# CHAPTER 12 MOLECULAR SYMMETRY

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1 CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical transformation which turns a molecule into an indistinguishable copy of itself is called a symmetry operation. A symmetry operation can consist of a rotation about an axis, a reflection in a plane, an inversion through a point, or some combination of these. The Ammonia Molecule We shall introduce the concepts of symmetry and group theory by considering a concrete example the ammonia molecule NH 3. In any symmetry operation on NH 3, the nitrogen atom remains fixed but the hydrogen atoms can be permuted in 3!=6 different ways. The axis of the molecule is called a C 3 axis, since the molecule can be rotated about it into 3 equivalent orientations, 120 apart. More generally, a C n axis has n equivalent orientations, separated by 2π/n radians. The axis of highest symmetry in a molecule is called the principal axis. Three mirror planes, designated σ 1, σ 2, σ 3, run through the principal axis in ammonia. These are designated as σ v or vertical planes of symmetry. Ammonia belongs to the symmetry group designated C 3v, characterized by a three-fold axis with three vertical planes of symmetry. Let us designate the orientation of the three hydrogen atoms in Fig. 1 as {1, 2, 3}, reading in clockwise order from the bottom. A counterclockwise rotation by 120, designated 1

2 H 2 H 3 C 3 H 1 Figure 1. Two views of the ammonia molecule. by the operator C 3, produces the orientation {2, 3, 1}. A second counterclockwise rotation, designated C3, 2 produces gives {3, 1, 2}. Note that two successive counterclockwise rotations by 120 is equivalent to one clockwise rotation by 120, so the last operation could also be designated C3 1. The three reflection operations σ 1, σ 2, σ 3 applied to the original configuration {1,2, 3} produces {1,3,2}, {3,2,1} and {2,1,3}, respectively. Finally, we must include the identity operation, designated E, which leaves an orientation unchanged. The effects of the six possible operations of the symmetry group C 3v can be summarized as follows: E {1, 2, 3} = {1, 2, 3} C 3 {1, 2, 3} = {2,3,1} C 2 3 {1, 2, 3} = {3, 1, 2} σ 1 {1,2,3} = {1,3,2} σ 2 {1,2,3} = {3,2, 1} σ 3 {1, 2, 3} = {2, 1, 3} We have thus accounted for all 6 posssible permutations of the three hydrogen atoms. 2

3 The successive application of two symmetry operations is equivalent to some single symmetry operation. For example, applying C 3, then σ 1 to our starting orientation, we have σ 1 C 3 {1,2,3} = σ 1 {2,3,1} = {2,1,3} But this is eqivalent to the single operation σ 3. This can be represented as an algebraic relation among symmetry operators σ 1 C 3 = σ 3 Note that successive operations are applied in the order right to left when represented algebraically. For the same two operations in reversed order, we find C 3 σ 1 {1, 2, 3} = C 3 {1, 3, 2} = {3, 2, 1} = σ 2 {1, 2, 3} Thus symmetry operations do not, in general commute A B B A (1) although they may commute, for example, C 3 and C 2 3. The algebra of the group C 3v can be summarized by the following multiplication table. 1 st E C 3 C3 2 σ 1 σ 2 σ 3 2 nd E E C 3 C3 2 σ 1 σ 2 σ 3 C 3 C 3 C3 2 E σ 3 σ 1 σ 2 C3 2 C3 2 E C 3 σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 E C 3 C3 2 σ 2 σ 2 σ 3 σ 1 C3 2 E C 3 σ 3 σ 3 σ 1 σ 2 C 3 C3 2 E Notice that each operation occurs once and only once in each row and each column. 3

4 Group Theory In mathematics, a group is defined as a set of g elements G {G 1, G 2... G h } together with a rule for combination of elements, which we usually refer to as a product. The elements must fulfill the following four conditions. (i) The product of any two elements of the group is another element of the group. That is G i G j = G k with G k G (ii) Group multiplication obeys an associative law, G i (G j G k ) = (G i G j )G k G i G j G k. (iii) There exists an identity element E such that EG i = G i E = G i for all i. (iv) Every element G i G i G 1 i = G 1 i has a unique inverse G 1 i G i = E with G 1 i G., such that The number of elements h is called the order of the group. Thus C 3v is a group of order 6. a set of quantities which obeys the group multiplication table is called a representation of the group. Because of the possible noncommutativity of group elements [cf. Eq (1)], simple numbers are not always adequate to represent groups; we must often use matrices. The group C 3v has three irreducible representations, or IR s, which cannot be broken down into simpler representations. A trivial, but nonetheless important, representation of any group is the totally symmetric representation, in which each group element is represented by 1. The multiplication table then simply reiterates that 1 1 = 1. For C 3v this is called the A 1 representation: A 1 : E = 1, C 3 = 1, C 2 3 = 1, σ 1 = 1, σ 2 = 1, σ 3 = 1 (2) A slightly less trivial representation is A 2 : A 2 : E = 1, C 3 = 1, C 2 3 = 1, σ 1 = 1, σ 2 = 1, σ 3 = 1 (3) 4

5 Much more exciting is the E representation, which requires 2 2 matrices : E = ( ) C 3 = ( ) C3 2 1/2 3/2 = 3/2 1/2 ( ) 1/2 3/2 σ 2 = 3/2 1/2 ( ) 1/2 3/2 3/2 1/2 ( ) 1 0 σ 1 = 0 1 ( ) 1/2 3/2 σ 3 = 3/2 1/2 The operations C 3 and C 2 3 are said to belong to the same class since they perform the same geometric function, but for different orientations in space. Analogously, σ 1, σ 2 and σ 3 are obviously in the same class. E is in a class by itself. The class structure of the group is designated by {E,2C 3, 3σ v }. We state without proof that the number of irreducible representations of a group is equal to the number of classes. Another important theorem states that the sum of the squares of the dimensionalities of the irreducible representations of a group adds up to the order of the group. Thus, for C 3v, we find = 6. The trace or character of a matrix is defined as the sum of the elements along the main diagonal: (4) χ(m) k M kk (5) For many purposes, it suffices to know just the characters of a matrix representation of a group, rather than the complete 5

