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Review on Point Groups and Symmetry

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Review on Point Groups and Symmetry Review Notes on Point Groups and Symmetry from undergraduate Inorganic Chemistry I course I. Introduction The major difference between organic and inorganic molecules is that organic molecules contain carbon and hydrogen atoms. Inorganic molecules are all compounds that do not contain carbon and hydrogen. Some points regarding inorganic molecules: • They often contain transition metals • Valence electrons in d-orbitals in transition metals are involved in bonding • s, p, d orbitals can be used in bonding (hybridization) • More bonds and geometries are possible around the central atom compared to bonds around a carbon atom Greater geometric complexity in inorganic molecules (about the central atom) related GEOMETRY SYMMETRY Symmetry plays a role in the physical properties of molecules, such as • Bonding- which orbitals interact to form bonds • Absorption spectra - Energy of transitions (position) - Transitions allowed or forbidden (intensity) • Magnetic properties- number of unpaired electrons • Packing of molecules in crystal lattice determines solid state structure and properties II. Symmetry of Objects and Molecules Compare a square to a rectangle. Which is more symmetrical? Why? A 90° rotation from the center about an axis 90o perpendicular to the paper leaves cube unchanged, but No change is apparent rotation not the rectangular object. In general, the square has Square more rotations & reflections that leave it unchanged, there are not as many for the rectangle. This makes 90o the square more symmetrical than then rectangle. Change is apparent rotation We need to relate these symmetry attributes to Rectangle molecules. 1 In general, we define Symmetry: invariance to transformation Transformation: movement of molecule (rotation, reflection, etc.) For example, compare the rotation of an equilateral triangle by 120o with that of a trigonal 120o planar molecule, BF3. When the triangle is No overall rotated, no overall change is apparent. Although rotation change the F’s were interchanged in BF3, we cannot tell because all F’s are equivalent, therefore, if we had not numbered the F atoms, we would say that the 2 o 1 molecule was left unchanged. F 120 F No overall Therefore, for BF , a 120° rotation to the 3 ⊥ change B rotation B plane of the molecule leaves the molecule F F 1 3 F F unchanged. We say that this transformation is a 3 2 symmetry operation of the BF3 molecule. Symmetry operation: a movement of a molecule that leaves the object or molecule unchanged Symmetry element: feature of the molecule that permits a transformation (operation) to be executed which leaves the object or molecule unchanged. Each symmetry operation has a symmetry element associated with it. The ones will be concerned with here are listed below. Operation Element Rotation Axis of rotation Reflection Mirror plane Inversion Center of inversion Improper rotation Axis of improper rotation The symmetry of a given molecule depends which type and how many operations leave it unchanged. Before we go over the symmetry of molecules we will discuss all the operations and their mathematical forms (handout on symmetry operations, matrices). In general, an operation can be thought of as a black box that moves or does something to an object ORIGINAL TRANSFORMED OBJECT OBJECT IN OUT Mathematical Function 2 A. Symmetry Elements and Operations 1. Mirror Plane, Reflection operation (σ) How many mirror planes are there in H2O? Total: 2 σ O • 1 σ bisecting the H-O-H bond (⊥ to paper) H H • 1 σ in the plane of the molecule (contains plane of paper) How about NH3? Total: 3 σ N H H H • 1 σ contains each N-H bond and bisects H-N-H BCl3 (planar molecule)? Total: 4 σ Cl B • 3 σ ⊥ to plane of the molecule along each B-Cl bond Cl Cl • 1 σ in the plane of the molecule (contains all atoms) 2– Planar [PtCl4] ? Total 5 σ; similar to BCl3 We can describe reflections are mathematically, since they are mathematical operations. For example, using Cartesian coordinates, one can ask where does a point (a,b,c) end up after reflection through xz- plane? z z (2,–1,3) (2,1,3) !xz y y or !y x x such that !xz (2,-1,3) (2,1,3) or !y or, in general, !xz (a,b,c) (a,-b,c) or !y 3 Similarly, (a,b,c) !yz (-a,b,c) or !x ! (a,b,c) xy (a,b,-c) or !z Each operation can be