J . Phys. Chem. 1989, 93, 627-631 exp(T) - Ello) along the ((ai] vanish. Nevertheless, the F functional yields a S E energy regardless of what (fa) amplitudes are used. We therefore define the amplitudes corresponding to the (In)]and {le))spaces as follows:
te = Ce/Co
n = 1, ..., M - 1
t , = C,/Co,
(8b)
where
co5 CCIG I
(8c)
c, cc,q I
(8d)
are obtained by “transforming” the C I coefficient vector (CI)from the C S F basis (IZ)) to the (lo),ln))basis
I*)
=
Cc#) + CCele) CC,ln) + Cola) + CCele) I
e
e
n
(9)
Given the set of (fn,te) = {t,) amplitudes defined in eq 8, the F2 functional of eq 7 can be evaluated. By making use of the fact that (Co,C,, and Ce)obey the C I eigenvalue equations
EoCo + CH&n n
+ CH&e =
(loa)
e
HaoCo + CHanC, + CHaeCe = &IC, N
(lob)
e
for (a)= (m,e),the expression for F2 can be rearranged to yield
F2({G1)= p ( 1 - C C / c )+ E c d l + = &1(1
cC/c)-EO
(114
a
+ Cc/c)- E ° C C / c a
627
CI correlation energy (EcI - EO) above that contained in EO. Recall that C,, the expansion coefficient of lo) in the C I wave function I$), is evaluated as in eq 8d in terms of the CSF-space C I vector (CI)and the CAS function’s C S F amplituded (G]. Alternatively, eq 1 l b expresses the S E energy as the “larger” C I energy EcI plus a correction equal to the correlation energy (EcI - Eo) above Eo multiplied by the ratio ( 1 -
c)/c.
Discussion The results given in eq 11b and 12 provide a S E estimate (because we truncated the expansion of F at the F2 level) of the total electronic energy in terms of the S E reference energy EO, the correlation energy above EO, (EcI -.EO), and which is the square of the overlap of $ and lo). This quite simple expression can be viewed as a generalization of the well-known6 “Davidson correction” which applies to CI calculations based on singleconfiguration reference functions. The expression given in eq 12 agrees, in form, with that put forth on semiempirical grounds by Brown and Truhlar7 and may even be viewed as a justification for the procedure, which they showed is capable of yielding excellent results. When written as in eq 1Ib, it is of the form used by Feller and Davidsod and by Bauschlicher and Tay10r.~ It is also similar in spirit to the result of Prime et a1.I0 It is applicable only when the truncation of the expansion of F a t the F2 level is accurate; this will be the case whenever the ita]amplitudes are small. It does not give the “best” S E energy estimate within the {lZ),le)]C S F space because the Ita] amplitudes employed do not cause exp(-T) H exp(t)lo) = E J o )to be obeyed in any optimal sense.
e,
Acknowledgment. The financial support of the National Science Foundation (CHE-85 11307) is gratefully acknowledged.
a
(6) Langhoff, S. R.; Davidson, E. R. Int. J. Quantum Chem. 1974,8,61; J . Chem. Phys. 1976, 64, 4699. (7) Bfown, F. B.; Truhlar, D. G. Chem. Phys. Lett. 1985,117,307. These authors introduced a parameter F through E = iP (Ecl - iP)/F and used empirical means to evaluate F. (8) Feller, D.; Davidson, E. R. J. Chem. Phys. 1985, 82, 4135. These authors used the functional form E = Ea (Eel - EO)(1 in their studies of electronic affinities of C and 0. (9) Bauschlicher, C. W.; Taylor, P. R. J. Chem. Phys. 1986, 85, 6510. They also used E = Ecl (Ecl - F)( 1in their study of the singlettriplet splitting in CHI. (10) Prime, S.; Rees, C.; Robb, M. A. Mol. Phys. 1981, 44, 173.
