CHAPTER 2 - APPLICATIONS OF GROUP THEORY

Transcription

1 36 HAPTER 2 APPLIATIONS OF GROUP THEORY 2 How Group Theory Applies to a Variety of hemical Problems The classification of molecules according to their symmetry point groups, provides a rigorous method for predicting optical activities For a molecule to exhibit optical activity, it must belong to a point group that does not possess an inversion center, mirror plane, or improper rotation axis The possibility of racemic mixtures must, of course, be considered Molecules that belong to point groups, such as 4 and other pure rotation groups, can exhibit optical activity if resolved into one optical isomer Another helpful symmetry rule in the analysis of diastereomeric protons in NMR spectroscopy is that chemically equivalent atoms (and hence protons with equivalent chemical shifts) must be interchanged by a symmetry operation of the point group For example, in Figure 2 the two protons H a and H b in structure A are equivalent because they are interchanged by the A B H H H 3 H 3 H 5 6 H 3 H a H b H a H b l l Figure 2 Illustration of a structure, A, where H a and H b are interchanged by a symmetry operation of the point group This would give rise to a single peak in the NMR spectrum In structure B, where no such operation exists, H a and H b would give rise to separate peaks in the proton NMR spectrum mirror plane operation, which contains the bond in the plane of the paper and lies perpendicular to the plane of the page It is important to recognize that rotation around the single bond does not interchange H a and H b The conformation produced by this bond rotation is not equivalent to the original one The l substituent now lies above the plane of the paper and the

2 37 molecule resides in a different rotomeric configuration (eclipsed vs the original staggered) Only the mirror plane operation interchanges H a and H b to produce a configuration indistinguishable from the original one In structure B of Figure 2 the substitution of a phenyl group for a methyl group destroys the mirror plane operation that interchanges H a and H b Now H a and H b are no longer equivalent and they give rise to separate peaks in the proton NMR spectrum Further examples will appear in hapter 7 that show the power of this approach for the analysis of chemical shift nonequivalence in complex structures The preceding examples concern straightforward conclusions derived from a consideration of symmetry operations and equivalent configurations; however, the real power of group theoretical methods results from their application to equivalent functions in a molecule Just as group theory can categorize equivalent atoms in a structure, it can also categorize equivalent functions The main applications concern electronic and vibrational wavefunctions in molecules For example, the two s atomic functions on hydrogen atoms H a and H b are equivalent in Figure 2 A The two H bond stretches (vibrational wavefunctions) to these atoms are also equivalent functions Since all of spectroscopy involves the transition between two states characterized by wavefunctions of one kind or another, it is possible to apply group theory widely This chapter describes how a set of n equivalent functions can be rearranged into a set of n linear combinations that take advantage of molecular point group symmetry This requires knowledge about group representations The payoff will be that this permits the prediction of spectra, selection rules, and molecular orbital diagrams without the need for detailed quantum mechanical calculations 22 Matrix Representations of Symmetry Groups Symmetry operations R acting on the point (x,y,z) are defined generally in eqn 22 R(x,y,z) (x, y, z ) (22) Because R preserves the size and shape of objects, it satisfies the requirements for a linear operator shown in eqn 222 and 223 Therefore, it is natural to apply the matrix methods of linear algebra in the description of symmetry operations R(ax, ay, az) ar(x, y, z) (222) R (x + bx 2, y + by 2, z + bz 2 ) R (x, y, z ) + br (x 2, y 2, z 2 ) (223) The action of R on a vector a a x ī + a y j + a z k (ī, j, k are the usual unit vectors for a righthanded orthogonal coordinate system) can be represented as in eqn 224

3 38 b Ra Because R is a linear operator, eqn 224 can be written as 225 b x i + b y j + b z k a x Ri + a y Rj + a z Rk (224) (225) The length of b must also equal the length of a ( b ā ) for a linear operator The symmetry transformed vectors R ī, R j and R k are new unit vectors in three dimensional space They can be expressed as some linear combination of ī, j, k, which are basis vectors for this space Ri Rj Rk r ī r 2 i r 3 i r 2 j r 22 j r 23 j r 3 k r 32 k r 33 k Substituting in eqn 225 above and collecting terms yields eqn 227 b x i + b y j + b z k (a x r + a y r 2 + a z r 3 )i + (a x r 2 + a y r 22 + a z r 23 )j + (a x r 3 + a y r 32 + a z r 33 )k (226) (227) Equating coefficients of ī, j, and k yields the set of equations of 228 b x a x r + a y r 2 + a z r 3 b y a x r 2 + a y r 22 + a z r 23 (228) b z a x r 3 + a y r 32 + a z r 33 In matrix notation, the set of equations b R ā can be written as shown in 229 Remember, to multiply a matrix times a column vector one multiplies each matrix row times the column b x r r 2 r 3 a x b y r 2 r 22 r 23 a y (229) b z r 3 r 32 r 33 a z The 3 x 3 matrix is called the matrix representation of the linear transformation R onsider the specific example of a fourfold rotation around the z axis The effect of this operation on x,y,z is given by eqn 22 4 (z) (x,y,z) (y,x,z) (22)

