J. Phys. Chem. 1984,88, 4688-4692
4688
useful one, and it appears to be quite well established that solids and liquids behave differently. From this point of view we can add the following generalizations to the two that were drawn from the earlier work' (and designated in the Introduction as (a) and (b)): (c) strongly displacing solids adsorbing from relatively concentrated solutions of weakly displacing liquids exhibit concave isotherms, but with significantly reduced limiting capacities; and (d) weakly displacing solids are essentially completely excluded in relatively concentrated solutions of strongly displacing liquids. If we consider the totality of the available evidence, we find that none of the alternative models is completely applicable, at least not without some additional postulates for which there is as yet no good evidence. In considering the various models, we first consider the miscible nonuniform model. This model previously' gave the best account of the adsorption of liquids from solutions of solids. However, as we have seen here, it failed badly to account for the essential exclusion of weak solids by strong liquids. Moreover, the miscible model contains an inner contradiction, in that it assumes complete miscibility of the organic adsorbate, regardless of any bulk immiscibility, but it does not assume miscibility of the organic adsorbate in (adsorbed) water. Of the two alternative models that allow for adsorbate immiscibility, the immiscible adsorbate model, with its assumption of an impervious solid adsorbate, may be safely ruled out as a general model because of its repeated failure to account for the data for those liquid-solid systems that test it adequately. The M H model accounts in quite straightforward fashion for the exclusion of weak solids by strong liquids. For
the adsorption of solids from solutions of less strongly displacing liquids it would require an empirical adjustment factor in the adsorbate volume in order to give a reasonable fit to the data. This sort of adjustment was applied by Manes and HoferZto their data on the adsorption of solids from (anhydrous) organic liquids, and one would expect the adsorption from saturated aqueous solutions to at least resemble the adsorption from the corresponding pure liquids. However, the absence of any means of predicting the adjustment factor is an important limitation. Moreover, the essential failure of the M H model to account for the many linear isotherms thus far observed remains a powerful bar to its acceptance. From a practical point of view the nonuniform miscible adsorbate model presently appears to be the best available model for predicting multicomponent adsorption, although it may be expected to fail for some systems involving solids. The problem of finding a physically satisfying model that applies universally to solid-containing systems must be considered as unresolved. Acknowledgment. This work was supported by the Calgon Corporation and by the National Science Foundation (Grant CME7909247). Registry No. PNP, 100-02-7;PHD,87-41-2; BZD, 55-21-0; carbon, 7440-44-0;coumarin, 91-64-5;phenylalanine, 63-91-2;thiourea, 62-56-6; methionine, 63-68-3. Supplementary Material Available: Tables I, 111, and IV containing the absorption data for the various models ( 5 pages). Ordering information is available on any current masthead page.
Nuclear Spin Species, Statistical Weights, and Correlation Tables for Weakly Bound van der Waals Complexes K. Balasubramaniant Department of Chemistry, Arizona State University, Tempe, Arizona 85287
and Thomas R. Dyke* Department of Chemistry, University of Oregon, Eugene, Oregon 94703 (Received: April 13, 1984)
Nuclear spin species, spin statistical weights, and correlation tables are obtained for weakly bound van der Waals complexes studied by various molecular beam techniques. These complexes undergo large-amplitude internal motions which split the rotational levels into tunneling levels. In this paper we obtain the spin species and statistical weights adapted to the symmetry group of the nonrigid complex and correlate them with the rigid equilibrium structure. The results derived in this paper would be very useful in interpreting intensity patterns and hyperfine and tunneling patterns in the microwave and Stark spectra of these complexes.
