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Ch12

X - Symmetry

Symmetry is important in quantum mechanics for determining molecular structure andfor interpreting spectroscopic information.

In addition of being used to simplify calculations, two properties directly depend onsymmetry: optical activity and dipole moments.

We consider equilibrium configurations, with the atoms in their mean positions.

Symmetry elements and operations

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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 correspondingsymmetry operations results in a configuration indistinguishable from the initialconfiguration.

Example of molecules with center of symmetry:

benzene, methane, carbon dioxide, staggered ethane, ethylene, hexafluoro sulfide.

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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 verticaland 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; xaxis horizontally and to the right, y axis vertically and up.

Benzene has a C6 axis perpendicular to the molecular plane and 6 distinct C2 axes onthe molecular plane; 3 C2 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.

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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 andcuts the molecular plane horizontally.

Applied twice, we get .

We distinguish three types of symmetry planes, depending of their location with respect tothe principal axis:

1. horizontal (h) if it is perpendicular to the principal axis.Example: eclipsed ethane has a C3 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 C2 axes. Example: staggeredethane. The principal axis is a C3 axis going through the C-C bond.But it also has 3 C2 axes perpendicular to the C-C bond that project aH atom of one methyl group into a H atom of the other methyl group.There are 3 d planes containing the C3 axis.

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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.

S1 is the same as . S2 is the same as i. n times Sn is

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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 indistinguishableconfigurations.

Different collections of symmetry operations are organized into groups.

These 11 groups were developed by Schoenflies.

C1:only identity. Example: CHBrClF

Cs:only a reflection plane. Example: CH2BrCl

Ci:only a center of symmetry. Example: staggered 1,2-dibromo-1,2-dichloroethane.

Cn:only a Cn center of symmetry.

Example of C2: hydrogen peroxide (not coplanar)

Cnv:only n-fold axis and n vertical (or dihedral) mirror planes.Example of C2v: water; of C3v: ammonia

Cnh:only n-fold axis, a horizontal mirror plane, a center of symmetry or animproper axis.Example of C2h: trans dichloroethylene; of C3h: B(OH)3.

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Dn:Only a Cn and C2 perpendicular to it (propeller):

Dnd:A Cn, two perpendicular C2 and a dihedral mirror plane colinear

with the principal axis. D2d Allene: H2C=C=CH2.

Dnh:A Cn, and a horizontal mirror plane perpendicular to Cn. D6h benzene

Sn:A Sn axis. S4 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: Td

(a cube is Th)

Octahedral molecules: Oh

Icosahedron and dodecahedron molecules: Ih

A sphere, like an atom, is Kh

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Decision tree:

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Dipole moments, optical activity, and Hamiltonian operators

For a molecule to have a permanent dipole moment, the dipole moment cannot beaffected in direction or magnitude by any symmetry operation.

Molecules with i, h, Snh, C2Cn cannot have dipole moments.

Molecules that have permanent dipole moment are Cn, Cs or Cnv.

A molecule is optically active if it is has a non superimposable mirror image. Arotation followed by a reflection converts a right-handed object into a left-handedobject. If the molecule has an Sn axis, it cannot be optically active. Molecules thatdo not have Sn axis but can internally rotate, they could have optically activeconformations but they cannot be detected.

The Hamiltonian operator of a molecule must be invariant under all symmetryoperations of the molecule.

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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 x1, y1, z1, and it goes to x2, y2, z2, such that:

x2 = -1x1 + 0y1 + 0z1

y2 = 0x1 - 1y1 + 0z1

z2 = 0x1 + 0y1 - 1z1

In matrix notation:

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Counterclockwise rotation about the z axis:

z remains unchanged.  B

OA goes to OB

OA angle  to x axis O

OB angle  to OA

xA = OA cos 

yA = OA sin 

xB = OB cos (+) = OB (cos cos - sin sin) = OA (cos cos - sin sin)

= (xA/cos) (cos cos - sin sin)

= xA (cos) - (xA sin)(sin/cos) = xA (cos) - xA (sin/cos) (sin) =

= xA (cos) - yA (sin)

since xA/yA = cos/sin

yB=OB sin (+) = OB (sin cos + cos sin) = OA (sin cos + cos sin)

= yA/sin() (sin cos + cos sin) = yA (cos/sin) (sin) + yA (cos)

= xA (sin) + yA (cos)

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Suppose  = 120o, that is a C3 rotation,

cos(120o) = -1/2;

sin(120o) = 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 oneoperation followed by another.

For example, performing i followed by i gives E:

One can perform all the possible multiplications for every pair of symmetry elementsand generate what is called the matrix multiplication table for a representation.

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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 theirreducible representation. The labels at the right are given to visualize the irreduciblerepresentation.

Example C2v:

Molecule in xz plane. H2Opzpxdxz

If the symmetry of a molecule is that of a given point group, the wavefunctionsmust transform like one of the irreducible representations. Symmetry can beused to determine if a molecule will undergo certain transition.

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Symmetry in MOPAC

For example, CH4:

AM1 SYMMETRY

CH4

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