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Symmetry

Symmetry. Which of these two compounds is more symmetrical? Symmetry is a concept familiar to us. Can quantitate symmetry. A symmetry operation is the act of physically doing something to an object so that the result is indistinguishable from the initial state.

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Symmetry

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  1. Symmetry • Which of these two compounds is more symmetrical? • Symmetry is a concept familiar to us. • Can quantitate symmetry. • A symmetry operation is the act of physically doing something to an object so that the result is indistinguishable from the initial state. • Even if we do nothing to the object, it still possesses a symmetry element, a geometrical property which generates the operation. • There are 5 symmetry elements: E, Cn, σ, I, and Sn. H COOH C=C HOOC H H Cl C=C H Br

  2. Symmetry Elements • E = identity; no change in the object (needed for mathematical completeness). • Cn = proper rotation axis; n = order of rotation, 360/n • for objects with more than one Cn axis, the one with the largest value of n = principle axis. If different axes with same order, designate as C2, C2’, C2’’, etc. 1 3 2 3 2 1 2 1 3 Each indistinguishable C3 C3 2 ● ● ● 3 = E C3 Note that there are 3 C2 axes. 2 1 3 2 1 3 1 2 3 C2

  3. Symmetry Elements • σ = reflection (aka a mirror plane) • exchanges every point on one side of the mirror with every point on the other side. • nomenclature: σv = plan includes principle axis (v = vertical) • σh = plane is perpendicular to principle axis (h = horizontal) • σd = plane in between C2 axes perpendicular to principle axis (d = dihedral) • all σ2 = E, so usually not considered.

  4. Symmetry Elements • i = inversion (about a center). • each point moves through the center of the object to an identical position opposite the original. • cannot be physically carried out using models. • again, i2 = E • Sn = improper rotation (n = order of axis). • rotation by 1/n turn, followed by reflection in a plane perpendicular to the axis. • neither rotation nor reflection need be present on their own.

  5. Symmetry Elements • Example. What symmetry elements are present in the following molecules? O H H N H H C═C H H Instead of having to write out all symmetry elements each time, use a short hand symbolism for the sets; these setsare called Point Groups. H H C═C HOOC COOH

  6. Determining Point Groups • Find the highest order rotation axis = n. • Are there n C2 axes perpendicular to this principle axis? • Is there a mirror plane perpendicular to the principle axis? • Are there dihedral mirror planes? 4. Are there vertical mirror planes? yes no Dn set Cn set yes no Dnh yes no Cnh yes no Cnv yes no Dnd Dn 5. Is there a S2n? yes no S2n Cn

  7. Determining Point Groups. Examples. • Determine the Point Group for each of the following molecules: Assume eclipsed configuration. 3 4 1 2

  8. Special Cases of Low and High Symmetry High Symmetry Low Symmetry C∞v : linear, with no perpendicualr C2 axes. C1 : no symmetry other than E. D∞h : linear, with perpendicualr C2 axes and σh. Cs: only one mirror plane. Td : pure tetrahedral symmetry. NOT just geometry. Ci: only one inversion center. Oh : pure octahedral symmetry. NOT just geometry.

  9. Character Tables. • The collection of symmetry elements represented by a given Point Group is listed as part of the Character Table; see Appendix C. • To understand the construction and use of Character Tables, need to look at matrices, which are the basis for the tables.

  10. Representations. • Quantitate the description of molecular symmetry by using numbers to represent symmetry operation; these numbers are called Representations. • The C2v Point Group consists of the following elements: E C2σ(xz) σ(yz) • What is the effect of C2 rotation on a px orbital? • What single number might represent this transformation? C2 px = -1px How about σ(xz)? σ(xz) px = +1px And σ(yz)? σ(yz) px = -1px And E? E px = +1px • We say that x belongs to the B1 representation of C2v because this set of numbers represents the effect of the group operations on a px orbital. z z — - + → x C2 ← + - — C2v E C2σ(xz) σ(yz) B1 1 -1 1 -1 x

  11. Representations • What set of numbers represent the effect of the operations on a py? • How about a pz? • The full set of representations is included in the Character Table of the group (see Appendix C). • The numbers in this table formally called The Characters of the Irreducible Representations. [NOT irreproducible!] C2v E C2σ(xz) σ(yz) B2 1 -1 -1 1 y C2v E C2σ(xz) σ(yz) A1 1 1 1 1 z s orbital is totally symmetric and always belongs to the A1 representation. d orbitals. rotations about an axis.

