Geometries and Vibrational Frequencies

FIGURE 41. A comparison of the distances in (A) for the phenoxyl radical and its para-halogenated derivatives. The entries are X = H (upper), F, Cl and Br (lower). Values were obtained from UB 3LYP/6-311 -h - -G(d,p) optimizations 

FIGURE 42. Comparison of the bond distances (A) and angles (deg) of three lowest-lying electronic states of phenoxyl radical. Values obtained from UMP2/6-31G(d,p) optimizations 

The C2v symmetry of PhO leads to 21 in-plane modes (11 fli and 10 B2) and 9 out-of-plane modes (3 02 and 6 b ). The assignment of the associated frequencies was also the subject of considerable discussion among experimental and theoretical °5-5is in particular as regards the location of the CO 

A correlation between the CO bond properties in the closed-shell molecules (single and double bonds) was proposed to estimate the bond lengths and stretching frequencies of open-shell phenoxyl radicals. Nevertheless, while it is possible to estimate the CO force constants using the Badger-type relations, it is difficult to relate them to the experimental frequencies that do not represent the stretching of a single bond. 

The CC and CO vibrations are also sensitive to the molecular environment by virtue of electrostatic and hydrogen bonding interactions. The frequencies of phenoxyl and tyrosyl radicals complexed by macrocyclic hgands and generated in wVo were measured by resonance Raman and FTIR techniques. Thus a selective enhancement of the vibrational CC and CO stretch modes of the phenoxyl chromophores in metal-coordinated radical 

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Polarization functions are functions of a higher angular momentum than the occupied orbitals, such as adding d orbitals to carbon or / orbitals to iron. These orbitals help the wave function better span the function space. This results in little additional energy, but more accurate geometries and vibrational frequencies. 

In general it is found that GGA methods often give geometries and vibrational frequencies for stable molecules of the same or better quality than MP2, at a computational cost similar to HE For systems containing multi-reference character, where MP2 usually fails badly, DFT methods are often found to generate results of a... 

Since, by assumption, the anion and neutral molecule do not differ significantly in their geometries (and vibrational frequencies) along the coordinates contributing to the FC factor (otherwise, the anion-neutral energy gap would depend substantially on these modes), the FC factor is probably close to unity in magnitude. Hence, for the remainder of this paper, we will focus only on the active-mode part of this expression, and will do so assuming only one such mode is operative (i.e., we treat one active mode at a time). 

Halgren, T. A. Merck Molecular Force Field. HI. Molecular geometries and vibrational frequencies for MMFF94. J Comput Chem 1996,17 553-86. 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

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. 

The force fields used for our purposes (simultaneous calculation of relative enthalpies, geometries and vibrational frequencies) are all of the type given in Eq. (2) or (4) (or a mixture of both types) with one important additional term which represents so-called nonbonded interactions (Vnb ). Two analytical formulations are in use for... 

The structure of the ground state of linear polyenes has been the subject of several theoretical studies12-37. Molecular geometries and vibrational frequencies for polyenes up to C18H20 have been reported. Much emphasis has been placed on the calculation of force constants that can be used in the construction of force fields. 

B3LYP/cc-pVTZ geometries and vibrational frequencies, as well as total energies at all levels of theory, are available on the World Wide Web at http //theochem.weizmann.ac.il/web/papers/aniline.html... 

Although we may not be able to use statistical mechanical results directly to estimate S and Cp, because of our lack of knowledge of precise molecular geometry and vibrational frequencies, we still can make use of these formulations to make systematic corrections to the thermochemistry of model compounds or to estimate these properties at different conditions. For example, based on the statistical mechanical formulas presented in Table VI, a working equation for estimating entropies from model compounds can be derived as... 

The thermodynamic properties were computed with the molecular geometry and vibrational frequencies given above assuming an ideal gas at 1 atm pressure and using the harmonic-oscillator rigid-rotor approximation. These properties are given for the range 0-2000°K in the Appendix (Table AI). 

C. Equilibrium Geometries and Vibrational Frequencies XII. Concluding Remarks... 

Piris and Otto (PO) achieved a reconstruction functional D[ D] satisfying the most general properties of the 2-RDM [58]. They kept the spin structure from Refs. [52, 53], but introduced a new spatial dependence in the correction term of the 2-RDM. Calculated values for polarizabilities [59], ionization energies, equilibrium geometries, and vibrational frequencies [60] in molecules were... 

Rotational and vibrational partition functions can be computed from the geometry and vibrational frequencies that are calculated for a molecule or TS. The entropy can then be obtained from these partition functions. Thus, electronic structure calculations can be used to compute not only the enthalpy difference between two stationary points but also the entropy and free energy differences. 

Slipchenko, L. V. and Krylov, A. I. 2003. Electronic Structure of the Trimethylenemethane Diradical in its Ground and Electronically Excited States Bonding, Equilibrium Geometries, and Vibrational Frequencies , J. Chem. Phys., 118, 6874. 

Most attention has been focused on the computation of Eeiec, because even fairly modest levels of theory can compute molecular geometries and vibrational frequencies sufficiently accurately to give good ZPVEs and thermal contributions, particularly if the frequencies are scaled by an appropriate factor (see Section 9.3). The simplest approach to improved feiec estimation is to scale it as a raw value as well, and tliis is the formalism implicit in the PCI-80 and SAC methods described in Section 7.7.1. 

Within the approximation of considering the complexes in the gas phase, the optimum geometry and vibrational frequencies of bi- and mono-dentate complexes have been obtained from a full geometry optimisation. To the best of our knowledge, no precise geometrical parameters are available for these complexes. Next, the vibrational frequencies have been computed as commented above and comparison with the experimental data is used to validate the particular theoretical approach chosen to carry out the present study. 

Results in parentheses use B3LYP/6-31G geometries and 0.96 scaled frequencies. As can be seen, the predicted ionization potentials, electron affinities, proton affinities, and enthalpies are not highly sensitive to the molecular geometries and vibrational frequencies adopted. 

Comparison of various forcefields for geometry (and vibrational frequencies) Halgren TA (1996) J Comp Chem 17 553... 

DFT-based methods are also proved to be quite reliable for calculations of both geometry and vibrational frequencies of CAs (see, for example, references341-343). 

Tucker SC, Truhlar DG, Ab initio calculations of the transition-state geometry and vibrational frequencies of the SN2 reaction of chloride with chloromethane, J Phys Chem 93, 8138-8142 (1989), and refs therein... 

Halgren TA (1996) Merck molecular force field. III. Molecular geometries and vibrational frequencies for Mmff94, J Comp Chem, 17 553-586... 

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