+
+ caC,’,,= 1 , we then obtain F2 = Eo + (EcI - E o ) / c
Using the fact that Cz
c)
+
(12)
Equation 12 expresses a S E energy in terms of the S E Eo (recall that, by assumption, lo) is a CAS-type wave function) and the
+
e)
Polarizability Anisotropy, Magnetic Anisotropy, and Quadrupole Moment of Cyclohexane Ian E. Craven,’” Mark R. Hesling,” Derek R. Laver,lb Philip B. Lukins,lb Geoffrey L. D. Ritchie,**’” and Julian Vrbancichlb Department of Chemistry, University of New England, New South Wales 2351, Australia, and School of Chemistry, University of Sydney, New South Wales 2006, Australia (Received: May 13, 1988)
Measurements of the Rayleigh depolarization ratio of cyclohexane vapor at 441.6 nm, with rigorous exclusion of the spurious Gontribution from vibrational Raman scattering, are reported and analyzed to provide a reliable value of the polarizability anisotropy ( 1040Aa/C m2 V-’ = -1.93 0.03) of this molecule. In addition, observations of the magnetic birefringence of the vapor and the field gradient birefringence of the liquid are interpreted in conjunction with the polarizability anisotropy to yield the magnetic anisotropy (1029Ax/J T2= 19.2 2.4) and the quadrupole moment (1040B/C m2= 3.0 f 1.0) of cyclohexane. The anisotropy in the second moment of the electronic charge distribution and other fundamental electric and magnetic properties of cyclohexane are also evaluated and discussed.
*
*
Introduction Reliable measurements of fundamental electric and magnetic characteristics of small molecules such as cyclopropane2 and (1) (a) University of New England.
(b) University of Sydney.
0022-3654/89/2093-0627$01.50/0
cyclohexane are currently of interest. The Rayleigh depolarization ratio of a gas provides information as to the anisotropy in the molecular polarizability, one of the most important of these (2) Lukins, P. B.; Laver, D. R.; Buckingham, A. D.; Ritchie, G. L. D. J . Phys. Chem. 1985,89, 1309-1312.
0 1989 American Chemical Society
628
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989
properties. However, it has become apparent) that if the molecule is only weakly anisotropic, the depolarization ratio obtained by conventional methods4 is likely to be an overestimate unless spurious contributions from depolarized vibrational Raman scattering are excluded. For this reason, among others, we have constructed state-of-the-art equipment for studies of Rayleigh light scattering by gases and vapors, and we show here that the previously reported depolarization ratio for cy~lohexane,~ like that of c y c l ~ p r o p a n eis, ~about ~ 20% too high. In addition, we have used the derived polarizability anisotropy in conjunction with the techniques of magnetic field induced birefringence (the CottonMouton effect) and electric field gradient induced birefringence to determine the magnetic anisotropy and the quadrupole moment of cyclohexane. A knowledge of these two quantities makes it possible to evaluate the anisotropy in the second moment of the electronic charge distribution and other properties that collectively provide considerable insight into the nature of the molecular charge distribution. The investigation of cyclohexane is, of course, greatly simplified by the fact that at normal temperatures only the chair (D3J conformation, which has a %fold axis of symmetry, needs to be considered; even at 400 K, the temperature at which the magnetic birefringence measurements were performed, at least 99% of the molecules remain in this form.5
Theory Relevant theory has been given elsewhere; symbols and other details not explicitly mentioned here are as in earlier report^.^,^.^ For a species that possesses a 3-fold or higher order rotation axis, so that the molecular polarizability, magnetizability, and quadrupole moment are axially symmetric (Le., cy, = ayU# aZ2, etc.), the relationship between the Rayleigh depolarization ratio p o = G / c and the molecular polarizability 5po(3 - 4po)-' = ( A c u ) ~ / ~ c u '
Craven et al. field, which also excites a paramagnetic current through the molecular framework by mixing in higher energy states. Similarly, the molar electric field gradient birefringence constant, ,Q, is expressible as6
where eq 4a is a measure of the birefringence induced in the fluid by the electric field gradient, E', and eq 4b provides the theoretical interpretation in terms of molecular properties. In these two equations n and t, are the refractive index and relative permittivity of the medium; V, is the molar volume; the tensor b describes the dependence of the polarizability on the electric field gradient; the factor f arises from a consideration of the quadrupole moment induced in a molecule by its own reaction field gradient;* and 0 (=e,, = -20,,) is the unique molecular quadrupole moment. The quadrupole moment is defined as 8 = (OICej(zj2 - xj2)lO)
