J. Phys. Chem. 1982, 86, 4668-4674
4868
TABLE 111: Photocatalytic Oxidations of Sz-, S,03'-, and SO,'. starting solutionb
TiO, catalystC
Na,S (pH 12.4)
none anatase
104(productconcn), mol/50 mL sZ-
S-0,'-
1.1 2.1 2.1
rutile none anatase rutile none
Na2S203
(pH 6.0)
anatase rutile
Na,S03 (pH 9.8)
0 0 0 0 0 0
0 0 0
SO,2- SO,'-
0.5 0.8
0.3 5.3
0.4
0.6
0 0 0
0.5 1.0 0 0.6 7.9 5.5
UV 29 cutoff filter, in the presence of 0, for 1 h. 10 x mo1/50 mL. 0.02 g.
a
on the light intensity (Figure 5a, and the reaction order of OH- is 1(Figure 4)). However, we do not conclude other reactions paralleling those stated above. In these reactions, the intermediates, for example, S2-,S2O?-, S032-, etc., will be produced. Table 111shows the photocatalytic oxidations
of these ions on Ti02 in 02-bubbled solution. A large is produced from S2-and SO-: solutions amount of SO-: on anatase. Thus, S2-and SO?- have been easily oxidized to SO-: on anatase, even if these ions were produced as intermediates as in the case of eq 8. According to the above photocatalytic mechanism, 4 / 3 photons (eq 5 and 6) or 4 photons (eq 7) are required for the production of one molecule of SO-: in water, and 3 photons are required in alkaline solution (2 photons are required in the oxidation of S032-according to the local cell mechanism). The quantum efficiencies, therefore, were calculated to be about 0.3% (for 4/3 photons) or 0.9% (for 4 photons) in water, and 0.7% in 0.1 M NaOH for photons having the Ti02 band-gap energy in the present experimental system. In this calculation, a rutile single-crystal electrode whose quantum efficiency has been already determined was used in order to determine the quantity of Ti02 band-gap photons from the light source. Under sunlight illumination for 12 h (Utsunomiya University, which stands at lat. 36'33' N, long. 139'55' E, at 9:00-15:00, Dec 3 and 4,1981), the amount of sulfuric acid photocatalytically produced by Ti02 in water was about 8X M.
Topological and Group Theoretical Analysls in Dynamic Nuclear Magnetic Resonance Spectroscopy K. Balasubramanlan Department of Chemistry and Lawrence Berkeley Laboratory, Universtty of California, Berkeley, California 94720 (Received: May 12, 1982; I n Final Form: July 27, 1982)
A method is developed for constructing NMR reaction graphs which give the NMR signal and intensity ratio patterns. This is done by generating the irreducible representations spanned by nuclei (whose NMR are of interest) with generating functions obtained from group character cycle indices (GCCI's). The method generates the symmetry species spanned by the nuclei in both the rigid and nonrigid groups without having to know the character of the representation spanned by nuclei. For nonrigid molecules which exhibit internal rotations the GCCI's can be obtained without having to know their character tables. We outline a double-coset technique to obtain the equivalence classes of nuclei and thus to construct NMR reaction graphs. Using this technique one can obtain the NMR signal and intensity ratio patterns. We introduce the concept of restricted character cycle indices (RCCI's) which generate the irreducible representations spanned by a given equivalence class of nuclei. This in turn enables the prediction of coalescence,splitting patterns, and intensity ratios of NMR signals in dynamic processes. Applications to spontaneous generation of chiral signals are outlined, where a nonrigid molecule which possesses no chiral signals suddenly possesses chiral signals (which can be resolved with chiral shift reagents) at lower temperatures.
1. Introduction
Topological and group theoretical analysis of dynamical systems such as molecules exhibiting large-amplitude nonrigid motions has been of considerable interest in recent years.'-13 Several of these topoligical schemes de(1)K. Balasubramanian, Theor. Chim.Acta, 51,37 (1979). (2)K. Balasubramanian, Theor. Chim.Acta, 53, 129 (1979). (3)K. Balasubramanian, J. Chem. Phys., 72,665 (1980). (4)M. RandiE, and D. J. Klein, Int. J. Quantum Chem., in press. (5)M. RandiE and B. C. Gerstein, J.Magn. Reson., 43,207 (1981). (6)M. RandiE, J. Chem. Phys., 60, 3920 (1974). (7)M. RandiC, Int. J. Quantum Chem. Symp., 14,557(1981);Chem. Phys. Lett., 42,283 (1976). (8) M. RandiC, Int. J . Quantum Chem., 21,647 (1982).
