Research: Science and Education
Trigonometric Basis Set Functions: Their Application to the C–H Stretching and Deformation Motions of Benzene and to Orbital Symmetry G. Bor Swiss Federal Institute of Technology (ETH) Zurich, Switzerland Sidney F. A. Kettle* School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, UK;
[email protected] Two of the most important applications of group theory in undergraduate courses are to concepts of orbital symmetry and to the study of molecular vibrations. In the present treatment we concentrate on the latter but briefly discuss the former. It is well recognized that a relatively simple group theoretical treatment gives not only the symmetry species of the vibrational modes of a molecule but also their infrared and Raman activities. Much more difficult is showing the form of the corresponding symmetry coordinates themselves and yet, without them, the discussion can appear rather abstract. In fact, in most cases there is a simple way of obtaining such symmetry coordinates and it is the primary purpose of the present communication to draw attention to this. Basis Functions The method hinges on the presentation of the character tables used in the group theoretical analyses. Almost invariably, the basis functions given alongside individual irreducible representations are simple Cartesian functions. An example is provided by the C2v character table, shown as Table 1. These basis functions have the advantage that x, y, and z all indicate infrared activity, while x2, y2, z2, xy, yz, and zx all indicate Raman activity. However, they fail to explicitly reveal an important distinction between the irreducible representations, namely, that they differ in the nodal patterns inherent in each. In a recent communication, one of us has shown how, by a redefinition of the concept of stereographic projections, these nodal patterns may be represented very simply (1). Basically, the larger the group (the more symmetry operations), the greater the number of different nodal patterns (irreducible representations) spanned. Using the stereographic projection approach, it is easy to give a pictorial representation of an
Table 1. The Conventional C2 v Character Table Augmented by Trigonometric Basis Functions
C2 v E C 2
σv
σv ′ Translational or Rotational
Basis Function Algebraic
Trigonometric h cos 0θ (= h)
A1
1
1
1
1
Tz
z;z ;x ;y
A2
1
1
{1
{1
Rz
xy
2
2
2
h sin 2θ
B1
1 {1
1
{1
Ty ; Rx
y;zx
h cos θ
B2
1 {1
{1
1
Tx ; Ry
x;yz
h sin θ
irreducible representation for which a basis function is only provided by some high-order spherical harmonic, so that no simple algebraic basis function is available. In Figure 1 these nodal patterns are shown for the irreducible representations of a simple case, that of the C2v group. Although for many teaching purposes diagrams such as those in Figure 1 are entirely adequate, there can be occasions on which it is helpful to have corresponding basis functions. It is the purpose of the present article to present this extension and to demonstrate its utility. Given the simple way in which the nodal patterns of Figure 1 evolve with increase in the order of (number of operations in) a group, it is not surprising that there is also a simple way in which these nodal patterns can be included in the basis functions associated with the character table. This is because the nodal patterns are described by simple trigonometric functions: in the case of Figure 1 by cos 0 (A1), cos θ (B1), sin θ (B2), and cos 2θ (A2), where θ is an angle around the z axis (the molecular C2) measured from either of the mirror planes (which then becomes that labeled σv). These basis functions have been included in Table 1 as an additional column and where, for reasons that will become apparent, they have been multiplied by a horizontal (with respect to the twofold axis) vector h, which is directed radially from the center of the relevant figure (an example involving such a vector will be given shortly).1 Once these functions have been added to the character table, the generation of symmetry coordinates for the level of problem encountered by students becomes a simple matter. To demonstrate this we will consider a case which, although of fundamental importance, would almost certainly hitherto have been considered too difficult to be tackled by students. This is the problem of the vibrations of the benzene molecule, of which for the moment we consider only a problem related to the C–H stretching vibrations (2).
A1
Figure 1. Diagrammatic representations of the irreducible representations of the C2v point group. In order, these are pictorial representations of the trigonometric functions cos 0 (A1), cos θ (B1), sin θ (B2), and cos 2θ ( A 2 ), where θ is an angle around the z axis.