1. Introduction
High-resolution spectroscopy of molecular complexes in the gas phase is an area of considerable activity. The growth in this field has been primarily caused by linking various high-resolution spectral techniques to supersonic nozzle sources of molecular beams. These sources provide reasonable number densities of complexes with low-temperature internal state distributions. Rotational spectra of hydrogen-bonded dimers' and van der Waals complexes have been extensively studied by molecular beam electric and magnetic resonance s p e c t r o s c ~ p ymicrowave ,~~~ absorption spectro~copy,~ and pulsed molecular beam, Fourier transform microwave spectro~copy.~Molecules such as (HF)2,2 (H20)2: C3H6.HF' (A-HF), Ar-HCl? and H2CO-HFhave been studied in addition to many others. Recently, infraredI0J1and Raman techniques'* have been applied to molecular complexes to give resolved rotation-vibration
spectra. A large number of van der Waals complexes have been studied by laser-induced fluorescence, in a few cases with rotational (1) (2) 2442. (3) (4)
Dyke, T. R. Top. Curr. Chem. 1983, 120, 85. Dyke, T. R.; Howard, B. J.; Klemperer, W. J . Chem. Phys. 1972, 56,
Verbene, J.; Reuss, Chem. Phys. 1981, 54, 189. Millen, D. J. J. Mol. Struct. 1978, 45, 1. (5) Balle, T. J.; Campbell, E. J.; Keenan, M. R.; Flygare, W. H. J . Chem. Phys. 1980, 72, 922. (6) (a) Dyke, T. R.; Mack, K. M.; Muenter, J. S.J. Chem. Phys. 1977, 66, 498. (b) Odutola, J. A.; Dyke, T. R. J . Chem. Phys. 1980, 72, 5062. (7) Buxton, L. W.; Aldrich, P. D.; Shea, J. A.; Legon, A. C.; Flygare, W. H.J . Chem. Phys. 1981, 75, 2681. (8) Novick, S.E.; Davies, P.; Harris, S.J.; Klemperer, W. J. Chem. Phys. 59. 2273. --. - - --(9) Baiocchi, F. A.; Klemperer, W. J. Chem. Phys. 1983, 78, 3509.
1973. -.
>
(10) Pine, A. S.;Lafferty, W. J. J. Chem. Phys. 1983, 78, 2154. (11) KyrB, E.; Warren, R.; McMillan, K.; Eliades, M.; Danzeiser, D.; Shoja-Chaghervand, P.; Lieb, S.G.; Bevan, J. W. J. Chem. Phys. 1983, 78, 5881. ..
(12) Godfried, H. P.; Silvera, I. F. Phys. Reu. Lett. 1982, 48, 1337.
?Alfred P. Sloan fellow.
0022-3654/84/2088-4688$01.50/0
0 1984 American Chemical Society
Group Theoretical Study of van der Waal Complexes analysis as for He.12.13 A number of lower-resolution techniques have given vibrational and structural information for molecular clusters. Molecules such as (HF),, (H20),, (NH,),, (C2H&, and (C6H6)2 have been examined by various photodissociation techn i q u e ~ ’ and ~ ’ ~by electric deflection Hydrogen bonds typically have energies of 5 kcal/mol and van der Waals bonds 1 kcal/mol and less. The weakness of these interactions allow the constituent monomers of a complex to undergo large-amplitude internal motions. Quite frequently, the problem involves multiple-minima tunneling with consequent large perturbation from rigid rotor, harmonic oscillator energy levels, unusual selection rules and nuclear spin statistics, reflecting symmetry groups larger than the geometric symmetry group of the complex. Such effects are seen prominently in the spectra of complexes such as (HF)2,1Jo(H20)2,6’21and Ar.S02.22a Even in molecules as small as (HF), and (H20),, group theoretical techniques are valuable for the assignment and interpretation of high-resolution spectra. As suggested above, highresolution spectroscopic methods are likely to be brought to bear on substantially larger molecules such as various dimers and larger complexes involving benzene, ethylene, ammonia, and water. For these molecules, systematic group theoretical procedures will be important for assigning the complex spectra that are likely to be found. Even for medium-resolution experiments in which tunneling-rotational splittings are not resolved (high barrier limit), the effects of nuclear spin statistics may be observed in low frequency, “intermolecular” vibrational modes of the complexes being studied; Le., modes that correlate with rotations of monomers. For theselreasons, we have examined some general group theoretical techniques for obtaining nuclear spin species and nuclear spin statistics for molecular complexes. We also show how correlation tables can be constructed which give consistent results in moving from low-barrier to high-barrier limits with respect to internal motions. In section 2 we discuss our general methodology of obtaining nuclear spin species and statistical weights of rotational levels. In section 3 correlation tables, allowed transitions, and nuclear spin statistics of van der Waals complexes such as the benzene dimer, water trimer, etc., both in deuterated and normal forms, are considered.