scribe interrelationships among a set of isomers which get interconverted into one another by nonrigid motions. They (9)A. T. Balaban, Ed., "Chemical Application of Graph Theory", Academic Press, New York, 1976. (10)A. T. Balaban, Rev. Roum.Chim., 18,841 (1973). (11)A. T. Balaban, Reu. Roum. Chim., 2 , 733 (1978). (12)R. W.Robinson, F. Harary, and A. T. Balaban, Tetrahedron, 2, 355 (1976). (13)K. Balasubramanian, Int. J. Quantum Chem., 22, 385 (1982). (14)K. Balasubramanian J. Magn. Reson., 48,165 (1982). (15)S. G.Williamson, J. Combinational Theory, 11A,122 (1971). (16)R. Merris, Linear Algebra and Its Applications, 29,255 (1980). (17)K.Balasubramanian, presented at the SIAM Conference on Application of Discrete Mathematics, Troy, NY, 1981. (18)K.Balasbramanian, Int. J. Quantum Chem., in press.
0022-3654/82/2086-4668$0l.25/00 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982 4669
Topological and Group Theoretical Analysis
provide insight not only into pathways and mechanisms which interconvert such isomers but also into several important experimental applications such as spontaneous generation of optical activity. Even though the area of isomerization reactions and reaction graphs obtained thus seems to be quite well studied, this is not the case in the area of dynamic NMR spectroscopy. Dynamic NMR is potentially useful in understanding large-amplitude nonrigid motions and interconversions that take place in molecules as a function of temperature at experimentally feasible conditions.22 Experimentally one observes coalescence and splitting of certain signals which is a consequence of some dynamic exchanges that are so rapid that at high temperatures one sees only an average effect (coalescence of signals). While rapid experimental progress has been accomplished in this field, theoretical understanding and representations of these dynamic phenomena seem to be in preliminary stages. We undertake the present investigation with the intent of understanding dynamic NMR coalescence and intensity patterns with a topological scheme. We develop generating-function techniques which provide for not only the number of NMR signals (by way of generating the number of totally symmetric representations spanned by nuclei) but also all irreducible representations spanned by the nuclei present in the molecule whose NMR is of interest. Using the method of subduced representations, one can correlate the symmetry species spanned by the nuclei from a larger group to its subgroup which enables the prediction of the splitting patterns of NMR signals. We also develop double-coset methods which provide the intensity patterns of both coalesced and split signals in dynamic NMR spectroscopy. The methods developed here should be applicable to not only dynamic systems but also systems which exhibit distortions such as Jahn-Teller distortions. We also apply the techniques developed here to an interesting problem, namely, spontaneous generation of chiral NMR signals in dynamic processes. In section 2 we outline theoretical methods; section 3 discusses double-coset methods for intensity patterns; in section 4 we obtain generators for symmetry species in the equivalence classes of nuclei using the concept of RCCI’s; section 5 outlines an application of the techniques developed here to the phenomenon of spontaneous generation of chiral signals; and in the last section we give an example.
2. Theoretical Methods A. Preliminaries and Definitions. Let G be the rotational subgroup or the full point group of the molecule. Note that we will let G be the rotational subgroup if NMR can differentiate chiral nuclei by appropriate chiral shift reagents. Otherwise G is the point group. Let D be the set of nuclei dl, d2,d3, ... whose magnetic resonance is under consideration. Let [Dl denote the number of elements in D. Let R be a set containing just two labels which we denote by czl and a2. Consider a map f i from D to R defined as follows: if i # j f i ( d j )= a1 f j ( d j )= a2 if i = j d j E D , j = 1, 2, ..., ID1
(2.1)
To illustrate we could consider the five 19Fnuclei as the D set. Then, for example, the maps f l and f 2 are as shown (19)G.PBlya, Acta Math., 68,145 (1937). (20)H. C.Longuet-Higgins, Mol. Phys., 6, 445 (1963). (21)K.Balasubramanian, J. Chem. Phys., 75,4572 (1981). (22)L. M.Jackman and F. A. Cotton, ‘Dynamic Nuclear Magnetic Resonance Spectroscopy”, Academic Press, New York, 1975.
