3.12 12.70 3 .a8 4 ” 13 3.53 3.25 3.81
5 6 6 8 9 10 32 14 15 35 16 17 18 18 I9 19
5 23 22 22 22 22 22 22
In KBr matrix. In KI matrix.
TABLE VII: Cubic Bond-Bond Angle Derivatives for Bent Triatomics
.
Molecule
D3T/bRibR2bB
baV/bRlhRnhB
Cl02
2.67 --11.59 .- 15.19
HzS RzO NO2
0.68 1.02
-3.56 10.10 15.35 -0.86 -0.23 -2.19
so2 0 3
I .oo
To error
9 98 152
Ref
10 12 14 16 17 23
exception, GeF2, niay be quite large. Hence we conclude that conventional arguments which attribute the large (a2T/aR18R2> 0, a2V/aRlaR2 < 0). Seventeen of the data bond angle in F12C to proton-proton repulsions38 are not sets in Table IV obey this rule. All of the data sets which totally correct; the unshielded protons may indeed cause disobey the rule, two partially and three completely, OSthe large bond angle, but only indirectly, by shaping the sess large or unknown dispersions. Hence this observation electron distribution which determines aT/a0. 2. I n linear triatomics V favors bending while T opposes appears to be of quite general validity, applying to molecules with both positive and negative force constants. bending (a2T/aB2a 0, a2V/aO2 < 0, (d2T/a021 > ia2V/ This result provides a new basis for the rule of Linnett This rule is also well supported by the data, as and H ~ a r e ,which ~ ~ , states ~ ~ that molecules with localized shown in Table II. Five molecules obey the rule, in two bonding electrons have negative force constants and molecases (BOz- and NCX-) KBr and KI matrix results agree cules with nonlocalized bonding electrons have positive with the rule, and in one case the rule is disobeyed. When force constants. The positive kinetic derivative should taken together, rules 1 and 2 support an interpretation of T a5 the kinetic ‘energy of a “ p a r t i c l e - i n - a - ~ e c t o r ” ~ ~ have - ~ ~ a larger magnitude when there are nonlocalized (any motion which i.ncreases the sector size decreases T ) . bonding electrons. 5. For bent triatomics, a3TlaRlza0 > 0 and d3V/dR12a0 oreover, rules ‘1 and 2 suggest that a model in which in< 0. A11 six molecules in Table V support this rule. Note creased bending leads to increased attractive electrostatic that for most molecules, a2T/aRla0 and. a3T/aR12a0 have interactions might be appropriate for V. A purely pointthe same sign; ordinarily a differentiation leads to a sign charge model would predict d2V/d02to be positive, howreversal. ever, so higher terms are necessary. 6. For both linear and nonlinear triafomics, a3T/aRaO2 3. In bent symmetric triatomics t h e kinetic bond-angle interaction is positive ja2rlaRa0 > 0 ) while t h e respective < 0 and a3V/8Rd02 > 0. Fourteen of the nineteen intrractions displayed in Table VI obey this rule. The hydride potential interaction is negative. As shown in Table 111, molecules HCN, H20, and H2S seems to be exceptions, (eight molecules obey this rule, while C102 and Se02 do but the estimated errors are quite large in these cases. As not. The uncertainty in the SeOz derivative is so great in rule 3, ClOz is an exception. There generally is a sign (210%), however, that this value is meaningless. The Cl02 reverhal between d2TlaRa0and d3T/dR8OZ. data are for an excited state, which may be atypical; no !statistical dispersions were given for this molecule. This 7 For nonlinear triatomics, a3TlaRlaRdO > 0 and rule is also in accord with the “particle-in-a-sector” idea. a3V/aRlaR$0 < 0. Four of the six cases listed in ‘Table VI1 4 . T m a k e s a positirie contribution t o t h e bond-bond insatisfy this relation. O3 and C102 do not, but both values teraction force comtaizt; V makes a negative contribution are presumed to have large uncertainties. The Journal of Physical Chemisfry, Vol. 78, No. 10, 1974
Gary Simons and Jay
99 2
8. Individual bond stretches follow the same pattern as in diatomics: dTIdR1 < 0, dV/dR1 > 0; d2T/dR12 > 0, iY2V/aRi2 < 0; d37j/dRi3 < 0, d3VlaR13 0; \dnT/aR1ni 2 I dnV/dRln1. This rule is, of course, not surprising. It is
in accord with earlier observations16 and values for particular derivatives are available from the authors. Attempts to obtain a structural rule for d3/dR12dRz derivatives failed, as the data are scattered and expected uncertainties are quke large. The structural rules proposed here should not, of course, be accepted as “final.” It is quite possible that new and improved data could modify some of the conclusions, and that molecules with unusual bonding patterns may he exceptions. Nevertheless, we believe that the proposed rules are baskally correct, and that they will apply t o the great majority of triatomic molecules.
