CHEM 442 Lecture 19 Problems 19-1. Show that the product of

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CHEM 442 Lecture 19 Problems

19-1. Show that the product of spherical harmonics Y

10

Y

10

is the linear combination of spherical harmonics Y

20

and Y

00

.

19-2. Show that the product of spherical harmonics Y

11

Y

11

is proportional to spherical harmonics Y

22

.

19-3. Extending the finding of 19-1, justify the selection rules of hydrogenic atomic transitions with z -polarized light:

D l

= ±

1;

D m l

=

0;

D m s

=

0 .

19-4. Extending the finding of 19-1 and 19-2, justify the selection rules of hydrogenic atomic transitions with nonpolarized or circularly polarized light:

D l

= ±

1;

D m l

=

0,

±

1;

D m s

=

0 .

19-5. List the orbitals to which a 4 s electron can make an optical transition.

19-6. The 3 d

3 z

2 r

2

orbital has the quantum numbers n

=

3, l

=

2, m l

=

0 . Consulting with the tables of radial and angular parts of the hydrogenic wave functions, show that it is

( z

2

r

2

) e

-

3 a

0 r

.

19-7. Using 19-6, show that the transition dipole moments in the Cartesian coordinates for the 1 s to 3 d

3 z

2 r

2

transition with z -polarized light is an odd function in the z coordinate and is forbidden. Show the same for x - and y -polarized lights.

19-8. Using 19-6, show that the transition dipole moments in the Cartesian coordinates for the 2 p z

to 3 d

3 z

2 r

2

transition with z -polarized light is an even function in all of the x , y , and z coordinates and is allowed. Show that the corresponding transition dipole moments for y - and z -polarized lights are forbidden.

19-9. Express the energy of a photon with wavelength of 400 nm in the units cm

−1

, eV,

K, and kcal/mol.

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