Lecture 4

Symmetry and group theory 2

Natural symmetry in plants 3

Symmetry in animals 4

Symmetry in the human body 5

The platonic solids 6

Symmetry in modern art M. C. Escher 7

Symmetry in arab architecture La Alhambra, Granada (Spain) 8

Symmetry in baroque art Gianlorenzo Bernini Saint Peter’s Church Rome 9

Native American crafts Symmetry in Native American crafts 10

7th grade art project Silver Star School Vernon, Canada 11

Re2(CO)10 12

C60 C2F4 13

Symmetry in chemistry Molecular structures Wave functions Description of orbitals and bonds Reaction pathways Optical activity Spectral interpretation (electronic, IR, NMR) ... 14

Molecular structures A molecule is said to have symmetry if some parts of it may be interchanged by others without altering the identity or the orientation of the molecule 15

Transformation of an object into an equivalent or indistinguishable Symmetry Operation: Transformation of an object into an equivalent or indistinguishable orientation C3, 120º Symmetry Elements: A point, line or plane about which a symmetry operation is carried out 16

5 types of symmetry operations/elements Operation 1: Identity Operation, do nothing. Identity: this operation does nothing, symbol: E 17

Operation 2: Cn, Proper Rotation: Rotation about an axis by an angle of 2/n = 360/n C2 C3 NH3 H2O How about: NFO2? 18

Returns molecule to original oreintation The Operation: Proper rotation Cn is the movement (2p/n) The Element: Proper rotation axis Cn is the line 180° (2p/2) Applying C2 twice Returns molecule to original oreintation C22 = E C2 19

Proper rotation axes C3, 120º C2 180º NH3 H2O How about: NFO2? 20

Rotation angle Symmetry operation 60º C6 120º C3 (= C62) 180º 240º C32(= C64) 300º C65 360º E (= C66) 21

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 C2 22

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 C4 23

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 C2 24

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 C2 25

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 C2 26

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 C2 27

Cn = Rotation about an axis by an angle of 2/n Proper Rotation: Cn = Rotation about an axis by an angle of 2/n PtCl4 Can perform operation several times. C2, C4 C2 C2 C2 C2 Rotation 2m/n 28

Operations can be performed sequentially Can perform operation several times. Means m successive rotations of 2p/n each time. Total rotation is 2mp/n m times Observe

The highest order rotation axis is the principal axis C3 axis Iron pentacarbonyl, Fe(CO)5 The highest order rotation axis is the principal axis and it is chosen as the z axis What other rotational axes do we have here? 30

Let’s look at the effect of a rotation on an algebraic function Consider the pz orbital and let’s rotate it CCW by 90 degrees. px proportional to xe-ar where r = sqrt(x2 + y2 + z2) using a coordinate system centered on the nucleus y y C4 x o x o px C4 px How do we express this mathematically? The rotation moves the function as shown. The value of the rotated function, C4 px, at point o is the same as the value of the original function px at the point o . The value of C4 px at the general point (x,y,z) is the value of px at the point (y,-x,z) Moving to a general function f(x,y,z) we have C4 f(x,y,z) = f(y,-x,z) Thus C4 can be expressed as (x,y,z) (y,-x,z). If C4 is a symmetry element for f then f(x,y,z) = f(y,-x,z)

Thus C4 px (x,y,z) = C4 xe-ar = ye-ar = py According to the pictures we see that C4 px yields py. Let’s do it analytically using C4 f(x,y,z) = f(y,-x,z) We start with px = xe-ar where r = sqrt(x2 + y2 + z2) and make the required substitution to perform C4 y y C4 x o x o px C4 px Thus C4 px (x,y,z) = C4 xe-ar = ye-ar = py And we can say that C4 around the z axis as shown is not a symmetry element for px

Operation 3: Reflection and reflection planes (mirrors) s s 33

 (reflection through a mirror plane) s NH3 Only one s? 34

H2O, reflection plane, perp to board s What is the exchange of atoms here? 35

H2O another, different reflection plane s’ What is the exchange of atoms here? 36

the principal axis it is called sv Classification of reflection planes B F If the plane contains the principal axis it is called sv B F If the plane is perpendicular to the principal axis it is called sh Sequential Application: sn = E (n = even) sn = s (n = odd) 37

Operation 4: Inversion: i Center of inversion or center of symmetry (x,y,z)  (-x,-y,-z) in = E (n is even) in = i (n is odd) 38

Inversion not the same as C2 rotation !! 39

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Figures with center of inversion Figures without center of inversion 41

Operation 5: Improper rotation (and improper rotation axis): Sn Rotation about an axis by an angle 2/n followed by reflection through perpendicular plane S4 in methane, tetrahedral structure. 42

Some things to ponder: S42 = C2 Also, S44 = E; S2 = i; S1 = s 43

Summary: Symmetry operations and elements proper rotation axis (Cn) improper rotation axis (Sn) reflection plane (s) inversion center (i) Identity (E) 44

Successive operations, Multiplication of Operators Already talked about multiplication of rotational Operators Means m successive rotations of 2p/n each time. Total rotation is 2mp/n But let’s examine some other multiplications of operators C4 2 1 1 1 4 C4 s 2 2 3 4 3 4 3 C4 s’ s We write s x C4 = s’, first done appears to right in this relationship between operators.

Translational symmetry not point symmetry

See: Chemical Applications of Group Theory by F. A. Cotton Symmetry point groups The set of all possible symmetry operations on a molecule is called the point group (there are 28 point groups) The mathematical treatment of the properties of groups is Group Theory In chemistry, group theory allows the assignment of structures, the definition of orbitals, analysis of vibrations, ... See: Chemical Applications of Group Theory by F. A. Cotton 47

To determine the point group of a molecule 48

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Groups of low symmetry 50

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