in eq 2.2, where we represent d j by j . Two maps f i and f j “2
fZ(1) = a1
f l ( 2 ) = “1
f d 2 ) = a2
f i ( 3 ) = 01
f2(3) =
f i ( 4 ) = “1
fz(4) = “1
flu)
=
“1
f i ( 5 ) = “1 f i ( 5 ) = “1 (2.2) are said to be equivalent if there exists a g in G such that f i ( d ) = fj(gd) for every d E D (2.3) For example, if one considers the rotational group D3 of the rigid PF6molecule and if 1,2, and 3 are equatorial and 4 and 5 are axial nuclei, then the maps f l and f 2 are equivalent since they are transformable into one another by 3-fold rotation. It can be seen that, if two maps f i and f j are equivalent, then the nuclei i and j are also magnetically equivalent. Thus, the group G divides D into equivalence classes, the number of equivalence classes gives the number of NMR signals, and the ratios of the number of elements in various classes give the intensity ratios of these signals. B. Generating-Function Techniques. The problem is to develop a general method which will give the number of equivalence classes and the ratios of the number of elements in various equivalence classes. I developed a method for the former problem using cycle indices and P6lya’s theorem.lJ4 In this paper we consider more general techniques for studying coalescence and intensity patterns in dynamic NMR. With each element r f R let us associate a weight w(r) which is just a formal symbol used to bookkeep the number of al’s and a2’sin any function f i . For example, we may associate a weight cy1 with a1and a weight a2with a2. Define the weight W of any map f i to be
(2.4) It can be easily seen that
W(fi)= f f 1 n - 1 f f 2 (2.5) where n is the number of nuclei. Define the generalized character cycle index (GCCI) P G X of a group G corresponding to the irreducible representation r whose character is x as (2.6) where a g E G has bl cycles of length 1,b2 cycles of length 2, etc. Equivlently the cycle type of g is (bl, b2, ...). Then using the projection operator methods, Williams o d 5and more recently MerrislG(for higher dimensional representations) proved that the generating function GFx for the number of irreducible representations in F , the set of maps from D to R , is given by
where the arrow stands for replacing every x k by C ( w ( r ) ) k . The coefficient of a term alnlaZn* in GFx gives the number of times that x occurs in the set of maps from D to R with the weight alnla2x. In particular, the coefficient of aln-1a2 in GFx gives the number of times that x occurs in the representation spanned by the set of nuclei. This is indeed quite elegant and advantageous over conventional techniques in that it does not require the character of the representation spanned by the nuclei. Further GCCI’s for the symmetry groups of nonrigid molecules can be ob-
4870
Balasubramanian
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982
TABLE I: Total Generating Function for the Nonrigid PF,'
A1 A* Gl
Gz
TABLE 11: Total Generating Function for the Rigid PF, Moleculeu
1
1
1
0 0 0 0
0
0
1
1 0 1 0 0
0 0 0 0
Hl 0 Hz I 0 The coefficient of c ~ under ~ each ~ irreducible c ~ ~ representation generates the number of that irreducible representation in the equivalence classes of nuclei. The coeffi4 1 the ~ A , representation generates the numcient of 0 1 ~ ~ in ber of NMR signals.
2 1 1
3 1 3 Note that the coefficient of a l 4 a Zof A , in Table I1 is 2 while it was 1 in Table I indicating the splitting of NMR signals. A,
1
2
0 0
4-5 Figure 2. NMR reaction graph of rigid PF, in 0, symmetry. Number of components in the graph gives number of ''F signals and the ratio of the number of vertices corresponds to the intensity ratio.
Figure 1. NMR reaction graph of nonrigld PF,. All NMR reaction graphs are &joint unions of complete graphs. The graph In the figure is K5 (the complete graph on five vertices).