6 matrices. For example, the characters for the E representation of C 3v in Eq (4) are given by χ(e) = 2, χ(c 3 ) = 1, χ(c 2 3) = 1, χ(σ 1 ) = 0, χ(σ 2 ) = 0, χ( σ 3 ) = 0 (6) It is true in general that the characters for all operations in the same class are equal. Thus Eq (6) can be abbreviated to χ(e) = 2, χ(c 3 ) = 1, χ(σ v ) = 0 (7) For one-dimensional representations, such as A 1 and A 2, the characters are equal to the matrices themselves, so Eqs (2) and (3) can be read as a table of characters. The essential information about a symmetry group is summarized in its character table. We display here the character table for C 3v C 3v E 2C 3 3σ v A z z 2, x 2 + y 2 A E (x, y) (xy, x 2 y 2 ), (xz, yz) The last two columns show how the cartesian coordinates x, y, z and their products transform under the operations of the group. Group Theory and Quantum Mechanics When a molecule has the symmetry of a group G, this means that each member of the group commutes with the molecular Hamiltonian [Ĝ i, Ĥ] = 0 i = 1... h (8) 6

7 where we now explicitly designate the group elements G i as operators on wavefunctions. As was shown in Chap. 4, commuting operators can have simultaneous eigenfunctions. A representation of the group of dimension d means that there must exist a set of d degenerate eigenfunctions of Ĥ that transform among themselves in accord with the corresponding matrix representation. For example, if the eigenvalue E n is d-fold degenerate, the commutation conditions (2) imply that, for i = 1... h, Ĝ i Ĥ ψ nk = Ĥ Ĝi ψ nk = E n Ĝ i ψ nk for k = 1... d (9) Thus each Ĝi ψ nk is also an eigenfunction of Ĥ with the same eigenvalue E n, and must therefore be represented as a linear combination of the eigenfunctions ψ nk. More precisely, the eigenfunctions transform among themselves according to Ĝ i ψ nk = d m=1 D(G i ) km ψ nm (10) where D(G i ) km means the {k, m} element of the matrix representing the operator Ĝ i. The character of the identity operation E immediately shows the degeneracy of the eigenvalues of that symmetry. The C 3v character table reveals that NH 3, and other molecules of the same symmetry, can have only nondegenerate and two-fold degenerate energy levels. The following notation for symmetry species was introduced by Mulliken: (i) One dimensional representations are designated either A or B. Those symmetric wrt rotation by 2π/n about the C n 7

8 principal axis are labelled A, while those antisymmetric are labelled B. (ii) Two dimensional representations are designated E; 3, 4 and 5 dimensional representations are designated T, F and G, respectively. These latter cases occur only in groups of high symmetry: cubic, octahedral and icosohedral. (iii) In groups with a center of inversion, the subscripts g and u indicate even and odd parity, respectively. (iv) Subscripts 1 and 2 indicate symmetry and antisymmetry, respectively, wrt a C 2 axis perpendicular to C n, or to a σ v plane. (v) Primes and double primes indicate symmetry and antisymmetry to a σ h plane. For individual orbitals, the lower case analogs of the symmetry designations are used. For example, MO s in ammonia are classified a 1, a 2 or e. For ammonia and other C 3v molecules, there exist three species of eigenfunctions. Those belonging to the classification A 1 are transformed into themselves by all symmetry operations of the group. The 1s, 2s and 2p z AO s on nitrogen are in this category. The z-axis is taken as the 3-fold axis. There are no low-lying orbitals belonging to A 2. The nitrogen 2p x and 2p y AO s form a two-dimensional representation of the group C 3v. That is to say, any of the six operations of the group transforms either one of these AO s into a linear combination of the two, with coefficients given by the matrices (4). The three hydrogen 1s orbitals transform like a 3 3 representation of the group. If we represent the hydrogens by a column vector {H 1, H 2, H 3 }, 8

9 then the six group operations generate the following algebra E = C 3 = C3 2 = σ 1 = σ 2 = σ 3 = (11) Let us denote this representation by Γ. It can be shown that Γ is a reducible representation, meaning that by some unitary transformation the 3 3 matrices can be factorized into blockdiagonal form with 2 2 plus 1 1 submatrices. The reducibility of Γ can be deduced from the character table. The characters of the matrices (11) are Γ : χ(e) = 3, χ(c 3 ) = 0, χ(σ v ) = 1 (12) The character of each of these permutation operations is equal to the number of H atoms left untouched: 3 for the identity, 1 for a reflection and 0 for a rotation. The characters of Γ are seen to equal the sum of the characters of A 1 plus E. This reducibility relation is expressed by writing Γ = A 1 E (13) The three H atom 1s functions can be combined into LCAO functions which transform according to the IR s of the group. Clearly the sum ψ = ψ 1s(1) + ψ 1s(2) + ψ 1s(3) (14) 9

10 ψ transforms like A 1. The two remaining linear combinations which transform like E must be orthogonal to (14) and to one another. One possible choice is = ψ 1s(2) ψ 1s(3), ψ = 2ψ 1s(1) ψ 1s(2) ψ 1s(3) (15) Now (14) can be combined with the N 1s, 2s and 2p z to form MO s of A 1 symmetry, while (15) can be combined with the N 2p x and 2p y to form MO s of E symmetry. Note that no hybridization of AO s is predetermined, it emerges automatically in the results of computation. 10

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