J. phys. Chem. wa3, 87, 3383-3387
be combined in a number of interesting ways to conduct high-resolution spectroscopy of ions. There remain many important problems on excited electronic states of transient molecules. Most of these states are perturbed by highly excited vibrational states associated with lower electronic states. The perturbative interactions govern dynamical properties of molecules in excited states. High-resolution spectroscopy will make clear various aspects of relaxation processes which have remained in a chaos of speculation. Metastable states of carbenes already mentioned are such examples. Acknowledgment. The results presented in this article have been obtained by my co-workers: Shuji Saito, Chikashi Yamada, Yasuki Endo, Kentarou Kawaguchi, Masao
3383
Kakimoto, Keiichi Nagai, and Tetsuo Suzuki. We are much indebted to many scientists who stayed at the IMS under the Joint Research Program. I especially mention the name of Jon Hougen, who has cleared for us theoretical aspects of complicated free-radical spectra. Finally, but not least, I express deep gratitude to Professor S. Nagakura, the Director General of the Institute for Molecular Science, for giving me an opportunity to write the present article. Re&tW NO.NO,10102-43-9;NS,12033-56-6; PO,14452-66-5; CF, 3889-75-6; CC1, 3889-76-7; SiF, 11128-24-8;OC1,14989-30-1; SF, 16068-96-5; HO2, 3170-83-0; HSO, 62470-71-7; FSO, 5366405-4; ClSO, 53169-26-9; HCO, 2597-44-6; FCO, 1871-24-5;CH3, 2229-07-4; CHZF, 3744-29-4; CF3, 2264-21-3.
ARTICLES Fluorine Atom Addition to Acetylene and Ethylene in Cancrinite: Electron Paramagnetic Resonance of CHFCH and *CH,FCH, at 77 K P. Raghunathan' and S. K. Sur Department of Chembtty, Indlan Institute of Technology, Kanpur-208018, Indla (Received: November 5, 1982)
Fluorovinyl (CHF'CH) and fluoroethyl(CHZFCHd radicals have been produced and stabilized by F atom addition to acetylene and ethylene, respectively, in a synthetic cancrinite matrix at 77 K. Analysis of the EPR powder spectra and computer simulation of the observed line shapes lead to the following "best-fit" anisotropic EPR spin Hamiltonian parameters: (i) C,HFC3H g, = 2.0030 f O.OOO1, gw = 2.0026 f O.OOO1, g,, = 2.0025 f O.OOO1; A,(H,) = 0.004091 f O.ooOo2 cm-', A,(H,) = 0.00405 f O.ooOo2 cm-', A,(H,) = 0.004352 f O.ooOo2 cm-'; A,,(F) = 0.002045 f O.ooOo2 cm-', A,,(F) = 0.002073 f 0.00002 cm-l, AJF) = 0.002755 f 0.00002 cm-'I A,,(H,) = 0.000308 f O.ooOo2 cm-', A,,(H,) = 0.000616 f O.ooOo2 cm-l, AJH,) = 0.002232 f O.ooOo2 cm-'; (ii) C,HzFC,H2: g,, = 2.0038 f O.OOO1, ,g = 2.0037 f O.OOO1, g,, = 2.0030 f 0.0001; A,,(F) = 0.000467 f 0.00002 cm-', A,,(F) = 0.001476 f 0.00002 cm-', AJF) = 0.002718 f 0.00002 cm-'; A,,(H& = 0.003549 f 0.00002 cm-', A,(H,) = 0.003213 f 0.00002 cm-', A,,(HB) = 0.003400 f 0.00002 cm-'; A,,(H,) = 0.002102 f 0.00002 cm-l, A,(H,) = 0.002036 f O.ooOo2 cm-', A,(H,) = 0.002597 f O.ooOo2 cm-'.For both species, spin density distributionsestimated from semiempirical MO calculations have been compared with those derived from EPR data.