An exact model force field will automatically predict :proper values for derivatives of T and V. If, however, a model is employed which does not predict all force constants precisely, then one may decompose the model force field into its kinetic and potential components to determine which terms are being treated accurately and which are not. In the following we report the results of such analyses of three model functions. A . The Parr-Brown Model. Parr and Br0wn~8.2~ have proposed that a. Born-Oppenheimer potential function of the form
W
=
+
+
W , / R , -1- W,/R, Wl1/Rl2 W22/91Z2-1- WIiIR1Rz W112/Ri2Rz+ gjY122/RlR2’ + W Gcsc (012YRiR‘z (6)
+
‘be employed to describe linear triatomics. They determined the W Coefficients by a least-squares fit to the force Iconstants for 6 0 2 , OCS, HCN, and N20.29The WI, Wz, W11, and W22 coefficients were interpreted using a pointcharge and particle-in-a-box model earlier found useful for diatornic~;40,~1 the W , term was justified by a particlein-a-sector argument. Smith and Overend have ~ u g g e s t e d ~ Othat 3 ~ ~ eq 6 be modified to W I= W1/Ipl 4- W 2 / R 2 W I 1 / R l 2
+
+ TN2,/R224- W 1 6 / R l -tG W t / R 2 G+ W s tan’ [1/2(7r
+
W12/R1R2
- 19)]/R1~Rp~( 7 )
They used this function to study COz, CS2, OCS, HCN, CICN, and N20. Both the original and modified functions yielded force constants in reasonable agreement with experimental values. By inspecting the T and V derivatives obtained from the original Parr-Brown potential function for COZ, OCS, HCN, and NzO, we have found that out of 84 distinct T and W derivatives, nine disagree in sign with the experimental values (three for CO2, two for OCS, one for HCN, and three for N2O). Some of the discrepancy is due to dez 83V/dR~2dR2; we rivatives of the type F r l d R ~ ~ d Rand regard the experimental values for these quantities as unreliable. In other cases, calculated derivatives which disagree with experiment but are in accord with the rules outlined in section I11 may, in fact, be more reliable than the experimental ones. Thus we distinguish between the I1 “Unconfirmed” discrepancies for these molecules, and the four “confirmed” discrepancies, cases in which calculated values disagree both with experiment and with the The Journal of Physical Chemistry, Vol. 76, No. 70, 7974
L. Novick
rules of section 111. For the potential function of eq 6, confirmed sign discrepancies occur for a2V/dB2 of OCS, a2V/ 802 of C02, a3V/aRN_oa02 of N20, and for a3V/dRN-NdfI2 of N20. There are no confirmed discrepancies for any kinetic derivatives or for any non-angle-dependent potential derivatives. In the Smith and Overend studies of these same molecules, there are 18 unconfirmed discrepancies, but only one confirmed discrepancy (d2V/dR1aR2 for HCN). The high number of unconfirmed discrepancies is due either to questionable terms involving d3/aR~2dR2or to cases in HCN and OCS where the predicted derivatives agree with the structural rules but the experimental values do not. Thus the modified potential function eq 7 is somewhat more accurate than eq 6, and supports the structural rules of section 111. This function is also successful in treating CS2 and ClCN, where there are six unconfirmed but only one confirmed discrepancy. B. The Anderson Potential Function. Anderson has recently p r o p o ~ e da~potential ~ , ~ ~ function of the form
W(Ri,R,, 0)
WD(RJ + WD(RA+ A/&” - B/(Ri
+