2. Group Theoretical Methods for the Nuclear Spin Statistics of Weakly Bound Complexes Symmetry Groups of van der Waals Complexes as Wreath Product Groups. Molecular Hamiltonians are invariant to the group of all permutations of identical particles and inversion of all spatial coordinates (as well as other operations). In the case of rigid molecules, this group is needlessly complex, and subgroups which are isomorphic to the usual point-group classifications supply the necessary, nonredundant symmetry labels for spectroscopic applications. For molecules exhibiting large amplitude motion such as internal rotations, twisting, and inversion, a permutation-inversion group larger than the point group, but often still a subgroup of the full permutation-inversion group, is more useful for spectroscopicapplications, as suggested by Longuet-Higgins.22b (13) Smalley, R. E. Levy, D. H.; Wharton, L. J. J. Chem. Phys. 1976,64, 3266. (14) Vernon, M. F.; Krjanovich, D. J.; Kwok, H. S.; Lisy, J. M.; Shen, Y. R.; Lee, Y. T. J. Chem. Phys. 1982, 77, 47. (15) Lisy, J. M.; Tramer, A.; Vernon, M. F.; Lee, Y. T. J . Chem. Phys. 1981, 75, 4733. (16) Hoffbauer, M. A. Leu, K.; Geise, C. F.; Gentry, W. R.; J. Chem. Phys. 1983, 78, 5567. (17) Casassa, M. P.; Bomse, D. S.; Janda, K. C. J. Chem. Phys. 1981,74, 5044. (18) Steed, J. M.; Dixon, T. A.; Klemperer, W. J. Chem. Phys. 1979,70, 4940. (19) Dyke, T. R.; Muenter, J. S.J. Chem. Phys. 1972,57, 5011. (20) Odutola, J. A.; Dyke, T. R.; Howard, B. J.; Muenter, J. S.J . Chem. Phys. 1979, 70, 4884. (21) Odutola, J. A.; Dyke, T. R., to be published. (22) (a) DeLeon, R. L.; Yokozeki, A.; Muenter, J. S. J. Chem. Phys. 1980, 73, 2044. (b) Longuet-Higgins, H. C. Mol. Phys. 1963, 6, 445.
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4689
Ip81
a
A
1A~1E~11)12)131~41
E
@ A
/Bs
lAE)113~124)
E
A
B
A
(A)(E)lI2)(34)
IA8)(141123)
Figure 1. Particle-in-a-box model for the permutation group of the nonrigid (H,O),. This group is the wreath product of the group of boxes ( S I )and particles (S2).
Operations which exchange identical particles and allow inversion are included if such exchanges are “feasible”, Le., occur rapidly enough to lead to observable spectral splittings. These latter symmetry groups can be expressed in terms of wreath product groups or generalized wreath products.23 Groups of NMR spin Hamiltonians, certain graphs, and trees of chemical interest also have generalized wreath product structure^.^^-^^ Several topics related to this problem can be found in the rev i e w ~and ~ ~the , ~recent ~ book on this topic.29 For details on the development of the various methods for the symmetry groups of nonrigid molecules the readers are referred to these reviews and the books by Altmann30 and Bunker31 and another review by Serre.32 The use of wreath products for the symmetry groups of van der Waals complexes was discussed by Odutola et al.32 We now briefly outline the wreath product formalism. Consider the water dimer as an example. The complex is weakly bound by the hydrogen bond between the two monomers. This complex can be modeled by a particle-in-a-box model. Let us associate a box with each oxygen atom and let the protons attached to that oxygen atom be the particles in that box. The weakness of the interaction between two water monomers allows large-amplitude motions which switch the protons of a given monomer and exchange the two monomers. In Figure 1 we show the eight permutations generated by these large-amplitude motions, using this model. Suppose G is the group which permutes the boxes (in this case it is S2 containing 2! elements) and H is the group which permutes the particles in each box (in this case H = S 2 ) , Then the group of all the permutations which include permutations of the particles in the boxes and of the boxes themselves is the wreath product of G with H denoted by G [ a . In this example it is S2[S2]. G where ~ lBl is~ The number of elements in C [ a is simply ~ the number of boxes. An element of G [ H l is of the form ( g ; r ) with r : B H where r is a map from B to H . The advantage of the wreath product and generalized wreath product formalism for the symmetry groups of nonrigid molecules has been discussed in several paper^.