(5)
I
in which the summation is over all nuclear and electronic charges, eJ, and z, and x j are components of the position vector locating thejth charge. If the nuclear and electronic contributions to 0 are separated, eq 5 can be rewritten as 0 = 0,,,
+ eel== eCZ,(z? n
- x?) - e(OlC(z; i
- x:)10)
(6)
where e is the protonic charge, 2, is the atomic number of the z~ is the anisotropy in the second nth nucleus, and ( O l ~ i ( -$)IO) moment of the electronic charge distribution. Since the diamagnetic anisotropy, Axd, can be expressed in terms of the electronic contribution to 0 as
(1)
in which A a = aZ2- axxis the polarizability anisotropy and a = (az2 2ax,)/3 is the mean polarizability. If the species is also diamagnetic, the low-density molar Cotton-Mouton constant, ,C, is (in SI units)'
+
+ 2)2]-1[(nll- n , ) B 2 ] B , o = (N~po'/270to)[A7 + (2/3kT)AaAx]
,C = 2nV,p2[3(n2
(2a) (2b)
where eq 2a describes the refractive index difference, nil - nl, induced in the gas by the magnetic induction B a n d eq 2b is the theoretical connection between the birefringence and fundamental molecular properties; A7 and Ax = xZ2- xxxare the anisotropies in the magnetic hyperpolarizability and the magnetizability, respectively. From perturbation theory, Ax is expressible in the form Ax = Axd +
AxP
(3)
in which the two terms on the right-hand side represent the oppositely signed, and rather finely balanced, contributions from the diamagnetizability and the temperature-independent paramagnetizability. The diamagnetizability originates in the ability of a magnetic field to cause electron circulation within the undistorted molecular orbitals, so that a magnetic moment and a field in opposition to that applied are established. The paramagnetizability arises from the perturbing effect of the magnetic (3) (a) Bridge, N. J.; Buckingham, A. D. Proc. R . SOC.London, Ser. A 1966,295,334-349. (b) Murphy, W. F. J . Chem. Phys. 1977,67,5877-5882. (c) Monan, M.; Bribes, J.-L.; Gaufrss, R. J . Raman Spectrosc. 1982, 12, 190-193. (d) Haverkort, J. E. M.; Baas, F.; Beenakker, J. J. M. Chem. Phys. 1983, 79, 105-109. (4) Bogaard, M. P.; Buckingham, A. D.; Pierens, R. K.; White, A. H. J . Chem. Soc., Faraday Trans. 1 1978, 74, 3008-3015. (5) Eliel, E. L.; Allinger, N . L.; Angyal, S. J.; Morrison, G. A. Conformational Analysis; Interscience: New York, 1966; Chapter 2, p 40. ( 6 ) Vrbancich, J.; Ritchie, G. L. D. J . Chem. Soc., Faraday Trans. 2 1980, 76. 648-659. '(7) Buckingham, A. D.; Pople, J. A. Proc. Phys. SOC.London, Secr. B 1956, 69, 1133-1138.
a knowledge of the molecular geometry makes possible the separation of the magnetic anisotropy into the diamagnetic and temperature-independent paramagnetic terms. Finally, it may be noted also that the connection between the molecular g value, which relates the rotational angular momentum to the magnetic moment, the magnetic anisotropy, and the quadrupole moment is gzJ22
- gxJxx
= (-m,/e)[(4me/e)Ax
+ 01
(8)
where ZZz and Zxx are the moments of inertia.
Experimental Section Rayleigh Depolarization Ratio. The apparatus developed in Armidale to measure the Rayleigh depolarization ratio of cyclohexane and other gases and vapors is similar to earlier arr a n g e m e n t ~ and , ~ ~only ~ ~ its main features are noted here. Light from a 10-mW He-Cd laser (441.6 nm) is chopped and passed through a Glan-Thompson prism to ensure vertical polarization. The beam then traverses a silica scattering cell and impinges on a photodiode, the output of which is used to monitor the laser power and as a gating signal for a photon-counting system. The scattered light passes through a rotatable GlanThompson prism, a wedge depolarizer, a filter, and a lens, which focuses it on to the photocathode of a cooled photomultiplier tube. Two circular stops limit the divergence of the scattered light collected by the detector to less than 3 O , so that negligible error is intr~duced.)~ The output signal from the photomultiplier tube is amplified and passed to a microcomputer-controlled photoncounting system. Observations were made with and without inclusion of the filter, which had a uniform 50% transmittance, centered at 441.6 nm, (8) (a) Buckingham, A. D.; Graham, C. Mol. Phys. 1971, 22, 335-340. (b) Buckingham, A. D. MTP Int. Rev. Sci.: Phys. Chem., Ser. I 1972, 2, 241-264.
