Electronic structure computations harmonic frequencies

The latest developments of time-dependent methods rooted in the density functional theory, especially by the so-called range separated functionals like LC-coPBE or LC-TPSS are allowing computation of accurate electronic spectra even for quite large systems. Moreover, the recent availability of analytical gradients for TD-DFT " allows an efficient computation of geometry structures and harmonic frequencies (through the numerical differentiation of analytical gradients) also for excited electronic states. 

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... 

In general, the accuracy of a simulated spectrum depends on the quality of the description of both the initial and the final electronic states of the transition. This is obviously related to the proper choice of a well-suited computational model a reliable description of equilibrium structures, harmonic frequencies, normal modes, and electronic transition energy is necessary. In the study of the A Bj Aj electronic transition of phenyl radical the structural and vibrational properties have been obtained with the B3LYP/TDB3LYP//N07D model, designed for computational studies of free radicals. Unconstrained geometry optimizations lead to planar... 

For many reactions the calculated structures for potential energy minima are as accurate as those found experimentally. Ab initio and experimental harmonic vibrational frequencies usually agree to within 10-15% at the Hartree-Fock level and 5% at the MP2 level (Hehre et al., 1986). It has been found that Hartree-Fock harmonic frequencies computed with a medium-size basis set ean be scaled by the factor 0.9 to give approximate anharmonic n = 0 — 1 transition frequencies (Hehre et al., 1986). A detailed study has been made of how the computed ab initio frequencies for benzene depend on the size of the basis set and the treatment of electron correlation (Maslen et al., 1992). 

For small molecules it is possible to perform accurate computations with post-Hartre—Fock approaches, and in this respect harmonic frequencies computed at the CCSD(T) (coupled clusters with single, double, and perturbative inclusion of triple excitation [27]) level, with basis sets of at least triple- quality reach an overall accuracy of 15—20cm for closed-shell systems [e.g., 28, 29]. For radicals, the situation is not so well assessed, but some recent investigations confirm that analogous accuracy can be reached [e.g., 30-34]. However, the unfavorable scaling of the CCSD(T) model with the number of active electrons limits its applicability to very small systems only. In addition, a simple reduction of the computational cost by combining correlated QM methods with small basis sets is not to be recommended due to the quite unpredictable accuracy of the results. Thus, the extension of computational studies to large systems requires cheaper and at the same time reliable electronic structure models. 

In view of some remaining unresolved problems (some of them to be discussed in the pfesent section), the construction of a completely new pair potential based on state-of-the-art ab initio computations was highly desirable. The recent benchmark studies of the structure, harmonic frequencies, and energetics of (HF)2 by Collins et al. and by Peterson and Dunning revealed the AO basis sets and levels of electron correlation treatment that ought to be considered for accurate ab initio calculations on this hydrogen-bonded species. 

Unfortunately, measured vibrational frequencies have some anharmonic component, and the vibrational frequencies computed in the manner above are harmonic. Thus, even the most accurate representation of the molecular structure and force constant will result in the calculated value having a positive deviation from experiment (Pople et al. 1981). Other systematic errors may be included in calculations of vibrational frequencies as well. For instance, Hartree-Fock calculations overestimate the dissociation energy of two atoms due to the fact that no electron correlation is included within the Hartree-Fock method (Hehre et al. 1986 Foresman and Frisch 1996). Basis sets used for frequency calculations are also typically limited (Curtiss et al. 1991) due to the requirements of performing a full energy minimization. Thus, errors due to the harmonic approximation, neglect of electron correlation and the size of the basis set selected can all contribute to discrepancies between experimental and calculated vibrational frequencies. 

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