Density functional theory DFT methods

The DFT method [16] rests upon a number of concepts and assumptions that will be briefly summarized in the following with a minimum amount of technical detail. Walter Kohn received the Nobel prize in 1998, jointly with John Pople. 

The Hohenberg-Kohn theorem states that the energy of a system of electrons in an external potential V (R) is a functional of the electron density (a fiinetion is a mathematical entity that takes a certain value for each specified value of a variable, a functional is a mathematical entity that takes a certain value for each specified value of another function). Thus, the ground state energy of a many-electron system is uniquely determined by the ground-state density. 

In the Kohn-Sham assumption, for a system of interacting electrons moving in a potential V(R) (i.e. a molecular electron cloud in the field of the nuclei) a local potential, Vks(R), can be introduced, such that a system of non-interacting electrons moving in the Vks (R) field will have the same density as the exact density of the interacting electron system. 

The Kohn-Sham local potential contains a term for the electron-nuclei coulombic attraction, a term for the electron-electron coulombic repulsion, and an exchange term, which incorporates all the exchange and electron correlation effects  

The Kohn-Sham hamiltonian operator (compare with the HF hamiltonian of equation 3.39) and the stationary state equation are then written  

---

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-... 

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... 

However, even the best experimental technique typically does not provide a detailed mechanistic picture of a chemical reaction. Computational quantum chemical methods such as the ab initio molecular orbital and density functional theory (DFT) " methods allow chemists to obtain a detailed picture of reaction potential energy surfaces and to elucidate important reaction-driving forces. Moreover, these methods can provide valuable kinetic and thermodynamic information (i.e., heats of formation, enthalpies, and free energies) for reactions and species for which reactivity and conditions make experiments difficult, thereby providing a powerful means to complement experimental data. 

Group 2 complexes are formally electron deficient and conformationally floppy only small energies (often only 1-2 kcal mol-1) are required to alter their geometries by large amounts (e.g., bond angles by 20° or more). In such cases, the inclusion of electron-correlation effects becomes critical to an accurate description of the molecules structures. Both HF/MP2 and density functional theory (DFT) methods have been applied to organoalkaline earth compounds. DFT approaches, which implicitly incorporate electron correlation in a computationally efficient form, are generally the more widely used. Molecular orbital calculations that successfully reproduce bent... 

Currently, the density functional theory (DFT) method has become the method of choice for the study of reaction mechanism with transition-metals involved. Gradient corrected DFT methods are of particular value for the computational modeling of catalytic cycles. They have been demonstrated in numerous applications for several elementary processes, to be able to provide quantitative information of high accuracy concerning structural and energetic properties of the involved key species and also to be capable of treating large model systems.30... 

It is seen from this comparison that the geometrical parameters obtained by the IPM and by the density functional theory (DFT) methods are close. It is enough to multiply re by 1.44 to obtain the same C H O distance for the TS calculated by the DFT method. The following parametric equations were proposed for the estimation of the C—H, O—H, and C—O distances in the reaction center of TS for reactions ROO + RH in the IPM method [34,35],... 

During recent years the combination of matrix IR spectroscopy — using highly sensitive F1IK spectrometers — and the accurate calculation of vibrational data with ab initio or density functional theory (DFT) methods has... 

Hqm is the Hamiltonian of the QM region and might be based on semiempirical, ah initio molecular orbital or density functional theory (DFT) methods. Hqm/mm represents the interactions between the QM and MM regions, Hmm is the Hamiltonian of the purely MM region, and Hboundary is Hamiltonian for the boundary of the system, if this contribution is included. The corresponding total energy ( tot) of the QM/MM system is ... 

