Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate system wherein the z-axis is coincident with the C 4 axis and the x- and y-axes lie perpendicular to the C 4 axis and coincident with the C 2 axes. (a) Construct matrices for each symmetry operation (i.e., construct a representation) using three p orbitals that sit on the origin as a basis: 1 p x : ; p : 1 y ; p : z 1 (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1 d z 2 : 1 ; d xz : ; d yz : 1 ; d x 2 y 2 : 1 ; d xy : 1 (c) By examining the representations you constructed in parts a and b, confirm that properties that divide the symmetry operators into classes are obeyed. Also, discuss the general form of any subblock structure you can identify. (2) To account for the nonlinear and anisotropic nature of the Zeeman splitting induced in a paramagnetic molecule by an applied magnetic field, H, an effective Hamiltonian is defined that includes a so-called g-tensor: H eff = µ B H g S = µ B H x H y H z g xx g xy g xz g yx g yy g yz g zx g zy g zz For a molecule with one unpaired electron, for example, this expression accounts for dependence of the splitting of the spin-up and spin-down spin states on the direction of the applied magnetic field with respect to the molecular coordinate system. (Anisotropy occurs because the electron may have some orbital magnetic moment that adds to or subtracts from the spin magnetic moment and the orbital magnetism depends on the shape of the molecule.) Were there no anisotropy in the g-tensor, the matrix would be diagonal, with g xx = g yy = g zz = g e = 2.23 and the Zeeman splitting would be independent of the applied field direction. Similarly, an induced electric dipole moment, µ also depends on the angle that the applied electric field, E, makes with the molecular axes. This is quantified in the molecule s polarizability tensor, α: µ elec (induced) = αe or µ x µ y µ z α xx α xy α xz = α yx α yy α yz α zx α zy α zz E x E y E z S x S y S z
Because the electronic distribution of the bonding electrons in a molecule is not isotropic, the induced dipole does is not generally aligned with the direction of the applied field. Both g and α are examples of second-rank tensors and their forms must conform to certain symmetry constraints. Let s use the label A to represent a general second-rank tensor; A is a 3 3 matrix. If we apply a symmetry operation to a molecule, we move it to a physically indistinguishable position, and the form of A must be unchanged. Therefore, the 3 3 matrix representing a symmetry operation,, must commute with A: A = A A = 1 A. Using this expression, we can find the symmetry constraints on the form of A for an O h - symmetry molecule. Let s start with a C 2 rotation along the z-axis: A = C 1 2 (z)ac 2 (z) A xx A xy A xz 1 A xx A xy A xz 1 A yx A yz = 1 A yx A yz 1 A zx A zy A zz 1 A zx A zy A zz 1 1 A xx A xy A xz A xx A xy A xz = 1 A yx A yz = A yx A yz 1 A zx A zy A zz A zx A zy A zz Comparing the first and last forms enables us to conclude A zx = A zy = A xz = A yz =. Applying the same procedure with the C 2 (x) or C 2 (y) rotations, we conclude that A yx = A xy =, too. So far then, we know A xx A = A zz Finally, we can use one of the C 3 rotations: A = C 1 3 (xyz)ac 3 (xyz) A xx 1 A xx 1 1 A xx = 1 1 = 1 A zz 1 A zz 1 1 A zz A xx 1 = A zz A xx = = A zz ( A) A = A xx A 1 A xx A xx 1 In O h symmetry then, all second rank tensors (like g-tensors and polarizabilities) are isotropic (diagonal with identical elements). Finally, your problem: find the general form of second-rank tensors for molecules that are (a) D 4, (b) D 5h, and (c) C 3h (use a conventional coordinate system in which the principal axis is along the z-direction).
(3) Class Operators. For any point group, we can define operators called class operators that are formed by summing point group operators within each class and dividing by the number of operators in the class. For example, for the D 4 group we have five class operators (for example, draw the symmetry elements for La(acac) 4, in which the oxygen atoms form an approximate square antiprism): Ω E = E ; Ω 4 = 1 2 (C 4 + C 4 3 ) ; Ω 2 = C 2 (= C 42 ) Ω 2 = 1 ( C + C 2 x ) y ; Ω = 1 ( C 2 + C 2 x= y x= ) y (Note that the number of operators in the class divides each sum. Note also that the set of class operators does not constitute a group.) (a) The elements of a group in a given class are the set {X 1,, X n }, the corresponding class operator is Ω N = 1 / n {X 1 + + X n }. First, show that if Y is a member of the group, Y 1 Ω N Y = Ω N. (Hint: Y 1 Ω N Y will involve a series of n terms, each of the form Y 1 X i Y; is it possible that any of the terms are the same?) (b) Use the result to (a) to show that all the class operators must commute with each other. (Hint: You can use the result of (a) to show that any element Y must commute with any class operator the rest is easy.) (a) Using the multiplication properties of the point group operations, construct a multiplication table based on class operators for D 4 (i.e., Fill in the table. A multiplication table for D 4 will be handy!): D 4 Ω Ε Ω 4 Ω 2 Ω 2 Ω 2 Ω Ε Ω 4 Note that since the class operators commute, the table should be symmetric about the diagonal. Ω 2 If a set of operators mutually Ω 2 commutes, then they share a common set of Ω 2 eigenfunctions. In this D 4 case, let us consider Φ to be an arbitrary eigenfunction of Ω E, Ω 4, Ω 2, Ω 2, and Ω 2 with eigenvalues λ E, λ 4, λ 2, λ 2, and λ 2, respectively. First, since Ω E = E is just the identity operator, it is clear that λ E = 1 for any eigenfunction Φ. Symbolically, Ω E Φ = 1 Φ; Ω 4 Φ = λ 4 Φ; Ω 2 Φ = λ 2 Φ; Ω 2 Φ = λ 2 Φ; Ω 2 Φ = λ 2 Φ (c) Using the multiplication table and the fact that all the class operators mutually commute, you can establish all the possible sets of eigenvalues for the class operators. elationships between the λ' s follow from the relationships between the Ω s. It is easy to check that one acceptable set of eigenvalues is λ E = λ 4 = λ 2 = λ 2 = λ 2 = 1. The other sets of possible eigenvalues can be worked out using the multiplication table.
