6.2 Diatomic Molecules
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• symmetric if the nuclei have integer (or zero) nuclear spin (bosons);
• antisymmetric if the nuclei have half-odd-integer nuclear spin (fermions).
We shall write the total wavefunction for a diatomic molecule symbolically as
the product
total (1, 2) = ψ
total (1, 2)ψ nuclear−spin (1, 2) ,
(6.2.81)
with 1 and 2 referring to the two nuclei, ψ
total (1, 2) to the combined translational,
rotational, vibrational, and electronic degrees of freedom for the molecule, and
ψ nuclear−spin (1, 2) to the nuclear spin pair-states for the molecule. The overall
nuclear interchange symmetry
total (2, 1) = ± total (1, 2)
required by the Pauli Principle will be determined by the product of the interchange
symmetries of ψ
total (1, 2) and ψ nuclear−spin (1, 2).
It will also be useful for us to write the wavefunction ψ
total (1, 2) as the product
of wavefunctions for the molecular degrees of freedom as
ψ
total (1, 2) = ψ trans (1, 2)ψ rot (1, 2)ψ vib (1, 2)ψ el (1, 2) ,
and then consider the effect of the nuclear interchange on each factor separately.
The translational wavefunction, ψ trans (1, 2), depends only upon the coordinates
of the centre-of-mass of the diatomic molecule, and is hence not affected by the
interchange of the two nuclei. Similarly, the vibrational wavefunction, ψ vib (1, 2),
depends only upon the magnitude |R−R e | of the separation between the two nuclei,
and is thus unaffected by the interchange process. The behaviour of the electronic
wavefunction, ψ el (1, 2), under the nuclear interchange is more complicated, and
depends upon the nature of the electronic state in which the molecule is found:
we shall see shortly how the nuclear interchange symmetry can be obtained from
the spectroscopic term symbol assigned to the electronic state. As most molecules
in thermal equilibrium are found in their ground electronic states at normal
temperatures, our main concern will be with molecules in their ground electronic
states.
Interlude: Electronic State Nuclear Interchange Symmetry
Let us review briefly the symmetry operations relevant to a homonuclear diatomic
molecule. Molecular symmetry operations in general consist of rotations about axes,
reflections across planes, and inversion, all relative to the centre of the molecule,
plus any other distinct combined operations. These operations are represented
mathematically by a set of symmetry operators that, together with the identity
operator (which leaves all atoms in a molecule unchanged), form a mathematical
285
• symmetric if the nuclei have integer (or zero) nuclear spin (bosons);
• antisymmetric if the nuclei have half-odd-integer nuclear spin (fermions).
We shall write the total wavefunction for a diatomic molecule symbolically as
the product
total (1, 2) = ψ
total (1, 2)ψ nuclear−spin (1, 2) ,
(6.2.81)
with 1 and 2 referring to the two nuclei, ψ
total (1, 2) to the combined translational,
rotational, vibrational, and electronic degrees of freedom for the molecule, and
ψ nuclear−spin (1, 2) to the nuclear spin pair-states for the molecule. The overall
nuclear interchange symmetry
total (2, 1) = ± total (1, 2)
required by the Pauli Principle will be determined by the product of the interchange
symmetries of ψ
total (1, 2) and ψ nuclear−spin (1, 2).
It will also be useful for us to write the wavefunction ψ
total (1, 2) as the product
of wavefunctions for the molecular degrees of freedom as
ψ
total (1, 2) = ψ trans (1, 2)ψ rot (1, 2)ψ vib (1, 2)ψ el (1, 2) ,
and then consider the effect of the nuclear interchange on each factor separately.
The translational wavefunction, ψ trans (1, 2), depends only upon the coordinates
of the centre-of-mass of the diatomic molecule, and is hence not affected by the
interchange of the two nuclei. Similarly, the vibrational wavefunction, ψ vib (1, 2),
depends only upon the magnitude |R−R e | of the separation between the two nuclei,
and is thus unaffected by the interchange process. The behaviour of the electronic
wavefunction, ψ el (1, 2), under the nuclear interchange is more complicated, and
depends upon the nature of the electronic state in which the molecule is found:
we shall see shortly how the nuclear interchange symmetry can be obtained from
the spectroscopic term symbol assigned to the electronic state. As most molecules
in thermal equilibrium are found in their ground electronic states at normal
temperatures, our main concern will be with molecules in their ground electronic
states.
Interlude: Electronic State Nuclear Interchange Symmetry
Let us review briefly the symmetry operations relevant to a homonuclear diatomic
molecule. Molecular symmetry operations in general consist of rotations about axes,
reflections across planes, and inversion, all relative to the centre of the molecule,
plus any other distinct combined operations. These operations are represented
mathematically by a set of symmetry operators that, together with the identity
operator (which leaves all atoms in a molecule unchanged), form a mathematical