Introduction With the increasing use of zeolitic framework structures for free-radical entra~ment,'-~ it has become possible not only to isolate novel and short-lived radical species, but also to characterize their EPR parameters extensively. In our laboratory, we have successfully employed polycrystalline synthetic cancrinite (N~A&Si60244!aC03.2H20), an aluminosilicate framework structure with P63 space group symmetry and hollow enclosures$5 as a convenient (1) P. H. Kasai and R. J. Bishop, Jr., in 'Zeolite Chemistry and Catalysis", J. A. Rabo, Ed.,American Chemical Society; Washington, DC, 1976, Chapter 6,p 350. (2) W. N.Delgass, G. L. Haller, R. Kellerman, and J. H. Lunsford, 'Spectroscopy in Heterogeneous Catalysis",Academic Press, New York, 1979, p 282. (3) M. Che in "MagneticResonance in Colloid and Interface Science", J. P. Fraissard and H. A. Resing, Ed., Reidel, Holland, 1980, p 79. (4) 0. Zarchow, Z.Kristallogr., 122, 407 (1965). 0022-3654/83/2087-3383$01.50/0
matrix for stabilizing a variety of paramagnetic systems! In this paper, we report the first complete EPR line shape analyses leading to an assessment of the g- and A -tensor anisotropies of two fluoro radicals, namely, fluorovinyl (CHFCH) and fluoroethyl (CH2FCH2),produced by F atom addition to C2H2and CzH4,respectively, in the cancrinite matrix at 77 K. Although several EPR investigations have been reported on the trapping of simple fluorohydrocarbon radicals produced in different media,' particularly in inert gas matrices at liquid helium temperatures,*~~ studies of fluoro radical intermediates sta(5) J. V. Smith in ref 1, Chapter 1, p 3. (6) S.K. Sur, Ph.D. Thesis, Indian Institute of Technology, Kanpur, India, 1982. (7) For a review, we M. Iwasaki in "Fluorine Chemistry Reviews", Vol. 5, P. Tarrant Ed., Marcel Dekker, New York, 1975, p 1. (8) J. Maruani, C. A. McDowell, H. Nakajima, and P. Raghunathan, Mol. Phys., 14,349 (1968).
0 1983 American Chemical Society
3384
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
Raghunathan and Sur
TABLE I: The “Best-Fitting’’EPR Hamiltonian Parameters for the .CHFCH Radical Spectrum in Cancrinite at 77 K A tensor
X
lo4 cm“~~
splitting from
g tensor g,, g,, g,=
= 2.0030 = 2.0026
2.0025
f
0.0001
:$
t
0,0001
‘Ha
f
0.0001
A xx 40.91 20.45 3.08
0.2 0.2 0.2
i f
i
AYY 40.54 i 0.2 20.13 f 0.2 6.16 r 0.2
~
A 22 43.52 i 0.2 27.55 t 0.2 22.32 t 0.2
bilized in zeolitic cage structures have been very sparse.1° Our present work demonstrates the usefulness of cancrinite as a matrix material in this respect.
Experimental Section Research grade trifluoromethyl hypofluorite (CF30F) was obtained from Ozark Mahoning Co., Oklahoma. Acetylene and ethylene gases were Matheson CP grade (-99.5% pure). All the gases were used in a vacuum system after trap-to-trap distillation. CF30F mixed with C2H2or C2H4(2:l v/v) was sorbed on “activated” synthetic cancrinite6J1in powdered form and kept sealed in a quartz sample tube. Fluorine atom addition to the unsaturated centers was induced by photolyzing CF30F in the sorbed system at 77 K by UV irradiation from a medium-pressure Hanovia mercury lamp. The EPR spectra were recorded at X band on a Varian E-109 spectrometer. The amount of gaseous precursors sorbed into the solid matrix was optimized by several trial runs until strong, well-resolved EPR spectra resulted upon photolysis. Results and Discussion Rifluoromethyl hypofluorite (bp 95 K)dissociates easily into CF30. and F. by UV irradiation and is thus a very good source for in situ generation of F atoms.12 The photolyzed mixture of (CF30F + C2HJ or (CF30F+ C2H4) in cancrinite at 77 K yielded highly anisotropic multiline EPR spectra, which are shown in Figures 1 and 2, respectively. These spectra, although complicated, are strong and well-resolved, thus demonstrating a high efficiency of radical production as well as isolation in the cancrinite matrix. The spectrum shown in Figure 1 clearly exhibits anisotropies in both gand hyperfine (A) tensor components. Appearing at the bottom of this figure is our assignment of the EPR lines. The assignment is consistent with splittings from three inequivalent I = 1/2 nuclei. The main 1:l doublet is split in succession by two inequivalent I = 1/2 nuclei, thus giving ultimately a pattern of 24 lines, resolvable into eight component hyperfine transitions for each principal direction. However, the characteristic dipolar line widths of the spectral components cause some of the lines to overlap. On the basis of the splitting pattern discussed above, the spectrum is assigned to the fluorovinyl radical, CHFCH, produced by direct F atom addition to the acetylene molecule. In analogy with the well-established EPR analysis of the vinyl radical,13 we assign the predominant splitting to the b-proton, the further smaller splittings being ascribed to the F atom and a-proton. A more thorough check of our line shape analysis is afforded by a trial-and-error simulation of the entire spectrum, and this was done on a DEC-1090 computer using a program (9)M. Jinguji, K. C. Lin, C. A. McDowell, and P. Raghunathan, J. Chem. Phys., 65,3910 (1976). (10)P. Svejda, J.Phys. Chem., 76,2690 (1972). (11) R. M. Barrer, J. F.Cole, and H. Sticher, J. Chem. SOC.A , 2475 (1968). ~~. ._, (12)N. Vanderkooi, Jr., and W. B. Fox, J. Chem. Phys., 47, 3634 (1967). (13)E.L. Cochran, F.J. Adrian, and V. A. Bowers, J. Chem. Phys., 40,213 (1964).
