1. Ch121 X - Symmetry Symmetry is important in quantum mechanics for determining molecular structure and for interpreting spectroscopic information. In addition of being used to simplify calculations, two properties directly depend on symmetry: optical activity and dipole moments. We consider equilibrium configurations, with the atoms in their mean positions. Symmetry elements and operations

2. Ch122 Inversion and center of symmetry The operation transforms x,y,z into -x,-y,-z. In the picture, the center of symmetry is at the center of the cube. Applied twice, we get. We say that a molecule has a certain symmetry element if the corresponding symmetry operations results in a configuration indistinguishable from the initial configuration. Example of molecules with center of symmetry: benzene, methane, carbon dioxide, staggered ethane, ethylene, hexafluoro sulfide.

3. Ch123 Rotation and symmetry axis The operation transforms x,y,z differently depending on the location of the axis. If the axis of rotation corresponds to the z axis, and it is a two-fold rotation: Applied twice, we get. Example: water. The two-fold axis of rotation is vertical and up; x axis horizontally to the right; y axis toward the back. If the axis of rotation corresponds to the z axis, and it is a three-fold rotation: Applied three times, we get. Example: ammonia. The z axis is perpendicular to the molecular plane and going up; x axis horizontally and to the right, y axis vertically and up. Benzene has a C 6 axis perpendicular to the molecular plane and 6 distinct C 2 axes on the molecular plane; 3 C 2 axes through opposite atoms, 3 through opposite bonds. Linear molecules have a C  containing the molecular axis. The principal axis is the axis of rotation of highest order.

4. Ch124 Reflection and symmetry plane Reflection in the xz plane transforms x,y,z into x,-y,z. The picture is on the xy plane. The xz plane is perpendicular to the molecular plane and cuts the molecular plane horizontally. Applied twice, we get. We distinguish three types of symmetry planes, depending of their location with respect to the principal axis: 1. horizontal (  h ) if it is perpendicular to the principal axis. Example: eclipsed ethane has a C 3 axis. It has a symmetry plane going perpendicular to that axis. 2. vertical (  v ) if it contains the principal axis. Examples: ammonia, with the plane going through one H atom. Ammonia has three  v axes. 3. dihedral (  d ) if it bisects angles formed by C 2 axes. Example: staggered ethane. The principal axis is a C 3 axis going through the C-C bond. But it also has 3 C 2 axes perpendicular to the C-C bond that project a H atom of one methyl group into a H atom of the other methyl group. There are 3  d planes containing the C 3 axis.

5. Ch125 Improper rotation and improper axis The improper rotation is the product of two operations, in a given order: 1. Rotation by 2  /n radians about an axis, followed by 2. Reflection through a plane perpendicular to the axis. S 1 is the same as . S 2 is the same as i. n times S n is

6. Ch126 Point groups Point groups are a way of classifying molecules in terms of their internal symmetry. Molecules can have many symmetry operations that result into indistinguishable configurations. Different collections of symmetry operations are organized into groups. These 11 groups were developed by Schoenflies. C 1 :only identity. Example: CHBrClF C s :only a reflection plane. Example: CH 2 BrCl C i :only a center of symmetry. Example: staggered 1,2-dibromo-1,2-dichloroethane. C n :only a C n center of symmetry. Example of C 2 : hydrogen peroxide (not coplanar) C nv :only n-fold axis and n vertical (or dihedral) mirror planes. Example of C 2v : water; of C 3v : ammonia C nh :only n-fold axis, a horizontal mirror plane, a center of symmetry or an improper axis. Example of C 2h : trans dichloroethylene; of C 3h : B(OH) 3.

7. Ch127 D n :Only a C n and C 2 perpendicular to it (propeller): D nd :A C n, two perpendicular C 2 and a dihedral mirror plane colinear with the principal axis. D 2 d Allene: H 2 C=C=CH 2. D nh :A C n, and a horizontal mirror plane perpendicular to C n. D 6h benzene S n :A S n axis. S 4 1,3,5,7-tetramethylcyclooctatetraene Special: Linear molecules: C  v :if there is no axis perpendicular to the principal axis D  h :if there is an axis perpendicular to the principal axis Tetrahedral molecules: T d (a cube is T h ) Octahedral molecules: O h Icosahedron and dodecahedron molecules: I h A sphere, like an atom, is K h