^^-^^ In a nutshell several properties of G [ a can be obtained in terms of the corresponding properties of G and H without having to expand G [ H l explicity. These properties include conjugacy class structures and generalized character cycle indices (GCCI’s). In this paper we will be making use of GCCI’s to generate nuclear spin species and nuclear spin statistical weights. -+
(23) Balasubramanian, K. J. Chem. Phys. 1980, 72, 665. (24) Balasubramanian, K. J . Chem. Phys. 1980, 73, 3321. (25) Balasubramanian, K. Znt. J. Quantum Chem. 1982, 22, 385. (26) Balasubramanian, K. J . Chem. Phys. 1983, 78, 6358. (27) Balasubramanian, K. In “Studies in Theoretical Chemistry”;Elsevier: Amsterdam, 1983; Vol. 23, p 142. (28) Balasubramanian, K. In ”Chemical Applications of Topology and Graph Theory”; Elsevier: Amsterdam, in press. (29) Maruani, J.; Serre, J., Eds. “Studies in Theoretical Chemistry”; Elsewer: Amsterdam, 1983; Vol. 23. (30) Altmann, S . L. “Induced Representation in Molecules and Crystals”; Academic: New York, 1977. (3 1) Bunker, P. R. “Molecular Symmetry and Spectroscopy”; Academic: New York, 1979. (32) Serre, J. Adu. Quantum Chem. 1974, 8, 1. (33) Odutola, J. A.; Alvis, D. L.; Curtis, C. W.; Dyke, T. R. Mol. Phys. 1981, 42, 267.
~
~
4690 The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
In the next section we describe the use of GCCI's for enumerating nuclear spin species. Generalized Character Cycle Indices (GCCl's). Let D be the set of nuclei that have a net nonzero spin in the weakly bound complex under consideration. Let B be the symmetry group of the complex. B consists of permutation and inversion operations. Suppose an element g of 0 generates bl cycles of length l , b z cycles of length 2, ..., b, cycles of length n upon its action on the nuclei in the set D. For example, for (H20), the permuation (12)(34) generates two cycles of length 2 upon its action on the four protons. Equivalently, the cycle type of g is ( b , ,bz, ..., bn). One can then where x k stands for a cycle associate with this a term xIblxZ~-x,bn of length k and the powers are simply the number of such cycles in a given permutation. To illustrate, with the permutation (12)(34), the associated cycle representation would be x?. Define a polynomial corresponding to an irreducible representation I' of B with character x as
P l thus defined was called GCCI by one of the authors34(K.B.) and its use for nuclear spin statistics was demonstrated in ref 34. For the wreath products G[M (and generalized wreath product) the GCCI's can be obtained in terms of the GCCI's of G and H (see ref 35). For large groups, this permits a substantial simplification in the calculation. We now show the use of GCCI's for the nuclear spin statistics of weakly bound complexes. Let us associate a weight with each nuclear spin state of the nuclei in D. For example, a is the weight associated with the spin a and p is the weight associated with the spin 8. Then the weight of a nuclear spin function is simply the product of the weights of the various spin states comprising the spin function. To illustrate, the protonic spin function aaPa of (H20), would have the weight a3p. The Z4 proton spin functions of (H20)2span a representation of dimension 24 in the wreath A generating function for the number of product group Sz[S2]. irreducible representations of Sz[S2]in the set of nuclear spin functions with a given weight can be obtained by the following substitution:
-
+
GFY = Pgr(xk ak pk) (2) It is obtained by replacing every x k in P,' by ak4- pk. To illustrate, consider the A, (totally symmetric) representation of the permutation group (S2[S2]) of the water dimer. The cycle index of A, and GFAl are given by (3) and (4), respectively.