tained from GCCI's of much smaller g r o u p ~ . ~ ' J ~ We will now illustrate eq 2.7 with the nonrigid PF, molecule which exhibits Berry pseudorotation. The pseudorotation subgroup of the nonrigid PF, is Sg. If one considers 19FNMR of PF,, then the set D consists of five 19Fnuclei. Consider, for example, the G1 representation of this molecule. The corresponding GCCI and GFx are shown in eq 2.8 and 2.9. The coefficient of a14a2in eq 2.9 GCCIGl=
+
-[4X15 + 2 0 ~ 1 ~ x 220x12x3- 20x2x3- 24x5] (2.8) 120 1 GFGl = -[4(a1 + a2)5+ 20(al + a2)3(a12 + a& + 120 20(a1 + cr2)2(a13 + a23)- 20(a12+ a22)(a13 + a23)24(a15+ a25)] 1
+
= a14a2 al3aZ2 + cyl2az3+ a1a24
(2.9)
is 1,implying that there is one G1representation spanned by the five 19F nuclei. Table I shows the total generating function for the nonrigid PF5molecule. In this table we follow the notation for the irreducible representations of an earlier paper of mine.13 By collecting all the coefficients of a14a2one can immediately conclude that the representation spanned by the 19Fnuclei is given by eq 2.10. rls,= A~ + G, (2.10) This is, of course, a trivial example but we used this for the sake of illustration. The number of Al (totally symmetric) representations in the representation spanned by nuclei gives the number of NMR signals by Pblya's theorem. This is because the number of Al representations is obtained from GCCI which is also just the ordinary cycle index defined by Pblya19for the enumeration of equivalence classes. Thus,the number of Al representations gives the number of equivalence classes and thus the number of NMR signals. I introduced a topological description for
the interconversion of nuclei in dynamic processes.2 In that scheme nuclei are represented by vertices and the possible interconversions among them by edges. A NMR reaction graph is a graph in which two nuclei are connected if they are transformable into one another by an operation in the rotational subgroup. Such a diagram for the 19Fnuclei of the nonrigid PF, molecule is shown in Figure 1. In the next section we will consider the splitting patterns of NMR graphs by way of correlation of the symmetry species spanned by these diagrams. C. Correlation of Symmetry Species and NMR Reaction Graphs. The symmetry group of the rigid PF5molecule is D3h and its rotational subgroup is D3. In this example there are no enantiotopic protons in both the rigid and nonrigid molecular groups so that one can treat both cases-by rotational subgroup. The set of irreducible representations spanned by the nuclei in the rotational group of the rigid molecule can be found by two methods. One method is to obtain the generating functions in the group D3. Such a generating function is shown in Table 11. Note that by collecting the coefficients of a t a 2in various irreducible representations we find rlq in D3 to be riq = 2A1 + A2 + E (2.11) The same expression can also be obtained by correlating the symmetry species of Ssto D3 by way of subduced representation. The Al representation of S5 correlates to Al of D3 and G1 splits into Al Az E so that expression 2.10 correlates to expression 2.11. Note that there are two Al representations in rF in the group D3 indicating that there are two equivalence classes for the rigid molecule. This establishes the fact that a single NMR signal of the nonrigid molecule splits into two signals in dynamic NMR at lower temperatures. The NMR reaction graph of the rigid molecule is shown in Figure 2, which contains two components. The number of components in the NMR reaction graph corresponds to the number of NMR signals and the ratio of the number of vertices in various components gives the intensity ratios of the corresponding NMR signals. Thus, we can infer that the single NMR peak of the nonrigid molecule splits into two signals with the intensity ratio 3:2. The method outlined here, even though it enables the prediction of the splitting and coa-
+ +
Topological and Group Theoretical Analysis
The Journal of Physical Chemistty, Vol. 86, No. 24, 7982 4671
lescence patterns of the NMR signals, does not quite enable construction of the N M R reaction graph (since it gives only the number of components and not the number of vertices in each component) and thus cannot predict the intensity patterns in dynamic processes. For this purpose we outline a double-coset method in the next section.
