36  university of california, san h bond stretches (vibrational wavefunctions) to these atoms are...
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CHAPTER 2  APPLICATIONS OF GROUP THEORY
2.1 How Group Theory Applies to a Variety of Chemical 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 C4 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 2.1 the two protons Ha and Hb in structure A are equivalent because they are interchanged by the
BA
Figure 2.1 Illustration of a structure, A, where Ha and Hb 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, Ha and Hb would give rise to separate peaks in the proton NMR spectrum.
C
C
CH3
Cl
H
C
HbHa
CH3
Ha
C
H5C6
H
Cl
CH3
Hb
mirror plane operation, which contains the CC 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 CC single bond does not interchange Ha and Hb. The conformation produced by this bond rotation is not equivalent to the original one. The Cl substituent now lies above the plane of the paper and the
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molecule resides in a different rotomeric configuration (eclipsed vs. the original staggered). Only the mirror plane operation interchanges Ha and Hb to produce a configuration indistinguishable from the original one. In structure B of Figure 2.1 the substitution of a phenyl group for a methyl group destroys the mirror plane operation that interchanges Ha and Hb. Now Ha and Hb are no longer equivalent and they give rise to separate peaks in the proton NMR spectrum. Further examples will appear in Chapter 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 1s atomic functions on hydrogen atoms Ha and Hb are equivalent in Figure 2.1 A. The two C 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.
2.2 Matrix Representations of Symmetry Groups
Symmetry operations R̂ acting on the point (x,y,z) are defined generally in eqn. 2.2.1. R(x,y,z) (x1, y 1, z 1) (2.2.1)^
Because R̂ preserves the size and shape of objects, it satisfies the requirements for a linear operator shown in eqn 2.2.2 and 2.2.3. Therefore, it is natural to apply the matrix methods of linear algebra in the description of symmetry operations.
R(x1, bx2, y 1 + by2, y3 + by3) =
R(ax, ay, az) = aR(x, y, z)^ ^
^
(2.2.2)
(2.2.3)^R̂(x1, y 1, z 1) + bR(x2, y 2, z 2)
R̂ (x1 + bx2, y1 + by2, z1 + bz2) = R̂ (x1, y1, z1) + bR̂ (x2, y2, z2) (2.2.3)
The action of R̂ on a vector a = axi  + ay
j + az k (i,
j , k are the usual unit vectors for a righthanded orthogonal coordinate system) can be represented as in eqn 2.2.4.
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(2.2.4)^b = Ra Because R is a linear operator, eqn 2.2.4 can be written as 2.2.5.
^^^  (2.2.5)bxi + byj + bzk = axRi + ayRj + azRk
The length of b must also equal the length of a (˙  b˙ = ˙ a˙ ) for a linear operator. The symmetry
transformed vectors R̂i, R̂ j and R̂k are new unit vectors in three dimensional space. They can be
expressed as some linear combination of i, j , k , which are basis vectors for this space.
R̂j
Ri^
=
=
=
r11i 
r12i
r13i
r21j
r22j
r23j
r31k
r32k
r33k
+
+
+
+
+
+
(2.2.6)
R̂k
Substituting in eqn 2.2.5 above and collecting terms yields eqn 2.2.7.
(2.2.7)
+ (axr31 + ayr32 + azr33)k
bxi + byj + bzk = (axr11 + ayr12 + azr13)i + (axr21 + ayr22 + azr23)j    

Equating coefficients of i, j, and k yields the set of equations of 2.2.8.
bx = axr11 + ayr12 + azr13 by = axr21 + ayr22 + azr23 (2.2.8)
bz = axr31 + ayr32 + azr33 In matrix notation, the set of equations
 b = R̂ a can be written as shown in 2.2.9. Remember, to
multiply a matrix times a column vector one multiplies each matrix row times the column.
bx
by
bz
=
r11 r12 r13
r21 r22 r23
r31 r32 r33
ax
ay
az
(2.2.9)
The 3 x 3 matrix is called the matrix representation of the linear transformation R. Consider the specific example of a fourfold rotation around the z axis. The effect of this
operation on x,y,z is given by eqn 2.2.10.
C4(z) (x,y,z) (y,x,z) (2.2.10) ^
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A point also defines a vector from the origin of the coordinate system. The action of the counterclockwise Ĉ4(z) rotation on the vector
a = (x,y,z) therefore yields  b = (y,x,z), which can be
pictured as follows:
bx = y = 0x + (1)y
by = x = 1x + 0y
bz = z = 0x + 0y
+ 0z
+ 0z
+ 1z
(2.2.11)
y
x (y,x,z) (x,yz)•
•
Thus eqn 2.2.10 can be abbreviated in matrix form as 2.2.12.
(2.2.12)
z
y
x 0 1 0
=
z
x
y
1 0 0
0 0 1
The matrix form of the operator C4(z) is the matrix shown in eqn 2.2.13.
0 0 1
1 0 0
0 1 0
(2.2.13)C4(z) =
It can be shown that matrices representing symmetry operations are real and orthogonal. Therefore, the transpose of matrix R gives the inverse matrix (R1 = transpose of R). Recall that the transpose of a matrix is constructed by interchanging corresponding elements across the diagonal of the matrix (rij Æ rji). 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 C4(z) leads is depicted below.
j
i •
i´
j´ •
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
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stationary point, from the reference frame of the coordinate system, as if the "stationary point" were rotated clockwise (the inverse transformation).
Consider the matrix representations for common symmetry operations. Counterclockwise rotation (Cn) 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:

i cos a + j sin a 



 
 i sin a + j cos a =
=
k´ = k
j´
i´=
==Cn(z)k
Cn(z)j =
Cn(z)i
i
i´ j
j́ ^
^
^
 
 
Therefore we can express Ĉn(z), where a = 360/n, as in eqn 2.2.14
C360/a(z) ^


 i
j
k

1
0
0
00
cos a
sin a
sin a
cos a

=
k
j
i
(2.2.14)
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.
z
y
xx
y
z
=
cos a
sin a
sin a
cos a
0 0
0
0
1
(2.2.15)Ĉ360/a(z)
For the corresponding operations Ŝn, where the Sn axis lies along z, reflection i