Simply to look at the literature is to convince yourself of the importance that density functional theory (DFT) methods have attained in molecular calculations. But there is among the molecular physics community, it seems to me, a widespread sense of unease about their undoubted successes. To many it seems quite indecent that such a cheap and cheerful approach (to employ Peter Atkins s wonderful phrase) should work at all, let alone often work very well indeed. I think that no-one in the com-mimity any longer seriously doubts the Hohenberg-Kohn theo-rem(s) and anxiety about this is not the source of the unease. As Roy reminded us at the last meeting, the N— representability problem is still imsolved. This remains true and, even though the problem seems to be circumvented in DFT, it is done so by making use of a model system. He pointed out that the connection between the model system and the actual system remains obscure and in practice DFT, however successful, still appears to contain empirical elements And I think that is the source of our present unease. 

Vibrational frequencies of 1,4-benzodioxin using the density functional theory (DFT) method, as well as the conventional HF and MM3 force-field methods, were calculated to evaluate the frequency prediction capability of each computational method and get a better understanding of the vibrational spectra . 

Density functional theory (DFT) methods they share the same shortcoming of Hartree-Fock in that the mono-electronic character and the inadequacy of density functional formulations prevent the description of polarization/disper-sion effects. Adoption of such methods requires a careful choice of the proper density functional [45],... 

The conformations of various cyclic phosphate derivatives of 2-deoxyribose (e.g., compound 6, Section 10.09.3.2) have been studied using density functional theory (DFT) methods <2005JOC7107>. 

These workers also calculated the relative stability of the tautomers lOa-c in the gas phase by ab initio and density functional theory (DFT) methods and in solution using several continuum solvation models such as self-consistent reaction fields (SCRF) and the Poisson-Boltzmann method. These results showed good agreement between the experimental and theoretical approaches. 

The first way has been followed in what has become known as Car-Parrinello molecular dynamics (CPMD) (9). A solute and 60-90 solvent molecules are considered to represent the system, and the QM calculations are performed with density functionals, usually of generalised gradient approximation type (GGA), such as the Becke-Lee-Young-Parr (BLYP) (10) or the Perdew-Burke-Enzerhofer (PBE) (11,12) functionals. It is clear that the semiempirical character of concurrent density functional theory (DFT) methods and the use of these simple functionals imply a number of error sources and do not really provide a method-inherent control procedure to test the reliability of results. Recently it has been shown that these functionals even do not enable a correct description of the solvent water itself, as at ambient temperature they will describe water not as liquid but as supercooled system... 

The homolytic bond dissociation energies (BDEs) of phenohc O—H bonds has been the subject of a computational study focusing on substituent effects by ab initio and density functional theory (DFT) methods.6 Consistent overestimation of the BDEs by MP2 and MP4 calculations was associated with spin contamination in the reference UHF wave functions, whilst the DFT calculations (particularly the B3LYP/6-31G level of theory) were relatively unaffected. Ab initio calculations of the photosensitized C—C BDEs of /f-phenethyl ethers has revealed a significant configurational... 

The influence of the dihedral angle on the natural bite angle of Ru complexes bearing biaryl units such as Ci-TunePhos 7, as chiral ligands, has been studied by MM2 <2000JOC6223, 2003TL823> and density functional theory (DFT) methods <2005CCL1555>. 

These studies, which employed density functional theory (DFT) methods (B3LYP/LANL2DZ/Gaussian 98) proposed that the reactions of all alkali metal phenoxides with C02 followed a similar ground mechanism that comprised three intermediates and three transition states. In step 1, C02 must first be activated by an alkali metal phenoxide. In the case of the sodium phenoxide [24a], C02 can only attack at the polarized O-Na bond to form a Ph0Na/C02 complex as the first intermediate (structure 4). The calculation definitely rules out a direct C-C bond formation at the aromatic ring. 

Implicit in the above discussion was the assumption that the geometry of the molecules under study are also determined to sufficient accuracy. Thus, we generally also need correlated methods for the geometry optimizations prior to NMR calculations. Because many of the systems we study are large, we have found that density functional theory (DFT) methods are often very useful, providing the effects of electron correlation in a computationally expedient fashion. In the following we provide several examples that demonstrate the utility of NMR calculations in the interpretation of experimental studies in catalysis. 

---