4 39 A point also defines a vector from the origin of the coordinate system The action of the counterclockwise 4 (z) rotation on the vector ā (x,y,z) therefore yields b (y,x,z), which can be pictured as follows: (y,x,z) y (x,yz) x b x y x + ()y b y x x + y b z z x + y Thus eqn 22 can be abbreviated in matrix form as z + z + z (22) y x x y (222) z z The matrix form of the operator 4 (z) is the matrix shown in eqn (z) (223) It can be shown that matrices representing symmetry operations are real and orthogonal Therefore, the transpose of matrix R gives the inverse matrix (R transpose of R) Recall that the transpose of a matrix is constructed by interchanging corresponding elements across the diagonal of the matrix (r ij Æ r ji ) It is important to remember that the matrix for a coordinate axis transformation is the inverse of the corresponding transformation for the point (x,y,z) For example, counterclockwise rotation of the coordinate system by 4 (z) leads is depicted below j i i The reason for the inverse relationship between the matrices of coordinate axes and points is easy to visualize Rotation of a coordinate system counterclockwise produces the same effect on a j

5 4 stationary point, from the reference frame of the coordinate system, as if the "stationary point" were rotated clockwise (the inverse transformation) onsider the matrix representations for common symmetry operations ounterclockwise rotation ( n ) about the z axis by angle a requires a computation of the rotated unit vectors i and j in terms of their projections on i and j Simple trigonometry yields: j j i i n (z)i n (z)j n (z)k i j k k Therefore we can express n (z), where a 36/n, as in eqn 224 i cos a + j sin a i sin a + j cos a i cos a sin a i 36/ a (z) j sin a cos a j (224) k k The transformation of the point or vector (x,y,z) uses the inverse of the transformation matrix for the coordinate system, which is just the transpose (interchange elements off the diagonal) of the preceding matrix x cos a sin a x 36/ a (z) y sin a cos a y (225) z z For the corresponding operations S n, where the S n axis lies along z, reflection in the x,y plane inverts z (ie z' z) The matrix for corresponding S 36/a operations are the same, except r 33 For s v in a plane containing z and making an angle b with the x axis, s v is related to s v (xz) by the similarity transformation that involves b

6 4 s v y b x (226) s v b s(xz) b cos b sin b cos b sin b s v (b) sin b cos b sin b cos b (227) cos b sin b cos b sin b sin b cos b sin b cos b (228) ( ) Ê (cos 2 b sin 2 b) 2 cosb sinb ˆ Á 2sinb( cosb) (sin 2 b cos 2 b) Á Ë ( 229) But sin 2a 2 sin a cos a and cos 2a cos 2 a sin 2 a This leads to eqn 222, when the mirror plane contains z and makes an angle b with the x axis cos 2b sin 2b s v sin 2b cos 2b (222) 23 haracter Tables and Symmetry Group Representations

7 42 We have defined a symmetry operation R acting on the point x, y, z and transforming the point to some new equivalent location x',y',z' as in eqn 23 R(x,y,z) (x,y,z ) (23) Applications to quantum mechanics require knowledge of the symmetry operation O R, which acts on a wavefunction, y(x,y,z) The operator O R acts on a function so that the new function O Ry evaluated at x',y',z' has the same value as (x,y,z) O R yr(x,y,z) O R y(x,y,z ) y(x,y,z) (232) Left multiplying by O R and transposing yields eqn 233 O R yr(x,y,z) y(x,y,z ) (233) This definition might seem backwards when compared with the definition of R; however, and the opposite convention (ie, O RyR(x,y,z) y(x',y',z')) can be adopted if one is consistent Because the set of inverses of all the elements of a group give the group back, the convention makes no real physical difference We will use the convention of eqn 2346, which can also be O R y(x,y,z ) y(x,y,z) (234) written as 235 O R y(x,y,z ) y(r x,y,z ) (235) Since the primes are arbitrary, let q represent (x,y,z) and one can use the expression 236 O R y(q) y(r q) (236) This latter form is most useful in applications A symmetry group can be regarded as an ndimensional function space The behavior of a general function y, under the operations of a group O R can be expressed as a combination of the behavior of "basis functions" that span the possible behaviors of a function belonging to group G The group s (s in the yz plane) and the behavior of various functions f(x) with respect to reflection illustrates this point sx x sx 2 (x) 2 x 2 scos(x) cos(x) cos(x) s(x 2 + x) x 2 x s(x 4 + x 3 ) x 4 x 3 antisymmetric fn or odd fn symmetric fn or even fn symmetric fn linear combination of a symmetric and antisymmetric fn linear combination of a symmetric and antisymmetric fn A function is symmetric if it is unchanged when acted on by s and antisymmetric if the function goes into minus itself The properties of symmetry and antisymmetry with respect to s define the fundamental behavior of functions in the s group All linear functions can be decomposed into a

