Chapter 6

Systematic Classification of Molecular Symmetries 1

Point groups are frequently incomplete to afford fuH symmetry information about moleculesj various molecules belong to the same point group. For example, a list of C 211 molecules contains water, hydrogen sulfide, formaldehyde, phosgene, chlorobenzene, fluorobenzene, 1,2- and 1,3-difluorobezenes, 1,2-benzoquinone, 1,4-dichloronaphthalene, 1,4-naphthoquinone, pyridine, 4-chloropyridine, 2,6-dichloro­pyridine, pyridine N-oxide, furan, thiophene, cyclopropanone, cyclobutanone, cy­clopentanone, tetrahydrofuran, tetrahydrothiophene, dichloromethane, difluoro­methane, oxirane, phenanthrene, bicyclo[2.2.1!heptane, basketane, adamantanone, noradamantane and so on. Such a list should be classified into several categories in the light of a rational criterion. For this purpose, Pople!l) has proposed the concept of "framework group". By this method, difluoromethane is designated as C 211 [C2(C), ull(F2), u:(H2 )!. Flurry(2) pointed out that the framework group is related to local (site) symmetries and proposed his notation based on the local sym­metries. Thus, the difluoromethane is designated as C 2 .. [C211(C), C.(F2 ), C:(H2)!. Although Flurry's method has a potential applicability, it is not so easy to deter­mine such local symmetries especially in the cases of complex molecules. We here" propose the SCR (set-of-coset-representation) notation for classifying molecular symmetry.

6.1 Assignment of Coset Representations to Orbits

Atoms in a molecule can be classified into several equivalence classes in the light of molecular symmetry (a point group). In order to obtain a more versatile method of classification beyond a point group, we should develop an effective tool of char­acterizing such classes. In the preceding chapter, we have referred to them as

'Reprinted in part with permission from S. Fujita, Bull. ehern. So<. Jpn., 63, 315-327 (1990). @(1990) The Chemical Society of Japan.

S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry© Springer-Verlag Berlin Heidelberg 1991

64 CHAPTER 6. SYSTEMATIC CLASSIFICATION OF MOLECULAR SYMMETRIES

orbits; and furthermore, we have clarified one-to-one correspondence between an orbit and a coset representaiton (CR). This correspondence provides us with a characterization tooI.

Assignment of a CR to an orbit is accomplished by means of two methods. Since we have discussed the principles of these methods in Chapter 5, we will here describe recipes for carrying out the assignment.

Method I. The 1st method consists of the following steps:

1. Find a point group (G) to describe a given molecule.

2. Partition atoms of the molecule into orbits (sets of equivalent atoms).

3. Count fixed atoms to produce an fixed-point vector (FPV) with respect to each of the orbits.

4. Compare the FPV with each of the rows of a mark table for the G group; an identical row is a CR to be assigned.

These steps should be repeated over all of the orbits that have been found in the 2nd step. Step 4 requires a mark table for the G point group. Such mark tables are found in Appendix A. Example 5.1 (Chapter 5) has exemplified Method 1. We examine an additional example as follows.

Example 6.1 Let us examine an oxirane molecule (1). This molecule belongs to C211 point group, which has SSG = {Ch C2, C., C:, C211}.

top vi ew

By inspection, we find three orbits, i.e., a four-membered orbit ß 1 (H4), a two-membered orbit ß2 (C2) and a one-membered orbit ß3 (0). When we apply all the operations of each subgroup of the C 211 point group, we o'btain the following FPVs:

6.1. ASSIGNMENT OF COSET REPRESENTATIONS TO ORBITS

Table 6.1: Mark table of C 2v

coset subgroup reprentation Cl C 2 C. C. C 2" C 2,,(fCl) 4 0 0 0 0 C2,,(fC2 ) 2 2 0 0 0 C2,,(fC.) 2 0 2 0 0 C2,,(fC:) 2 0 0 2 0 C2,,(fC2,,) 1 1 1 1 1

(4 0 0 0 0) for the orbit (~d of H., (2 0 2 0 0) for the orbit (~2) of C2, and (1 1 1 1 1) for the orbit (~3) of O.

65

By the comparison of these FPVs with the mark table of C2" (Table 6.1), we have C2,,(fCl ) for the ~l orbit, C2,,(fC.) for the ~2 orbit, and C2,,(fC2,,) for the ~3 orbit. Note i C 2" I/I Cl I = 4/1 = 4, I C 2" I/I C, I = 4/2 = 2, and I C 2" I/I C 2" I = 4/4 = 1, which are equal to the lengths of the respective orbits.

