1. Identify the irreducible representations for all of the vibrations for [PdCl_{4}]^{2–}. Which of these vibrations are allowed by IR absorption spectroscopy?

First, draw the structure and label the important symmetry operations: The point group is D

_{4h}.

D

_{4h}E

2C

_{4}C

_{2}2C

_{2}'2C

_{2}"i

2S

_{4}σ

_{h}2σ

_{v}2σ

_{d}

a

_{1g}1

1

1

1

1

1

1

1

1

1

x

^{2}+ y^{2}, z^{2}a

_{2g}1

1

1

-1

-1

1

1

1

-1

-1

R

_{z}

b

_{1g}1

-1

1

1

-1

1

-1

1

1

-1

x

^{2}–y^{2}b

_{2g}1

-1

1

-1

1

1

-1

1

-1

1

xy

e

_{g}2

0

-2

0

0

2

0

-2

0

0

(R

_{x}, R_{y})(xz, yz)

a

_{1u}1

1

1

1

1

-1

-1

-1

-1

-1

a

_{2u}1

1

1

-1

-1

-1

-1

-1

1

1

z

b

_{1u}1

-1

1

1

-1

-1

1

-1

-1

1

b

_{2u}1

-1

1

-1

1

-1

1

-1

1

-1

e

_{u}2

0

-2

0

0

-2

0

2

0

0

(x, y)

Γ

_{coordinates}15

1

-1

-3

-1

-3

-1

5

3

1

All coordinates transform as a

_{1g}+ a_{2g}+ b_{1g}+ b_{2g}+ e_{g}+ 2a_{2u}+ b_{2u}+ 3e_{u}. The translations transform as a_{2u}+ e_{u}, the rotations transform as a_{2g}+ e_{g}, which leaves the irreducibile representations for the vibrations as a_{1g}+ b_{1g}+ b_{2g}+ a_{2u}+ b_{2u}+ 2e_{u}. Only the a_{2u}+ 2e_{u}are IR active.

2. Determine the irreducible representations of the CO stretching vibrations in Ni(CO)_{4}. Which of these vibrations are allowed by IR absorption spectroscopy?

First, draw the structure and label the important symmetry operations: The point group is T

_{d}.

T

_{d}E

8C

_{3}3C

_{2}6S

_{4}6σ

_{d}

a

_{1}1

1

1

1

1

x

^{2}+ y^{2}+ z^{2}a

_{2}1

1

1

-1

-1

e

2

-1

2

0

0

(2z

^{2}–x^{2}–y^{2}, x^{2}–y^{2)}t

_{1}3

0

-1

1

-1

(R

_{x}, R_{y}, R_{z})

t

_{2}3

0

-1

-1

1

(x, y, z)

(xz, yz, xy)

Γ

_{CO}4

1

0

0

2

The CO stretches transform as a

_{1}+ t_{2}. Only the t_{2}vibration is IR active.

3. Find the irreducible representations for the σ orbitals in PF_{5}. What is the hybridization on the P atom? What atomic orbitals are used to create the hybrid orbits? Be specific.

First, draw the structure and label the important symmetry operations: The point group is D

_{3h}.

D

_{3h}E

2C

_{3}3C

_{2}'σ

_{h}2S

_{3}3σ

_{v}

a

_{1}'1

1

1

1

1

1

x

^{2}+ y^{2}, z^{2}a

_{2}'1

1

-1

1

1

-1

R

_{z}

e'

2

-1

0

2

-1

0

(x, y)

(x

^{2}–y^{2}, xy)a

_{1}"1

1

1

-1

-1

-1

a

_{2}"1

1

-1

-1

-1

1

z

e"

2

-1

0

-2

1

0

(R

_{x}, R_{y})(xz, yz)

Γ

_{σ}5

2

1

3

0

3

The σ bonds transform as 2a

_{1}' + e' + a_{2}".The hybrid is sp

^{3}d with components of 3s (a_{1}') + 3p_{z}(a_{2}") + (3p_{x}, 3p_{y}) (e') +3d_{z}2 (a_{1}').