Vibrational isotope effect of linear and planar molecules: Deuterated bromoethenes

Linear (planar) molecules A and B which are identical except for isotopic s ubstitutions at the atomic sites tau are considered, Stretching (bending, o ut-of-plane) frequencies omega (k) and normal modes Psi (k) of the isotopic ally perturbed molecule B are expressed in terms of stretching (bending, ou t-of-plane) frequencies v(i) and the corresponding normal modes Phi (i) of the unperturbed molecule A. Complete specification of the unperturbed norma l modes is not required. All that is needed are stretching (bending, out-of -plane) amplitudes (tau\Phi (i)) of the normal modes Phi (i) at those sites tau that are affected by isotopic substitution The rule which interlaces f requencies omega (k) of molecule A with frequencies v(i) of molecule B is d erived. Given two isotopic molecules A and B-tau that differ by a single is otopic substitution at site tau, the inversion relation is derived. This re lation expresses unperturbed stretching (bending, out-of-plane) amplitudes at the site tau in terms of stretching (bending, out-of-plane) frequencies of molecules A and B-tau. As an example, out-of-plane vibrations of deutera ted bromoethene were considered. In the simplest method 12 out-of-plane fre quencies of four polydeuterated bromoethenes were calculated from 12 out-of -plane frequencies of bromoethene and three monodeuterated bromoethenes. St andard deviation of thus calculated frequencies from experimental frequenci es is Delta = 2.74 cm(-1). In another method, 15 out-of-plane frequencies o f four polydeuterated bromoethenes and selected monodeuterated bromoethene are calculated from 9 out-of-plane frequencies of bromoethene and the remai ning two monodeuterated bromoethenes. Depending on which monodeuterated bro moethene is selected (1-, cis-or trans-), standard deviation of thus obtain ed frequencies from experimental frequencies is Delta1 = 2.84 cm(-1), Delta c = 2.96 cm(-1) and Deltat = 2.72 cm(-1).