The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 629
Polarizability Anisotropy of Cyclohexane TABLE I: Depolarization Ratios, Average Polarizabilities, and Polarizability Anisotropies of Cyclohexane
X/nm ~~~
1OOon
1040a/C m2 V-I
1040Aa/C m2 V-'
12.5 12.5 12.4 12.3 12.2
-1.93 f 0.03 -2.10 f 0.04 -1.95 -1.93 -1.74
~
441.6 441.6 488.0 514.5 632.8
0 . k 0.188 0.165 0.163 0.136
0.0050 f 0.007b f 0.003' f 0.003' f 0.003c
a Present work, vibrational Raman contribution excluded. Present work. vibrational Raman contribution included. Reference 4.
1
I
2.2 i l-
I
>
hl
E
J . -
0
2.0
Ii I
I
A I
/
I
TABLE II: Analysis of the Rayleigh Depolarization Ratio, Molar Cotton-Mouton Constant, and Molar Field Gradient Birefringence Constant of Cyclohexane Property value 0.158 f 0.005" 12.5 f O.lb -1.93 f 0.03 11.2 f 0.1 13.1 f 0.1 -1.78 f 0.10' 0.0 f 0 . 2 d 19.2 f 2.4 -1 13.4 f 0.5' -101 f 2 -120 f 1 -3.0 f 0 . 9 -1.23 f 0.188 3.0 f 1.0 -785.9 788.9 -346.9 366.1 -0.49 k 0.06
"See Table I . bExtrapolated from data in ref 4. ' T = 400.2 f 1.0 K, X = 441.6 nm (He-Cd laser). dAssumed value. elandoltBornstein. Zahlenwerte und Funktionen, Magnetische Eigenschaften 11; Springer-Verlag: Berlin, 1967; I1 Band, 10 Teil, p 89. fLiquid cyclohexane, T = 298 K, X = 632.8 nm (He-Ne laser). #Effective polarizability anisotropy for liquid state (see text).
1.4 1.5
2.5
3.5 -29
2
4.5
-2
v Is Figure 1. Frequency dependence of the polarizability anisotropy of cys - ~ (A = 441.6 nm) were clohexane. Values for v2 = 4.609 X obtained in the present work, the smaller with exclusion of the vibrational Raman contribution and the larger with inclusion of the vibrational Raman contribution; other data are from ref 4. 10
over the range of the rotational Raman spectrum of cyclohexane; however, it excluded the vibrational Raman spectrum except for approximately 10% of a low-intensity band (depolarization ratio =0.08) at 386 ern-': so that negligible error arose from this source. The depolarization ratio of cyclohexane was measured at several pressures up to about three-fourths of the room-temperature vapor pressure, and no pressure dependence was detected; analogous measurements on argon gave a zero depolarization ratio over the pressure range 0-1 atm. The procedure was as follows: measurement of the background (vacuum) intensity; measurement of the polarized (or the depolarized) component of the scattered intensity for a chosen pressure of cyclohexane vapor; rotation of the analyzer through 90";measurement of the depolarized (or the polarized) component; measurement of the background intensity for the alternative orientation of the analyzer. Integration times of 50 and 500 s typically were required for the polarized and depolarized components; and the count rates, with the filter in place and at the maximum pressure, were approximately 22000 and 120 countsfs, respectively. The background intensity contributed much less than 1% to the polarized component and about 50% to the depolarized component of the scattered intensity. Table I summarizes the depolarization ratios observed with and without the filter, the mean polarizabilities, and the derived polarizability anisotropies for 441.6 nm. It also includes previously reported data, from which the vibrational Raman contributions were not excluded, for other wavelengths! Obviously, exclusion of this contribution results in a depolarization ratio and a polarizability anisotropy that are significantly lower than would otherwise have been obtained. For frequencies, u, small relative to absorption frequencies, A a should be a linear function of v2.Io (9) (a) Snyder, R. G. J . Mol.Spectrosc. 1970, 36, 204-221. (b) Dollish, F. R.;Fateley, W. G.; Bentley, F. F. Characreristic Raman Frequencies of Organic Compounds; Wiley-Interscience: New York, 1974; p 306.