Method 11. On the other hand, the 2nd method contains the following steps:

1. Find a point group (G) to describe a given moleeule.

2. Count fixed atoms to produce an fixed-point vect~r (FPV) with respect to all of the atoms to be examined.

3. Multiply the FPV by the inverse of the mark table for the G group.

4. Refer the the resulting vector to SSG = {Gb G 2 , ••• , G.} and find the mul-tiplicity of the CR G(fGi) to be assigned.

Method n requires the inverses of mark tables for point groups. They are found in Appendix B. Example 5.2 (Chapter 5) explains Method II by using D 2d group. The following example is concerned with Ca" group.

Example 6.2 For illustrating the 2nd method, let us examine the same oxirane moleeule (1). When we apply all the operations of each subgroup of the C 2" point group to the seven atoms, we obtain an FPV, (7 1 3 1 1). This vector is multiplied by the inverse of the mark table, i. e.,

[

1/4 -1/4 1/2

(71311) -1/4 0 -1/4 0 1/2 -1/2

o

1~2 ~ ~ 1 = (1 0 1 0 1), o 1/2 0

-1/2 -1/2 1

(6.1)

66 CHAPTER 6. SYSTEMATIC CLASSIFICATION OF MOLECULAR SYMMETRIES

wherein the second 5 x 5 matrix is the inverse. The resulting vector contains respective multiplicities of coset representations (CRs). The multiplicities are aligned in the order of a set of CRs, i.e., SCR = {C 2,,(fC1), C2,,(JC2), C2,,(fC.), C2,,(JC:), C2,,(fC2,,)}. Hence, we have a formal expression,

(6.2)

We can easily assign these CRs to the sets of equivalent atoms, i.e., C2,,(fCd to the 6.1 orbit (H4), C2,,(fC.) to the 6.2 orbit (C2), and C2,,(fC2,,) to the 6.3 orbit (0).

It should be noted that there is arbitrariness in selecting C. or C:. Ac­cordingly, an alternative assignment of C21J(fC:) to the 6.2 orbit (C2) is possible. However, this arbitrariness is not essential in discussing symmetrical properties.

6.2 SCR Notation

The discussions in the preceding section provide a basis for characterizing the symmetry of a given moleeule. Atoms contained in the moleeule are partitioned into several orbits or sets of equivalent atoms, which correspond to an appropriate sum of CRs, Ei=1 Cl.iG(fGi), where G is the point group of the moleculej Gi is a subgroup of Gj and the symbol CI.; represents the multiplicity of each CR (Theorem 5.7 in Chapter 5). Let the symbol 6.;" denote an CI.-th orbit that is subject to the CR G(fG;), where CI. = 1,2, ... ,CI.; for each i. Suppose that the orbit (6.;,,) is occupied by the atoms (M;"» of the same kind and that the number of the atoms is equal to r = 16.;" I = I G 1I1 Gi I· Then, the moleeule is represented by

G[····/G·(A(il) A(;2) A(;".»· ... ] , a .. r , r+ ' r " , (6.3)

'" which contains each CR after a slush (f) and its members in parentheses. We call this symmol an SCR (set-ol-CR) notation. Since the orbits of the oxirane (1) is represented by eq. 6.2, the SCR notation of 1 is obtained as C 211 [fC1(H4)j IC.(C2)i IC21J (0)]. This SCR notation means that four hydrogen atoms of 1 construct an orbit subject to C 2,,(fC1)i that two carbon atoms construct an orbit subject to C 211 (fC.)i and that one oxygen atom constructs an orbit subject to C 211(fC211 ).

Any moleeules can be specified in terms of eq. 6.3. Table06.2 lists the SCR notations of moleeules of C 2" symmetry, where several structures are shown.

6.2. SCR NOTATION

Table 6.2: SeR Notations of C2v-molecules

moleeule water hydrogen sulfide formaldehyde phosgene chloro benzene fluorobenzene 1,2-difluorobezene 1,3-difluorobezene 1,2-benzoquinone 1,4-dichloronaphthalene 1,4-naphthoquinone pyridine 4-chloropYJ:idine 2,6-dichloropyridine pyridine N-oxide furan thiophene cyclopropanone cyclobutanone cyclopentanone tetrahydrofuran tetrahydrothiophene dichloromethane difluoromethane oxirane (1) phenanthrene (2) iceanedione (3) noradamantane (4)

bicyclo[2.2.1]heptane (5)

basketane (6) adamantanone (7)

l,l-difluorocyclopropane (8) l,l-difluoroallene (9)

