Functional Theory Methods

It turns out to be useful to separate the energy functional into three different parts, which can be recognized from the Schrodinger equation a kinetic energy part 1 p], the attraction between nuclei and electrons , [Pl and the electron-electron repulsion [/ ] (the nuclear-nuclear repulsion adds a constant energy term to the total energy in the Born-Oppenheimer approximation). The electron-electron repulsion energy is further divided into two parts the classical electron-electron repulsion term J p, i. e. 

In order to make DFT a practical and accurate approach, Kohn and Sham reintroduced the use of orbitals in computational DFT. The reason for this is twofold most importantly, the kinetic energy term can be calculated exactly from the orbital approach (but will clearly only give the right number if the orbitals themselves are exact), and the electron density can be simply obtained by summing the square of the orbital wave functions  

To this end, Kohn and Sham assumed that the electrons in these so-called Kohn-Sham orbitals are non-interacting, such that the total electronic wave function can be written as a Slater Determinant. This allows the kinetic energy functional to be split into two parts, one of which, Ts, can be evaluated exactly, in a fashion very similar to the way it is done in Hartree-Fock theory, and a small correction term, which is formally absorbed in the exchange-correlation energy term. Thus, a general DFT energy expression can be written as 

If EofT is the exact energy, this equation may be regarded as the definition of the exchange-correlation energy. 

The exact meaning of the exchange-correlation energy is a difficult one, partly because the DFT definitions of exchange and correlation are not exactly the same as those used in wave-function methods. As mentioned in the previous section, electron correlation arises from the correlated behavior between electrons that is not accounted for in the mean-field Hartree-Fock approach. The exchange energy is the total electron-electron repulsion minus the Coulomb repulsion, and is basically a consequence of the Pauli principle, which states that no two electrons can have the same quantum numbers, i. e. two electrons in the same orbital must have opposite spin. 

In the last few years, methods based on Density Functional Theory have gained steadily in popularity. The best DFT methods achieve significantly greater accuracy than Harttee-Fock theory at only a modest increase in cost (far less than MP2 for medium-size and larger molecular systems). They do so by including some of the effects of electron correlation much less expensively than traditional correlated methods. 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself 

Exploring Chemistry with Electronic Structure Methods 

A variety of functionals have been defined, generally distinguished by the way that they treat the exchange and correlation components  

Local exchange and correlation functionals involve only the values of the electron spin densities. Slater and Xa are well-known local exchange functionals, and the local spin density treatment of Vosko, Wilk and Nusair (VWN) is a widely-used local correlation functional. 

DFT electron correlation, with parameters adjusted to provide the best fit with specific experimental data. The dynamic interaction between electrons is theoretically included by these density functional methods. This gives these methods the benefit of including electron-electron correlation for a computational expense similar to HF, giving DFT methods the major advantage of low computational cost compared to accuracy (Hohenberg and Kohn, 1964 Kohn and Sham, 1965 Parr and Yang, 1989 Foresman and Frisch, 1996). 

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Fattebert J-L and Bernholc J 2000 Towards grid-based 0(N) density-functional theory methods optimized nonorthogonal orbitals and multigrid acceleration Phys. Rev. B 62 1713-22... 

Some density functional theory methods occasionally yield frequencies with a bit of erratic behavior, but with a smaller deviation from the experimental results than semiempirical methods give. Overall systematic error with the better DFT functionals is less than with HF. 

We win run this job on methane at the Hartree-Fock level using the 6-31G(d) basis our molecule specification is the result of a geometry optimization using the B3LYP Density Functional Theory method with the same basis set. This combination is cited... 

The stability of SCF solutions for unknown systems should always be tested. Stability considerations apply to and may be tested for in calculations using Density Functional Theory methods as well. 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

Jursic, B. S., 1996, Computing Transition State Structures With Density Functional Theory Methods in Recent Developments and Applications of Modem Density Functional Theory, Seminario, J. M. (ed.), Elsevier, Amsterdam. 

Jursic, B. S., 1999, Study of the Water-Methanol Dimer With Gaussian and Complete Basis Set Ah Initio, and Density Functional Theory Methods , J. Mol. Struct. (Theochem), 466, 203. 