3. Double-Coset Method for Constructing the NMR Reaction Graph and Prediction of Coalescence and Intensity Patterns Let S, denote the symmetric group containing n! permutations of nuclei. Since we are choosing n - 1 a1labels and 1 a2label to label the nuclei, the group S,-l X S1 (Sl is the group containing just the identity element) is known as the label subgroup L which is isomorphic to S,-.l L is, in general, a subgroup of S,. Recall that G is the rotational subgroup or the point group of the molecule. For an element s E S, the set LsG is known as the double cosetz3 of L and G. Any map f i defined in section 2A can be considered as an element in S,. This is because we can consider the identity element of S, as the map f, which maps first n - 1 nuclei to a1and the nth nucleus to a2by definition. Then any other map can be obtained by applying s E S, to f,. To illustrate consider PFp The map f, labels the nuclei C Y ~ C Y ~ C Y Then, ~ ~ ~ C Yfor ~ . example, the permutation (12345)generates the following labeling of nuclei: Q ~ C Y ~ C Y ~ L YThus, ~ C Y ~ . the map f l can be considered as an element in S5. Two elements s1 and s2 are said to be in the same double coset if there exists I E L and g E G such that s1 = IS& (3.1) Then it can be seen that the maps corresponding to s1 and sz are equivalent and, consequently, the corresponding nuclei are also equivalent. It can be proved that eq 3.1 is an equivalence relation and thus the double cosets generate equivalence classes of nuclei. The elements of a set S = {s1,s2, ...,s,) are said to be distinct representatives of the double cosets or the equivalence classes of nuclei if the group S, can be obtained as a disjoint decomposition of the double cosets formed by the elements of S. In symbols, the elements of S are distinct representatives if m
S, = L' LsiG
(3.2)
i=l
(LsiG) 0 (LsjG) = 4 if i # j where 4 is the null set. The number of elements in S, M, can be found by using the generating-function techniques outlined in section 2B by way of generating the number of Al representations in the set of nuclei. The number of elements in any double coset LsG is given by eq 3.3. Thus, using eq 3.3,one can obtain the
lLsGl = ~ L I I G ~ / L~ s -n ~ GI = ILIIGI/IL
n sGs-ll
(3.3)
number of elements in the double coset without having to construct the coset. This is indeed quite advantageous because for our present purpose we only need to know the number of nuclei in each equivalence class (in fact, the ratio of nuclei in various classes) for the prediction of intensity ratio patterns. The number of elements in a double coset, in general, is not equal to the number of elements in the corresponding equivalence class of nuclei. This is because two elements in a double coset can generate the same map. To illustrate for the PF5 molecule the identity in S5 and the permutation (1234)generates the (23) E. Ruch, W. Hiiaselbarth, and B. Richter, Theor. Chim. Acta, 19, 288 (1970).
same map f5. Nevertheless, in general, the ratio of the number of elements in various double cosets is exactly the ratio of the number of elements in the corresponding equivalence classes. This is indeed very useful since we are interested in the intensity ratios of various peaks which correspond to the ratio of the number of elements in the corresponding equivalence class. Thus, we arrive at the following important relation. For a set of distinct representatives S = {s1,s2,...,sm),which generate the double-coset decomposition of S, we have lLslGl: lLs2Gl: :ILs,GI = I L I I G I /n L s , ~ s l - l i :I L I I G I / n IL s2~s2-1l: :I L I I G I / In L s,Gs,-~~ (3.4)
...
...
Thus, the intensity ratios of various signals is given by eq 3.5. Also the number of elements in an equivalence class 1 1 1 . . (3.5) IL n s1Gsl-l( * IL n s,Gs,-~I* * * * I L n S,GS,-~I of nuclei corresponding to the double coset LsiG is given by eq 3.6. number of elements in an equivalence class of nuclei = ILsiGln/lS,I (3.6) The double cosets of nuclei also facilitate elegant correlation of equivalence classes of nuclei in dynamic processes such as large-amplitude motions. Each double coset generates an equivalence class of nuclei. The number of totally symmetric representations in any double coset is unity. The splitting patterns of equivalence classes of nuclei in dynamic processes can be found by correlating the symmetry species in the equivalence classes of nuclei. We will now illustrate the above method of double cosets with the example of PF,. For the nonrigid molecule the rotational group is S5. Thus G = S, and L = S4. This is a trivial case where G is isomorphic to S,. Thus, any s E S, is a distinct representative and there is only one such double coset since for any s E S,, LsG = S5. Thus, all five nuclei are equivalent and they span a single equivalence class C = { 1,2,3,4,5). The irreducible representations contained in C are A, and G (cf. eq 2.10). Thus, when one considers a rigid molecule whose rotational group G = D,, the class C splits into two classes C1 and C2. In terms of the double cosets the single double coset splits into two double cosets. We will now look at this in detail as an illustration. The group G is
G = D3 = {e,(123),(132),(12) (45),(13)(45),(23)(45)l
L = S4 = {e,(l2),(13),(14),(23),(24),(34),(123!,(132), (124),(142),(234),(243),(134),(143),(1234),(1243),(1324), (1342),(1423),(1432),(12)(34),(14)(23),(13)(24)~ We have to look for S = {s1,s21 such that LslG U L s ~ G= S5 LslG n Ls2G = 4 One obvious choice is s1 = e E S, where e is the identity. The number of elements in this double coset is