SCR notation C 2v [/C.(H2)j jC2v(0)] C 2v[/C.(H2)j jC2v (S)] C 2v[/Cs (H2)j jC2v (C,0)] C 2v [/C.(Ch)j jC2v(C,0)] C 2v [/C.(2H2,2C2)j jC2v (H,2C,Cl)] C 2v [/C.(2H2,2C2)j jC2v (H,2C,F)] C2v[/Cs(2H2,3C2, F2 ») C2v[/C.(H2,2C2, F2)j jC2v(2H,2C)] C2v[/Cs(2H2,3C2, O2)] C2v [/C.(3H2,5C2, Ch») C2v [/C.(3H2,5C2, O2)] C 2v[/C.(2H2,2C2)j jC2v(H,C,N)] C 2v[/C.(2H2,2C2)j jC2v (C,N,Cl)] C2v[/Cs(H2,2C2,Ch)i jC2v (H,C,N)] C2v [/C.(2H2,2C2)i jC2v(H,C,N,0)] C 2v [/C.(2H2,2C2)j jC2v ( 0)] C2v [/C.(2H2,2C2)i jC2v (S)] C 2v [/C1(H4)i jC.(C2)i jC2v(C,O)] C 2v [/C1(H4 )j jCs(C2)i jC~(H2)i jC2v(2C,O)] C 2v [/C1(2H4)j jC.(2C2)i jC2v(C,0)] C2v [/C1(2H4)j jCs(2C2)j jC2v(O)] C 2v [/C1(2H4)j jC.(2C2)i jC2v(S)] C 2v[/C.(H2)j jC:(Ch)j jC2v(C)] C 2v [/C.(H2)j jC~(F2)j jC2v(C)] C 2v [/C1(H4)j jC.(C12)j jC2v (0)] C2v [/Cs(7C2, 5H2)]

C2v[/Cl(2C4,3H4)i jCs(2C2,H2,02)] C2v[/Cl(C4,2H4)j jC.(C2, 2H2); jC~(C2' H2);

jC2v(C)] C2v[/Cl(C4,2H4)j jCs(C2, H2); jC~(H2);

jC2v(C)] C2v[/Cl(C4,2H4); jC s (3C2 , 2H2)] C2v[/Cl(C4,2H4); jC.(C2, 2H2)i jC~(C2,H2);

jC 2v(2C,O)] C 2v [/C 1(H4 ); jC.(C2); jC:(F2)i jC2v(C)] C2v [/C.(H2); jC~(F2); jC2v(3C~

67

68 CHAPTER 6. SYSTEMATIC CLASSIFICATION OF MOLECULAR SYMMETRIES

(0)°(0> 2 3

Ö tb 4) 4 5

6

0 F H F-\j-H H F~·~C=C=C/ F~ \H

H H 9 8

7

Example 6.3 Let us now examine molecules of D 3h symmetry. For the D 3h point gr~up, we obtain SSG = {Cl. C 2, C .. C~, C3, C 2", C3", C3h, D 3, D 3h}'

Let us work out an X3 Y2 derivative (10) of phosphorane. When we count fixed points for every subgroup of the SSG, we have an FPV = (6 2 4 4 3 2 3 1 1 1). The FPV is multiplied by the inverse of a mark table for D 3h to afford

(6244323111)x

6.2. SOR NOTATION 69

1/12 0 0 0 0 0 0 0 0 0 -1/4 1/2 0 0 0 0 0 0 0 0 -1/4 0 1/2 0 0 0 0 0 0 0 -1/12 0 0 1/6 0 0 0 0 0 0 -1/12 0 0 0 1/4 0 0 0 0 0

1/2 -1/2 -1/2 -1/2 0 1 0 0 0 0 1/4 0 -1/2 . 0 -1/4 0 1/2 0 0 0 1/12 0 0 -1/6 -1/4 0 0 1/2 0 0 1/4 -1/2 0 0 -1/4 0 0 0 1/2 0

-1/2 1/2 1/2 1/2 1/2 -1 -1/2 -1/2 -1/2 1 (0 0 000 1 1 0 0 1).

Note that the second 10 X 10 matrix is the inverse. The resulting vector indicates

PD3A = D3h(fC2v) + D3h(fC3v ) + D3h(fD3h), (6.4)

since a set of CRs is represented by SCR = {D3h(fC1), D3h(fC2), D3h(fC.), D3h(fC:), D3h(fC3), D3h(fC2v ), D3h(fC3v ), D3h(fC3h), D3h(fD3), D3h(fD3h)}.

We can easily assign these CRs to the sets of equivalent atoms, i.e., D 3h(fC2v) to the 6,1 orbit (X3), D3h(fC3v) to the 6,2 orbit (Y2), and D3h(fD3h) to the 6,3 orbit (P). This result is summarized by the SCR notation, D3h[fC2v(X3}; /C3v(Y2);/D3h(P)].