The interaction of carbon disulfide as a substrate in carbonic anhydrase model systems has been studied using density functional theory methods. A higher activation energy of CS2 compared to C02 in the reaction with [L3ZnOH]+ was due to the reduced electrophilicity of CS2. The reversibility of the reaction on the basis of these calculations is questionable with [L3ZnSC(0)SH]+ as intermediate.572... 

Unlike an orbital, the electron density of a molecule is a physical observable that can be obtained by experiment and also by calculation using ab initio or density functional theory methods. 

In particular, the reactions of electrophilically activated benzonitrile A-oxides with 3-methylenephthalimidines with formation of 2-isoxazolines and oximes and the cycloaddition between alkynyl metal(O) Fischer carbenes and nitrones leading to 4-isoxazolines have been investigated by density functional theory methods <06JOC9319 06JOC6178>. 

In this case, single point energies on HF/3-21G geometries were evaluated at the B3LYP/6-31G level, a density functional theory method. 

Density functional theory methods using the hybrid B3LYP functionals have been performed to study geometries and energetics of several intramolecular [2+3] dipolar cycloadditions of azides to nitriles (Section 11.06.6.1) toward fused tetrazole formation, including tetrazoles 14 and 15 <2003JOC9076>. 

A review of literature would show that a suite of QC descriptors have also been used in QSARs for biological and toxicological correlations. Such indices have been derived both from semiempirical and ab initio (Hartree Fock and density functional theory) methods. In particular, in our QSAR studies, we have used the following levels of QC indices local and global electrophilicity indices [11],... 

This mechanistic question is one of the examples of the success of density functional theory methods in organometallic chemistry. Earlier work on the reaction mechanism could not discriminate between the two alternatives. Analysis of the different orbitals based on extended Hiickel calculations came to the result that the [3+2] pathway is more likely, but could not exclude the possibility of a [2+2] pathway [13]. Similar conclusions where obtained from the results of Hartree-Fock calculations in combination with QCISD(T) single point calculations [21], Attempts to use Ru04 as a model for osmium tetraoxide indicated that the formation of an oxetane is less favorable compared to the [3+2] pathway, but still possible [22, 23],... 

Most published cytochrome P450 catalytic cycles will show the end-on peroxo coordination as shown in Figure 7.14 (4,5a, 5b). Shaik and co-workers have used density functional theory methods to propose a two-state theory... 

Y. Zhao and D. G. Truhlar, Hybrid Meta Density Functional Theory Methods for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions The MPW1B95 and MPWB1K Models and Comparative Assessments for Hydrogen Bonding and van der Waals Interactions, J. Phys. Chem. A 108 (2004), 6908. 

G. Pacchioni, Modeling Doped and Defective Oxides in Catalysis with Density Functional Theory Methods Room for Improvements, J. Chem. Phys. 128... 

The NEB method has been applied successfiilly to a wide range of problems, for example studies of diffusion processes at metal surfaces,28 multiple atom exchange processes observed in sputter deposition simulations,29 dissociative adsorption of a molecule on a surface,25 diffusion of rigid water molecules on an ice Ih surface,30 contact formation between metal tip and a surface,31 cross-slip of screw dislocations in a metal (a simulation requiring over 100,000 atoms in the system, and a total of over 2,000,000 atoms in the MEP calculation),32 and diffusion processes at and near semiconductor surfaces (using a plane wave based Density Functional Theory method to calculate the atomic forces).33 In the last two applications the calculation was carried out on a cluster of workstations with the force on each image calculated on a separate node. 

John Slater, bom Oak Park, Illinois, 1900. Ph.D. Harvard, 1923. Professor of physics, Harvard, 1924-1930 MIT 1930-1966 University of Florida at Gainesville, 1966-1976. Author of 14 textbooks, contributed to solid-state physics and quantum chemistry, developed X-alpha method (early density functional theory method). Died Sanibel Island, Florida, 1976. 