+ 2xI2x2+ 3xz2+ 2x4) (3) GFAl = !/*[(a+ p)4 + 2(a + p)2(a2+ p2) + 3(a2+ p2)z + 2(a4 + ,3411 = a4 + 2 p + 2a2p2 + p.2 + p 4 (4) GCCIAl = '/(xI4
The significance of the above generating function GFr is that the coefficient of a typical term aQP*gives the number of times the irreducible representation I? occurs in the set of spin functions containing a, a's and az B's. Thus, for (HzO), there is one A, in the set containing all a's, one A, in the set containing three a's and one 8, two A,'s in the set containing two a's and two Ps, etc. Note that this information was obtained without having to find the character of the representation spanned by the 24 spin functions of (HzO)z.. All we need is the set of GCCI's. By examining the coefficients in the generating function one can immediately generate the nuclear spin multiplets. Since a typical term aalP2represents nuclear spin functions with the total MF = (a, - a2)/2 one can sort the coefficients in GFr in accordance to M Fand generate the spin multiplets. For example, for the A, representation of (Hz0)2,one A, representation occurs with M F of 2, 1, 0, -1, -2 corresponding to and we are simply left with an A, with MF = 0, corresponding to 'A,. Consequently, from the generating function one can immediately obtain the nuclear spin species. (34) Balasubramanian, K. J. Chem. Phys. 1981, 74, 6824. (35) Balasubramanian, K.Inr. J. Quantum Chem. 1982, 22, 1013.
Balasubramanian and Dyke TABLE I: The GCCI's of (H20)3
xI6 A A2 A, A4 E E2 TI
T2 T, T,
l
l
1 1 1 l
-1 5 2
2 3 3 3 3
xz3 I -5
2 -2 -9 3 9 -3
~ 1 ~ x 2xl2xZz 3 9 3 -3 -3 9 -3 -3 6 6 -6 6 3 3 -9 3 -3 3 -3 -9
~ 2 x 4 x12x4 x,Z x, 6 6 8 8 -6 -6 8 8 6 -6 8 -8 -6 6 8 -8 0 0 -8 -8 0 0 - 8 8 -6 6 0 0 6 - 6 0 0 0 0 -6 -6 6 6 0 0
TABLE 11: Proton Nuclear SDin Swcies of the Water Trimer
For the wreath product G[W one can obtain all the GCCI's in terms of a simple substitution of the GCCI of G into those of H or by simple products of GCCI's of G and H. The readers are referred to ref 35 and 36 for detailed discussion of these methods and several illustrative examples. Computer programs have also been developed to generate the nuclear spin multiplets and statistical weights from GCCI's (see ref 37). In the next section we describe the applications of these methods to weakly bound van der Waals complexes.
3. Applications We now consider the nuclear spin species and nuclear spin statistical weights of a few weakly bound van der Waals complexes that have symmetry groups of large order. First consider the water trimer. The symmetry group of this complex is S3[Sz]X I where Z is the group containing identity and inversion operations. S3 is the symmetric group containing 3! elements. The order of this group is 6 0 2 ~ ~Since 2 . the permutation-inversion group of this complex is a direct product of permutation and inversion groups it is enough if one considers the permutation group S3[S2]to determine the nuclear spin statistical weights. In this case the nuclear spin statistical weights are unaffected by the inversion operations. The GCCI's of S,[Sz] can be obtained in terms of the GCCI's of S3and S2 (see ref 35 and 36). The GCCI's of S,[Sz] are shown in Table I. The first row indicates the various types of terms occurring in the GCCI's. For each representation the coefficient corresponding to a term is shown under that column. The generating functions for the proton species are obtained by replacing every x k by ak Ok. The protonic spin species thus obtained from the generating function for (HZO), are shown in Table 11. By collecting all the spin species of a given irreducible representation one can easily see that
+
rHspln = lOAl + Az + A, + 8El + 6Tl + 3Tz + 3T3 Since the overall species is A, the spin statistical weights are Al(l), (36) Balasubramanian, K. J . Chem. Phys. 1981, 75, 4572. (37) Balasubramanian, K.J. Comput. Chem. 1982, 3, 69, 75.
