Harmonic vibrational frequencies stationary points

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. 

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... 

Each stationary point found was characterized as minimum or first-order saddle point by computing the harmonic vibrational frequencies. 

The optimized geometries and resulting energies for the reactants, entrance complex, transition state, exit complex, and products were predicted. Harmonic vibrational frequencies predicted at the same level (up to cc-pVQZ and cc-pVQZ-PP basis sets) were used for the characterization of stationary points and zero-point vibrational energies (ZPVE). The CFOUR program of Stanton, Gauss, Harding, Szalay, and coworkers was used for the coupled-cluster computations [24]. Most DZ results are not shown in text, but they are available in Supplementary Material. 

We consider first the nonrelativistic predictions of this research. Shown in Fig. 1 and Table 1 are the geometrical structures and relative energies for the reactants, entrance complex, transition state, exit complex, and products involved in the title reaction, as optimized at the CCSD(T)/ cc-pVnZ, (n = D, T, Q) levels of theory. The harmonic vibrational frequencies (cm ) for all the stationary points of the Br + H2O -> HBr + HO potential energy surface are reported in Table 2. For comparison, the limited available experimental results are also shown in Tables 1 and 2 [25-28]. Our ZPVE value for water (13.52 kcal/mol) is easily tested. Barletta et al. [29] have determined the exact ZPVE of water to be 4,638.31 cm , or 13.26 kcal/ mol. This means that there is an error of (13.52-13.26 =) 0.26 kcal/mol in our water ZPVE. 

Table 2 Nonrelativistic harmonic vibrational frequencies (cm ) and zero-point vibrational energies (kcal/mol) for the stationary points of the Br -b H2O -> HBr -b OH reaction ...

Pseudopotentials (PP) were originally proposed to reduce the computational cost for the heavy atoms with the replacement of the core orbitals by an effective potential. Modern pseudopotentials implicitly include relativistic effects by means of adjustment to quasi-relativistic Har-tree-Fock or Dirac-Hartree-Fock orbital energies and densities [35]. In the present research, we adopted Peterson s correlation-consistent cc-pVnZ-PP (n — D, T, Q, 5) basis sets [23] with the corresponding relativistic pseudopotential for the Br atom. The corresponding cc-pVnZ (n = D, T, Q, 5) basis sets were used for the O and H atoms. The optimized geometries and relative energies for the stationary points are reported in Table 1 and Fig. 3, and the harmonic vibrational frequencies and zero-point vibrational energies are reported in Table 4. 

By combining the force constants, k, with the reduced masses, fi, for the vibrational modes, we obtain the set of 3N-6 harmonic vibrational frequencies as documented in Section 8.2.1. As this process involves taking the square root of k this means that a vibrational frequency calculation performed on a structure that is a minimum on the PES will generate a set of 3N-6 real-number vibrational frequencies. If the stmcture is a saddle point or a maximum, one or more vibrational frequencies will be imaginary numbers the remainder will be real numbers. In this way, we can readily differentiate between the different types of stationary points on the PES. 

Parameters of the molecular geometry, electronic structure and thermodynamic properties of the benzoyl peroxide (BPO) molecule and its symmetrical derivatives were calculated by the GAUSSIAN09 [9]. The molecular geometry optimization of all objects was carried out at the first stage of the work. The calculation of harmonic frequencies of vibrations and thermodynamic parameters were performed after that. The stationary points obtained after the molecular geometry optimization were identified as minima, as there were no negative values of analytic harmonic vibration frequencies for them. The reaction center of the peroxide compounds is a peroxide bond -0-0-. Therefore, selection criterion for the quantum chemical calculation method was the best reproduction of the peroxide moiety molecular geometry. 

Harmonic vibrational frequencies have several uses. They are approximations to the fundamental frequencies observed in vibrational spectroscopy the number of imaginary frequencies identifies a stationary point as a local minimum (0 imaginary frequencies), transition state (1), or a higher order saddle point... 

To verify the nature of the two states, the harmonic vibrational frequencies at the minimum and transition state stationary points are reported in Table VI. For these states, the calculated frequencies in die harmonic approximation may be expected to be similar to those obtained from a full Jahn-Teller description, since the Jahn-Teller distortion in this case is relatively small. The deeper-lying Ui state is demonstrated to be a true minimum in all directions. The slightly less stable (0.006 eV) 82 state retains a single imaginary frequency in one component of the former e asymmetric bend mode and is thus a transition state, in this case to pseudorotation between the equivalent minima. On the full potential energy surface, the three equivalent minima and three equivalent... 

Free energy second derivatives are mainly used to analyse the nature of stationary points on the PES, and to compute harmonic force constants and vibrational frequencies to perform such calculations in solution, one needs analytical expressions for Qa second derivatives with respect to nuclear displacements (the alternative of using numerical differentiation of gradients is far too much expensive except for very small molecules). 

After the quenching, the character of stationary points found is determined by performing harmonic vibrational analysis. Because the harmonic frequencies of the cluster studied are very low (especially in the case of benzene...Arjj cluster), it was not easy to determine the nature of the stationary point. We believe, however, that the total population of the stationary point represents a better characterization of the point. If this population is not negligible, the point probably corresponds to the minimum. 

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