Rydberg states can be defined as those which fit Eq. (1) and whose wavefunctions are appropriate to generate such a series. Excitation energies are often reported as the difference between the Rydberg state and the corresponding ionization limit; these are referred to as ``term values'' in this context. For theoretical work, it may be more desirable to compute these term values than the actual excitation energies because the Rydberg state should look more like the ion than the ground state; orbital relaxation and correlation effects are more likely to cancel.
While any particular valence transition may also fit Eq. (1), it will not belong to a series with increasing quantum number n. An electron in a polyatomic molecule will fit such a formula if it is very diffuse, so that the core appears as a point charge. Of course this will never be entirely true, so the parameter is added to correct for the extent to which the distant electron penetrates the core. The ground state of an atom may have an electron with a large radius and may thus be referred to as a Rydberg state; for neutral polyatomics, this never happens. Rydberg transitions may be split by core asymmetries, but the splittings decrease with increasing n because the core becomes better modeled as a point charge as the electron radius increases.
For second-row atoms and molecules, the general trends in are:
The preceeding analysis is of course not directly applicable to polyatmoic molecules. Indeed, a nonpenetrating AO on one center may still penetrate another center, suggesting that Rydberg orbitals in polyatomics are generally more penetrating. The recapitulation idea is now not nearly as simple, since the Rydberg orbital must be orthogonalized against other occupied orbitals which may have no overlap with the precursor. Generally still holds, but now quantum mechanical exchange has an effect.
So far, the discussion of penetration has implicitly assumed an effect
of the form
For polyatomics, the Rydberg states can mix heavily. For NO, 3s is unperturbed by mixing, but above that ns mixes strongly with (n-1)d, making the nd series have a negative . Actually this makes sense in that for nd is about 0 whereas for ns is about 1.0, so ns and (n-1)d orbitals are about degenerate. The ns orbitals are a bit lower, so mixing causes the (n-1)d orbitals to be slightly raised in energy and hence they acquire a negative (their binding energy becomes slightly smaller than in hydrogen).
The singlet-triplet splitting of a pair of Rydberg states should be small because, at the SCF level, this splitting is roughly 2K, and for Rydberg states the exchange integral should be relatively small. In most polyatomic molecules, the singlet-triplet splitting of Rydberg states is typically less than 5000 cm-1. One exception to this rule is found found for ethylene, where the singlet Rydberg configuration is strongly mixed with the singlet valence configuration . However, the correspoding triplet configurations are not mixed, and thus the singlet-triplet splitting is not given by something like 2K.
For polyatomics there is an additional difficulty in determining whether an orbital is Rydberg or unoccupied valence; for example, in methane the carbon 3s orbital has the same nodal structure as the antibonding orbital. Such orbitals are called a conjugate pair. Mulliken claims that these are alternative descriptions of the same orbital; however, Robin argues that in the above example the 3s orbital is nonbonding while is antibonding and that numerous calculations support the idea of separate valence and Rydberg states. In ethylene, the valence shell . and Rydberg configurations have the same nodal patterns, and the V state has been attributed to each of these separately. Now the prevailing view is that the state does not fit the Rydberg formula and is primarily a valence transition with some Rydberg character mixed in. In the triplet manifold, the analogous configurations give rise to distinct valence and Rydberg states. If mixing of the conjugate pair is large, then it can in fact add a node to the Rydberg orbital, which makes it possible to describe the orbital as n+1 instead of n (and this is the practice of some authors). However, the size of the orbital and the frequency from Eq. (1) is much closer to that expected for n.
Oscillator strengths of Rydberg transitions are governed by the transition
moment matrix element
The following generalizations are given for Rydberg transitions converging on ionization potentials other than the first:
Experimentally, one way to distinguish between valence and Rydberg excitations is that the Rydberg excitations can be sensitive to a ``perturber gas.'' This is rationalized by the diffuse nature of the excited electron. Dramatic effects can be produced by perturber gasses with partial pressures of only atm. Of great value is the fact that the perturber gas causes pressure broadening of the absorber only to higher frequencies. In contrast, using higher concentrations of the absorber gas (the perturber is the absorber) causes pressure broadening to higher and lower frequencies of both Rydberg and valence excitations. In cases where the excitations appear as continuous bands, the pressure effect is useless unless it is greatly magnified; this can be accomplished by trapping the absorber in a low-temperature rare-gas matrix, by dissolving it in a transmitting solvent, or by forming a polycrystalline film of the neat absorber at low temperatures. The lowest Rydberg excitation energy of a dilute guest molecule in a rare-gas host will increase by 2000-5000 cm-1 compared to the gas-phase spectrum, and the vibronic bandwidths will be 200-500 cm-1. By contrast, most valence shell excitations will experience a redshift of 1000-3000 cm-1. If placed in a low-mobility organic matrix, valence excitations behave similarly but Rydberg excitations can broaden sufficiently as to disappear. The V state of ethylene behaves as a valence excitation under perturbations. For diatomics and presumably for polyatomics, in condensed phases coupled Rydberg and valence states can become uncoupled.