Density functional theory DFT

DFT is based on the notion that the electron density is uniquely defined by the external potential, which for a molecule or condensed solid is simply the interaction potential with the constituent nuclei. This was put forward as the Hohen-berg-Kohn theorem [9] which states that the ground-state electron density po(r) minimizes the energy functional, T[p(r)], which for a system of electrons interacting with nuclei is given by  

Coulomb and exchange integrals increases rapidly with the number of electrons considered. 

The form of F[p(r)] is cmcial for any successful appHcation of DFT. Kohn and Sham postulated [10] that F[p(r)j can be written  

Progress is usually made by writing the electron density in terms of a set of one-electron wavefunctions, xj/, such that  

These equations map the problem of a system of interacting electrons onto a system of non-interacting electrons moving in an effective potential. The behavior of the exchange-correlation potential as a function of the electron density is a complex problem. The interactions it represents can be long-range in nature, so even if the correct form of f xc were known, its calculation would be as complex as the post-HF methods mentioned at the end of the previous section. To make DFT practical and efficient, approximations for Fxc are introduced. 

5 Independent-electron models 5.5 Density functional theory (DFT) 

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 is based on the fact that the average energy of an electron in an atom or a molecule is a function of its probability density p in the vicinity of the nucleus or nuclei, and p is a function of its position in space. A function of a function is a functional. In DFT, the energy of an electron is expressed as a sum of three classical energies, a kinetic energy T, a nuclear charge repulsion potential Vn, and an electron charge repulsion potential Ve, plus one quantum mechanical 

The Kohn-Sham equations are very like the Schroedinger equations 

The classical terms are readily computed but difficulty is encountered in calculating Exc called the exchange-correlation energy. 

The exchange-correlation functional can be divided into two parts, the exchange energy Ex, and the correlation energy Ec 

Several schemes exist for calculating Exc. All are largely empirical, containing adjustable parameters, though each rests on a reasonable theoretical model. One of the most accurate functionals for hydrocarbon thermochemistry, as indicated by comparison of calculated Af//298 values with experiment, was devised by Becke (1988) and Lee, Yang, and Parr (1988), hence the combination is called a BLYP theory. The B3LYP method uses a hybrid functional containing 3 parameters. 

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To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

In formulating a mathematical representation of molecules, it is necessary to define a reference system that is defined as having zero energy. This zero of energy is different from one approximation to the next. For ah initio or density functional theory (DFT) methods, which model all the electrons in a system, zero energy corresponds to having all nuclei and electrons at an infinite distance from one another. Most semiempirical methods use a valence energy that cor-... 

Density functional theory (DFT) has become very popular in recent years. This is justified based on the pragmatic observation that it is less computationally intensive than other methods with similar accuracy. This theory has been developed more recently than other ah initio methods. Because of this, there are classes of problems not yet explored with this theory, making it all the more crucial to test the accuracy of the method before applying it to unknown systems. 

This technique has been applied occasionally to orbital-based methods, where it is called seam searching. The rest of the techniques mentioned in this chapter are applicable to semiempirical, density functional theory (DFT), and ah initio techniques. 

Davidsou-Fletcher-Powell (DFP) a geometry optimization algorithm De Novo algorithms algorithms that apply artificial intelligence or rational techniques to solving chemical problems density functional theory (DFT) a computational method based on the total electron density... 

Another approach to calculating molecular geometry and energy is based on density functional theory (DFT). DFT focuses on the electron cloud corresponding to a molecule. The energy of a molecule is uniquely specified by the electron density functional. The calculation involves the construction of an expression for the electron density. The energy of the system is then expressed as... 

Let us underline some similarities and differences between a field theory (FT) and a density functional theory (DFT). First, note that for either FT or DFT the standard microscopic-level Hamiltonian is not the relevant quantity. The DFT is based on the existence of a unique functional of ionic densities H[p+(F), p (F)] such that the grand potential Q, of the studied system is the minimum value of the functional Q relative to any variation of the densities, and then the trial density distributions for which the minimum is achieved are the average equihbrium distributions. Only some schemes of approximations exist in order to determine Q. In contrast to FT no functional integrations are involved in the calculations. In FT we construct the effective Hamiltonian p f)] which never reduces to a thermo-... 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

