Vibrational frequencies harmonic

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

As examples of molecular properties we will look at how the dipole moment and harmonic vibrational frequencies converge as a function of level of theory. 

Extensive comparisons of experimental frequencies with HF, MP2 and DFT results have been reported [7-10]. Calculated harmonic vibrational frequencies generally overestimate the wavenumbers of the fundamental vibrations. Given the systematic nature of the errors, calculated raw frequencies are usually scaled uniformly by a scaling factor for comparison with the experimental data. 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

The computational prediction of vibrational spectra is among the important areas of application for modem quantum chemical methods because it allows the interpretation of experimental spectra and can be very instrumental for the identification of unknown species. A vibrational spectrum consists of two characteristics, the frequency of the incident light at which the absorption occurs and how much of the radiation is absorbed. The first quantity can be obtained computationally by calculating the harmonic vibrational frequencies of a molecule. As outlined in Chapter 8 density functional methods do a rather good job in that area. To complete the picture, one must also consider the second quantity, i. e., accurate computational predictions of the corresponding intensities have to be provided. 

Finley, J. W., Stephens, P. J., 1995, Density Functional Theory Calculations of Molecular Structures and Harmonic Vibrational Frequencies Using Hybrid Density Functionals , J. Mol. Struct. (Theochem), 357, 225. 

Scott, A. P., Radom, L., 1996, Harmonic Vibrational Frequencies An Evaluation of Hartree-Fock, Moller-Plesset, Quadratic Configuration Interaction, Density Functional Theory, and Semiempirical Scale Factors , J. Phys. Chem., 100, 16502. 

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. 

Overall, the trends in the calculated XY properties are satisfactorily reproduced with all four approaches MP2, CCSD(T), B3LYP, and BH HLYP. For the harmonic vibrational frequencies, the CCSD(T) data are far superior. 

Table 4 Calculated harmonic vibrational frequencies and infrared intensities of the dihalogens as obtained with different methods applying the aug-cc-pVTZ basis set3 [35]... 

CH3C1 and CHjBr, are 1.06 and 1.01 times larger than the respective experimental harmonic vibrational frequencies. One of the most interesting properties of the vibrational frequencies is the increase in the C-H stretch frequencies in going from the reactants (or products) of the SN2 reaction to the central barrier. 

Table 7. Harmonic Vibrational Frequencies for the Complexes and Central Barrier3...

Harmonic vibrational frequencies for s-trans butadiene have also been calculated at several levels of calculation19,21,23 24 31 35. Table 2 presents the computed values of some of the vibrational frequencies. 

The MP2/TZDP optimized structures were then used to calculate the stationary state geometry force constants and harmonic vibrational frequencies, also at the MP2 level. These results serve several purposes. Firstly, they test that the calculated geometry is really an energy minimum by showing all real frequencies in the normal coordinate analysis. Secondly, they provide values of the zero-point energy (ZPE) that can be used... 

The C=C harmonic vibrational frequency is calculated at 1671 cm-1 in free ethylene and is infrared (IR) forbidden. Its IR intensity is therefore expected to remain low in the vinyl series of compounds. The C=C stretch energy is calculated to be 1687 cm-1 in propene and then decline to 1629 4 cm-1 for X = Si - Pb. As in the equilibrium bond distance, there is also a very small counter-trend change in the vibrational frequency going from X = Sn to X = Pb that indicates a slight strengthening of the C=C bond. 

From the eigenvalues of this matrix, the harmonic vibrational frequencies can be obtained, and the corresponding eigenvectors describe the vibrational modes. 

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