8 43 linear sum of symmetric and antisymmetric parts A character table for s, which summarizes these conclusions is shown below s E s s E s symm A antisymm A The symmetric and antisymmetric types of functions are more conventionally denoted A and A These A and A function types are called irreducible representations They are the two fundamental types of functions one needs to describe symmetry behavior in the twodimensional group s The characters ± show how operations of the group change functions that transform like these irreducible representations For example, the functions x and x 3 are A and cos x or x 2 are A in the s point group A linear function, such as x + x 2 + x 3, can be decomposed into A + 2A irreducible parts This intuitive development can be formalized Earlier we showed that matrices could be used to represent symmetry operators of a group It follows that matrix representations multiply just like the symmetry group operations, and form a group isomorphic to the point group of symmetry operations In fact, any set of square matrices that multiply like the elements of a group form a representation for that group The order of the matrices defines the dimension of the representation The correspondence between matrices and symmetry operations need not be onetoone A matrix can correspond to more than one group element For example, let the x matrix () all the elements of a group Thus for s E s and E x s () () () x () () This trivial onedimensional representation satisfies all the requirements for a group and is a valid group representation The many to less correspondence between the group elements and matrices is called a homomorphism, and the matrix representation is termed unfaithful When a oneone correspondence (isomorphism) exists, the representation is called faithful When dealing with matrices, it is convenient to define the trace or character of a matrix R that is the sum of the diagonal elements in eqn 237 Tr(R) Sa ii i (237) For symmetry operator matrices that represent a group isomorphically R 2 R R 2 defines a similarity transformation of coordinates From matrix theory it can be shown that R 2 R R 2 defines R with respect to a new set of basis vectors In other words, the operation is the same but just defined in another basis This makes sense, since the equivalent symmetry operations in a conjugate class

9 44 consist of similar kinds of operations that differ only in spatial orientation The trace or character of a matrix does not change after a similarity transformation With this aside, return to the problem of representations + For a group of order n, it is possible to find a set of n matrices that faithfully represent the group By similarity transformations, the set of matrices can be arranged in the most reduced form (block diagonal form), as shown above The number of elements in the submatricies for each operation will equal the order of the group The submatrices are themselves unfaithful representations of the group For most purposes, enough information is contained in the trace or character of these matrix representations Furthermore, group elements belonging to a class can be grouped together because they have identical characters The character table for the 3v group has the form shown below 3v E 2 3 (z) 3s v A z, z 2, x 2 + y 2 A 2 R z E 2 (x,y), (R x,r y ), (x 2 y 2, xy), (xz, yz) The irreducible representations named A, A 2, and E are the fundamental ways that a function can be classified in the 3v point group This example provides some new complexities as compared to s The simultaneous effect of 3 and s operations influences the classification of symmetry type Both the A and A 2 representations are unchanged (ie, symmetric) under 3 or 3 2 rotations While A also does not change under a s operation, the A 2 representation is antisymmetric For the two dimensional E representation, the characters alone do not tell us about symmetric or antisymmetric behavior, because they represent the composite trace of 2 x 2 matrices The rightmost column provides transformation properties for several functions relevant to physical problems The x, y, and z coordinates behave just like atomic p x, p y, and p z orbitals or the x, y, and z components