Figure 1, which includes the values of p o for 441.6 nm observed with and without the filter, confirms this expectation; and it shows, furthermore, that the present measurements, made with only a IO-mW laser, are reliable and consistent with other results. Magnetic Field Induced Birefringence (CottoeMouton Effect). Apparatus and procedures as previously described2*" were used to determine the vapor-state magnetic field induced birefringence of cyclohexane. The detection system, which is based on a lock-in amplifier, integrator, and programmable current supply, was operated in a closed-loop, continuous-null configuration; the current through the compensator coil was measured by using a microcomputer and a chart recorder to monitor the voltage, V,, across a current-sensing resistor in series with the coil. Because of the relatively high cell window temperatures (=90-100 "C) required, and the smallness of the effect in cyclohexane, it was necessary to employ an additional noise reduction technique. Values of the voltage, V,, obtained over 8, 16, or 32 cycles of the magnetic field (=0.0125 Hz) were digitized and smoothed by a Fourier transform averaging technique on a PDP- 1 1/03 microcomputer. Under favorable circumstances this procedure allowed rad. the detection of effects as small as 1 X The range of temperature over which accurate data could be obtained was limited by the inadequate vapor pressure at lower temperatures and the diminution of the effect, as well as the increased optical noise, at higher temperatures. In consequence, a full temperature dependence study was impractical and measurements were performed over a range of gas pressures (12.9-1 13.4 kPa, 11 pressures) at a single optimum temperature (400.2 f 1.0 K). The definition of the molar Cotton-Mouton constant in terms of experimental observables is (9)
in which j B z dl = 3.289 f 0.045 T2m; @' = (2nI/X)9n(n2 + 2)-2(nll- n l ) is a measure of the birefringence induced in the gas by the magnetic field; and V, is the molar volume. Our result, shown in Table 11, has attributed to it an uncertainty based on the standard deviation derived from the least-squares fitting of (IO) Alms, G . R.; Burnham, A. K.;Flygare, W. H. J. Chem. Phys. 1975, 63, 3321-3326. (1 1) Lukins, P. B.; Buckingham, A. D.; Ritchie, G. L. D. J . Phys. Chem. 1984, 88, 2414-2418.
630 The Journal of Physical Chemistry, Vol. 93, No. 2, 1989 a straight line to the density dependence data, together with appropriate allowance for systematic errors. No other determinations of the vapor-state Cotton-Mouton constant of cyclohexane have been reported, in fact, the maximum magnetic birefringence exhibited by this species under these X lod rad) is comparable with the resolution of conditions (4 earlier versions of the equipment.I2 However, the molar Cotton-Mouton constants (expressed here as 1027,C/m5 A-2 mol-I) for cyclohexane in the vapor and liquid states can be compared. The value for the vapor (-1.78 at 400.2 K and 441.6 nm) is considerably more negative than that previously reportedI3 for the liquid (-0.70 a t 298 K and 632.8 nm). Field Gradient Induced Birefringence. The equipment and procedures used to determine the electric field gradient induced birefringences of liquids have been fully detailed e l s e ~ h e r e . ~ . ' ~ For the present measurements on cyclohexane at T = 298 K and X = 632.8 nm, the two-wire field gradient arrangement and the Kerr cell detection system were used. In terms of the actual experimental observables, eq 4a can be rewritten as
,Q = [6XnVm(3c,+ 2)/5c,(n2 + 2)2](k,/*)(lE,/Vo)-'V~ (10) in which ko = (2.787 f 0.027) X lo-" rad V2is the Kerr cell calibration constant; (IE,,/Vo) = -(3.25 f 0.06) X lo4 m-l is a measure of the electric field gradient; and VDc = +3.8 f 0.6 V is the required Kerr cell nulling voltage. The molar field gradient birefringence constant calculable from eq 10 is included in Table 11.
The cyclohexane used for the observations of the Rayleigh depolarizaton ratio was distilled from P2O5 and subjected to several freeze-pumpthaw-distill cycles in the vapor-handling system immediately prior to the measurements; gas chromatographic analysis gave the purity as 99.8%. Samples for the magnetic field and electric field gradient birefringence experiments were similarly purified.
