Physical Chemistry Third Edition

742 17 The Electronic States of Atoms. I. The Hydrogen Atom

of̂Lz. The complex orbitals are more useful in discussing properties that relate to

angular momentum, and the real orbitals are more useful in discussions of chemical

bonding.

The set of states with the same value ofncorrespond to the same value of the energy

and are called ashell. This term is used because the expectation value of the distance

of the electron from the nucleus is roughly the same for all states in a shell. The shells

are labeled with the value ofn, the principal quantum number. There is also an older

notation in which the first (n1) shell is called theKshell, the second (n2) shell

is called theLshell, and so on. Within a given shell, the states with a given value ofl

constitute asubshell. Thel0 state of a shell is called itsssubshell. The three states

withl1 constitute ap subshell.Ad subshellconsists of the fivel2 states. Anf

subshellconsists of the sevenl3 states. Any further subshells are given the lettersg,

h,i, and so on (in alphabetical order afterf). The letterss,p,d, andfcame from the

old spectroscopic terms “sharp,” “principal,” “diffuse,” and “fundamental,” but these

names have no connection with the present usage. There arensubshells in thenth shell.

The first shell has only the 1ssubshell, while the seventh shell has the 7s,7p,7d,7f,

7 g,7h, and 7isubshells, and so on.

Exercise 17.9

Give the value of each of the three quantum numbers for each state of the fourth (n4) shell.

Table 17.3 contains formulas for the real hydrogen-like orbitals for the first three

shells. The complex wave functions can be generated by replacing the realΦfunctions

by the complexΦfunctions. As indicated in the table, we can use the letter of the

subshell as a subscript instead of the value of the subscriptl. The 211 orbital can be

called the 2p1 orbital, the 21xorbital can be called the 2pxorbital, and the 210 orbital

can be called the 2p0 orbital or the 2pzorbital. The 3dz 2 orbital is the same as the

320 orbital. The 3dxzand 3dyzorbitals are linear combinations of the 321 and 32,− 1

orbitals, and the 3dxyand the 3dx (^2) −y 2 orbitals are linear combinations of the 322 and
32,−2 orbitals.

EXAMPLE17.4

Write the formula forψ 211.

Solution

Φ 1 (φ)

1

√

2 π

eiφ

Θ 11 (θ)

√

3

4

sin(θ)

R 21 (ρ)

(

Z

a

) 3 / 2

1

2

√

6

ρe−ρ/^2

Ψ 211 

(

Z

a

) 3 / 2

1

8

√

π

ρe−ρ/^2 sin(θ)eiφ

(

Z

a

) 3 / 2

1

8

√

π

Zr

a

e−Zr/^2 asin(θ)eiφ