Relationship between bond stretching frequencies and internal bonding for [¹⁶O₄]- and [¹⁸O₄]phosphates in aqueous solution

A normal-mode analysis is used to show that the fundamental stretching frequency, defined as v² = (v_s² + n_av_a²)/(n_a + 1) where v_s and v_a are the symmetric and asymmetric modes, respectively, and n_a the degeneracy of the asymmetric modes, depends almost entirely on the force constant and reduced mass of a P-^..O (or P-OH) bond in phosphoric acid or its anions to a good approximation. On the other hand, v_s or v_a, separately, depend not only on these parameters but also on molecular geometry. Thus, P-^..O (or P-OH) bond order is more closely related to the fundamental frequency than to the symmetric or asymmetric stretching frequencies. Similar conclusions apply to V-^..O (or V-OH) bonds in vanadate molecules. The frequencies for the four different P-^..O bonds and three different P-OH bonds in [¹⁸O₄] phosphoric acid and its three anions were measured by Raman and FT-IR spectroscopy. The measured values, plus those for [¹⁶O₄] phosphates, were correlated with P-^..O valence bond order, by using a modification of the Hardcastle-Wachs procedure (Hardcastle, F. D.; Wachs, I. J. Phys. Chem. 1991, 95, 5031). The bond order/stretching frequency correlations thus produced are expected to hold accurately over a wide range, for both [¹⁶O] and [¹⁸O] phosphates. Thus, P-^..O bond order and bond length in phosphates can be determined from vibrational spectra by using the derived bond order/stretching frequency correlation and the bond length/bond order correlation of Brown and Wu (Acta Crystallogr. 1976, B32, 1957). The accuracy is very high. The error in these relationships is estimated to be within ±0.04 vu and ±0.004 Å for bond orders and bond lengths, respectively, as judged by examining the affects of neglecting small terms in normal-mode analysis and by comparisons of derived bond lengths of P-^..O bonds from frequency data to the results from small-molecule crystallographic data.