Vibrational frequencies equilibrium

This factor can be obtained from the vibration partition function which was omitted from the expression for the equilibrium constant stated above and is, for one degree of vibrational freedom where vq is the vibrational frequency in the lowest energy state. 

Summarizing, in order to calculate rate and equilibrium constants, we need to calculate and AGq. This can be done if the geometry, energy and force constants are known for the reactant, TS and product. The translational and rotational contributions are trivial to calculate, while the vibrational frequencies require the ftill force constant matrix (i.e. all energy second derivatives), which may involve a significant computational effort. 

The subscript zero in this expression refers to the equilibrium configuration. The normal coordinate Q is also a function of vibrational frequency i i and of time t. 

Kaplan and Thornton (1967) used three different sets of vibrational frequencies to estimate the zero-point energies of the reactants and products of the equilibrium, which provided three different isotope exchange equilibrium constants 1-163, 1-311 and 1-050. The value 1-311 is considered to be most reasonable, whereas the others are rejected as unrealistic for the case in hand. Calculations using the complete theory led to values that varied from 1-086 to 1-774 for different sets of valence-force constants for the compounds involved. 

The most important calculated and experimental monomer data, such as equilibrium distances, dipole moments, polarizabilities, and the harmonic vibrational frequencies of the dihalogens XY, are reported in Tables 1-4. 

The active space used for both systems in these calculations is sufficiently large to incorporate important core-core, core-valence, and valence-valence electron correlation, and hence should be capable of providing a reliable estimate of Wj- In addition to the P,T-odd interaction constant Wd, we also compute ground to excited state transition energies, the ionization potential, dipole moment (pe), ground state equilibrium bond length and vibrational frequency (ov) for the YbF and pe for the BaF molecule. 

The equilibrium bond length (R,.) and ground state vibrational frequencies (( ,) computed at the DF and RASCI levels, are compared with experiment and with other calculations in Table IV. It is evident from Table IV that the RASCI method offers a more accurate estimate of R(, than the DF approximation, while the later method yields a more accurate estimate of the vibrational frequency ooe. However, the minuscule error in ( , (at the DF level) is perhaps fortuitous given the larger error in Rf of 2.8%. 

Florian and Johnson50 calculated vibrational frequencies in isolated formamide using the DFT calculations at the LDA (SVWN) and post-LDA (B88/LYP) levels. The DFT frequencies were compared with the ones derived from the Hartree-Fock and MP2 calculations, and from experiment. The authors found that the DFT(B88/LYP) frequencies were more in line with experiment then the MP2 ones. The DFT(SVWN) calculations led to geometry, force constants, and infrared spectra fully comparable to the MP2 results. The equilibrium geometry and vibrational frequencies of formamide were also the subject of studies by Andzelm et al.51. It was found that the DFT(B88/P86) calculations led to frequencies in a better agreement with experiment than those obtained from the CISD calculations. 

---