A2 +
+
+
–
+
+
–
+
–
+
–
–
–
+
+
+
B1
B2
NOTE: The angle θ is measured from the σv plane; h is defined in the note.1
JChemEd.chem.wisc.edu • Vol. 76 No. 12 December 1999 • Journal of Chemical Education
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Research: Science and Education Table 2. The C6 v Character Table with Trigonometric Basis Functions
C6 v
E
2 C6
A1
1
1
2 C3 C2 1
A2
1
1
B1
1
{1
B2
1
{1
E1
2
E2
2
3σv
3σv ′
1
1
1
Trigonometric Basis
1
1
{1
{1
h sin 6θ
1
{1
1
{1
h cos 3θ
1
{1
{1
1
h sin 3θ
1
{1
{2
0
0
h sin θ; h cos θ
{1
{1
2
0
0
h sin 2θ; h cos 2θ
h cos 0θ (= h)
NOTE: The angle θ is measured from a σv mirror plane; h is defined in the note.1 Note the relationship between the integers multiplying θ in the trigonometric functions and the 6 in C6v.
The C–H Vibrations of Benzene In the graphical presentation, of course, we should consider the C–H vibrations themselves; but, although the approach we adopt is simple, the diagrams then become somewhat complicated. We therefore consider a case that admits of simpler diagrams, that of a single six-membered ring of atoms. The legends to the figures discuss the relationships to the C–H case. In Table 2 is given the character table of the C6v point group, the simplest in which the problem can be tackled. As basis functions only the trigonometric basis functions are shown. It is important to recognize that these basis functions cover every irreducible representation; the unifying aspect of their use is evident. In the usual presentation of the C6v character table there is no function given for the A2 irreducible representation, and algebraic functions associated with two of the electronic f-orbitals have to be used to span the B1 (the function x[x2 – 3y 2]) and B2 (the function y[3x2 – y 2]). Table 2
Figure 2. The totally symmetric breathing mode of 6 atoms in a hexagon. The dotted circle represents the rest position of each atom; the arrows are an exaggerated indication of the extremity of its motion in one phase (the other phase would be represented by a reversal of the arrows). In benzene, the carbon and hydrogen atoms would move with opposite phases, the center of mass of each C–H unit remaining unmoved. It is then the A1 (A1g) C–H stretching vibration of the benzene molecule in C6v (D6h) symmetry. This behaves like h cos 0 = h; all C–H amplitudes are the same.
Figure 4. The (Raman-active) symmetric in-plane deformation mode of 6 atoms in a hexagon. The dotted circle represents the equilibrium position of each atom; the arrows are an exaggerated indication of the extremity of the motion in one phase (the other phase would be represented by a reversal of the arrows). In benzene, the carbon and hydrogen atoms would move with opposite phases. This then becomes a representative E2 (E2g) C–H stretching vibration of the benzene molecule in C6v (D6h) symmetry. The one shown here behaves like h cos 2θ. The (orthogonal) second component behaves like h sin 2θ.
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makes it evident that the greater the order of the group (i.e., the larger the number of operations in the group), the more values of n appear in the cos(n θ) and sin(n θ) functions (although n is always such that it divides exactly into the foldness of the highest rotational axis; here this is 6 in C6, and so n = 1, 2, 3, and 6). It is helpful to note that the value of n gives the number of nodal planes in a particular function and this reappears when diagrams are drawn, as will be seen in the figures in this paper. Throughout this paper we make the assumption of a knowledge of the simple group theory of vibrational problems. Thus, we assume that it has already been shown that the stretching vibrations of the C–H groups of the benzene molecule span the A1 + B1 + E1 + E2 irreducible representations of the C6v group. In Figures 2–5 we plot the functions given as trigonometric basis functions for these irreducible representations (we give only one for each degenerate pair). To do this we plot the appropriate basis function(s) in Table 2 on a circle passing through the carbon atoms of the benzene molecule. Adding the radial, horizontal, vectors h gives the form of one extreme of amplitude of the symmetry coordinate in a very easily visualized way. The reader may reasonably object that the point group of the benzene molecule is D6h, not C6v. We have followed what we believe to be good teaching practice in introducing a new topic stepwise and as simply as possible. C6v is perfectly adequate for the introduction of trigonometric basis functions; the use of D6h adds nothing beyond requiring that which we have tacitly assumed—that the C–H stretching motions take place in the plane of the benzene ring. Where the D6h group becomes essential is in the treatment of the outof-plane motions (we will consider the out-of-plane C–H
Figure 3. The (infrared-active) antisymmetric inplane deformation mode of 6 atoms in a hexagon. The upper dotted circle represents the equilibrium position of each atom; the arrows are an exaggerated indication of the extremity of the motion in one phase (the other phase would be represented by a reversal of the arrows). The motion shown involves a translation of the center of gravity of the hexagon and so would be a nongenuine vibration. However, in benzene the carbon and hydrogen atoms would move with opposite phases, the center of mass of the molecule and of each C–H unit remaining unmoved. For benzene, this is a representative E1 (E1u) C–H stretching vibration of the benzene molecule in C6v (D6h) symmetry. It behaves like h cos θ. The (orthogonal) second component behaves like h sin θ.
