First-principles calculations defined

A commonly used quantity to present the information obtained from a first-principles calculation based on the density-functional method is the local density of states (LDOS) at every energy value below the Fermi level at zero absolute temperature. Because every state has an energy eigenvalue, the information with both spatial and energetic distributions is important for many experiments involving energy information. The LDOS p(r, ) at a point r and at an energy level E is defined as... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

Density functional theory thus offers precise definitions of previously poorly defined chemical quantities, enabling their first principles calculation. This part of DFT is termed by Parr as conceptual DFT [6]. 

New experimental methods have been developed to growth well defined films of oxides on metal supports in UHV conditions. This approach allows in principle to overcome some difficulties connected to the use of electron spectroscopies for the study of insulating materials, as most of the oxides are. Recently, some very good review articles have been published on this subject [3-10]. It is no surprise that the increasing experimental activity has stimulated a parallel computational activity based on high-quality first principle calculations. 

The calculation of Slot and Fi is a simple and fast numerical procedure that does not involve quantum mechanical calculations, although the latter are often used when generating potentials. Potential functions contain fitting parameters, which are adjusted to give desired properties of the material known from experiment or first-principles calculations. Once the fitting procedure is complete, the parameters are not subject to any further changes and the potential thus defined is used in all subsequent simulations of the material. The underlying assumption is that a... 

Note that here individual blobs with a size of f is defined as portions of the confined chain whose center of gravity undergoes the Brownian motion under the agitation of the thermal energy. Such a dependence (Eq. 5.8) was supported by the recent first principle calculation [36, 37]. 

For large systems the parametrisation method has to be efficient and provide fast convergence. The reference values can be obtained either by higher level first principles - calculations or, preferably, from the experimental measurements of certain properties. Useful reference quantities are energy differences among frontier molecular orbital levels, the electric dipole moment, the symmetry and eharge distribution of specifie orbitals, fragment populations, etc. For instanee, the cost (or penalty) function can be defined as... 

Although the absorption of a gas in a gas-liquid disperser is governed by basic mass-transfer phenomena, our knowledge of bubble dynamics and of the fluid dynamic conditions in the vessel are insufficient to permit the calculation of mass-transfer rates from first principles. One approach that is sometimes fruitful under conditions where our knowledge is insufficient to completely define the system is that of dimensional analysis. 

In this chapter, we have summarized (recent) progress in the mechanistic understanding of the oxidation of carbon monoxide, formic acid, methanol, and ethanol on transition metal (primarily Pt) electrodes. We have emphasized the surface science approach employing well-defined electrode surfaces, i.e., single crystals, in combination with surface-sensitive techniques (FTIR and online OEMS), kinetic modeling and first-principles DFT calculations. 

It is not easy to calculate oscillator strengths from first principles except in some very simple cases. On the other hand, the oscillator strength distribution must fulfill certain sum rules, which in some cases help to unravel their character. Referring the (dipole) oscillator strength for the transition from the ground state with excitation energy n to state n as fn, a sum may be defined by... 

In the most elementary models of orbital structure, the quantities that explicitly define the potential V are not computed from first principles as they are in so-called ab initio methods (see Section 6). Rather, either experimental data or results of ab initio calculations are used to determine the parameters in terms of which V is expressed. The resulting empirical or semi-empirical methods discussed below differ in the sophistication used to include electron-electron interactions as well as in the manner experimental data or ab initio computational results are used to specify V. 

When the limiting conditions of the friction approximation are not valid, e.g., there is strong non-adiabatic coupling or rapid temporal variation of the coupling, there is at present no well-defined first principles method to calculate the breakdown in the BOA. The fundamental problem is that DFT cannot calculate excited states of adsorbates and quantum chemistry techniques, that can in principle calculate excited states, are not possible for extended systems. 

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