932 22 Translational, Rotational, and Vibrational States of Atoms and Molecules
eigenvalue+1 (are “gerade”), and for odd values ofJthey are eigenfunctions with
eigenvalue−1 (are “ungerade”). The same eigenvalues apply to a rotation of 180◦
around an axis perpendicular to the bond axis, because this operation has the same
effect onθandφas does inversion. The rotational factor is unchanged by the operations
applied only to the electrons, so the three symmetry operations must change the sign
of the electronic factor ifJis odd and must not change it ifJis even.Exercise 22.5
Show that the spherical harmonic functionY 00 is an eigenfunction of the inversion operator with
eigenvalue 1, while the spherical harmonic functionY 11 is an eigenfunction with eigenvalue
−1. In spherical polar coordinates the inversion operator replacesθbyπ−θand replacesφby
π+φ. Show that rotation of 180◦around an axis perpendicular to the bond axis gives the same
result.The electronic factor is unaffected by the rotation of the entire molecule, since the
electrons simply follow the nuclei according to the Born–Oppenheimer approximation.
We have already discussed the effect of aσvoperation and an inversion operation
on the electronic factor in Chapter 20. For aΣterm the superscript+is used to
denote eigenfunctions of̂σvwith eigenvalue+1, and a superscript−is used to denote
eigenfunctions of̂σvwith eigenvalue−1. The electronic wave functions are denoted
by g if they are eigenfunctions of the inversion operator with eigenvalue+1 and by
u if the eigenvalue is−1. Functions with eigenvalue+1 are said to haveeven parity,
and those with eigenvalue−1 are said to haveodd parity. The reflection and inversion
operations change the sign of the electronic factor for a wave function that is+and u
or for a wave function that is−and g. Otherwise the sign does not change.EXAMPLE22.7
Find the permitted wave functions for H 2 in its electronic ground state.
Solution
The^1 H nuclei are protons withI 1 /2, so the entire wave function must change sign if
the nuclei are exchanged. The ground term symbol is^1 Σ+g. The ground-state electronic
wave function does not change sign under the three symmetry operations. If the nuclear spin
function is the antisymmetric singlet function, the rotational wave function must correspond
to even values ofJ. If the nuclear spin function is one of the triplet functions, which are
symmetric, the rotational wave function must correspond to odd values ofJ.
The form of hydrogen with triplet nuclear spin states and odd values ofJis calledortho-
hydrogenand the form with singlet nuclear spin state and even values ofJis calledpara-
hydrogen. This terminology can be remembered from the fact that in theparaform the nuclear
spins point in opposite directions, just as do two groupsparato each other on a benzene ring.
For values ofIgreater than 1/2, the spin functions are more complicated and we will
not discuss them. However, for any specific nuclear spin state of a homonuclear diatomic
molecule, either even values ofJor odd values ofJare permitted. As with hydrogen, only
half of the values ofJcan occur. Both even values ofJand odd values ofJcan occur for a
heteronuclear diatomic molecule, because the nuclei are not identical.Exercise 22.6
For^200 Hg,I0 and the electronic ground state is g and+. What values ofJcan occur with(^200) Hg 2?