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Chapter 6

Molecular Symmetry

This chapter will deal with the symmetry characteristics of individual molecules, i.e., how

molecules can be rotated or imaged along certain axes and be indistinguishable form a non-

rotated/imaged molecule.

Symmetry Operation: an operation performed on an object which leaves it in a configuration

that is indistinguishable from, and superimposable on, the original configuration.

EX. BF3

Rotations (Cn)

If the object can be rotated about an axis, we say there is rotational symmetry

EX. BF3—this can be rotated by ⅓ of a circle, so it has C3 symmetry

EX. PtCl42-

EX. CCl4

The axis with the highest n is called the principal axis.

EX. BF3 EX. SF6

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Mirror Planes (σ)

A plane of symmetry (mirror plane) exists if reflection of all parts of a molecule through this

plane results in an indistinguishable configuration

EX. BF3

A σv occurs if a σ contains the primary axis (“v” for vertical).

A σh occurs if a σ is perpendicular to the primary axis (“h” for horizontal).

A σd occurs if a σ contains both the primary axis and bisects two adjacent 2 fold axes (“d” for dihedral).

Center of Inversion (i)

If reflection of all parts of an object through the center of the object produces an

indistinguishable configuration, the object has a center of symmetry aka center of inversion

EX. Which of the following have an i?

BF3

C6H6

CO2

H2O

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Improper axis of rotation (Sn) If you rotate an object about an axis (a Cn operation), then follow it by a reflection through a

plane perpendicular to the axis—and the molecule comes back indistinguishable, you have an Sn

operation.

EX. BF3

Does [PtCl6]2- have an Sn operation?

Note: S1 is the same as a mirror plane (σh), but we just use σh, not the S1 operation

What is the S2 operation equivalent to?

Identity Operator (E)

This is, in essence, a 360º rotation—leaving the molecule unchanged. At the very least, ALL

molecules have this type of symmetry element.

EX. Can you think of a molecule that has NO Cn, Sn, i, or σ(h, v, d)?

Linking operations together If an object has a Cn, then if you perform Cn n-times, you rotate the object back to the original

position.

EX. BF3

Similarly, combining a C3 with a σh is an S3 operation: S3 = C3 × σh

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Assigning point groups

Sometimes (particularly in spectroscopy and MO theory) we need to know all of the symmetry

operations that pertain to a certain chemical. Rather than try to figure these out for every species,

a flow chart (your book calls this a “decision tree”) allows us to assign a “point group” which

will characterize all of the symmetry elements. Note: it isn’t necessary to find ALL symmetry

elements when assigning a point group. Additional flow charts are at the end of this chapter

notes. The point group is indicated with a Schoenflies symbol.

Assign point groups to the following:

BF3

CHBrClF

H2O

HCN

acetylene

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Special point groups (Td, Oh, Ih)

Certain molecular shapes have a high degree of symmetry and have special designations:

Td = tetrahedral (>1 primary axis)

Oh = octahedral (>1 primary axis)

Ih = icosahedral (12 five-fold axes)

Character tables The appendix has a list of common character tables—these will show all of the symmetry

elements for a particular point group

EX.

D4h E 2C4 C2 2C2’ 2C2” i 2S4 σh 2σv 2σd

The left column has the symmetry labels:

A and B refer to singly degenerate modes or orbitals

E refers to doubly degenerate modes or orbitals

T refers to triply degenerate modes or orbitals

“h” refers to the order—the number of symmetry operations

The right columns refer to functions:

x, y, z, Rx, Ry, Rz give information relevant to IR spectroscopy

More complicated functions (x2, xz, xy, etc.) are relevant to Raman spectroscopy

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Applications of symmetry

Polarity Molecules are polar if there is a net dipole moment. Chemicals with an i can’t be polar. Why?

Additionally, molecules can’t have dipole moments perpendicular to a mirror plane or perpendicular to

any axis of rotation.

EX. From point group designations, which of the following are polar?