Discussion Table I1 summarizes the analysis of the measured Rayleigh depolarization ratio, molar Cotton-Mouton constant, and molar field gradient birefringence constant of cyclohexane in terms of fundamental electric and magnetic properties. From eq 1 it can be seen that the depolarization ratio and the known mean polarizability yield the magnitude but not the sign of the polarizability anisotropy, Aa, of this molecule. Obviously, the sign of A a must be determined in order to use eq 2b and 4b to evaluate the magnetic anisotropy and the quadrupole moment, respectively. Fortunately, the dilemma can be resolved through a qualitative argument that uses the bond-additivity approximation to model the polarizability of both cyclohexane and ethane and relies also on the known signs of the Cotton-Mouton constant and the magnetic anisotropy of the latter molecule, as follows. As is well-known,ls the polarizability anisotropies of alicyclic or aliphatic hydrocarbon molecules with regular tetrahedral bond angles can be expressed in terms of the irreducible bond polarizability parameter, rU,defined as where the subscripts 11 and I refer to directions parallel and perpendicular to the axes of the C C and C H bonds. With the same labeling of molecular axes as previously (Le., z coincident with the 3-fold rotation axis), it is easily established that for cyclohexane Aa(C6Hl2) = -2r, while for ethane Aa(C2H6) = Fa, so that, in the bond-additivity approximation, Aa(c6Hl2)/ (12) Buckingham, A. D.; Prichard, W. H.; Whiffen, D. H. Tram. Faraday SOC.1967,63, 1057-1064. (13) Battaglia, M. R.; Ritchie, G. L. D. J . Chem. SOC.,Faraday Trans. 2 1977. 73. 209-221. (14j VrkncichTJ.; Bogaard, M. P.; Ritchie, G. L. D. J. Phys. E 1981, 14, 166-169. (15) Le Fbre, R. J. W.; Orr, B. J.; Ritchie, G. L. D. J . Chem. Soc. B 1966, 273-280.
Craven et al. Aa(C2H6) = -2. The sign of Aa(C2H6) is, however, unambiguously known: from eq 2b ,C is proportional to AaAx, and since for ethane both ,C and Ax are negative,16 it follows that Aa(C2H6) must be positive. In consequence, F a must also be positive, so that Aa(C6HI2)is necessarily negative in sign. Combination of the polarizability anisotropy so deduced with the mean polarizability for X = 441.6 nm yields the individual elements, arrand axx,of the polarizability. It may be noted here that although the conclusion drawn above is undoubtedly correct, the bond-additivity approximation apparently is less than adequate as a basis for quantitative comparison of the anisotropies of cyclohexane and ethane. The value of ra (expressed as lO4Or,/C m2 V-l) here derived from the Rayleigh depolarization ratio of cyclohexane is obviously 0.97, whereas the analogous value for ethane, extrapolated from the data of Baas and Van den Hout,I7 is 0.78; and the ratio of Aa(CSH12)/Aa(C2H6)is -2.5, rather than -2.0 as would have been expected if ru were a transferable constant. For reasons already explained, the temperature dependence of the vapor-phase Cotton-Mouton constant of cyclohexane was not measured. To use eq 2 to derive the magnetic anisotropy, it therefore was assumed that for this species the contribution to the observed effect from the magnetic hyperpolarizability anisotropy, AT, is zero; however, an uncertainty considered to span the range of possible values, as estimated from data for cyclopropane and other molecules, was attributed to this term. The magnetic anisotropy, Ax, the mean magnetizability, x, and the individual components, xzzand xxx, of the magnetizability of cyclohexane are shown in Table 11. It is of interest to consider the validity of the bond-additivity approximation in relation to the magnetizability, as well as the polarizability, of cyclohexane and ethane. The analogous bond magnetizability parameter, r x ,is
rx
= (XIICC- XICC)- 2(XIICH- XICH)
(12)