8. Ch128 Decision tree:

9. Ch129 Dipole moments, optical activity, and Hamiltonian operators For a molecule to have a permanent dipole moment, the dipole moment cannot be affected in direction or magnitude by any symmetry operation. Molecules with i,  h, S nh, C 2  C n cannot have dipole moments. Molecules that have permanent dipole moment are C n, C s or C nv. A molecule is optically active if it is has a non superimposable mirror image. A rotation followed by a reflection converts a right-handed object into a left-handed object. If the molecule has an S n axis, it cannot be optically active. Molecules that do not have S n axis but can internally rotate, they could have optically active conformations but they cannot be detected. The Hamiltonian operator of a molecule must be invariant under all symmetry operations of the molecule.

10. Ch1210 Matrix representation Identity: x,y,z goes to x,y,z. In matrix notation: The 3x3 matrix is called the transformation matrix for the identity operation. Inversion: x, y, z goes to -x, -y, -z. We start with x 1, y 1, z 1, and it goes to x 2, y 2, z 2, such that: x 2 = -1x 1 + 0y 1 + 0z 1 y 2 = 0x 1 - 1y 1 + 0z 1 z 2 = 0x 1 + 0y 1 - 1z 1 In matrix notation:

11. Ch1211 Counterclockwise rotation about the z axis: A z remains unchanged.  B OA goes to OB  OA angle  to x axis O OB angle  to OA x A = OA cos  y A = OA sin  x B = OB cos (  +  ) = OB (cos  cos  - sin  sin  ) = OA (cos  cos  - sin  sin  ) = (x A /cos  ) (cos  cos  - sin  sin  ) = x A (cos  ) - (x A sin  )(sin  /cos  ) = x A (cos  ) - x A (sin  /cos  ) (sin  ) = = x A (cos  ) - y A (sin  ) since x A /y A = cos  /sin  y B =OB sin (  +  ) = OB (sin  cos  + cos  sin  ) = OA (sin  cos  + cos  sin  ) = y A /sin(  ) (sin  cos  + cos  sin  ) = y A (cos  /sin  ) (sin  ) + y A (cos  ) = x A (sin  ) + y A (cos  )

12. Ch1212 Suppose  = 120 o, that is a C 3 rotation, cos(120 o ) = -1/2; sin(120 o ) = 0.866 =  3 / 2 One can write the transformation matrices for each operation of a point group. Multiplying the transformation matrices have the same effect as performing one operation followed by another. For example, performing i followed by i gives E: One can perform all the possible multiplications for every pair of symmetry elements and generate what is called the matrix multiplication table for a representation.

13. Ch1213 Character tables We often work with character tables, containing information for all point groups. The labels at the top refer to the symmetry elements. The labels going down represent the irreducible representations. These correspond to fundamental structures in the configurations. The numbers tell us what is going to happen when an operation is performed for the irreducible representation. The labels at the right are given to visualize the irreducible representation. Example C 2v : Molecule in xz plane. H 2 Op z p x d xz If the symmetry of a molecule is that of a given point group, the wavefunctions must transform like one of the irreducible representations. Symmetry can be used to determine if a molecule will undergo certain transition.

14. Ch1214 Symmetry in MOPAC For example, CH 4 : AM1 SYMMETRY CH4 C H 1.0 11 H 1.0 1 109.0 11 2 H 1.0 1 109.0 1 120.0 11 3 2 H 1.0 1 109.0 1 120.0 11 4 2 5 1 2 3 4 Atom 5 as a reference, code 1 = bond, other related atoms 5 2 3 4Atom 5 as a reference, code 2 = angle, other atoms 5 3 4Atom 5 as a reference, code 3 = dihedral, other atoms ATOM CHEMICAL BOND LENGTH BOND ANGLE TWIST ANGLE NUMBER SYMBOL (ANGSTROMS) (DEGREES) (DEGREES) (I) NA:I NB:NA:I NC:NB:NA:I NA NB NC 1 C 2 H 1.11157 * 1 3 H 1.11157 * 109.48506 * 1 2 4 H 1.11157 * 109.48506 * 119.99814 * 1 3 2 5 H 1.11157 * 109.48506 * 119.99814 * 1 4 2