Figure 1. Observed EPR spectrum of the CHFCH radical produced in cancrinite at 77 K. The stick diagram indicates our assignment of the EPR parameters.
developed by us based on a rapid and elegant method due to 1 ~ a s a k i . l ~ The computer calculation assumes a spin Hamiltonian of the form Hs = /3,H*@S+ 1.A.S where the principal axes of g and A are assumed to be collinear, Be and H are, respectively, the Bohr magneton and the applied magnetic field. The calculated resonance field positions are fitted with a Gaussian line shape function and the resulting simulation is shown by the dotted line of Figure 3 (the solid line is the experimental spectrum, reproduced here for comparison). It is noticed that our simulation leads to a generally good agreement with the overall features of the experimental spectrum. Minor discrepancies, however, do still exist between parts a and b of Figure 3, and the question arises as to whether this is indeed not characteristic of polycrystalline spectra of fluoro radicals whose magnetic tensors have noncollinear principal axes, as illustrated in a couple of our earlier investigation^.^*^ In several additional simulations tried by us, the possibility of noncollinear g and A tensor axes for the fluorovinyl radical has been taken into account, and it could be established that the most critical feature is the definition of a common principal z axis along the C-C bond direction, which at least is quite in keeping with what one would expect from the symmetry of this radical. Deviations of the z principal axis components of the fluorine and proton hyperfine interactions by even as little as 5 - 7 O from the correspondingg-tensor axis (defined along the C-C bond) actually worsen the agreement between observed and computed line shapes. The g and A values that led to the “best-fit”of the experimental spectrum are presented in Table I. (14)M.Iwasaki, J.Magn. Reson., 16,417 (1974).
EPR of GHFCH and -CH,FCH,
The Journal of phvsical Chemistry, Vol. 87, No. 18, 1983 3385
LIZ-
r----- I
-1. ----1 : - -1 1
-----1
---1
(F)
(Hp)
(%)
Figure 2. Observed EPR spectrum of the .CH2FCH2radical produced in cancrinhe at 77 K. The partial stick pattern for the z component of the EPR tensors is shown at the bottom.