ILeGl = IL(IGI/IL n e-'Gel = ILllG(/lL
n GI = 24.6/3 = 48
Thus, the number of elements in the corresponding equivalence class of nuclei is given by nlLeGI/IS51 = 5.48/5! = 2 The elements of the equivalence class C1can be generated by first operating e on f 5which is f 5 and any other element
4672
Balasubramanian
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982
RCCIA2(C,) = Y6(x13- 3XlX2
in G or the double coset LeG. Thus, it can be seen that C, = (f5,f4). By choosing s2 to be (12345) we obtain the second double coset. We choose s2 = (12345) since this is not present in the product LG. The number of elements in this double coset is
lLs2GI = ILIIGI/IL
n s2-'Gs21 = 24.6/2
RCCIE(C,) = '/S(2xl3- 2x3) RCCIA1(C2)= '/6(3X12+ 3x2) = ' / ( X 1 2
The second class is easily constructed by operating (12345) on f 5 and generating the other elements by operating the resulting map by elements of G. It can be seen that C2 = (fl,f2,f3]. One can immediately obtain an NMR reaction graph by connecting all elements in any C,. The graph thus obtained is shown in Figure 2. Of course, to predict the nature of intensity and splitting patterns of NMR signals it is not necessary to explicitly construct all the double cosets. We need to know only the number of double cosets and the ratio of the number of elements in all double cosets which in turn give the number and intensity ratios of N M R signals, respectively. The number of double cosets can be found by generating the number of A , representations as shown in section 2C. The ratio of the number of elements in all the double cosets is obtained with formula 3.5. Thus, one immediately infers using these techniques that the single 19FNMR peak splits into two peaks in dynamic NMR with the intensity ratio (1/2):(1/3) or 3:2 at lower temperatures. Conversely, the peaks with this intensity ratio will coaleace into a single peak at higher temperatures. For several applications one needs to know the irreducible representations contained in each equivalence class of nuclei. A technique for this is in the next section. 4. Generation of Symmetry Species in Each Class
of Nuclei and the Assignment of Splitting and Coalescence Patterns In order to completely assign the splitting and coalescence pattern of each signal, it is necessary to know the symmetry species contained in each equivalence class of nuclei generated by the double cosets. In earlier sections we had a method to generate only the symmetry species spanned by all nuclei rather than equivalence classes of nuclei. However, in order to know whether a given signal will split or not (or conversely coalesce), it is necessary to know the irreducible representations spanned by the equivalence class of nuclei which gives rise to that NMR signal. For this purpose we define and use a new restricted character cycle index (RCCI) which generates symmetry species in each equivalence class. We define a RCCI as a character cycle index restricted to an equivalence class of nuclei. For an equivalence class C of nuclei we know that any g in the symmetry group G permutes nuclei only within the class C. Thus, the cycle product of any g E G over all nuclei can be separated and restricted only to the class C. Define a RCCI corresponding to a class C and an irreducible representation I' with character x as
Where C(b,), C ( b 2 ) ,... are the number of 1, 2 cycles generated in the equivalence class C when g is applied on C. To illustrate, the two equivalence classes of nuclei for PFB in D3 are C1 = (1,2,3}and C2 = (4,5}. The RCCI's for the various representations of D3 restricted to C1 and C2 are as follows: RCCIA1(C,)= 1/(x13+ 3XlX2 + 2x3)
+ X2)
RCCIA2(C2)= f/S(3Xi2- 3x2) = '/2(X12 - X2)
= 72
Consequently the number of elements in the second equivalence class is given by ~ ~ L s ~ G I /=I S5.72/5! ~[ =3
+ 2x3)
RCCIE(C2)= Y6(2x12- 2x12) = 0 The restricted generating function RGFX(C) restricted to a class C generates the number of irreducible representations whose character is x restricted to C. It is obtained as follows:
-
C (w(r))k)
RGFx(C) = RCCIX(xk
r€R
To illustrate let us obtain the generating functions corresponding to RCCI's of PFS in Dsh representations. RGFA1(C1)= 5/6[(a1+ a2)3+ 3(a1 + a2)(a12 + ~ 2+)2(a13+ CY:)]
+
+
+
= a13 a12a2 ala22 a23
RGFA2(C1)= 0
+
RGFE(C1)= a12a2 ala? RGFA1(C2)= l / [ ( a l + a2)2+ (a12+ a22)]= a12
+ ala2 + a22
RGFA2(C2)= RGFE(C2)= 0 The coefficient of a l l c ~a2 ~ in l RGFX generates the number of times that r occurs (whose character is x) in the equivalence class C, where lCi1 is the number of elements in the class Ci. Thus, by collecting the coefficient of al2aZ in RCCI's one can immediately infer that the nuclei in C1 span A, + E representations and similarly nuclei in C2span A , Az representations. The technique outlined above thus generates symmetry species in each class without having to know the character of the representation spanned by the nuclei present in that class. This is indeed very advantageous for large molecules which have several equivalence classes containing varied numbers of nuclei. In order to theoretically predict if a given signal will split in dynamic NMR at lower temperatures, all that we need to do is to correlate the symmetry species spanned by the nuclei which gave rise to that signal to rigid molecular group.