written in the form of a matrix. A 3x3 matrix is required for the transformation of an x,y,z point (a,b,c). For example, using the example above for a reflection through the xz plane, σxz, from point P at (a,b,c) to point P´ at (a,-b,c), we can write: P´ = σ xz (P), which means that the reflection operation on point P, σxz (P), results in P´. Since P = (a, b, c) and P´ = (a´, b´, c´) = (a, -b, c), we can write (a´,b´,c´) = σ xz (a,b,c) = (a,-b,c) Using matrices we can then write: a´ 1 0 0 a a b´ = 0 -1 0 b = -b c´ 0 0 1 c c P´ !xz P P´ Similarly, we can write the transformation matrices for σyz and σxy as follows. -1 0 0 1 0 0 !x = !yz = 0 1 0 !z = !xy = 0 1 0 0 0 1 0 0 -1 2. Inversion, center of inversion (i) Inversion operation: takes a point on a line through the origin to an equal distance on the other side For a point at x,y,z coordinates (2,-3,-4) inversion would move the point to (-2,3,4), such that i (2,-3,-4) = (-2,3,4) Therefore, in general, inversion of a point (a,b,c) results in a point at (-a,-b,-c) or i (a,b,c) = (-a,-b,-c) 4 The transformation matrix for inversion is given by -1 0 0 i = 0 -1 0 0 0 -1 If inversion operation is a symmetry operation of the molecule then we can say that: • the molecule possesses a center of symmetry • the molecule in centrosymmetric Do the following molecules have centers of inversion? F Cl 2– Br H Cl Pt Cl Cl M Cl C B Cl Cl H H H F F Cl Br Yes Yes No No 3. Rotation, Axis of Rotation (Cn) Cn = rotation about an axis of n-fold symmetry Cn (axis of rotation) ! An object has axial symmetry if it is invariant to rotation by θ, where n (n = 2π/θ) is an integer. n is the order of rotation θ is the angle of rotation Convention: clockwise rotation looking down axis m Cn means doing the Cn operation m times. n Cn takes the molecule back to starting position 5 Rotations in NH3 (top view): 2 1 3 2 o H H H H ! = 120 Starting C3 C3 C3 n = 3 Point N N N N C 3 H H H H H H H H 1 3 3 2 2 1 1 3 2 C3 3 C3 The general transformation matrix for a Cn rotation about the z-axis is given by cos(2!/ n) sin(2!/ n) 0 z Cn = –sin(2!/ n) cos(2!/ n) 0 0 0 1 So for C2 and C4 rotations about the z-axis cos(!) sin(!) 0 -1 0 0 C2 = –sin(!) cos(!) 0 = 0 -1 0 0 0 1 0 0 1 cos(!/ 2) sin(!/ 2) 0 0 1 0 C4 = –sin(!/ 2) cos(!/ 2) 0 = -1 0 0 0 0 1 0 0 1 The general matrices for Cn rotations about the x and y axes are given by: 1 0 0 cos(2!/ n) 0 sin(2!/ n) x y Cn = 0 cos(2!/ n) sin(2!/ n) Cn = 0 1 0 0 –sin(2!/ n) cos(2!/ n) –sin(2!/ n) 0 cos(2!/ n) All linear molecules have a C∞ axis along the axis of the molecule. They can be rotated by an infinitesimal angle and remain unchanged. 6 2– Example: List the rotational axes and operations present in square planar [PtCl4] . C2 C2´ Axes Operations Cl 2 3 C4 ! plane of molecule C4, C4 = C2, C4 C2 Cl Pt Cl 2 C2 contain Pt-Cl bonds 2 C2 Cl 2 C ´ bisect Cl-Pt-Cl C2´ 2 2 C2´ 4. Identity Operation: E Identity operation leaves a molecule unchanged. The operation E performed on a Cartesian point (a,b,c) results in (a,b,c). The matix for E is given by: 1 0 0 E = 0 1 0 0 0 1 Various operations performed successively result in placing the molecule in the original position, such as one reflection followed by another, inversion followed by inversion, and a Cn rotation performed n times, such that σ • σ = E i • i = E n Cn = E 5. Improper Rotation: Sn (element= axis of improper rotation) The improper rotation, Sn, is defined as a rotation (Cn) followed by reflection (σ) through plane perpendicular to the Cn axis. Cn P → P´ by C , P´ P n P´ → P´´ by σ ! Overall, Sn(P) = P´´ P´´ Sn = Cn × σ = σ × Cn ⇒ the operations commute 7 We can multiply the corresponding matrices for rotation (along z-axis) and reflection to arrive at the transformation matrix for the Sn operation. 1 0 0 cos(2!/ n) sin(2!/ n) 0 Sn = "xy x Cn = 0 1 0 –sin(2!/ n) cos(2!/ n) 0 = 0 0 -1 0 0 1 cos(2!/ n) sin(2!/ n) 0 = –sin(2!/ n) cos(2!/ n) 0 0 0 -1 Doing the Sn operation m times: m = Cn if m = even m m m Sn = Cn × σ = m = Cn × σ if m = odd = E if m = even n n Sn = Cn × σ = = σ if m = odd Example: S4 rotation and its repetitions on CH4 1 H S4 4 H C H H 2 3 Looking down the S4 axis (rotating the CH4 molecule so that S4 arrow points at you): 1 3 3 H H H C4 ! 3 H C H 4 2 H C H 1 2 H C H 1 H H H 2 4 4 S (single operation) 4 8 Notice that there is a C2 axis coincident with the S4 axis that arises from doing the S4 operation two times.
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