Figure 5. The (spectrally inactive) in-plane deformation mode of highest nodality of 6 atoms in a hexagon. The dotted circle represents the equilibrium position of each atom; the arrows are an exaggerated indication of the extremity of the motion in one phase (the other phase would be represented by a reversal of the arrows). In benzene, the carbon and hydrogen atoms would move with opposite phases. This is B1 (B1u) C–H stretching vibration of the benzene molecule in C6v (D6h) symmetry. It behaves like h cos 3θ. The function h sin 3θ has a value of zero at all atomic positions.
Journal of Chemical Education • Vol. 76 No. 12 December 1999 • JChemEd.chem.wisc.edu
Research: Science and Education Table 3. The D6 h Character Table with Trigonometric Basis Functions 2C6 2C3 C2 3C2 ′ 3C2 ′′
i
2S3 2S6 σh
3σd 3σd ′
Trigonometric Basis
D6 h
E
A1 g
1
1
1
1
1
1
1
1
1
1
1
1
A2 g
1
1
1
1
{1
{1
1
1
1
1
{1
{1
h sin 6θ
B1 g
1
{1
1
{1
1
{1
1
{1
1
{1
1
{1
v cos 3θ
B2 g
1
{1
1
{1
{1
1
1
{1
1
{1
{1
1
v sin 3θ
E1 g
2
1
{1
{2
0
0
2
1
{1
{2
0
0
v sin θ; v cos θ
E2 g
2
{1
{1
2
0
0
2
{1
{1
2
0
0
A1 u
1
1
1
1
1
1
{1
{1
{1
{1
{1
{1
A2 u
1
1
1
1
{1
{1
{1
{1
{1
{1
1
1
B1 u
1
{1
1
{1
1
{1
{1
1
{1
1
{1
1
h cos 3θ
B2 u
1
{1
1
{1
{1
1
{1
1
{1
1
1
{1
h sin 3θ
E1 u
2
1
{1
{2
0
0
{2
{1
1
2
0
0
h sin θ; h cos θ
E2 u
2
{1
{1
2
0
0
{2
1
1
{2
0
0
v sin 2θ; v cos 2θ
h cos 0θ (= h)
h sin 2θ; h cos 2θ v cos 0θ (= v) v sin 6θ
NOTE: In Tables 1 and 2, functions based on the v vectors of Table 3 could serve equally well as basis functions as those based on h; h and v are defined in the note.1
deformations). Again, for diagrammatic simplicity, the figures show the case of a hexagon of six atoms; again, the legends relate the diagrams to the benzene case. The D6h character table, along with trigonometric basis functions, is given in Table 3. In the normal presentation of this table, no fewer than seven irreducible representations are without basis functions; this number drops to four if f-orbital bases are included. In our presentation of this table every irreducible representation admits of a simple basis function. The basis functions alongside the irreducible representations that are symmetric with respect to reflection in the σh mirror plane (i.e. have the same characters under this operation as under the identity operation E ) are just those we have already met in Table 2. Those for the irreducible representations
that are antisymmetric with respect to reflection in the σh mirror plane (i.e., have the characters under this operation as under the identity operation E but with a change of sign) are similar—in a 1:1 correspondence, in fact—but with the horizontal vector h replaced by a vertical (parallel to the sixfold axis) vector v. The essential difference between h and v is that the former is symmetric and the latter antisymmetric with respect to reflection in the horizontal mirror plane σh. By definition, the out-of-plane C–H deformations are all antisymmetric with respect to reflection in σh. They are A1u + B1g + E1g + E2u (compare with those of the C–H stretching vibrations in C6v and D6h). Just as for the C–H stretching vibrations, in Figures 6–9 we give pictures of the out-of-plane C–H deformations that transform as these irreducible repre-
Figure 6. The (infrared-active) deformation mode of 6 atoms in a hexagon. The circle represents the rest position of each atom; the arrows are an exaggerated indication of the extremity of its motion in one phase (the other phase would be represented by a reversal of the arrows). The motion shown involves a translation of the center of gravity of the hexagon. However, in benzene the carbon and hydrogen atoms would move with opposite phases, the center of mass of the molecule and of each C–H unit remaining unmoved. For this molecule the diagram represents the A1 (A1u) C–H in-phase outof-plane deformation vibration of the benzene molecule in C6v (D6h) symmetry. This behaves like v cos 0 = v.