v R J ’ (8) where the WD are functions obtained from the respective diatomic molecules via a Poisson’s equation approach,*2 7 is a scale parameter, and N is taken to be either 2 or 4. He evaluated his potential function for COz, CS2, OCS, HCN, H20, SO2, and 03.For the two linear symmetric species, COz and CS2, a virial theorem analysis of his results shows that there are no confirmed or unconfirmed discrepancies; eq 8 seems to describe quite accurately this kind of molecule. These results are superior to those reported by Parr and Brown29and Smith and Overend.31 For the two nonsymmetric molecules, OCS and HCN, there are 17 unconfirmed and 4 confirmed discrepancies; these results are inferior to those reported in section IVA. Three of the four confirmed discrepancies involve derivatives of V. The three nonlinear molecules, H20, S02, and 03,are more difficult to model successfully. Anderson’s results contain 20 unconfirmed and 10 confirmed discrepancies. All of the confirmed discrepancies involve angle-dependent derivatives, five of T and five of V Thus the vhial theorem analysis suggests that Anderson’s function is quite accurate for symmetric linear species, somewhat less accurate for nonsymmetric linear molecules, and significantly less accurate for nonlinear triatomics. C. Point.Charge Point-Dipole Model. One of us has recently proposed a point-charge point-dipole model for V which contains two parameters: q, a bond charge, and p , a point dipole on the central atom.36 The suggested form of Vwas
VCR,, R2, 0) = -20(q2e2/9R1)- 20(q2e219R2) 10[qpe cos (8/2)/3Rt]- 10[qpe cos (d/2)/3R22]+ [22q2e2/9(R12 R22- 2R1R2cos 19)1’z1 [2q2e2/3I(Rl2 l!4RZ2 - RlR2 cos 8)1”2][2q2e2/3’(1/4RI2 RZ2- RlR2 cos 8)1/2] ( 9 )
+ +
+
The parameters were evaluated using approximate relations obtained from the virial theorem and their values were shown to be in accord with chemical intuition. Approximate values for d3W/dRl3 and d2Wld82were predicted.36 Now that experimental values for 3 V/aRI and d V/a0 are available, we have determined q and p parameters for
Viirial
Theorem Decomposition of Molecular Force Fields
CO2, CS2, S02, Qa, SeOa, C102, BOz-, H2S, HzO, HzSe, and OFn. The new q values are generally larger than those obtained previously (qoId(H2O)= 1.16, qnew(HzO) = 1.60; q o l a ( @ 0 2 ) --- 2.30, qlleW(C02)= 3-62), so the connection between q vidues and bond orders is now less satisfactory. Values for derivatives of V were calculated, and their signs were compared to those predicted by the structural rules. We found that the signs of a2V/dR12, a2V/aRla0, d"V/8R13, and d ~ ~ ~ ~ 6 ~are ~ predicted d ~ 2 a correctly, . ~ a the s i p s of d2VIdR18R2,6aV/d02,and a3V/aR12d6 are incorare ambiguous. Thus, our rect, and results for d"/dRld@2 conclusions are in agreement with those obtained by Ray ~ initial formulation of the model appears and P ~ . r r . *The to be chemically wseful at the simple quadratic valence fo'ree field level, but an improved expression for V, capab1.e of correctly predicting more derivatives, will require the inclusion of other electrostatic interactions, such as point charge point c p m h q " l e or point dipole point dipole terms. V. Cornclusion
The viriai theorem decomposition technique i s a useful pyocedure for extracting physical information from empirically determined force fields. By decomposing the force fields of a number of triatomic molecules, we have developed B set of rules regarding the contributions of 5" and V to moiecular structure. The rules have been shown to be of interpretive value, and can be further employed to study the strengths and weakness of model force fields.
Achszowlerlgmeat. The generous support provided by the Wichita State University Research Committee is gratefully ~ ~ ~ ~ o w l . e ( ~ ~ e d . References a n d Notes ( i ) Y. Morino, Pure Appi. Chem.. 18, 323 (1969). (2) J. Owerend, Ann. Rev. Phys. Chem.. 21, 265 (1970).
(3) T.Shimanouchi in "Physical Chemistry: An Advanced Treatise," Vol. 4, H. Eyring, D. Henderson, and W. Jost, Ed., Academic Press, New York,
N . Y , 1970, Chapter 6.
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The Journalof Physical Chemistry, roi. 78, No 70, 1974
