19.6 Atoms with More Than Two Electrons 819calculate a value for the effective nuclear charge felt by
the 3selectron in a sodium atom in its ground state. State
any assumptions.19.37Which of the elements in the first two rows of the
periodic table have electronic charge distributions that are
spherically symmetric?19.38 a.The Fe^2 +ion and the Cr atom are isoelectronic (have
the same number of electrons). Give the
lowest-energy electron configuration for each.
b.Give the lowest-energy term symbols for the Fe^2 +ion
and for the Cr atom.
19.39 a.The Fe^3 +ion and the V atom are isoelectronic (have
the same number of electrons). Give the
lowest-energy electron configuration for each.
b.Give the lowest-energy term symbols for the Fe^3 +ion
and for the V atom.
19.40For each of the first 18 elements of the periodic table,
give the number of unpaired electrons in the ground state.Summary of the Chapter
There are several commonly used approximation schemes that can be applied to the
electronic states of multielectron atoms. The first approximation scheme was the varia-
tion method, in which a variation trial function is chosen to minimize the approximate
ground-state energy calculated with it. A simple orbital variation trial function was
found to correspond to a reduced nuclear charge in the helium atom. This result was
interpreted to mean that each electron in a helium atom shields the other electron
from the full charge of the nucleus. A better variation trial function includes electron
correlation, a dependence of the wave function on the electron–electron distance.
In the perturbation method the Hamiltonian is written aŝH(0)+Ĥ′, whereĤ(0)
corresponds to a Schrödinger equation that can be solved. The perturbation termĤ′
is arbitrarily multiplied by a fictitious parameterλ, so thatλ1 corresponds to the
actual case. The method is based on representations of energy eigenvalues and energy
eigenfunctions as power series inλand approximation of the series by partial sums.
The method can be applied to excited states. In the helium atom treatment the electron–
electron repulsive potential energy was treated as the perturbation term in the Hamil-
tonian operator.
The third approximation scheme was the self-consistent field method invented by
Hartree and Fock. In this method an optimum orbital wave function is sought without
restricting the search to a single family of functions. The original method represented
the orbitals numerically, but the method was extended by Roothaan, who introduced
linear combination representations of the orbitals. For the helium atom the electron–
electron repulsive energy is represented by assuming the probability density for the
second electron to be given by an earlier approximate orbital and solving the resulting
integrodifferential equation by iteration.
The density functional method is based on a theorem of Hohenberg and Kohn,
which states that the ground-state energy of an atom or molecule is a functional of
the electron probability density. Schemes exist to obtain a usable approximation to the
desired functional and the necessary electron probability density.
Results of various calculations were presented. In the orbital approximation, the
energies of the orbitals in multielectron atoms depend on the angular momentum quan-
tum number as well as on the principal quantum number, increasing aslincreases.
The ground state of a multielectron atom is identified by the Aufbau principle, choos-
ing orbitals that give the lowest sum of the orbital energies consistent with the Pauli
exclusion principle.
Hund’s first rule is that the largest value ofScorresponds to the lowest energy in a
configuration. Hund’s second rule is that for fixed value ofS, the largest value ofL,