  12. Representation “Problem” • What are the representations of the z-axis in C4v? • What happens to an arrow along the y-axis when a C4 operation (clockwise) is performed? • it is converted into the x-axis (note also x → -y) • So, what simple number represents this transformation? • Can NOT be represented by a simple number, but by a matrix. C4v E 2C4 C2 4σv A1 1 1 1 1 z

  13. Matrices [ ] ( ) { } [ ] 1 -5 8 1 0 0 1 3 4 2 7 0 -1 6 9 1 6 3 9 10 17 2 • A matrix is an array of numbers enclosed within brackets. • In symmetry we are most interested in multiplying matrices. In order to do this, two matrices must be conformable. That is, the number of columns in the first is equal to the number of rows in the second. • The product of any two matrices is found by “RC”; an element in the rth row and cth column of the product is formed by multiplying together the elements of the rth row of matrix 1 by the cth column of matrix 2. A = (a * r) + (b * u) + (c * x) D = (d * r) + (e * u) + (f * x) B = (a * s) + (b * v) + (c * y) E = (d * s) + (e * v) + (f * y) C = (a * t) + ( b * w) + (c * z) F = (d * t) + ( e * w) + (f * z) ( ) r s t u v w = x y z ( ) ( ) a b c A B C d e f D E F

  14. Matrices • Evaluate the following matrices: • Note that matrix multiplication is non-communative. • Evaluate the following: ( ) ( ) 1 2 1 1 3 4 2 2 ( ) = = ( ) ( ) ( ) 1 1 1 2 2 2 3 4 [ ] [ ] [ ] 0 1 0 x -1 0 0 y 0 0 0 z =

  15. Back to the Representation • What effect does a clockwise C4 rotation have on the x and y axes? new x = old y new y = -old x • What is effect of σyz on x and y-axes? new x = -old x new y = old y • We will make use of one other result which comes from matrix algebra: the character of the matrix, which is simply the sum of the diagonal elements. • Notice that the character expresses the extent to which x is converted to itself and y is converted to itself in the original equations. • Remember Character Tables? The numbers in these tables are the characters of the matrixes which represent the group operations! ( ) ( ) expressed as a matrix ( ) 0 1 x y -1 0 y -x = ( ) ( ) -1 0 x -x 0 1 y y ( ) = ( ) ( ) 0 1 -1 0 -1 0 0 1 character = 0 character = 0

  16. Reducible Representations. • Is the set of numbers, 3 3 1 1, an irreducible representation of C2v? • No, but they are a reducible representation of C2v; they can be obtained by adding 2A1 + A2. • How can the reducible representation, 3 -1 -1 -1, be obtained? • Much of the use of Group Theory to solve real problems involves generating a reducible representation and then reducing it to its constituent irreducible representations. A1 1 1 1 1 A1 1 1 1 1 A2 1 1 -1 -1 2A1 + A2 3 3 1 1 the reducible representation can be reduced to its component irreducible representation, A2 +B1 + B2 A2 1 1 -1 -1 B1 1 -1 1 -1 B2 1 -1 -1 1 A2 +B1 + B2 3 -1 -1 -1

  17. Reducible Representations. • The number of times an irreducible representation occurs in the reducible representation is given by a reduction formula: (1/h) ∑ΧR * ΧI * N number of symmetry operations in the class order of the group (number of operations) character of the reducible representation character of the irreducible representation h = 1 + 2 + 3 = 6 Reduce the following in C3v: Г = 4 1 -2 C3v 1E 2C33σv A11 1 1 A21 1 -1 E 2 -1 0 #A1 = (1/6)[ (4 * 1 * 1) + (1 * 1 * 2) + (-2 * 1 * 3) ] = 0 #A2 = (1/6)[ (4 * 1 * 1) + (1 * 1 * 2) + (-2 * -1 * 3) ] = 2 #E = (1/6)[ (4 * 2 * 1) + (1 * -1 * 2) + (-2 * 0 * 3) ] = 1 So, Г = 2A2 + E

  18. Group Theory: Example • The use of Group Theory can be summarized in 3 rules: • Use an appropriate basis to generate a reducible representation. • Reduce it to an irreducible representation. • Interpret the results. • Example: What orbitals can be hybridized to produce a set of three trigonal planar σ-bonds in BF3? • What is the character of each matrix representing the 6 operations in D3h? (i.e. which of the 3 vectors are transferred to themselves or left unshifted by the operation?) a1 a2 a3 D3h point group 3 vectors as the basis D3h E 2C3 3C2σh 2S3 3σv Г

  19. Group Theory: Example D3h E 2C3 3C2σh 2S3 3σv Г 3 0 1 3 0 1 • This is a reducible representation, so use character table to reduce it. • We will use symmetry and group theory to construct Molecular Orbitals, for example. #A1’ = (1/12) [ 3 + 0 + 3 + 3 + 0 + 3] = 1 #A2’ = (1/12) [ 3 + 0 - 3 + 3 + 0 - 3] = 0 … #E’ = (1/12) [ 6 + 0 + 0 + 6 + 0 + 0] = 1 Г = A1’ + E’ What orbitals belong to these symmetry species? A1’ = s-orbital E’ = 2 p-orbitals Therefore, it is an sp2 hybrid orbital

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