Physical Chemistry Third Edition

19.6 Atoms with More Than Two Electrons 807

nuclear charges are used in the 1sand 2sorbitals. The minimum in the variation energy,

− 201 .2 eV, corresponds to effective nuclear charges of 2.686 protons for the 1sorbitals

and 1.776 protons for the 2sorbital.^18 The correct value of the ground-state energy is

− 203 .5 eV, so this variation function gives an error of 2.3 eV or 1.1%.

The effective nuclear charge seen by the 1selectrons is nearly the same as would

be seen by the 1selectrons in Li+, because the minimum in the variation energy of a

helium-like atom in Eq. (19.1-15) occurs atZ′ 2 .6875 ifZ3. The 1selectron in a

lithium atom apparently sees almost no shielding due to the 2selectron. This is reason-

able, since a 2selectron is on the average farther from the nucleus than a 1selectron.

The 2selectron sees considerable shielding from the 1selectrons, corresponding to

an effective nuclear charge of 1.776 protons, considerably less than the actual nuclear

charge of 3 protons. This difference in effective nuclear charge makes the average

distance from the nucleus for the 1sand 2selectrons even more different than if they

corresponded to the same effective nuclear charge.

EXAMPLE19.3

Find the value of〈r〉for a hydrogen-like 1sorbital withZ 2 .686.

Solution

〈r〉 1 s 4 π

(

Z

a

) 3 (

1

π

)∞∫

0

r^3 e−^2 Zr/adr 4

(

Z

a

) 3 (

a

2 Z

) 4 ∞∫

0

x^3 e−xdx 4

(

Z

a

) 3 (

a

2 Z

) 4

(6)



3 a

2 Z



3 a

5. 732

 0. 5585 a 2. 95 × 10 −^11 m 29 .5pm

For comparison, the value of〈r〉fora2sorbital withZ 1 .776 is 179 pm, whereas〈r〉for

a2sorbital withZ 2 .686 is 118 pm. (See Problem 19.27.)

Many modern calculations employ the self-consistent field (SCF) method. In SCF

calculations on any multielectron atom the 2porbitals are found to be higher in energy

than the 2sorbital. A 2pelectron is not on the average farther from the nucleus than

one in a 2sorbital with the same nuclear charge (see Problem 17.13). Figure 17.11b

shows that the radial probability distribution for the 2sorbital has a “hump” close to

the nucleus that the 2porbital does not have. An electron in a 2sorbital therefore

has a greater probability of being found close to the nucleus, where the shielding is

least effective. The 2pelectron is more effectively screened from the nuclear charge

than is a 2selectron, so its orbital energy is higher. We say that the 2sorbital is more

“penetrating” toward the nucleus than are the 2porbitals, and the same is true for the

sandpsubshells in other shells.

Thefirst ionization potentialis defined as the energy required to remove one electron

from an isolated neutral atom. If the orbitals for the other electrons are not significantly

changed by the removal of one electron, the ionization potential is nearly equal to

the magnitude of the energy of the orbital occupied by the outermost electrons.^19 The

ionization potential can therefore be used to obtain an estimate of the effective nuclear

charge seen by the outer electrons.

(^18) I. N. Levine,op. cit., p. 298ff (note 2).
(^19) I. N. Levine,op. cit., p. 503 (note 2).