Example 6.3 is based on Method 11. Of course, Method I is applicable to this· case, where the mark table listed in Appendix A is used. There are various molecules belonging to D 3h symmetry. Table 6.3 lists SCR notations for represen­tatives of such D3h-molecules.

Example 6.4 Let us now examine methane (16) that is simple but belongs to complicated Td symmetry. According to SSG = {Ch C 2, C., C3, S4, D 2, C2v ,

C3", Du, T, Td}, we count fixed points among 5 atoms. Then, we have FPV = (5 1 3 2 1 1 1 2 1 1 1). This FPV is multiplied by the inverse of a'mark table for T d to afford

(5 1 3 2 1 1 1 2 1 1 l)x

70 CHAPTER 6. SYSTEMATIC CLASSIFICATION OF MOLECULAR SYMMETRIES

Table 6.3: SCR Notations of D3h-molecules

moleeule SCR notation 10 D3h(fC2v(X3); /C311(Y2);/D3h(P)] 11 D3h(fC,(H6); /C2v(C3)] 12 D3h(fC,(C6,H6 )]

13 D3h (fC,(3C6 ,2H6 ); /C3v(C2 ,H2)]

14 D3h(fC1(H12); /C:(2Ca)] 15 D3h(fC2v(F3); /D3h(B)]

13

H

H A~~ H~H

11

14

12

15

6.2. seN NOTATION 71

1/24 0 0 0 0 0 0 0 0 0 0 -1/8 1/4 0 0 0 0 0 0 0 0 0 -1/4 0 1/2 0 0 0 0 0 0 0 0 -1/6 0 0 1/2 0 0 0 0 0 0 0

0 -1/4 0 0 1/2 0 0 0 0 0 0 1/12 -1/4 0 0 0 1/6 0 0 0 0 0 1/4 -1/4 -1/2 0 0 0 1/2 0 0 0 0 1/2 0 -1 -1/2 0 0 0 1 0 0 0 0 1/2 0 0 -1/2 -1/2 -1/2 0 1 0 0

1/6 0 0 -1/2 0 -1/6 0 0 0 1/2 0 -1/2 0 1 1/2 0 1/2 0 -1 -1 -1/2 1

= (0 0 0 0 0 0 0 1 0 0 1),

wherein the second 11 x 11 matrix is the inverse matrix. The resulting vector indicates

PTd = Td(fC3v ) + Td(fTd), (6.5)

because a set of CRs is SCR = { Td(fCd, T d(fC2), Td(fC.), Td(fC3 ), Td(fS4), Ti/D2), T d(fC 2v ), T d(fC3v ), Td(jD 2d ), Td(jT), Td(jTd)} in this case. We can easily assign these CRs to the sets of equivalent atoms, i.e., Td(fC3v ) to the ß[

orbit (H4) and Td(jTd) to the ß2 orbit (C). This result is summarized by the SCR notation, Td[jC3v(H4); /Td(C)].

Table 6.4 collects SCR notations for several T d-molecules. The compound (17) is named tetrahedrane after its geometrical form. The compound (18) is called adamantane because of its diamond structure (Greek: adamas). The ge­ometrical relationship between 17 and 18 will be discussed in Chapter 17. The compound (19) is derived by substituting nitrogen atoms for four methines of cubane. The substitution reduces the original Oh symmetry of cubane into Td.

The original version!3] of the SCR notation consists of type land II nota­tions. The type II notation is based on the subduction of coset representations, characterizing the symmetry of a moleeule as weIl as that of a parent skeleton. The type I notation is an abbreviation of the type I notation, where information about the skeleton is omitted. The present version essentially succeeds to the type I notation, whereas the distinction between moleeules and skeletons are not taken into consideration.

In Chapter 7, we will show that any point of a G(fGi ) orbit belongs to the local symmatry Gi. The G(jGi ) determines the permutational properties as weIl as the local symmetry of the orbit. The symbol (G(j Gi)) is convenient to indicate such inherent nature of the orbit.

72 CHAPTER 6. SYSTEMATIC CLASSIFICATlON OF MOLECULAR SYMMETRIES

Table 6.4: SeR Notations of Td-molecules

mole eule SCR notation 16 Td!lC311(H4)i jTd(C)] 17 Td!lC3v(C4 , H4)]

18 Td!lC.(H12)i jC2v(C6)i jC3v (C4 , H4 )]

19 Td!lC311(C4,H4,N4)]

H

I H _ ... C ______

-- 1 --H H

16 17 18

Bibliography

[1] J. A. Pople, J. Am. ehern. Soc., 102,4615 (1980).

[2] R. L. Flurry, Jr. J. Am. ehern. Soc., 103, 2901 (1981).

[3] S. Fujita, Bult. ehern. Soc. Jpn., 63, 315 (1990).