and the magnetic anisotropies are Ax(C6HI2)= -2rXand Ax(c2H6) = r x ,SO that A X ( C ~ H ~ ~ ) / A X ( C=~-2. H ~The ) value of r x(expressed as 10291'x/JT2)here deduced from the vapor-phase Cotton-Mouton constant of cyclohexane is -9.6, whereas that from the magnetic anisotropy of ethane, as inferred from a study of the microwave Zeeman effect in l,l,l-trideuterioethane by Huttner, Haussler, and Majer,16bis -9.7. The agreement between these two results provides some support for the bond-additivity approximation, at least in respect of the molecular magnetizability; and it confirms that the magnetic anisotropy of cyclohexane is entirely attributable to local rather than nonlocal effects. Other estimates of r x ,all of which appear to be much too small, have been summarized elsewhere.I3 In order to use eq 4b to derive the molecular quadrupole moment, 8,of cyclohexane from the molar field gradient birefringence constant, ,Q, of the liquid, it is necessary to assume, as previously,6q18that the contribution to the observed effect from the hyperpolarizability, b , is negligibly small. Available e~perimental'~ and theoretica120 information suggests that this is reasonable, although it may be less so for weakly quadrupolar molecules like cyclohexane. In addition, g2 is taken as unity, an assumption implying the absence or cancellation of correlational effects. The effective polarizability anisotropy, A d f f , which differs from the free-molecule value, Aa, because of the local field within the dielectric medium, was evaluated by means of the approximation (16) (a) Kling, H.; Geschka, H.; Hiittner, W. Chem. Phys. Lert. 1983,96, 631-63s. (b) Hiittner, W.; Haussler, H.; Majer, W. Chem. Phys. Lett. 1984, 109, 359-362. (17) Baas, F.; Van den Hout, K. D. Physica 1979, 95A, 597-601. (18) Vrbancich, J.; Ritchie, G. L. D. Chem. Phys. Lert. 1983, 94, 63-68. (19) (a) Buckingham, A. D.; Disch, R. L. Proc. R. SOC.London, Ser. A 1963,273,275-289. (b) Buckingham, A. D.; Disch, R. L.; Dunmur, D. A. J . Am. Chem. SOC.1968,90, 3104-3107. (c) Battaglia, M. R.; Buckingham, A. D.; Williams, J. H. Chem. Phys. Lett. 1981, 78, 421-423. (d) Battaglia,
M. R.; Buckingham, A. D.; Neumark, D.; Pierens, R. K.; Williams, J. H. Mol. Phys. 1981,43, 1015-1020. (e) Buckingham, A. D.; Graham, C.; Williams, J. H. Mol. P h p . 1983, 49, 703-710. (20) Amos, R. D. Chem. Phys. Lett. 1982, 85, 123-12s.
J . Phys. Chem. 1989, 93,631-634
A d f f = Aa[,K(liquid)/,K(vapor)]
I/*
in which &(liquid) and &(vapor) are the liquid- and vapor-phase molar Kerr constants of cyc1ohexane.l3J' Such a procedure yields A d f f = (-1.23 f 0.18) X lo-"' C m2 V-' and, with f = (3q 2)(2c, 3)/5(3q 2 4 = 1.13 and g2 = 1, 8 is as shown in Table 11. It is immediately apparent that the quadrupole moment so derived (+3.0 X lo-"' C m2) is in reasonable agreement, at least in magnitude, with a somewhat uncertain and unsigned value ( z k 5 . 6 X lo-"' C m2) obtained by Davies, Chamberlain, and DaviesZ2 from studies of liquid-state far-infrared absorption. However, it is in order-of-magnitude disagreement with an even more uncertain and also unsigned estimate (=f37 X lo4 C m2) deduced by Lalanne, Martin, and K i e l i ~ hfrom ~ ~ measurements of hyper Rayleigh scattering by liquid cyclohexane. The implications of these discordant values of 8 can be explored by considering a point-charge model of cyclohexane, in which charges q and -q/2 are attributed to the C and H atoms, respectively. If regular tetrahedral bond angles are assumed, the molecular quadrupole moment is simply expressible as
+
+
+
= 2q1CH(1CC - ICH)
(14)
in which Icc and lCH are the CC and C H bond lengths. Our value of 8, together with lcc = 153.5 pm and ICH= 109.5 pm, yields q = 3.1 X C, so that, in the point-charge approximation, the C atoms carry a positive charge of about 0.20e and the H atoms cary a negative charge of about -0.10e. Such charges, which correspond to a bond dipole moment of about 1.7 X C m (0.50 D), indicate that the bond polarization is +C-H-, in agreement with conclusions from recent theoretical and not -C-H+ as has been commonly accepted. Obviously, a quadrupole moment as large as that reported by Lalanne, Martin, and Kielich, and of either sign, would, at least in relation to the point-charge model, imply atomic charges that are physically unreasonable. (21) Buckineham. A. D.: Sutter. H. J . Chem. Phvs. 1976.64. 364-369. (22j Davies,"G. J.; Chamberlain; J.; Davies, M. Chem. Soc:, Faraday Trans. 2 1973,69, 1223-1236. (23) Lalanne, J. R.; Martin F.B.; Kielich, S. Chem. Phys. Lett. 1975, 30, 73-76. (24) (a) Hinchliffe, A,; Kidd, I. F. J . Chem. Soc., Faraday Trans. 2 1980, 76, 172-176. (b) Wiberg, K. B.; Wendoloski, J. J. J . Phys. Chem. 1984,88, 586-593.