R. Improta, V. Barone, G. Scalmani, M.J. Frisch, A state-specific polarizable continuum model time dependent density functional theory method for excited state calculations in solution. J. Chem. Phys. 125, 054103 (2006)... 

The chiral discrimination in the self-association of chiral l,3a,4,6a-tetrahydroi-midazo[4,5-d]imidazoles 3 has been studied using density functional theory methods [37], (Scheme 3.20). Clusters from dimers to heptamers have been considered. The heterochiral dimers (RR SS or SS RR) are more stable than the homochiral ones (RR RR or SS SS) with energy differences up to 17.5 kJ mol-1. Besides, in larger clusters, the presence of two adjacent homochiral molecules imposes an energetic penalty when compared to alternated chiral systems (RR SS RR SS...). The differences in interaction energy within the dimers of the different derivatives have been analyzed based on the atomic energy partition carried out within the AIM framework. The mechanism of proton transfer in the homo- and heterochiral dimers shows large transition-state barriers, except in those cases where a third additional molecule is involved in the transfer. The optical rotatory power of several clusters of the parent compound has been calculated and rationalized based on the number of homochiral interactions and the number of monomers of each enantiomer within the complexes. 

This review has provided an overview of the studies of pericyclic reaction transition states using density functional theory methods up to the middle of 1995. Since the parent systems for most of the pericyclic reaction classes have been studied, a first assessment of DFT methods for the calculation of pericyclic transition structures can be made. 

Zhao Y, Truhlar DG (2004) Hybrid meta density functional theory methods for thermochemistry, thermochemical kinetics, and noncovalent interactions The MPW1B95 and MPWB1K models and comparative assessments for hydrogen bonding and van der Waals interactions, J Phys Chem A, 108 6908-6918... 

A mode coupling theory is recently developed [135] which goes beyond the time-dependent density functional theory method. In this theory a projection operator formalism is used to derive an expression for the coupling vertex projecting the fluctuating transition frequency onto the subspace spanned by the product of the solvent self-density and solvent collective density modes. The theory has been applied to the case of nonpolar solvation dynamics of dense Lennard-Jones fluid. Also it has been extended to the case of solvation dynamics of the LJ fluid in the supercritical state [135],... 

Before any computational study on molecular properties can be carried out, a molecular model needs to be established. It can be based on an appropriate crystal structure or derived using any technique that can produce a valid model for a given compound, whether or not it has been prepared. Molecular mechanics is one such technique and, primarily for reasons of computational simplicity and efficiency, it is one of the most widely used technique. Quantum-mechanical modeling is far more computationally intensive and until recently has been used only rarely for metal complexes. However, the development of effective-core potentials (ECP) and density-functional-theory methods (DFT) has made the use of quantum mechanics a practical alternative. This is particularly so when the electronic structures of a small number of compounds or isomers are required or when transition states or excited states, which are not usually available in molecular mechanics, are to be investigated. However, molecular mechanics is still orders of magnitude faster than ab-initio quantum mechanics and therefore, when large numbers of... 

The following sections are purposely separated into specific structural classes of square planar Pt" complexes of the general formulae Pt(NAN)(C=CR)2, [Pt(NANAN) (OCR)]+, Pt(NANAC)(C=CR), rra s-Pt(PR3)2(OCR)2, and d.v-Pt(PAP)(( =CR)2, where NAN is a bidentate 2,2 -bipyridine, NANAN and NANAC are tridentate polypyridines, PR3 is a monodentate phosphine, and PAP is a bidentate phosphine ligand. The final section of this work is dedicated to recent electronic structure calculations on these molecules with an emphasis on the successful application of DFT (density functional theory) and TD-DFT (time-dependent density functional theory) methods towards understanding the absorption and emission processes of these chromophores. 

Fig. 18. Ground state electronic structures for cresyl and o-(methylthio)cresyl phenoxyl radicals. Isosurface representations of molecular orbitals solved by ab initio density functional theory methods for cresyl (ere) and o-(methylthio)cresyl (mtc) phenoxyl radicals. Eigenvalues are listed and for each the SOMO is identihed with an asterisk ( ).

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