TABLE 11: The “Best-Fitting’’ EPR Hamiltonian Parameters for the GH,FCH, Radical Spectrum in Cancrinite at 7 7 K
A tensor X lo4 cm-’ g tensor , g = 2.0038 f 0.0001 g, = 2.0037 t 0.0001 g, = 2.0030 f 0.0001
splitting from
F Ho
Ha
A,, 4.67 + 0.2 35.49 + 0.2 21.02 f 0.2
Although it is possible in theory to observe the species C F 3 0 produced from the direct dissociation of CF30-F, we did not observe any extra spectral lines from either .CF3012or .CF3,8*10J6 both of which are expected to give characteristic patterns with large hyperfine splittings. It is suggested, therefore, that the CF30. after generation either dimerizes to the diamagnetic CF300CFBor reacts with an oxygen atom of the matrix. If we accept that the latter mechanism is operative, the resultant CF300radical will give a well-known12 six-line EPR spectrum centered a t g = 2.0041 with atotal spectral width of about 7 G (-6.5 X lo4 cm-’) a t 77 K. Therefore, even if this species is present, its spectrum would be buried in the large line width of the main spectrum (Figure 1). The expected hyperfine line positions for .CF300 are indicated in Figure 1. The spectral features shown in Figure 2 for the fluoroethyl radical are again characterized by severe anisotropy. Indeed, such anisotropies in the EPR spectra of fluorine containing species are often o b ~ e r v e d *because ~~ the 2p orbitals of fluorine make important contributions to electron delocalization processes. A preliminary assignment of the line positions of Figure 2 was done by considering the highly anisotropic hyperfine interaction from an I = 1/2 nucleus, the doublet line being further split by two equivalent I = 1/2 nuclei which couple to act as I = 1 in the “high-field” approximation. These splittings, which are in turn split by another pair of equivalent I = 1/2 nuclei, ultimately lead to a spectrum ascribable to the fluoroethyl (.CH2FCH2)radical, consisting of eighteen hyperfine component lines for each principal direction. A partial splitting pattern for the nine low-field z component lines is shown a t the bottom of Figure 2. Again, the as(15) R. W. Fessenden and R. H. Schuler, J. Chem. Phys., 43, 2704
(1965).
A,, 14.76 f 0.2 32.13 f 0.2 20.36 f 0.2
A ZZ 27.18 f 0.2 34.00 t 0.2 25.97 * 0.2
3374.50
t
I
Figure 3. Experimental (a) and computer-simulated (b) EPR spectrum of GHFCH in cancrintte at 77 K. A Gaussian line width of 2.5 0 was used in the simulation.
signment was further checked by simulating the entire line shapes. The best-fitting simulation is displayed as a dotted curve in Figure 4 together with the experimental spectrum (solid line) for comparison. Here, again, the overall agreement between computed and experimental spectra is good, although some specific details of the intensity pattern of the latter are somewhat poorly reproduced. These will be further discussed below. The EPR Hamiltonian parameters assessed from our “Besbfit” are presented in Table II. Stabilization of CF30 and .CF3 radicals in the cancrinite cage is ruled out in this
Raghunathan and Sur
3386 The Journal of Physical Chemistry. Vol. 87, No. 18, 1983
TABLE 111: Comparison of the Experimental Isotropic Hyperfine Couplings (A,) of .CHFCH with Those Derived from MO Spin Densities for the Two Minimum Energy Configurations calcd A , X lo4 cm-’ ~
nucleus
exptl A , x io4 cm-’
PBD~
anti ( p = 150”) BDC
syn ( P = 210”) BDC
PBD~
DPd
DPd
10.53 6.25 8.24 a-proton 10.14 11.10 14.62 18.00 26.38 34.75 42.78 0-proton 41.66 60.46 79.65 98.08 263.78 281.76 105.70 fluorine 22.91 175.85 187.84 70.46 a From MO spin densities. Pople-Beveridge-Dobosh parametrization without considering annihilation of the quartet Beveridge-Dobosh parametrization involving annihilation of the quartet spin component. 23 Direct spin component. z 2 parametrization involving annihilation of the quartet spin component.*‘
model as well as “forbidden” transitions in our simulation procedure led to prohibitively long computer runs and therefore had to be abandoned. The experimental spin density distributions in the two radical systems, assessed by decomposing the proton and fluorine hyperfine values (Tables I and 11) in the usual way,l’ may be further examined in the light of IND0/2 molecular orbital calculations.18 For the latter calculations, the geometries of CHFCH and CH2FCH2were assumed to be derived from the parent diamagnetic molecules, viz., CHFCH219and CH2FCHP2O For the CHFCH radical our MO calculations indicate two energy minima corresponding to the anti and syn configurations, namely F
H
>,=/cy
Figure 4. Experimental(a) and computer-simulated (b) EPR spectrum of .CHpCH2 in cancrinite at 77 K. A Gausslan line width of 2.5 G was used in the simulation.
