DFT+U method

In this chapter, we show that computational simulations based on first-principles DFT calculations have been extensively performed to study the various geometric, electronic, and catalytic properties of bulk and surfaces of rare earth Ce02. In particular, the DFT calculations corrected by on-site Coulomb interaction have been shown to reproduce available experimental results, indicating that the DFT + U method is reliable to explain and predict the catalytic activities of rare earth Ce02-based materials. 

The formation of 5-methyl-l,2,3-oxadiazole 3-oxide 9 by the fixation of nitric oxide (NO) using propynyllithium has been investigated using ab initio (U)MP2 and DFT/(U)B3LYP methods (Scheme 1) <2005JOC5045>. 

Figure /. A/eaAi unsigned error over all 202 data vs. gradient cost for AMI (m), ab initio (U), CBS (k), G3 (O), MCCM (m), DFT (V), and hybrid DFT (A) methods. The figure includes all the methods that are included in Tables 3-5.

The fact that self-interaction errors are canceled exactly in HF calculations suggests that a judicious combination of an HF-like approach for localized states with DFT for everything else may be a viable approach for strongly correlated electron materials. This idea is the motivation for a group of methods known as DFT+U. The usual application of this method introduces a correction to the DFT energy that corrects for electron self-interaction by introducing a single numerical parameter, U — J, where U and J involve different aspects of self-interaction. The numerical tools needed to use DFT+U are now fairly widely implemented in plane-wave DFT codes. 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. U njust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

In defects on transition metal oxides DFT again fails, giving structures that show unlikely relaxations and tending to delocalize electrons associated with the defect into conduction band states. Hybrid functionals and DFT + U have also been used to correct the models in these cases, giving a localized picture of surface reduction. These methods are now able to give useful descriptions of reactions at these defect sites, including the transfer of electrons between surface and adsorbate required in redox chemistry. 

Although this intermediate situation is not the best scenario for DFT calculations, which tend to energetically favor the delocalized (metallic) state over the (magnetic) states with localized electrons, we will show that the consideration of electronic repulsion, even in an extremely simple way such as the DFT-I-U method [34], is able to explain the observed physical behavior in such a complex case as K4P3. 

The DFT-l-U approach [34] is a simple way to introduce the effects of electron repulsion in the framework of DFT. Although from a conceptual point of view it is quite easy to understand, the DFT-l-U method has, however, some disadvantages, the most serious one being the introduction of the electronic repulsion U as an empirical parameter of an unknown magnitude. Although, as mentioned, electron... 

Table 2. The multiplet energies of ruby (eV) calculated by the simple DFT-CI method and the DFT-CI-CC method, using the unrelaxed cluster (U) and the relaxed cluster (R) together with the peak positions in the observed absorption spectra of ruby reported by Fairbank et al . The calculated multiplet energies are averaged within each state in the O/, notation.

Density-functional Methods for Strongly Correlated Systems SIC DFT and DFT- -U Approaches... 

Such an approach introduces, in fact, the self-interaction correction for these orbitals. Therefore DFT-SIC and DFT-I-U methods give close results in many cases. 

The intramolecular radical cation 2 -I- 2-cycloadditions of bis(styrenes) were investigated by the DFT (U)B3LYP method. The key reaction pathway of the cycloaddition involves the formation of a five-membered ring intermediate in a stepwise manner. ... 

The substituent effects on the H-bonding in an adenine-uracil (A-U) base pair were studied for a series of common functional groups [99JPC(A)8516]. Substitutions in the 5 position of uracil are of particular importance because they are located toward the major groove and can easily be introduced by several chemical methods. Based on DFT calculation with a basis set including diffuse functions, variations of about 1 kcal/mol were found for the two H-bonds. The solvent effects on three different Watson-Crick A-U base pairs (Scheme 100) have been modeled by seven water molecules creating the first solvation shell [98JPC(A)6167]. 

The employment of NMR-active isotopes permits to access experimental parameters which are intrinsically difficult to measure, unless a significant concentration of the sugar is present in the NMR tube. For instance, aqueous solutions of N-acetyIncuraminic acid, labeled with 13C at Cl, C2, and/or C3, were analyzed to detect and quantify the various chemical species present in equilibrium at different pHs. In fact, in addition to the expected a and (3 pyranose forms, acyclic keto, keto hydrate and enol forms were identified on the basis of 13C NMR spectroscopic data. Besides, DFT methods were employed to predict the effect of enol and hydrate structure on the coupling constant values Jc,u and /c c involving C2 and C3, finding that 2/c2,h3 can be safely used to differentiate the cis and tram isomers of the enol forms.9... 

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