[A Worked Example: If one has a set of mutually commuting operators, then any algebraic relation that is true of the operators must also be true of the eigenvalues of the operators. For example, suppose that operators A, B, and C all mutually commute. Further suppose that we know the following multiplication table: A B C A A B C B B C C C 1 (A + B) C 2 1 (A + 2B) 3 Let us consider Φ to be an arbitrary eigenfunction of A, B, and C with eigenvalues λ A, λ B, and λ C, respectively. Inspection of the multiplication table shows that A is just the identity operator, therefore it is clear that λ A = 1, always. Symbolically, we can write AΦ = 1 Φ BΦ = λ B Φ CΦ = λ C Φ Using the multiplication table, we can write, which has the solutions λ B = ½, 1. Also, B 2 Φ = 1 (A + B)Φ λ 2 = 1 (1+ λ ) 2 B 2 B C 2 Φ = 1 (A + 2B)Φ λ 2 = 1 (1+ 2λ ) 3 C 3 B (emember, the equations involving the eigenvalues are true only because A, B, and C mutually commute.) Solving these two simultaneous equations, we conclude that the possible eigenvalues are λ B = 1 and λ C = ±1 or λ B = ½ and λ C = To summarize, we can construct a table with three distinct possible sets of eigenvalues (labeled Γ 1, Γ 2, Γ 3 ): A B C Γ 1 1 1 1 Γ 2 1 1 1 Γ 3 1 ½ Your task is to repeat the above procedure for the D 4 class operators.]
If you have worked out all the possible sets of eigenvalues, you should get three more solutions that do not contradict the multiplication table you initially constructed. Write down a table with four distinct possible sets of eigenvalues (labeled Γ 1, Γ 2, Γ 3, Γ 4, and Γ 5 ): C 5v λ E λ 4 λ 2 λ 2 λ 2 Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 (d) Take the z-axis to be the C 4 axis. Show that each of the following orbitals, centered at the origin is an eigenfunction of the class operators and assign each to one of these 5 sets: p x, p y, p z orbitals; d x 2-y2, d z 2, d xy, d xz, d yz orbitals. (e) You should be able to see obvious similarities between the eigenvalue table and character tables for the D 4 group (and this would work for any group you wanted to work through). Exploiting these similarities and using facts from lecture and/or outside reading, give an informative, coherent discussion of the implications of this problem. Your discussion should touch upon at least the following: (i) the relationships between the number of commuting operators, the number of irreducible representations, and the number of class operators; (ii) the connection between quantum numbers and symmetry; (iii) the relationship between symmetry operators and the Hamiltonian operator. (4) (a) Derive complete matrices for all the irreducible representations of the D 3h point group. Construct a table in which each of the symmetry operations is listed across the top and each of the representations forms a separate row (your table should look like a character table, except that matrices appear instead of just the characters). Hint: The matrices for one-dimensional I s are trivial. For higher dimensional representations it is easier to choose a convenient set of functions that must form a basis for the irreducible representation; various atomic orbitals located on the origin can sometimes be helpful see the character table. (b) Use your table to give one example of each of the properties discussed in lecture: (i) Vectors formed from matrix elements from the m th rows and n th columns of different irreducible representations are orthogonal (ii) [Γ i () mn ][Γ j () mn ] = if i j Such vectors formed from different row-column sets of the same irreducible representation are orthogonal and have magnitude h/l i :
(iii) (iv) (v) (vi) (vii) i l i 2 = l 1 2 + l 2 2 + l 3 2 + = h The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group, that is, [Γ i () mn ][Γ i () m n ] = (h li )δ m m δ n n The sum of the squares of the characters in any irreducible representation equals h, the order of the group: [χ i ()] 2 = The vectors whose components are the characters of two different irreducible representations are orthogonal, that is, χ i ()χ j () = when i j Matrices in the same class have equal characters The number of irreducible representations of a group is equal to the number of classes in the group. h