10 45 of the electric dipole moment operator, so far as their symmetry properties are concerned These behaviors can be derived using methods described in section 25 and 26 The following theorems hold for the group representations: ) The number of classes in the group Number of unique irreducible representations 2) Let G i i th irreducible representation and let X R (G i ) the character for the R th operator of the representation G i Then 2 S X (G order of the group g E i ) i This means that the number of elements contained in the matrix representations equals the order of the group 2 3) S X (G i ) order of the group for any G i (note S includes sum over R R so all elements R in each class must be included) above G i should be G i!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 4) SX R (G i )X R (G j ) gd ij (ie, rows are orthogonal) R 2 5) If S X R (G i ) g, t R then G i is an irreducible representation 6) The columns of the character table are orthogonal and normalized to g/n k (N k number of elements in the class for the k'th column of the table) onsider the application of these theorems to cyclic groups For 5, the five elements can be written as b, b 2, b 3, b 4, and b 5 e By application of similarity transforms, one can show that each element belongs to its own class Therefore by theorems and 2 there will be 5 onedimensional irreducible representations From Lagrange's Theorem (section 2) the order of the elements of the group must be or 5 Since the element of order is e, the others must obey the relation X b 5 (G j ) c e 2pi(n/g) n,2,3, g This equation defines the g th roots of unity, since c g [G(b)] e 2pi These roots are abbreviated as e, e 2,, e g (e e 2πi/g ) For the specific case of 5, where e e 2pi/5 :

11 G e e 2 e 3 e 4 e 5 G 2 e 2 e 4 e 6 e 8 e G 3 e 3 e 6 e 9 e 2 e 5 G 4 e 4 e 8 e 2 e 6 e 2 G 5 e 5 e e 5 e 2 e 25 Next use the fact that: e 3 e 2pi3/5 e 2pi(2/5) (e 2pi(2/5) ) * e 2* and e n5+j e j Here * denotes the complex conjugate, and we recognize that rotation by (+3/5)2p and (2/5)2p in the complex plane are equivalent ontinuing the process for the group 5 yields the character table: 5 E G 5 A G e e 2 e 2* e * E G 4 e * e 2* e 2 e G 2 e 2 e * e e 2* E 2 G 3 e 2* e e * e 2 The reason for pairing G, G 4 as E and G 2, G 3 as E 2 is that they are complex conjugate pairs For most physical problems involving real functions, the complex conjugate pairs are effectively degenerate (not so for magnetic problems like the Zeeman effect where y ~e imf ) The complex conjugate pairs can be added [recall e ia cos a + i sin a and e ia cosa i(sina) (e ia ) * ] and then only the real (cosq) part is retained This yields a 5 character table that is sufficient for solving problems with nonimaginary functions

12 47 5 A G + G 4 E G 2 + G 3 E 2 E 2 2 2cos 5 2 p 5 2cos 5 2 2cos 4 p 2 p 5 2cos p 4 p 2cos 5 5 2cos p 5 2cos 2 p 4 p 2cos Symbols for Irreducible Representations The Bethe nomenclature uses G, G 2, G 3 to specify irreducible group representations This notation is still used for double groups, as discussed in a later chapter Otherwise the Mulliken notation is usually employed by chemists For onedimensional irreducible representations, c(e) One dimensional representations are called A if symmetric with rotation around the principal axis of symmetry, or B if antisymmetric with rotation around the principal rotation axis of the group Subscripts and 2 denote symmetry or antisymmetry with respect to a 2, or a s v if no 2 exists Primes and double primes specify symmetry or antisymmetry with respect to s h In groups that contain an inversion center, g or u subscripts denote even (g) or odd (u) behavior with respect to inversion Twodimensional irreducible representations have c(e) 2 and are called E Threedimensional irreducible representations, for which c(e) 3, are named T For onedimensional irreducible representations, the character is the matrix and + denotes symmetry with respect to a given operation, while a character of means antisymmetry The totally symmetric representation is conventionally the first A representation in every character table and all of its characters are A function with this symmetry possesses the full symmetry of the point group All the other representations are called nontotally symmetric representations They represent functions of lower symmetry, since a character means the function does not go into itself on application of that symmetry operation haracter tables for the important point groups are collected in Appendix A of this chapter 25 Decomposing Reducible Representations The Great Orthogonality Theorem Given a set of characters X R (G i ) for some representation, G i, of a group, it is desirable to find out how to reduce the representation to a sum of irreducible representations The answer is provided by the great orthogonality theorem That is, given X R(G) we want to find out G ag i + bg j + The number of times the irreducible representation G i of the group is contained in G may be calculated by the formula: a G i /g S X R * (G i ) X R (G) ( a G i the number of times the G i representations R occurs in G, g order of group)

