Elements and Operations. A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed.

Transcription

1 Elements and Operations A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed. A symmetry operation is a movement of an object about a symmetry element such that the object's orientation and position before and after the operation are indistinguishable. A symmetry operation carries every point in the object into an equivalent point or the identical point.

2 Point Group Symmetry All symmetry elements of a molecule pass through a central point within the molecule. he more symmetry operations a molecule has, the higher its symmetry is. Regardless of how many or few symmetry operations a molecule possesses, all are examples of one of five types. Operation Element Element Construct Identity, E he object N/A Proper rotation, C n Reflection, σ Inversion, i Rotation-reflection Improper rotation, S n Proper axis, Rotation axis irror plane, Reflection plane Inversion center, Center of symmetry Improper axis, alternating axis line plane point line

3 Proper Rotation, C n If a molecule has rotational symmetry, rotation by 2π/n = 360E/n brings the object into an equivalent position. he value of n is the order of an n-fold rotation. If the molecule has one or more rotational axes, the one with the highest value of n is the principal axis of rotation.

4 Successive C 4 clockwise rotations of a planar 4 molecule about an axis perpendicular to the plane of the molecule ( A = B = C = D ). A D B C C4 D C 4 2 = C 2 C A B C 4 C 4 3 = C 4-1 C B D 4 C 4 = E A C4 B A C D C 4 A D B C

5 he C 2 ' and C 2 " axes of a planar 4 molecule. C 2 ' C 2 " C 2 " C 2 '

6 General Relationships for C n C n n = E C n/2 n = C 2 (n even) C n 1 1 n = C n C n+m n = C m n (m < n) Every n-fold rotational axis has n 1 associated operations (excluding C n n = E).

7 Reflection, σ For every point a distance r along a normal to a mirror plane there exists a point at r. -r r wo points, equidistant from a mirror plane σ, related by reflection For a point (x,y,z), reflection across a mirror plane σ xy takes the point into (x,y, z). Each mirror plane has only one operation associated with it, since σ 2 = E.

8 Horizontal, Vertical, and Dihedral irror Planes irror planes of a square planar molecule 4. h v v d d

9 Inversion, i If inversion symmetry exists, for every point (x,y,z) there is an equivalent point ( x, y, z). Each inversion center has only one operation associated with it, since i 2 = E. A B D F i C E E C F D B A Effect of inversion (i) on an octahedral 6 molecule ( A = B = C = D = E = F ).

10 Inversion Center of Ethane in Staggered Configuration H C H H H H C H Ethane in the staggered configuration. he inversion center is at the midpoint along the C-C bond. Hydrogen atoms related by inversion are connected by dotted lines, which intersect at the inversion center. he two carbon atoms are also related by inversion.

11 Rotation-Reflection (Improper Rotation), S n S n exists if the movements C n followed by σ h (or vice versa) bring the object to an equivalent position. If both C n and σ h exist, then S n must exist. Example: S 4 collinear with C 4 in planar 4. Neither C n nor σ h need exist for S n to exist. Example: S 4 collinear with C 2 in tetrahedral 4.

12 S 4 Rotation of etrahedral 4 D C B A C 4 C B D S 4 A h C B D A S 4 improper rotation of a tetrahedral 4 molecule ( A = B = C = D ). he improper axis is perpendicular to the page. Rotation is arbitrarily taken in a clockwise direction. Note that neither C 4 nor σ h are genuine symmetry operations of tetrahedral 4.

13 C D A B S 4 C C 2 B D A S 4 S 4 3 A B D C S 4 D A B C S 4-1 S 4 C D A B Successive S 4 operations on a tetrahedral 4 molecule ( A = B = C = D ). Rotations are clockwise, except S 4-1, which is equivalent to the clockwise operation S 4 3.

14 Representing a etrahedral 4 olecule in a Cube C 2, S4 C 2, S 4 C 2, S4 A C 2 axis, collinear with an S 4 axis, passes through the centers of each pair of opposite cube faces and through the center of the molecule.

15 Non-Genuine S n Operations: S 1 = σ S 2 = i (-x,-y,z) (x,y,z) C 2 + h (-x,-y, -z) (x,y,z) i (-x,-y, -z)

16 General Relations of Improper Axes Equivalences of successive S n operations: If n is even, S n n = E If n is odd, S n n = σ and S n 2n = E If m is even, S n m = C n m when m < n and S n m = C n m n when m > n If S n with even n exists, then C n/2 exists If S n with odd n exists, then both C n and σ perpendicular to C n exist.

17 Examples Find all symmetry elements and operations in the following: