Density functional theory calculating minimum energy

Ruthenium has long been known to be an effective catalyst for ammonia synthesis. However, compared to the traditional iron-based catalysts, studies on ruthenium-based catalysts are limited. The rate determining step of ammonia synthesis, the dissociative adsorption of dinitrogen, has been shown to extremely structure sensitive on both iron and mthenium catalysts. To study this structure sensitivity on ruthenium, density functional theory calculations were performed on Ru(OOl) and Ru(llO) clusters. End-on, side-on, and dissociated adsorption states were investigated on both surfaces. While the Ru(llO) cluster could stabilize aJl three adsorption modes, a minimum energy structure for the side-on adsorption on Ru(OOl) could not be found. It is likely that this side-on mode can provide a low energy pathway to the dissociated state, thereby resulting in faster dissociative adsorption on Ru(llO). 

Figure Cl. 1.6. Minimum energy stmctures for neutral Si clusters ( = 12-20) calculated using density functional theory witli tire local density approximation. Cohesive energies per atom are indicated. Note tire two nearly degenerate stmctures of Si g. Ho K M, Shvartsburg A A, Pan B, Lu Z Y, Wang C Z, Wacher J G, Fye J L and Jarrold M F 1998 Nature 392 582, figure 2.

Density-functional theory is best known as the basis for electronic structure calculations. A variant of this theory can be used to calculate the structure of inhomogeneous fluids [35] the free energy of the fluid is expressed as a functional of the density of the various components a theorem asserts that this functional attains its minimum for the true density profiles. 

The DECP model successfully explained the observed initial phase of the fully symmetric phonons in a number of opaque crystals [24]. The absence of the Eg mode was attributed to an exclusive coupling between the electrons photoexcited near the r point and the fully symmetric phonons. A recent density functional theory (DFT) calculation [23] demonstrated this exclusive coupling as the potential energy surface (Fig. 2.4). The minimum of the potential surface of the excited state shifted significantly along the trigonal (z) axis,... 

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

Quantum chemical calculations of the Ne potential-energy hypersurface have shown that the qualitative shape shongly depends on the choice of the theoretical method and basis set. All the geometries represented in Scheme 6 have been shown to be minima on the potential smface, but most of them do not possess minima at all the levels of theories applied. Hexaazabenzene (13), for example, has a minimum for a stmctme at the HF level of theory. However, this geometry is a second-order saddle point with the density functional theory (DFT) and also at the MP2 level of theory. D2 hexaazabenzene has a minimum structure at DFT, but at the CCSD(T)/aug-cc-pVDZ level, the D2 geometry resembles a van der Waals complex of two N3 units, whereas it is a minimum structure at the CCSD(T)/cc-pVTZ level of theory. Similar behavior has been observed for most of the other isomers. 

---