Physical Chemistry Third Edition

774 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

PROBLEMS

Section 18.4: Excited States of the Helium Atom

18.8 Draw a graph of a probability density for finding any
electron at a distancerfrom the nucleus of a helium atom

in the (1s)(2s) configuration, using the zero-order wave

function of Eq. (18.4-2a).

18.5 Angular Momentum in the Helium Atom

In any set of interacting particles the total angular momentum is the vector sum of the

angular momenta of all of the particles. The total angular momentum of an isolated

system of interacting particles isconserved, both in classical mechanics and in quantum

mechanics. That is, if the angular momentum of one particle in such a system changes,

the angular momentum of other particles must change so that the vector sum does

not change. A conserved quantity does not change in time and is called aconstant

of the motion. A quantum number determining the value of a conserved quantity is

called agood quantum number, and the quantity itself is sometimes referred to by

that name.

The total angular momentum of an atom includes the angular momentum of the

nucleus and all electrons. We assume a stationary nucleus and assume also that the

angular momentum of the nucleus does not change so that we can ignore it. The angular

momentum of an electron is the vector sum of its orbital and spin angular momenta.

The total electronic angular momentum of a multielectron atom is the vector sum of the

angular momenta of all electrons. It can also be considered to be the sum of the total

orbital angular momentum and the total spin angular momentum. For atoms in the first

part of the periodic table the total orbital angular momentum and the total spin angular

momentum can separately be assumed to be good quantum numbers to an adequate

approximation. This assumption is calledRussell–Saunders coupling, and we will now

assume it to be an adequate approximation.

We now use lower-case letters for the angular momentum quantities of a single

electron and use capital letters for the angular momentum quantities of an atom. Let

l 1 ands 1 be the orbital and spin angular momenta of electron 1, letl 2 ands 2 be the

orbital and spin angular momenta of electron 2, and so on. The total orbital angular

momentumLand total spin angular momentaSof the electrons of the helium atom

are vector sums of the contributions of the individual electrons:

Ll 1 +l 2 + ··· (18.5-1)

Ss 1 +s 2 + ··· (18.5-2)

The total electronic angular momentum of the atom is denoted byJ:

JL+S (18.5-3)