Physical Chemistry , 1st ed.

a spectrum illustrates). Since the energy of the transition depends on the value

ofnand not of, the different transitions ultimately have the same E.

Example 15.1

Which pairs of transitions [indicated by the (n,,m) quantum numbers] of

the hydrogen atom occur with the same value of E?

a.(2, 1, 1) →(3, 2, 2) and (3, 2, 2) →(4, 1, 2)

b.(3, 1, 0) →(5, 2, 1) and (3, 4, 0) →(5, 3, 0)

Solution

a.Since the two transitions are occurring between different values of the

principal quantum number, these two transitions do not have the same E

value and would be seen at different wavelengths in the spectrum.

b.Even though the and mquantum numbers are different, because the

transitions occur between wavefunctions having the same principal quantum

number, the energies of transition are the same.

The above example and selection rules are also applicable to hydrogen-like

ions, which have a single electron. However, such systems are in the vast mi-

nority of atomic species whose spectra need to be understood. Recall that one

of the final failings of classical mechanics was the inability to explain spectra.

Although quantum mechanics does not provide analytic solutions for wave-

functions of multielectron systems, it does provide tools for understanding it.

15.4 Angular Momenta: Orbital and Spin

In the discussion of the 2-D and 3-D rigid rotors, the concept of angular mo-

mentum arose, and in particular we used the fact that the angular momen-

tum of an object in some circular motion is related to its energy. For three

dimensions, the wavefunctions are the spherical harmonics, and the eigen-

value energies Eare dependent on an angular momentum quantum number

such that

E

(

2 I

1)^2



where is the angular momentum quantum number,is Planck’s constant di-

vided by 2, and Iis the moment of inertia. In the case of a 3-D rigid rotor,

the angular momentum is a well-understood classical property. In the applica-

tion of the 3-D rigid rotor to the hydrogen atom, the total electronic energy is

determined by the principal quantum number n, but the electron in the hy-

drogen atom has definite angular momentum values due to its orbitalangular

momentum.

An electron in a hydrogen atom also has spin. Spin acts like an angular mo-

mentum, so it is proper to speak not only of orbital angular momentum but

also ofspinangular momentum. The two different types of angular momenta

of an electron will each generate an intrinsic magnetic field, as will any charged

species that has angular momentum (that is, that accelerates by moving in

some sort of circular motion). These two intrinsic magnetic fields will interact

with each other in such a way that they combine to make an overallangular

momentum. It is this overall angular momentum, the combination of orbital

522 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure