Vibrational frequencies calculations

Statistical mechanics computations are often tacked onto the end of ah initio vibrational frequency calculations for gas-phase properties at low pressure. For condensed-phase properties, often molecular dynamics or Monte Carlo calculations are necessary in order to obtain statistical data. The following are the principles that make this possible. 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

Vibrational frequencies Calculated vibrational frequencies are larger than measured values, typically by about 12%. Systematic scaling of calculated frequencies (by 0.88) leads to values which are generally suitable for assignment and interpretation of experimental infrared/Raman spectra. 

The carbonyl complex formed on Cu(P8/T4)-K(P8/Tl) dual cation site is depicted in Figure 2. Both cations are localized in the FER cage Cu+ is in the P8/T4 site (that is the most stable Cu+ site when framework A1 is in T4, see Ref. [7] for details) and K+ is in the P8/T1 site (the most stable K+ site in the vicinity of framework A1 in Tl, see Ref. [2] for details). The vibrational frequency calculated for this complex is 2133 cm 1 in a very reasonable agreement with the experimental value for low-energy band (Figure 1). The stability of CO adsorption complex is -107 kj/mol, 4 kJ/mol less than the corresponding complex on the isolated P8/T4 site of Cu+ [7],... 

N. Nevins and N. L. Allinger, Molecular mechanics (MM4) vibrational frequency calculations for alkenes and conjugated hydrocarbons, J. Comp. Chem. 17 730 (1996). 

The linear structure of 93 was derived from experiments with labeled precursor molecules and by correlation of vibrational frequencies calculated from estimated force constants with the recorded IR absorptions. The three fundamentals were observed as well as the UV/VIS spectrum,131 which was resolved and analyzed by gas phase measurements.132 The predicted triplet ground state was confirmed by recording the ESR spectrum of 93 isolated in various matrices.131... 

All of the theoretical methods that allow for geometry optimization predict the bowl-shaped conformation of 8 to predominate over the planar alternative. However, the calculated energy differences between the two conformations differ drastically and span the range of 0.8 to 84 kcal/mol. Moreover, the calculated energy difference between the plants and bowl-shaped conformers is quite important since it represents the barrier for bowl-to-bowl inversion if (a) planar corannulene represents the transition state (TS) for the process, and (b) the entropic term in AG is negligible. Condition (a) seems to be fulfilled, since vibrational frequency calculations give one and only one imaginary frequency for the planar structure... 

Now that we understand how to get a well-converged vibrational frequency for CO from DFT, we can compare this result to experimental data. Experimentally, the stretching frequency of gas-phase CO is 2143 cm- This value is 20 cm-1 higher than our DFT result. This result is fairly typical of a wide range of vibrational frequencies calculated with DFT. The discrepancy between the DFT result and the true vibrational frequency arises in part because of our harmonic treatment of the vibrations, but is also due to the inexact nature of DFT in solving the Schrodinger equation. We return to this issue in the context of a more general discussion of the accuracy of DFT in Chapter 10. 

The MP2 geometry optimizations and the HF vibrational frequency calculations use the 641(d) basis set of Binning and Curtiss89 90 for Ga-Kr along with 6-31G(d) for first- and second-row atoms, referred to overall for simplicity as 6-31G(d). The same scale factor (0.8929) is used for the zero-point energies. 

The values of harmonic vibrational frequencies calculated at the MP2/6-31G(d,p) level of theory [33]... 

Let us now turn our attention to the calculated vibrational frequencies of H20,02F2, and B2H6. First of all, it should be mentioned that the calculation of these frequencies is a computationally expensive task. As a result, high-level calculations of vibrational frequencies are performed only for relatively small systems. When the calculated frequencies are examined and compared with experimental data, it is found that the former are often larger than the latter. Indeed, after an extensive comparison between calculation and experiment, researchers have arrived at a scaling factor of 0.8929 for the HF/6-31G(d) frequencies. In other words, vibrational frequencies calculated at this level are... 

Calculated interaction energies (corrected for the basis set superposition error and ZPE) are relatively large and amount to 9.8 kcal mol-1 by B3LYP and 8.9 kcal mol-1 by CCSD(T) methods. Harmonic vibration frequencies calculated for monomers and the complex are in reasonable agreement with experimental data241. Agreement is good for IR band shifts due to complexation and isotopic substitution. 

Both approaches include tunneling corrections and provide approximately the same accuracy. However, the variational transition state theory is computationally quite demanding, and at least 40 points on the path of the proton transfer should be available. In contrast, the instanton approach uses only vibrational frequencies calculated for local minima and transition states and corresponding values of energy. 

Hydroxyl (O-H) stretching vibration frequency calculations compared with infrared spectra... 

The optimum binding energies for each of the five positions described above are shown in Table II. It is clear that the formation of C-O-AI complex (position 5, a = 0°) is favored. The AI-0 distance is found to be 1.80 A. (We should stress here that the C-0 distance is not reoptimized after bonding to Al.) In Table 2, we also show the vibrational frequency calculated, using the harmonic oscillator approximation, for the AI-0 streching in position 5. Its value is 890 cm 1. Pireaux et al (15), using HREELS techniques to study the AI/PI interaction, found that the vibration at 1720 cm 1 associated with C = 0... 

Jacobsen, R. L. Johnson, R. D. Irikura, K. K. Kacker, R. N. Anharmonic vibrational frequency calculations are not worthwhile for small basis sets, 7. Chem. Theor. Comput. 2013, 9, 951-954. 

Table 4.10. The Si-O bond lengths (R(Si-O)] and vibrational frequencies calculated" using Hartree-Fock-Roothaan (SCF) molecular-orbital methods with different basis sets and compared with experimental properties of Si(OH)4 in Dj, symmetry...

Intramolecular nucleophilic substitution to form thiiranes was studied by means of ab initio MO computations based on the 6-31G basis set <1997JCC1773>. Systems studied included the anions SCH2CH2F and CH2C(=S)CH2F which would afford thiirane and 2-methylenethiirane, respectively (Equations Z and 3). It was important to include electron correlation which was done with the frozen-core approximation at the second-order Moller-Plesset perturbation level. Optimized structures were confirmed by means of vibrational frequency calculations. The main conclusions were that electron correlation is important in lowering AG and AG°, that the displacements are enthalpy controlled, and that reaction energies are strongly dependent on reactant stabilities. 

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