631
In order to use the molecular quadrupole moment to determine the anisotropy in the second moment of the electronic charge distribution (eq 6) and to separate the diamagnetic and temperature-independent paramagnetic contributions to the magnetic anisotropy (eq 7), the nuclear contribution to the quadrupole moment, e,,,, was evaluated from the molecular geometry. Regular bond angles and bond lengths as noted above. were assumed, so that e,,, is exactly expressible in the very simple form where e is, as before, the protonic charge. The results, shown in Table 11, highlight once again the fact that the large and oppositely signed nuclear and electronic contributions to the quadrupole moment are so finely balanced that, in general, neither the magnitude nor the sign of this important property can be reliably predicted. A similar balance occurs between the oppositely signed contributions to the magnetic anisotropy, although the sum of the terms in this case must be positive in sign. Summary The present study of the vapor-phase Rayleigh depolarization ratio and Cotton-Mouton constant together with the liquid-phase field gradient birefringence constant of cyclohexane has yielded important conclusions in relation to the charge distribution in this molecule. As has been found for other weakly anisotropic molecules, failure to exclude the effects of vibrational Raman scattering led to significant overestimation of the Rayleigh depolarization ratio and the derived polarizability anisotropy. A qualitative application of the bond-additivity approximation unambiguously established the sign of the polarizability anisotropy and, consequently, the magnetic anisotropy and the quadrupole moment. The quadrupole moment was found to be l order of magnitude smaller than the more recent of two previous estimates, but consistent with a qualitatively reasonable charge distribution. In addition, the anisotropy in the second moment of the electronic charge distribution and several other electric and magnetic properties of cyclohexane were evaluated and discussed. Acknowledgment. The award of a University of Sydney Special Ptoject Research Scholarship (to P.B.L.), financial support from the Australian Research Grants Scheme (to G.L.D.R.), and generous assistance from Dr. M. P. 3ogaard (University of New South Wales) are gratefully acknowledged. Registry No. Cyclohexane, 110-82-7.
Stretching Vibrational States In Germane Lauri Halonen Department of Physical Chemistry, University of Helsinki, Meritullinkatu I C, SF-00170 Helsinki, Finland (Received: May 13, 1988)
A curvilinear internal coordinate Hamiltonian based on coupled Morse oscillators is used to interpret the observed stretching vibrational band origins of germane. The eigenvalues are obtained variationally with a symmetrized Morse oscillator basis, and the potential energy parameters are optimized with the nonlinear least-squares method using experimental stretching vibrational term values as data. All observed stretching fundamental and overtone band origins (up to 17 000 cm-') of six isotopic species (70GeH4,'2GeH4, 73GeH4,74GeH4,76GeH4,and 74GeD4)are produced well with a single three-parameter potential energy surface. The GeH, isotopic species show typical local mode energy level patterns already at low energies whereas the "GeD4 molecule becomes local in its vibrational behavior at higher energies.
Introduction A simple local mode model with coupled Morse oscillators has been found to describe the stretching vibrational states of many tetrahedral molecules (CHI, SiH4, GeH4, and SiF,) This is particularly pleasing because this model offers a clear physical
picture. In spite of this success, the earlier work on germane's2 is limited due to the small amount Of experimental band origins (1) Halonen, L.; Child, M. S . Mol. Phys. 1982, 46, 239. (2) Halonen, L.; Child, M. S. Comput. Phys. Commun., in press.
0022-3654/89/2093-0631$01.50/00 1989 American Chemical Society