case also, since no extra prominent features warranting their presence were found in the spectrum. The causes underlying the intensity differences between the simulated and experimental spectra of -CH2FCH2may now be rationalized. First of all, as our preliminary analysis (Figure 2) indicates, the a-proton couplings in this ?r-radicalare nearly axial. In a motionally “rigid” model, the principal axes of the coupling tensors of these two a-protons will be staggered by 120°, and the consequent nonequivalence of these protons would result in the central component of the a-proton “triplet” patterns in the “high” and “low” field regions of the spectrum (the low-field pattern is shown in Figure 2) to be split and weakened in intensity. On the other hand, the model of the -OH2group undergoing torsional oscillations has been successfully invoked elsewhere9J6to explain not only the near axiality of a-proton tensors but also spectral intensity anomalies very similar to the ones observed here. In any event, in view of the &proton splitting (to be discussed below) and the finding that the fluorine hyperfine tensor is clearly orthorhombic (Table 11), free rotation about the C-C bond is ruled out. Secondly, nuclear “forbidden” transitions, which would almost certainly play some role in the overall pattern of EPR transitions of fluoro radi~als,~ have not been included in our line shape computations. We do note, however, that in the spectrum of CHFCH (Figure 1) “forbidden” transitions are either unobservably weak or are being masked by the component line widths. The simultaneous inclusion of calculations required for the above “torsional oscillation” (16) C.A. McDowell and K. Shimokoshi, J . Chem. Phys., 60, 1619 (1974).
>C=% H
H
anti,
F
9 =
150”
syn,
H‘
= 210”
The optimum angles of 150 and 210’ are quite reasonable21 considering that the C-H bond hybridization would be between sp and sp2in such radical systems. However, one intuitively expects the anti configuration to be preferred in order to minimize the direct spatial interaction of the electronegative F atom with the unpaired electron. The calculated difference in total energies between the two configurations [E(0= 210’) - E(O = 150’)] is as small as 0.00024 au or 0.151 kcal/mol. However, values of the isotropic a-proton, /3-proton, and fluorine hyperfine splittings calculated from the MO spin densities of the two energy-minimum configurations are more revealing. In Table 111, we compare the experimental isotropic splittings with corresponding values calculated by the Pople-Beveridge-Dobosh (PBD) parametrization method22as well as the Beveridge-Dobosh (BD)23and Chiu-Gilbert-Sutcliffe21direct parametrization (DP) methods which consider annihilation of the quartet spin component explicitl~.~~ As seen from Table 111, agreement with experiment, especially for a- and ,&protons in the anti configuration, is very good and enables us to conclude that in the cancrinite cage this configuration is favored. It is noteworthy (17) F. J. Adrian, J. Colloid Interface Sei., 26, 317 (1968). (18) J. A. Pople,D. L. Beveridge, and P. A. Dobosh, J. Chem. Phys., 47, 2026 (1967). (19) J. L. Carlos,Jr., R. R. Karl,Jr., and S.H. Bauer, J. Chem. SOC., Faraday Tram. 2,70, 177 (1974). (20) R. C.Bingham, M. J. S. Dewar, and D. H. Lo, J.Am. Chem. SOC., 97, 1285 (1975). (21) M. F. Chiu, B. C.Gilbert, and B. T. Sutcliffe, J . Phys. Chem., 76, 553 (1972). (22) J. A. Pople, D. L. Beveridge, and P. A. Dobosh, J. Am. Chem. SOC.,90, 4201 (1968). (23) D. L. Beveridge and P. A. Dobosh, J. Chem. Phys., 48, 5532 (1968). (24) A. T. Amos and L. Snyder, J. Chem. Phys., 39, 362 (1963).
