Physical Chemistry Third Edition

22.3 Nuclear Spins and Wave Function Symmetry 931

a multielectron atom can have different electronic spin states. Very large energies are

required to raise nuclei to excited states so that chemists ordinarily encounter nuclei

only in their ground states. We therefore regardIas fixed for a particular nucleus.

Table A.24 in Appendix A lists the spin quantum numbers for the nuclear ground states

of several nuclei. IfI0 there must be an even number of nucleons and the nucleus

is a boson. The total wave function must be symmetric with respect to interchange of

the nuclei. There is a single nuclear spin function, which is symmetric with respect to

interchange of the nuclei. The other factors must combine to give a symmetric function.

IfI 1 /2, as is the case with^1 H,^13 C, and some other nuclei, there must be an

odd number of nucleons. The nuclei are fermions and the total wave function must be

antisymmetric. Each nucleus can occupy nuclear spin functions like those of electrons,

denoted byαfor spin up andβfor spin down, and the diatomic molecule has singlet and

triplet nuclear spin states like those of two electrons. The triplet nuclear spin functions

for such a diatomic molecule are symmetric:

α(A)α(B), β(A)β(B),

√

1

2

[α(A)β(B)+β(A)α(B)]

where we denote the nuclei by A and B. The singlet spin function is antisymmetric:

√

1

2

[α(A)β(B)−β(A)α(B)]

The other factors of the total wave function must combine to form an antisymmetric

function if they are combined with a triplet nuclear spin function. They must form a

symmetric function if they are combined with the singlet spin function. These factors

could be symmetrized or antisymmetrized by constructing a two-term wave function, as

we did with a two-electron wave function. However, this is unnecessary since the wave

functions we have constructed are generally eigenfunctions of symmetry operators, and

this fact makes them either symmetric or antisymmetric with respect to exchange of

the nuclei. The following sequence of symmetry operations exchanges the nuclei and

puts the electrons back in their original positions:

1. Rotate the entire molecule by 180◦around an axis perpendicular to the internuclear
   axis.


2. Reflect the electrons through a plane perpendicular to the axis of rotation that
   contains the internuclear axis.


3. Invert the electrons through the origin.

Exercise 22.4

Assume that a homonuclear diatomic molecule is located with the nuclei at (0, 0,zn) and

(0, 0,−zn) and that one electron is at (x,y,z). Show that the above listed operations exchange

the nuclei and put the electrons back at their original locations.

We can determine the effect of these operations on each factor of the wave function.

The translational factor depends only on the coordinates of the center of mass of the

molecule, and is unaffected. The vibrational factor is unaffected because it depends only

onrAB, which is a positive scalar quantity that remains unchanged under inversion,

rotation, or reflection. The rotational factor of the wave function of a diatomic molecule

is a spherical harmonic function. For even values of the rotational quantum number

Jthe spherical harmonic functions are eigenfunctions of the inversion operator with