Physical Chemistry Third Edition

18.5 Angular Momentum in the Helium Atom 775

There are operators for all of these quantities and theirzcomponents. The eigenvalues

of thêJ^2 ,̂Jz,̂L^2 ,̂Lz,̂S^2 , and̂Szoperators follow the same pattern as other angular

momenta, as follows:

̂J^2 Ψh ̄^2 J(J+1)Ψ (18.5-4)

̂JzΨhM ̄ JΨ (18.5-5)

̂L^2 Ψh ̄^2 L(L+1)Ψ (18.5-6)

̂LzΨhM ̄ LΨ (18.5-7)

̂S^2 Ψh ̄^2 S(S+1)Ψ (18.5-8)

̂SzΨhM ̄ SΨ (18.5-9)

The orbital angular momentum quantum numberLis a non-negative integer. The total

spin angular momentum quantum numberSis a non-negative integer or half-integer,

and the total angular momentum quantum numberJis a non-negative integer or half-

integer. SinceJis the vector sum ofLandS, the largest value that the quantum number

J can have isL+S, and its smallest value is|L−S|. From the universal pattern of

angular momentum quantum numbers,MLranges fromLto−L,MSranges fromS

to−S, andMJranges fromJto−J. Figure 18.3 illustrates how angular momentum

vectors can add vectorially to produce some particular values of the quantum numbers

L,ML,S, andMS. In each diagram, the tail of the second vector is placed at the head

of the first vector, as is done in the geometric representation of vector addition.

In Russell–Saunders coupling, each energy level corresponds to a particular value

ofLand a particular value ofSand is called aterm.ARussell–Saunders term symbol

is assigned to each term. The main part of the symbol is a letter giving the value ofL,

as follows:

Value ofL Symbol

0S

1P

2D

3F

4G

etc.

After F the symbols are in alphabetical order. Notice the similarity with the subshell

designations for a hydrogen-like atom.

A left superscript with the value 2S+1 is attached to the term symbol. For two

electrons we can haveS0, giving a superscript of 1 (asinglet term), or we can have

S1, giving a superscript of 3 (atriplet term). For a single electron we can have only

S 1 /2, giving a superscript of 2 (adoublet term). IfS 3 /2, the superscript equals

4(aquartet term), and so on. The value of the left superscript is called themultiplicity

of the term. A right subscript can be attached to the term symbol to specify the value

ofJ, but is sometimes omitted.