Figure 7. The (Raman-active) out-of-plane deformation mode of 6 atoms in a hexagon. The circle represents the equilibrium position of each atom; the arrows are an exaggerated indication of the extremity of the motion in one phase (the other phase would be represented by a reversal of the arrows). The case shown is actually a nongenuine vibration, since the motion involves a rotation of the 6-membered ring. In benzene the carbon and hydrogen atoms would move with opposite phases and there would be no resultant rotation. It is then a representative E1 (E1g) C–H out-of-plane deformation vibration of the benzene molecule in C6v (D6h) symmetry. The one shown behaves like v cos θ. Its degenerate partner behaves like v sin θ.
Figure 8. The (spectrally inactive) out-ofplane deformation mode of 6 atoms in a hexagon. The circle represents the equilibrium position of each atom; the arrows are an exaggerated indication of the extremity of the motion in one phase (the other phase would be represented by a reversal of the arrows). In benzene the carbon and hydrogen atoms would move with opposite phases. This is then a representative E2 (E2u) C–H out-of-plane deformation vibration of the benzene molecule in C6v (D6h) symmetry. It behaves like v cos 2θ. Its degenerate partner behaves like v sin 2θ.
Figure 9. The (spectrally inactive) out-of-plane deformation mode of highest nodality of 6 atoms in a hexagon. The circle represents the equilibrium position of each atom; the arrows are an exaggerated indication of the extremity of the motion in one phase (the other phase would be represented by a reversal of the arrows). In benzene the carbon and hydrogen atoms would move with opposite phases. This then becomes the B1 (B1g) C–H out-of-plane deformation vibration of the benzene molecule in C6v (D6h) symmetry. This behaves like v cos 3θ. The function v sin 3θ has a value of zero at all atomic positions.
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Research: Science and Education
sentations, with just one example of each of the degenerate functions. The form of basis functions that have been the subject of this paper can be applied to all axial groups (i.e., groups that have one principal axis, any other axes being perpendicular to it). This excludes cubic and icosahedral groups. The general approach can be extended to these groups also, but it becomes necessary to add a second angular variable. This makes the extension unpalatable to most students. For such groups it is probably simplest either to use them to introduce the formal projection operator method or, if simplicity is held to be important, to use an ascent-in-symmetry analysis (3). Finally, we return to the other important application of simple group theory used by undergraduates: that of orbital symmetry. Thus, pictures of the pπ orbitals of the benzene molecule find a place in many texts, organic, theoretical, spectroscopic, and inorganic. The group theoretical derivation of the form of these orbitals is less commonly encountered. However, it is evident that the approach used in this paper can immediately be applied to generate them, the v basis set is appropriate to the generation of symmetry-adapted π func-
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tions, and the h basis to the generation of σ-type functions (such as the C–H bonding orbitals). Note 1. We use the symbols h (horizontal) and v (vertical) in this communication because these are the symbols used in the character tables. Some would prefer to use the symbols r (radial) and t (tangential) in their place.
Literature Cited 1. Kettle, S. F. A. J. Chem. Educ. 1999, 76, 675. 2. An approach very similar to that adopted in the present work is to be found in Bor, G; Holly, S. Az infravörös spektroszkópia alapjai (Fundamentals of Infrared Spectroscopy); Tankönyvkiadó: Budapest, 1965. The formal group theoretical treatment of a problem analogous to the present is to be found in Eyring, H.; Walter, J.; Kimball, G. E. Quantum Chemistry; Wiley: New York, 1944; and in Cotton, F. A. Chemical Applications of Group Theory; Wiley: New York, 1990. 3. For a relevant example, see Kettle, S. F. A. J. Chem. Educ. 1966, 43, 652.
Journal of Chemical Education • Vol. 76 No. 12 December 1999 • JChemEd.chem.wisc.edu