J. Phys. Chem. 1983, 8 7 , 3387-3400
that the isotropic hyperfine values estimated from the DP scheme correlate particularly well with our proton experimental coupling constants (Table 111). Comparison between observed and calculated fluorine hyperfine couplings is rather poor-a lo4 typical of several fluoro radHowever, the essentially axial anisotropy in the fluorine hyperfine interaction observed in our experiment, AAeXpt= 1/3(Azr - 1/2(Azz A,)) = 2.32 X cm-l, correlates nicely with that calculated from the p-orbital spin densities of the anti configuration @Amti = 1.93 X cm-') thereby supporting our earlier conculsion. In the case of .CH2FCH2,INDO/2 calculation reveals that the unpaired electron is mainly localized in the 2p, orbital of the a-carbon atom. In view of this, the observed splitting for the a-proton may be rationalized in terms of spin polarization of the C-H a-bond induced by the unpaired r-electron that is basically in the 2p, orbital of the trigonal a-carbon. Also, our calculation indicates a negative isotropic coupling of 20.13 X lo4 cm-' for the a-protons in -CH2FCH2,the calculated magnitude comparing extremely well with the experimentally derived value (22.03 x lo4 cm-'). We have stated earlier that free rotation about the C-C bond axis is not likely in CH2FCH isolated in the cancrinite matrix. This is supported by our observed 0-proton and fluorine splittings for this radical. For free rotation, the isotropic &proton hyperfine coupling should have a value that is nearly 28 Gel6Examination of the &proton couplings for this radical (Table 11)in terms of the wellknown relationship
+
Aop = B1
+ B2 cos2 4
where 4 is the dihedral angle between the axis of the odd-electron 2p, orbital and the C,-C,-H, plane and B1
-
-
3387
and B2 are constants (B, 0 G; Bz 55 G), reveals that the observed fl-proton splitting (36.27 G or 33.87 X cm-') is much larger than the 28 G (26.15 X cm-') expected for the free rotation of CH2Fabout the C-C bond. Furthermore, the &fluorine hyperfine interaction has been noted to be highly anisotropic and is comparatively much smaller than that of the fluorovinyl radical (A, = 15.54 X cm-' for .CH2FCH2 and A , = 22.91 X cm-' for CHFCH, see Tables I and 11). This reflects the fact that in CH2FCH2the anisotropic contribution to the hyperfine tensor from fluorine 2p orbital is much more effective than that in CHFCH radical. In other words, there is a direct overlap of the fluorine 2p orbital with the carbon 2p, orbitaLg This is most effective when the fluorine is almost in the nodal plane of the carbon 2p, orbital, i.e., when the dihedral angle of C-F bond is about 90°.
Conclusion We have effectively stabilized the fluorovinyl and fluoroethyl radicals in the cancrinite matrix at 77 K. The EPR powder spectra have enabled a fairly complete analysis of their spin Hamiltonian parameters. Spin density distributions estimated from semiempirical MO calculations compare favorably with those derived from the EPR data for both species. The finer details of the experimental spectrum of CHzFCHzare suggestive of a model in which torsional oscillations about the C-C bond may be involved, and further experiments on the temperature dependence of the EPR parameters for this radical are under way. Acknowledgment. We thank a referee for several helpful suggestions which have served to clarify our interpretations. Registry No. CHFCH, 86129-81-9; CH2FCH2,28761-00-4; acetylene, 74-86-2; ethylene, 74-85-1; fluorine atom, 14762-94-8.
Classical Solvent Dynamics and Electron Transfer. 1. Continuum Theory Daniel F. Calef*+ and Peter 0. Wolynes' Depertment of Chemishy, Harvard Universw, Cambrwe, Massachusetts 02138 (Received: May 24, 1982; In Final Form: January 14, 1983)
The importance of solvent dynamics in solution-phase chemistry is discussed. The role of solvent fluctuations in determining the rate of adiabatic electron-transfer reactions is investigated. In this first report, a classical continuum dielectric model of the solvent is used. Under certain circumstances,the transfer of charge between two centers reduces to one-dimensionaldiffusion over a barrier. The reaction coordinate is identified and the parameters in the resulting simple closed-form expression for the rate coefficient are evaluated in terms of properties of the reactants and the solvent. The limits of the adiabatic approach are discussed. The results of this investigation are compared with experiment.
I. Introduction Much significant chemistry takes place in condensed phases. It is therefore important to understand how the solvent affects chemical kinetics. Recently the role of the solvent as a heat bath for a dynamical event has been elucidated by many investigators.' In some chemical Current address: Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 01239. *Current address: School of Chemical Sciences, University of Illinois, Urbana, IL 61801.
reactions, however, the solvent plays a less passive part. Notable among these are electron-transfer reactions. In this paper we begin the examination of electron-transfer reactions in the light of modern liquid-state chemical physics. The transfer of an electron is an activated event. By this we mean there is a free energy barrier separating two
f
0022-3654/03/2007-3307~0 1.50/0
(1) (a) J. L. Skinner and P. G. Wolynes, J. Chem. Phys., 69,2143 (1978);(b) E.Helfand, ibid., 54, 4651 (1971);(c) D.Chandler, ibid., 68, 2959 (1978).
0 1903 American Chemical Society
