Density functional methods, DFT

The computational chemistry community has dedicated a lot of attention in recent years to the application of density functional methods (DFT) to molecular systems (for general reviews see ref. (1)). This is due, on one hand, to the accuracy that the more sophisticated of these methods can exhibit, usually comparable to highly-correlated post-Hartree-Fock methodologies. On the other hand, for the same degree of accuracy, DFT methods scale much more slowly with the size N of the system than conventional correlated ab initio methods and are therefore applicable to the study of larger systems. 

A more recent introduction to quantum chemistry is density-functional methods (DFT). These in fact pre-date the other methods but were for a long time used primarily by physicists and were applied to solids. The electron density is given by the square of the wavefunction, and is a measure of how much electron there is at any point. Working from the realization that all properties of a molecule in its lowest energy state can be calculated from the electron density, Walter Kohn and his co-workers re-wrote the wave equation in terms of electron density to show that... 

The connecting link between ab initio calculations and vibrational spectra is the concept of the energy surface. In harmonic approximation, usually adopted for large systems, the second derivatives of the energy with respect to the nuclear positions at the equilibrium geometry give the harmonic force constants. For many QM methods such as Hartree-Fock theory (HF), density functional methods (DFT) or second-order Moller-Plesset pertiubation theory (MP2), analytical formulas for the computation of the second derivatives are available. However, a common practice is to compute the force constants numerically as finite differences of the analytically obtained gradients for small atomic displacements. Due to recent advances in both software and computer hardware, the theoretical determination of force field parameters by ab initio methods has become one of the most common and successful applications of quantum chemistry. Nowadays, analysis of vibrational spectra of wide classes of molecules by means of ab initio methods is a routine method [85]. 

How can a theoretical method decide between proposed mechanisms, and how can the origin of the enzymatic power be identified This review will try to answer these questions for one particular theoretical approach, the one where an active site model is treated by accurate quantum mechanical (QM) methods. The main idea in the QM active site approach is to make sure that the computational results have the required accuracy. During the last decade the accuracy of density functional methods (DFT) has been dramatically improved, and in particular the hybrid B3LYP functional has achieved a remarkable accuracy [8, 9]. The use of DFT has also made it possible to treat dramatically larger molecular systems than can be done with conventional wave-function methods of similar accuracy. In spite of this important development, DFT models have usually been limited to 50-60 atoms, but more recently systems with more than 100 atoms have been treated efficiently. Still, even 100 atoms is a very small part of the total number of 8,300 atoms in yeast ODCase, not counting hydrogens or surrounding water molecules. Thus a very severe selection has to be made when the enzyme model is set up, and an important task is to select the residues required to solve the mechanism and to analyze all important contributions. 

The observable properties of solid materials are governed by quantum mechanics, as expressed by solutions of a Schrodinger equation for the motion of the electrons and the nuclei. However, because of the inherent difficulty of obtaining even coarsely approximate solutions of the full many body Schrodinger equation, one t5q)ically focuses on reduced descriptions that are believed to capture the essential energetic of the problem of Interest Tow main quantum mechanics method are Ab initio and density function method (DFT). ... 

The most widespread method, which can be applied to large systems, where the use of MC SCF or Coupled Clusters (an approximate singles-doubles CC2) methods is not possible, is density functional method (DFT). DFT is now an acronym for a variety of different approximations of the functional [90]. It has a comparatively low scaling with the system size and typically also has lower demands on the basis set. Its cost is similar to that of RHF model and nowadays it plays a dominant role in the theoretical investigations of the vibrational rotatory strength/ gi, go- The introduction of the DFT methods caused that the evaluation of rotatory strength became a routine procedure, now used to support experimental works. 

The majority of the undergoing theoretical (hyper)polarizability studies on semiconductor clusters rely on conventional ab-initio and density functional methods (DFT) As it is broadly... 

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... 

Recently, a third class of electronic structure methods have come into wide use density functional methods. These DFT methods are similar to ab initio methods in many ways. DFT calculations require about the same amount of computation resources as Hartree-Fock theory, the least expensive ab initio method. 

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. 

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... 

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... 

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