Polarizability volumes
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص624-625
2025-12-15
52
Polarizability volumes
Polarizability has the units (coulomb metre)2 per joule (C2 m2 J−1). That collection of units is awkward, so α is often expressed as a polarizability volume, α′, by using the relation
α′=
where ε0 is the vacuum permittivity. Because the units of 4πε0 are coulomb-squared per joule per metre (C2 J−1 m−1), it follows that α′ has the dimensions of volume (hence its name). Polarizability volumes are similar in magnitude to actual molecular volumes (of the order of 10−30 m3, 10−3 nm3, 1 Å3).
Some experimental polarizability volumes of molecules are given in Table 18.1. As shown in the Justification below, polarizability volumes correlate with the HOMO LUMO separations in atoms and molecules. The electron distribution can be distorted readily if the LUMO lies close to the HOMO in energy, so the polarizability is then large. If the LUMO lies high above the HOMO, an applied field cannot perturb the electron distribution significantly, and the polarizability is low. Molecules with small HOMO–LUMO gaps are typically large, with numerous electrons.
Justification 18.2 Polarizabilities and molecular structures When an electric field is increased by dE, the energy of a molecule changes by −µdE, and if the molecule is polarizable, we interpret µ as µ* (eqn 18.8). Therefore, the change in energy when the field is increased from 0 to E is
The contribution to the hamiltonian when a dipole moment is exposed to an electric field E in the z-direction is
H(1) =−µzE
Comparison of these two expressions suggests that we should use second-order perturbation theory to calculate the energy of the system in the presence of the field, because then we shall obtain an expression proportional to E2. According to eqn 9.65b, the second-order contribution to the energy is
where µz,0n is the transition electric dipole moment in the z-direction (eqn 9.70). By comparing the two expressions for the energy, we conclude that the polarizability of the molecule in the z-direction is
The content of eqn 18.10 can be appreciated by approximating the excitation energies by a mean value ∆E (an indication of the HOMO–LUMO separation), and supposing that the most important transition dipole moment is approximately equal to the charge of an electron multiplied by the radius, R, of the molecule. Then
α≈ 
This expression shows that α increases with the size of the molecule and with the ease with which it can be excited (the smaller the value of ∆E). If the excitation energy is approximated by the energy needed to remove an electron to infinity from a distance R from a single positive charge, we can write ∆E ≈ e2/4πε0R. When this expression is substituted into the equation above, both sides are divided by 4πε0, and the factor of 2 ignored in this approximation, we obtain α′≈R3, which is of the same order of magnitude as the molecular volume.
For most molecules, the polarizability is anisotropic, by which is meant that its value depends on the orientation of the molecule relative to the field. The polarizability volume of benzene when the field is applied perpendicular to the ring is 0.0067 nm3 and it is 0.0123 nm3 when the field is applied in the plane of the ring. The anisotropy of the polarizability determines whether a molecule is rotationally Raman active (Section 13.7).
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