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4691
Group Theoretical Study of van der Waal Complexes
TABLE V I Deuterium Spin Species of (CsD&
TABLE I V Correlation Table for Normal and Deuterated Water TrimeP
+ A3+(10)+ A4+(l) + T1+(63) + T2+(45) + T3+(36)+ T4+(18) (HzO), AI-(%) Al-(l) + Ar(0) + A3-(10) + Ad-(l) + Tl-(3) T2-(0) + T3-(6) + T4-(3) (D20), A1-(249) Al-(56) + A2-(20) + AC(10) + A4-(l) + TI-(63) + T2-(45) + Tc(36) + T4-(18) El+(O) + E2+(8) + TI"(3) + T2+(0) + (H20)3 E'(20) T3+(6) + T4+(3) EI'(70) + E2+(8) + TI'(63) + T2+(45) + (D20)3 E'(240) Tt'(36) + Ta'(18) El-(b)'+ E2-(8j + T;-(3) + TT(0) + (H20)3E-(20) T3-(6) + T433) EI-(70) + E2-(8) + T1-(63) + T2-(45) + (D20)3 E-(240) T3-(36) + T4-(18)
r
spin species
(D20)3A1+(249) AI+(56) + A2'(20)
+
'Correlation for C3to S 3 [ S 2 ]X Z is obtained by simply grouping + and - representations on the right side and deleting the + and - labels on the left. TABLE V Proton Species of the Benzene Dimer
r
soin sDecies
AAO), A3(10), .44(1), El(O), W),T1(3), T2(0), T3(6), and V 3 ) . The nuclear spin species of (D20)3 can be obtained by replacing every x k by Xk pk + vk in GCCI's where A, p , and v are the weights of the three nuclear spin states of D nuclei. By sorting the coefficients of various terms in the generating function in accordance to the MFquantum number one obtains the deuterium nuclear spin species shown in Table 111. Since D nuclei are Bosons the overall species is Al and hence the spin statistical weights of (D2% are A1(56), A2(20), A3(10), A d l h Ei(70), E2(8), T1(63), T2(45), T3(36), and T4(8). In order to obtain intensity patterns and tunneling splitting patterns one needs to obtain a correlation table which connects the rotational levels and statistical weights of the rigid equilibrium structure to the nonrigid structure. The electric deflection experiments for (H,O), indicate that the probable equilibrium geometry of (H2O)3 is cyclic. It is likely that it is also not planar, with either all of the hydrogens above the plane or two hydrogens above and one below the plane. In the former case the group is C3 and in the latter case the group would be identity. For the planar cyclic structure the group is C3h: Table IV shows the correlation of symmetry species and statistical weights for planar (H20), and (DZO),. The correlation table for a C3 (nonplanar) structure can be easily obtained from
+
Table IV. From this table one can easily construct a correlation diagram for the rotational levels. To illustrate, a J = 0, K = 0 rotational level has A,+ (A,) symmetry in a C3*(C,) group. This is split apart into A,', A2+,A3+,A4+,TI+,T2+,T3+,and T4+levels (+ and - signs indicate symmetric or antisymmetric with respect to inversion operations). The allowed electric dipole transitions are A,+ Al-, A2+ A2-, A,+ A,-, A4+ A4-, El+ El-, E2+ E2-, T TI-, T2+ T2-, T3+ T3-, and T4+ T4-. The statistical weights thus obtained for (HZO), and (D,O), would
-- -- -- - +
--
4692
J. Phys. Chem. 1984,88, 4692-4696
TABLE VII: Correlation Table for (Ca,), and (CsDs),
Al"(28)
(C6H6)Z Al'(1984)
+ A2'(21) + A5*(l) + A6*(0) +
Gz'(63) + G3'(33) + 3G4'(27) + G5*(11) + 3G6'(9) + 3G,'(143) Gg'(117) + 2G,'(66) + 2(310*(55) 2G11'(45) + 2Gl2'(36) + 4K*(99) A,*(741) + Ad'(703) + A7'(2701) + Ag'(2628) + El'(3496) + 2E2'(4232) + E3*(6716) + E4*(1748) + E6'(3358) + 3Glf(10672) + G2*(11408) + G3*(4408) + 3G4'(4712) + G5'(5336) + 3(36'(5704) + 3G7*(8468) + Gs'(9052) + 2G9"(6786) + 2Glof(6670) + 2G11'(7750) + 2G12'(7626) + 4K'( 14384)
+
(C6D6)2 Az'(264627)
be very useful in interpreting the hyperfine structure and intensity patterns of the Stark spectra of these complexes. As a second application consider the protonic and deuterated benzene dimer. The PI group of benzene dimer is s2[&] X I and the order of this group is 2-122.2. One can see the complexity involved in this problem. However, the wreath product structure of this group enables simplifications. The GCCI's of s2[06] can
