Physical Chemistry Third Edition

812 19 The Electronic States of Atoms. III. Higher-Order Approximations

predicted by the diagonal rule.^25 However, assigning a single configuration to an atom

is only an approximation. The correct wave function more nearly resembles a linear

combination of various configurations as in Eq. (19.6-1), and in assigning a single

configuration we are hoping only to identify the most prominent configuration.

The diagonal mnemonic rule does not necessarily apply to ions, because the shielding

is different for ions than for neutral atoms. For example, the iron atom has six 3d

electrons and two 4selectrons, in conformity with the diagonal rule. The Ni^2 +ion,

with the same number of electrons but with two additional protons in the nucleus, has

eight 3delectrons and no 4selectrons. The correct electron configuration for positive

ions can usually be obtained by finding the configuration of the neutral atom and then

first removing electrons from the outer shell, which is not necessarily the subshell to

which the last electrons were assigned by use of the diagonal mnemonic rule.

Hund’s Rules

For those elements with partially filled subshells, the values of the quantum numbers

L,S, andJof the ground level can be predicted by using three rules due to Hund, which

generally agree with experiment and with SCF calculations.Hund’s first ruleis:For a

given configuration, the level with the largest value ofShas the lowest energy. Hund’s

second ruleis:For a given value ofS, the level with the largest value ofLhas the

lowest energy. Hund’s third ruleis:For subshells that are more than half filled, higher

values ofJcorrespond to lower energies, and for subshells that are less than half filled,

lower values of Jcorrespond to lower energies.Hund’s second rule is applied only

after the first rule has been applied, and the third rule is applied only after the first

and second rules have been applied. The second rule can be regarded as a tiebreaker

for the first rule, and the third rule can be regarded as a tiebreaker for the second

rule.^26 Hund’s first rule has been explained by the idea that in the triplet spin state the

symmetric spin factor combines with an antisymmetric space factor, corresponding to

a lower probability that the electrons will be found close together. This interpretation

has been challenged, since calculations have indicated that the electron’s proximity to

the nucleus or nuclei is the more important property. It is best to regard Hund’s rules

as empirical rules based on observations. They are quite reliable for ground levels, but

less reliable for other levels.^27

In order to find the possible values of the quantum numbersS,L, andJwe determine

the possible values ofMS,ML, andMJas in Chapter 18. These quantum numbers are

algebraic sums:

ML

∑Z

i 1

mi (19.6-2)

MS

∑Z

i 1

msi (19.6-3)

MJMS+ML (19.6-4)

(^25) W. B. Jensen,J. Chem. Educ., 59 , 635 (1982).
(^26) T. S. Carlton,J. Chem. Educ., 83 , 477 (2006).
(^27) I. N. Levine,op. cit., p. 328ff (note 2).