Julian Heicklen

Relations among Vibrational Frequencies of Isotopically Substituted Molecules and the Determination of Force Constants

A simple derivation is given for the product, sum, and complete isotope rules. Additional sum rules and their implications are presented. In particular, it is shown that if equivalent atoms of a molecule are isotopically substituted, then one relationship (the product rule) exists between two isotopic species; two more relationships (the first‐order sum rules) exist among three isotopic species; two more relationships exist among four isotopic species; and for five or more isotopic species all frequencies are related. If the isotopic substitutions retain the full symmetry of the molecule, the aforementioned relationships exist for each symmetry block. Even if isotopic substitution reduces the symmetry, the number of relationships is not diminished. Furthermore, if equivalent atoms are isotopically substituted, then for the inplane vibrations of planar molecules, all the vibrational frequencies are related among four or more isotopic species; for the out‐of‐plane vibrations of planar molecules and for line...

Frequency shifts and geometrical changes in condensed phases—I: Theory for spherically symmetric pentatomic tetrahedral molecules

Abstract A theory is presented which predicts vibrational and rotational frequency shifts and changes in geometry in condensed phases for spherically symmetric tetrahedral molecules. The change in bond length, Δ r 0 , is: Δ r 0 = q 0 Δλ E /λ A 0 where q 0 is the interaction distance between the peripheral atoms of the molecule and the cavity within which the molecule resides, Δλ E is the change in the doubly degenerate frequency parameter, and λ A 0 is the frequency parameter of the totally symmetric vibration in the gas phase. Since the quantities on the right must all be positive under all conditions, the bond length is always longer in a condensed phase than in the gas phase. The changes in the frequency parameters are given by: where K and K are two constants which can be found from the intermolecular potential function, m F and m σ are the masses of the peripheral and central atoms, respectively, μ F and μ c are the corresponding reciprocal masses, f r,r 0 and f α,α 0 are the triply degenerate stretching and bending force constants, respectively, and i is either 3 or 4; if i is 3, then j is 4, and vice versa. The constants K and K can be computed from the observed values of Δλ A and Δλ E by using the first two equations. The values found in this way for CF 4 and TiCl 4 are consistent with those predicted from a 6–9 potential function. The computed value of Δλ 4 for CF 4 is in excellent agreement with that found experimentally. It is also shown that for any spherically symmetric tetrahedral molecule, Δλ E and Δλ Rot must always be positive, i.e. the gas-phase frequencies are smaller than the corresponding ones in the liquid or solid. On the other hand, Δλ A must be negative for a liquid, but can be either positive or negative for a solid. For a 6–9 intermolecular potential function, −7 ⩽ Δλ A /Δλ E ⩽ −2 for a liquid, but Δλ A /Δλ E ⩾ −2 for a solid.