Density functional theory procedure

Figure 6.35. Potential energy diagrams for adsorption and dissociation of N2on a Ru(0001) surface and on the same surface with a monoatomic step, as calculated with a density functional theory procedure. [Adapted from S. Dahl, A. Logadottir, R. Egberg, J. Larsen, I. Chorkendorff,...

Molecular frequencies depend on the second derivative of the energy with respect to the nuclear positions. Analytic second derivatives are available for the Hartree-Fock (HF keyword). Density Functional Theory (primarily the B3LYP keyword in this book), second-order Moller-Plesset (MP2 keyword) and CASSCF (CASSCF keyword) theoretical procedures. Numeric second derivatives—which are much more time consuming—are available for other methods. 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

A currently popular alternative to the ah initio method is density functional theory, in which the energy is expressed in terms of the electron density rather than the wave-function itself. The advantage of this approach is that it is less demanding computationally, requires less computer time, and in some cases—particularly for d-metal complexes—gives better agreement with experimental values than other procedures. 

The physical reasoning for why these densities were frequently employed in the earlier days of density functional theory was that in this way the degeneracy of the partially filled d-orbitals could be retained. A technical reason why these densities still have to be employed in some recent investigations is that calculations with integral orbital occupations simply do not converge in the self consistent field procedure (see, e. g., Blanchet, Duarte, and Salahub, 1997). Such densities correspond to a representation of a particular state 2S+1L with Mg = S and a spherical averaging over ML. 

We may ask now, whether the same procedure may be applied to density-functional theory, just by replacing the Fock operator by the corresponding Kohn-Sham operator. To this end we have to look at the minimization of the total energy with respect to the density of a multi-determinantal wavefunction 4. We write the density as ... 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

According to the Hohenberg-Kohn theorem of the density functional theory, the ground-state electron density determines all molecular properties. E. Bright Wilson [46] noticed that Kato s theorem [47,48] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density ... 

Density functional theory (DFT) [9] is becoming increasingly important in determining chemical properties. Typical methods involve the BLYP functional and the hybrid B3LYP procedure. DFT methods are attractive in that they are often highly cost effective and therefore offer the possibility of application to quite large systems, provided that they are suitably reliable. 

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