Fischer-type carbene complexes, generally characterized by the formula (CO)5M=C(X)R (M=Cr, Mo, W X=7r-donor substitutent, R=alkyl, aryl or unsaturated alkenyl and alkynyl), have been known now for about 40 years. They have been widely used in synthetic reactions [37,51-58] and show a very good reactivity especially in cycloaddition reactions [59-64]. As described above, Fischer-type carbene complexes are characterized by a formal metal-carbon double bond to a low-valent transition metal which is usually stabilized by 7r-acceptor substituents such as CO, PPh3 or Cp. The electronic structure of the metal-carbene bond is of great interest because it determines the reactivity of the complex [65-68]. Several theoretical studies have addressed this problem by means of semiempirical [69-73], Hartree-Fock (HF) [74-79] and post-HF [80-83] calculations and lately also by density functional theory (DFT) calculations [67, 84-94]. Often these studies also compared Fischer-type and... 

Quantum-chemical calculations which utilize the density functional theory (DFT) are now perhaps amongst the most frequently performed because of their relatively low cost and high accuracy. Structural results obtained from DFT based methods are often as good as those derived from MP2 calculations. It is well documented that DFT methods, especially those involving hybrid functionals such as B3LYP, B3P86 and B3PW91, yield reliable... 

The molecular interpretation of major topics in catalytic kinetics will be highlighted based on insights on the properties of transition-state intermediates as deduced from computational chemical density functional theory (DFT) calculations. 

Assuming that substituted Sb at the surface may work as catalytic active site as well as W, First-principles density functional theory (DFT) calculations were performed with Becke-Perdew [7, 9] functional to evaluate the binding energy between p-xylene and catalyst. Scalar relativistic effects were treated with the energy-consistent pseudo-potentials for W and Sb. However, the binding strength with p-xylene is much weaker for Sb (0.6 eV) than for W (2.4 eV), as shown in Fig. 4. 

A group of investigators recently suggested that the density-functional theory (DFT), which calculates IR and Raman spectra, is a useful tool for direct characterization of the structures of diamondoids with increasing complexity [66]. They applied DFT to calculate Raman spectra whose frequencies and relative intensities were shown to be in excellent agreement with the experimental Raman spectra for C26H30, thus providing direct vibrational proof of its existence. 

Stener and co-workers [59] used an alternative B-spline LCAO density functional theory (DFT) method in their PECD investigations [53, 57, 60-63]. In this approach a normal LCAO basis set is adapted for the continuum by the addition of B-spline radial functions. A large single center expansion of such... 

The authors carried out theoretical calculations on this system as well as on the [ (PMej) ] system to compare their reactivity with hexafluorobenzene. They found that the barrier for [ (liPr) ] is approximately 10 kJ/mol lower in energy than the corresponding barrier for the phosphine-bearing system. This value was in agreement with the different reactivity of both complexes but could not fully explain the large difference in reaction times. Density functional Theory (DFT) calculations also showed that the trans product is more stable than the cis product (total energies are respectively -130.9 and 91.1 kJ/mol), which was in agreement with the experimental values. 

The exact nature of the catalytically active Ni species in these reactions is yet to be conclusively established. Hydrodechlorination proves optimal with a NHC Ni ratio of 2 1 suggesting that 14-electron Ni(NHC)2 is involved, whereas the 1 1 NHC Ni ratio necessary for hydrodefluorination implies that it is the 12-electron mono-carbene adduct Ni(NHC) which is catalytically active [10]. Smdies by Matsubara et al. revealed that treatment of NKacac) with either one or two equivalents of IMes HCl 1 or SlMes HCl 2 in the presence of NaOHu formed the mono-NHC complex Ni(NHC)(acac)j which, upon reduction with NaH in the presence or absence of carbene, formed Ni(NHC)2 [11]. Density functional theory (DFT) calculations suggest that the strength of the Ni-NHC bond (ca. 50 kcal/mol) makes... 

Density Functional Theory (DFT) has shown that low-coordinated sites on the gold nanoparticles can adsorb small inorganic molecules such as O2 and CO, and the presence of these sites is the key factor for the catal5dic properties of supported gold nanoclusters. Other contributions, induced by the presence of the support, can provide parallel channels for the reaction and modulate the final efficiency of Au-based catalysts. Also these calculations extended for the adsorption of O and CO on flat and... 

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