+
5. Spontaneous Generation of Chiral Signals In this section we consider an important application of the techniques outlined in the earlier sections to spontaneous generation of chiral signals. A molecule which does not have any chiral nuclei at higher temperatures when it exhibits very large-amplitude motions may have chiral (enantiotopic) nuclei at lower temperatures. The chiral nuclei can be resolved with the use of chiral shift reagents. This phenomenon can be called spontaneous generation of chiral NMR signals. Let us consider the proton NMR of propane which exhibits this phenomenon. The rotational group of the nonrigid molecule can be represented by the generalized wreath product group C2[C3,E], where E corresponds to the identity group acting on the methylene protons. The rotational group of the rigid molecule, however, contains only the identity and a 2-fold rotation. Under the group C2[C3,E]there are two double cosets and two equivalence classes of nuclei denoted by C1 = (1,2,3,6,7,8},C2 = (431where 1,2,3,6,7,8are methyl protons and 4,5 are methylene protons. Chiral nuclei can be dif-
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982 4673
Topological and Group Theoretical Analysis
TABLE 111: Generating Functions for the Irreducible Representations Spanned by the Protons of the Nonrigid B(CH,), Molecule
I
r AI A1 A3 A4 El El
8 Figure 3. NMR reaction graph of nonrigid propane.
*6
I*
E3
E4
2*
(2)3*
G
*8
11
* 7 (8)
12 13 14
I,
-5
4.
16 17
Figure 4. NMR reaction graph of rigM propane in C , symmetry. The proton pairs (2,3)and (7,8) are chirai pairs. Enantiotopic protons will be denoted by bars.
18
Or18aa
a17ff21 f f 1 6 f f 2 3
ff19
alal
ff11ff27
ff15a*4
ff13a16 a14ffa5
1
1
2
3
3
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
0 0 0
0 0
0 0
1 3
1 4 0 1 0 1 3
0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
2 0 0 0 0
1 0
2 1 0 0
1 0
0
1 0 0
2 1 3 2
1 0
2 1
1 4 3 2 1 3 1
TABLE IV: Generating Function for the Protons of B(CH,), in Its Equilibrium Conformationa
1-6
'B: 7
ff19
At AEX+
-
E,+ ExEY-
2 1 2 2 1 1
7 5 7 7 5 5
17 13 15 15 12 12
23 19 23 23 19 19
a From the coefficient of a 1 7 a 1one can immediately infer that the nuclei span the representation 2A' + A - t 2EXt+ 2Eyt t E x - + Ey-,
5
4
1 0 0 0 0 0
Figure 5. NMR reaction graph of rigid propane in CPhsymmetry.
ferentiated in the rotational group. However, they become equivalent in the point group. I t can be easily seen that the class structure remains unaltered if inversion operations are included in the rotational subgroup of the nonrigid molecule. Thus, there are no chiral nuclei for the nonrigid molecule. The corresponding NMR graph is shown in Figure 3. The irreducible representations spanned by C1 and C2 are A, A2 G and A, Az where A, and A2 are totally symmetric and antisymmetric representations; G actually has two degenerate two-dimensional representations. The representations A,, A2, and G correlate to A,, A2, and 2(A1 A2) in the rotational group C2 of the rigid molecule so that the class C1 of the nonrigid molecule contains three A, representations in the rigid molecular group. Consequently, the class C1 of the nonrigid molecule splits into three classes. The doublecosets analysis shows that these are C1,1 = (1,6),C1.2 = (2,8) and C1,3 = (3,7). The class C2 of the nonrigid molecule remains unaltered. The corresponding reaction graph is shown in Figure 4. If we, however, introduce the inversion operation in the rigid group, then the group becomes CZh for the equilibrium conformation. The species A,, A2, and G correlate to A,, A2,and Al A2 + A3 Ad, representations in Cw. Thus the class C1 contains only two A, representation in C2h. The double-coset analysis with G = Ca gives C,,, = {1,6),C1,2= (2,8,3,7). Thus, the two classes C,,z and C1.3 in the group C2 coalesce into a single class in C2h indicating the presence of chiral protons. The proton pairs (2,3) and (7,8) are equivalent chiral pairs. We can thus indicate 3 as 2 (mirror image of 2) and S as 7. (See Figures 4 and 5.) The corresponding NMR graph is in Figure 5. The analysis outlined here thus shows that there are two NMR signals
+ +
+
+
+
+
for the nonrigid molecule at high temperature with the intensity ratio 3:l. The higher-intensity peak generates chiral signals at lower temperature two of which are chiral (and therefore can be differentiated only with chiral reagents) and the other is achiral. The over all spectrum has four signals with equal intensity. In the absence of chiral shift reagents the intensity pattern is 2:l:l. There are several such systems which exhibit this phenomenon of spontaneous generation of chiral NMR signals. 6.