13 48 For example the representation E 2 3 3s v 6 in 3v reduces to A + A 2 + 2E The calculation proceeds as follows: a A / 6 {(6) + ()2 + ()3} a A 2 / 6 {(6) + ()2 ()3} a E /6 {2(6) + ()2 + ()3} 2 Notice that because the sum occurs over all symmetry operations, the 2 3 operations in the same class and the 3 equivalent s v must be included explicitly in the calculation 26 The OH Stretches of Water: Understanding Irreducible Representations Significance of Irreducible Representations as Basis Sets for hemical Problems Suppose there exists a set of linearly independent functions f, f 2, f 3, f n onsider a symmetry operation, O R, that permutes the set of functions, or mixes them with one another to form a new linearly independent set of functions with dimension n The set of matrices for O R form a representation of the point group O R above!!!!!!!!! q R f f n a f i +b f j + reorg Any set of n independent equivalent functions that are transformed into linear combinations of one another by the symmetry operations of the group form a basis for an ndimensional representation of the group The trace of the representation matrix yields characters that allow the decomposition of the ndimensional representation into a sum of irreducible representations This procedure permits a classification of electronic, vibrational, rotational or spin wavefunctions of a molecule transform in terms of the irreducible representations of the molecular point group The degree of degeneracy of a molecular energy level equals the dimension of the irreducible representation to which its wavefunction belongs onsider what group theory has to say about a simple problem, the OH stretching vibrations in the water molecule The vibrational wavefunctions are classified according to the

14 49 various irreducible representations of the molecular symmetry group for water A naive approach might suggest that both OH bonds are equivalent, and therefore both would occur at the same stretching frequency in the infrared spectrum This simple view ignores the coupling of the two vibrations Stretching of one OH bond changes the degree of difficulty of stretching the other bond Group theory requires that two equivalent functions form the basis for a twodimensional group representation Refer back to Figures 4 and 5, which show that s(xz) and 2 interchange the two OH bonds, while s(yz) leaves them unchanged Each bond stretch can be depicted by a double headed arrow centered in each bond Therefore the symmetry operations affect the two OH bonds as follows: 2 OH a OH b OH b OH a s(xz) OH a OH b OH b OH a s(yz) OH a OH b OH a OH b The matrices for this twodimensional representation are: E 2 s(xz) s(yz) The characters for this representation, denoted G OH, are obtained by summing the diagonal elements and can be compared with the 2v character table shown below

15 5 2v E 2 (z) s v (xz) s v (yz) A A 2 B B 2 G OH 2 2 Thus G OH is a reducible representation, which by inspection (or the formula of section 25) consists of A + B 2 Because there are no degenerate representations in 2v symmetry, the two O H vibrational wavefunctions cannot be degenerate The water molecule should show two different OH stretching vibrations by infrared spectroscopy (more about selection rules later) This prediction of group theory is borne out experimentally, since two such vibrations occur at 3652 cm and 3756 cm in the gasphase IR spectrum of water The 2v character table requires that all wavefunctions of the water molecule be nondegenerate Any degeneracy that might occur must be by sheer coincidence and is termed accidental degeneracy Before proceeding, there are two significant observations First, in computing the character for the two OH stretches a value of occurs along the diagonal for each bond that goes into itself Offdiagonal elements of the matrices need not be considered, when calculating the characters Second, if the coordinate system were selected so the water molecule was in the xz plane, then the characters for the OH stretches would have been (2 2 ) A + B Either description of the O H stretches as A + B or A + B 2 is correct, they merely correspond to a different choice of coordinates Two scientists describing the IR spectrum of water could appear to be in disagreement, when the difference is merely one of coordinate axes This illustrates the importance of clarifying the coordinate system choice (and being consistent) to avoid confusion Neither OH a nor OH b alone behave as an irreducible representation in the 2v point group Now consider a more complex case where mixing of functions occurs For the functions x and y in 3v symmetry, the 3 operation mixes the two coordinates according to eqn 225 (a specific case of the general rotation matrix derived previously) Because of this mixing, both x and y must be considered together as a twodimensional representation E 3 (z) cos 2 sin 2 sin 2 cos 2 s v omputation of the s v matrix takes advantage of the option that the orientation of the x and y axes are arbitrary, as long as they lie perpendicular to z The calculation is simplified with the assumption that s v lies in the xz plane By reference to eqn 222, it is easy to see that the trace of the matrix would indeed be zero regardless of the value of b (ie, the orientation of s v with respect to the x

16 5 axis) Furthermore, the character is the same for equivalent elements in a class, and so only one of the 2 3 and 3s v operations need be considered The trace of these three matrices (2 ) verifies that x and y together form a basis for the E representation of 3v