CH2Cl2 PCl5 cis-[PdCl2(NH3)2]

What do we find? Molecules with a D-, T-, O-, or I-related point group cannot be polar.

Chirality What two simple symmetry operations prevent a molecule from being chiral? E, Cn, i, or σ(h, v, d)

Note that an improper axis of rotation is compatible with these and a better way to determine if a

molecule is nonsuperimposable on its mirror image is to look for an Sn operation. Chiral

molecules lack an Sn.

EX. Determine the point group of trioxalatoferrate(III). Will it be chiral?

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Note this molecule of tetrafluorospiropentane: Does it have an i? a σ? is it chiral?

Relating symmetry to vibrational spectroscopy For molecules to give rise to strong absorptions in the IR spectrum, they must change the

molecule’s dipole moment

An additional technique called Raman spectroscopy will give rise to strong signals if there is a

change in the polarizability of the molecule

Rule of mutual exclusion: for centrosymmetric molecules, IR active modes will NOT be

Raman active and vice versa

For linear molecules, the number of fundamental vibrational modes is: 3n-5 (n = # of atoms)

For nonlinear molecules, the number of fundamental vibrational modes is: 3n-6 (n = # of atoms)

EX. How many vibrational modes are in CO2? H2O?

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So how do character tables help us with this?

For H2O (C2v) the character table is

Consider the symmetry operations on H2O—how many bonds remain unchanged after each operation?

Looking at the C2v character table, which rows add up to give the reducible representation?

For the A1 we note all values are “1”—this means the molecule is left unchanged with this

operation. In B2, some values are -1, which means the molecule is reversed. Use this approach

for the stretching or bending modes.

How are the vibrational modes of the symmetric stretch affected by the symmetry operations in C2v?

Asymmetric stretch affected by symmetry operations?

Recall that H2O will have 3 modes—we’ve described two stretching, so the last one must be the

scissoring (bending).

If the symmetry label (A1, B2, etc) has an x, y, or z in the character table, the mode is IR active

If the symmetry label has a product term (z2, x2, xy, etc) in the character table, the mode is Raman active

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XY3 molecules with D3h symmetry Consider SO3: how many vibrational modes will it have?

What values do we find for the reducible representation:

D3h E C3 C2 σh S3 σv

Which rows sum up to be this reducible representation?

Are these IR-active, Raman-active, both, or neither?

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XY3 molecules with C3v symmetry Consider NH3: How many vibrational modes would it have?

What values do we find for the reducible representation:

C3v E C3 σv

Which rows sum up to be this reducible representation?

Are these IR-active, Raman-active, both, or neither?

Relating symmetry to molecular orbitals For non-linear molecules, it is helpful to consider outer atoms as a group—and look at their

symmetry as a unit. Your text calls this a SALC (symmetry-adapted linear combination), other

texts use the term LGO (ligand group orbitals). Your text provides pictures of various SALCs in

Resource Section 5.

SALCs that match the symmetry of atomic orbitals will overlap and form MOs.

For the H’s in NH3, perform symmetry operations like finding the reducible representation (but

only for the hydrogen atoms as a group), then determine the irreducible representations. This will

give A1 and E.

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Which ATOMIC ORBITALS on the nitrogen will match these symmetries?

Build the MO from corresponding overlaps.

Projection operators (deriving what the SALCs will “look” like)

Consider the chlorine sigma bonding SALC in [PtCl4]2-. What is the reducible representation?

E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh 2σv 2σd

E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh 2σv 2σd

A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2

A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz

B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2

B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy

Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)

A1u 1 1 1 1 1 -1 -1 -1 -1 -1

A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z

B1u 1 -1 1 1 -1 -1 1 -1 -1 1

B2u 1 -1 1 -1 1 -1 1 -1 1 -1

Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y)

What are the irreducible representations?

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Now perform a projection of how one wavefunction transforms to the other locations:

E C4 (z) C43 (z) C2 (C42) C'2 C'2 C''2 C''2 i S4 S43 σh σv σv σd σd

1

...and multiply by the characters in the irreducible representations.

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