An Example
In this section we consider the molecule B(CH& as an example to illustrate all the concepts outlined in earlier sections. This molecule was considered by Longuet-Higginsa as an illustration of a nonrigid molecule. The symmetry group of the nonrigid B(CH3)3is &[C3]. For the nonrigid molecule which has three methyl rotors exhibiting internal rotation it can be easily seen that there is only one class of protons. Generating functions for all the irreducible representations are shown in Table 111. Collecting the coefficients of q 8 a 2under various irreducible representations, we immediately infer that the protons of this nonrigid molecule span the representation Al El + 4. The molecule in its equilibrium configuration belongs to the point group C3h In this configuration three protons (one from each methyl group) are in one plane and the other two protons of each methyl group are in a plane perpendicular to this plane. The rotational subgroup is C3. Generating functions for all irreducible representations of C3h are shown in Table IV. From this table we infer that the protons now span a representation shown as follows:
+
C3h: 2A+
+ A- + 2E,+ + 2Ey++ E L + Ey-
4674
Balasubramanian
The Journal of Physical Chemistry, Vol, 86, No. 24, 7982
TABLE V : Generating Function for the Irreducible Representations of B(CH,), in the Group C,
r '41
Ex E,
elg a,q
1 0 0
al~az
a17a22
o(,*a23
a15a24
a1a28
a12a*'
a13a26
a14a25
3 3 3
12 12 12
30 27 27
42 42 42
6
5
Flgure 6, NMR reaction graph of nonrigid B(CH3),.
4A 7
V
Figure 7. NMR reaction graph of rigid B(CH,), in CBhgroup.
Equivalently, we could have obtained the above result by subducing the representations Al, El, and IIfrom &[C3] to C 3 h which correlate to A+, E,+ Eu+,and A+ + A- + E,+ + E,+ + E; + E;, respectively. However, the latter method needs the character table of the nonrigid molecule, while the former method does not need the character tables since the generating functions can be obtained without knowing the character table^.'^^^^ In this sense generating-function techniques are more elegant than the method of subducing representations. Note that protons of B(CH3)3span two totally symmetric representations in the group (2%. Consequently,there are two equivalence classes of nuclei. The double-coset analysis in (2% shows they are C1,l = (1,4,7]and Cl,2= {2,3,8,9,5,6).If one considers the rotational group of the molecule in its equilibrium configuration, the generating functions are shown in Table V. Thus, the nuclei span the following representations in the C3 group: C3: 3A1 3E, 3Es
a
+
+
+
There are three totally symmetric representations for the molecule in the C3 group. The three equivalence classes are C1,l = {1,4,7},Cl,z = (2,5,9},and C1,3 = {3,5,8).Since the number of equivalence classes in C3 and C 3 h differ, one immediately infers the existence of chiral nuclei for the rigid molecules. The nuclei pairs (2,3), (5,6), and (€49) are
8 (9) Flgure 8. NMR reaction graph of rigid B(CH3), in C3 group.
chiral pairs. The nonrigid molecule, however, did not have any chiral centers. Thus, this molecules is an example of a molecule which exhibits spontaneous generation of c h i d signal. The NMR reaction graph of this nonrigid molecule is shown in Figure 6. The corresponding reaction graphs of the rigid molecule in C%,C3groups are in Figures 7 and 8, respectively. A single NMR signal of the nonrigid molecule splits into three signals with equal intensity, two of which are chiral. The chiral signals coalesce in the absence of chiral shift reagents resulting in two signals with the intensity ratio 2:l. Acknowledgment. I am indebted to Professor Earl L. Muetterties for many illuminating discussions. I thank Professor Kenneth S. Pitzer for his encouragement. This research was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the US.Department of Energy, under Contract Number DE-AC03-76SF00098.