be obtained in terms of the GCCI's of Sz and D6. One can find the generating functions and nuclear spin species using the methods described in the earlier sections. Table V shows the proton nuclear spin species of the benzene dimer and Table VI shows the deuterium nuclear spin species of the deuterated benzene dimer [(c$6),]. The nuclear spin statistical weights of (C6H6)2 are A1(28), A2(21), A3(6), A4(3), A d l ) , A6(0), A7(91), A # ' 8 ) , Ei(21), E2(7), E3(91), E4(3), E5(39), E6(13), Gi(77), G2(63), G3(33), G4(27), G5(1 G6(9), G7(143), Gs(l17), G9(66), Gio(55), Gii(45), G12(36), K(99). One can obtain the nuclear spin statistical weights of (C6D6)2 in a similar manner. They are as follows: A1(4278), A2(4186), A3(741), A4(7O3), A5(1081), A6(1035), A7(2701), A,(2628), E1(3496), E2(4232), E3(6716), E4(1748), E5(2774), E6(3358), G1(10672), G2(11408), G3(4408), G,(4712), G5(5336), G6(5704), G,(8468), (38(9052), G9(6786), G10(6670), G11(7750), Giz(7626), K(14384). The benzene dimer in its equilibrium geometry seems to be T-shaped and thus has C , symmetry. In order to interpret the tunneling and nuclear spin patterns one needs to correlate the symmetry species of c,, to s2[&] X I . The correlation table of statistical weights and symmetry species is shown in Table VI1 for (C6Hs)Z and (CsD6)2. From the correlation tables one can infer the patterns of tunneling splittings caused by the large-amplitude nonrigid motions. For example, for the benzene dimer, the symmetric rotational level (of symmetry Al+) would be split into Al*, A2*, A5*, &*, El*, E,*, E4*, 2E5', E6', Glf, 3GZf,3G3', G4*, 3G5*, G6*, G7*,3G8*, 2G9*, 2Glo*, 2Gllf, 2G12', and 4K* tunnelling levels. The allowed electric dipole transitions would be between tunneling levels of same symmetry species but for the sign. The results that we obtained in this paper are expected to be very useful in interpreting the intensity patterns, hyperfine structure, and tunneling splittings of microwave spectra and Stark spectra of these weakly bound complexes. Acknowledgment. This work has been partially supported by the Air Force Office of Scientific Research (F49620-83-C-0007) and the Research Corporation. Registry NO. H20,7732-18-5; D@, 7789-20-0; C6H6, 71-43-2; 0 6 , 1076-43-3.
Nuclear Magnetic Relaxation in '%D2 and Similar Spin Groupings Larry Werbelow,* Alain Allouche, and Guy Pouzard Laboratoire de Methodes Spectroscopiques, Universite de Provence, Centre de St. Jerome, 13397 Marseille Cedex 13, France (Received: November 28, 1983; In Final Form: February 28, 1984)
The perturbation-response characteristics of the longitudinal magnetizations of 13CDzand analogous three-spin groupings are derived and discussed. It is demonstrated that this spin system can be completely described within a framework provided by 17 independent macroscopic kinetic variables. In anisotropic media, 10 of these variables correspond to observable parameters. However, beneath this superficial complexity, it is argued that in most situations of practical importance, the spin kinetics of the 13CDzspin grouping are exceedingly simple.
Introduction Although the nuclear spin-lattice relaxation experiment is a versatile probe of molecular structure and dynamics, there exist a large number of relatively simple spin groupings that lack detailed theoretical treatment. One such spin grouping is the system composed of two quadrupolar ( I = 1) and one dipolar ( I = 1 / 2 ) nuclei, e.g. 13CD2,I2CHDz,or I5ND2. Although this basic spin grouping is commonly encountered in various isotopomeric nuclear magnetic resonance (NMR) relaxation studies, extraction of the 0022-3654/84/2088-4692$01.50/0
informational content encoded within the dynamical behavior of this spin grouping has not been exploited. However, due to recent, remarkable advances in both NMR instrumentation and methodology, it is expected that this three-spin moiety will play an increasingly important role in the application of NMR relaxation techniques. In this work, the detailed relaxation characteristics of the longitudinal magnetizations of the A ( I = 1/2)Xz(I = 1) spin grouping will be developed. A magnetization mode (Le. product 0 1984 American Chemical Society
