Density-functional theory calculations

The development of DFT is based on Kohn and Hohenberg s mathematical theorem, which states that the ground state of the electronic energy can be calculated as a functional of the electron density [18], The task of finding the electron density was solved by Kohn and Sham [19]. They derived a set of equations in which each equation is related to a single electron wave function. From the single electron wave functions one can easily calculate the electron density. In DFT computer codes, the electron density of the core electrons, that is, those electrons that are not important for chemical bonds, is often represented by a pseudopotential that reproduces important physical features, so that the Kohn-Sham equations span only a select number of electrons. For each type of pseudopotential, a cutoff energy or basis set must be specified. 

A periodic slab calculation takes advantage of the symmetry of a surface. A supercell is set with atoms in a certain number of layers and vacuum space in the third dimension. The surface adjacent to the vacuum represents the active surface. Many DET codes employ periodic bonndary conditions in three dimensions. Eigure 8.3 presents a rendering of (111) and (211) surfaces of an fee metal and their respective supercells. To calculate the energy of a surface with, for example, four atomic layers, the bottom two layers of the slab are frozen in their initial position to mimic the bnlk phase and the top layers and adsorbates are allowed to relax. The initial positions are defined from the crystal structure using the corresponding computationally determined lattice constant. 

FIGURE 83 Rendering of (III) and (211) surfaces of an fee metal. The binding sites on (III) surface are shown. The surperceUs used in DFT ealculations are also presented. 

DFT packages that apply periodic boundary conditions use either the plane wave method (e.g., VASP [5, 25], DACAPO [7] and CASTEP [8]) or a linear combination of atomic orbitals (e.g., SIESTA [6]). Although the latter package is believed to be less accurate, it is computationally less demanding than the aforementioned plane wave methods, and offers a good balance between accuracy and computational cost. 

There are multiple methods for TS searches [26]. As an example, the one implemented in the SIESTA code [6] is the constrained optimization scheme [27, 28]. Initially, the distance between atoms participating in the bond that forms or breaks is constrained at an estimated value, and the total energy of the system is minimized with respect to all the other degrees of freedom. Then, this procedure is repeated with a new distance until the TS is found so that all forces on atoms vanish and the total energy is a maximum along the reaction coordinate but a minimum with respect to the remaining degrees of freedom. 

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A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

Density functional theory calculations have shown promise in recent studies. Gradient-corrected or hybrid functionals must be used. Usually, it is necessary to employ a moderately large basis set with polarization and diffuse functions along with these functionals. 

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... 

Density functional theory calculations on model diazenes of the type... 

Perform a low-level geometry optimization with a medium-sized basis set, for example, a Hartree-Fock or B3LYP Density Functional Theory calculation with the 6-31G(d) basis set. (For very large systems, a smaller basis set might be necessary.)... 

Ab initio Hartree-Fock and density functional theory calculations were performed to study the transition state geometry in intramolecular Diels-Alder cycloaddition of azoalkenes 55 to give 2-substituted 3,4,4u,5,6,7-hexahydro-8//-pyrido[l,2-ft]pyridazin-8-ones 56 (01MI7). 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

Thiourea Ugands can be bounded to the metal centre through one nitrogen atom, the sulfur atom, or the C = S double bond. These coordination modes were studied by density functional theory calculations for Rh-thiourea complexes (Scheme 13). No stable structure was attained by optimisation of the nitrogen coordination mode I but optimised geometries as trigonal-bipyramidal complexes were obtained for modes II and III. An coordination is determined for the latter complex through both S and C atoms. As this... 

Another stndy on binding to NHC complexes, that combined experiments and DFT (density functional theory) calculations was recently reported on a ruthenium system. This study shows the reversible binding of oxygen to the tetra-NHC complex [Ru(NHC) H)][BAr/] 6 (BAr/ = B (3,5-CF3) C H3 ), which leads to complex 7 (Scheme 10.2) [12]. Unexpectedly, the chemical shift of the hydride ligand undergoes a large downfield shift upon coordination to (from -41.2 ppm... 

Figure 1.22 Density Functional Theory calculations of the tetranuclear and dinuclear amidinate complexes at both the Gaussian 98 and ADF levels.

Cotton, F.A., Feng, X. and Timmons, D.J. (1998) Further study of very close non-bonded Cu -Cu contacts, molecular structure of a new compound and density functional theory calculations. Inorganic Chemistry, 37, 4066—4069. 

Ab initio methods allow the nature of active sites to be elucidated and the influence of supports or solvents on the catalytic kinetics to be predicted. Neurock and coworkers have successfully coupled theory with atomic-scale simulations and have tracked the molecular transformations that occur over different surfaces to assess their catalytic activity and selectivity [95-98]. Relevant examples are the Pt-catalyzed NO decomposition and methanol oxidation. In case of NO decomposition, density functional theory calculations and kinetic Monte Carlo simulations substantially helped to optimize the composition of the nanocatalyst by alloying Pt with Au and creating a specific structure of the PtgAu7 particles. In catalytic methanol decomposition the elementary pathways were identified... 

Electrocatalysis and Catalyst Screening from Density Functional Theory Calculations... 

Gong X-Q, Liu Z-P, Raval R, Hu P. 2003. A systematic study of CO oxidation on metals and metal oxides Density Functional Theory calculations. J Am Chem Soc 126 8-9. 

Skulason E, Karlberg GS, Rossmeisl J, Bligaard T, Greeley J, Jonsson H, Nprskov JK. 2007. Density functional theory calculations for the hydrogen evolution reaction in an electrochemical double layer on the Pt(lll) electrode. Phys Chem Chem Phys 9 3241-3250. 

Watson GW, Wells RPK, Willock DJ, Hutchings GJ. 2001. A comparison of the adsorption and diffusion of hydrogen on the 111 surfaces of Ni, Pd, and Pt from density functional theory calculations. J Phys Chem B 105 4889-4894. 

The assignment of the TR spectra were based on the known photochemistry of the aryl azides and comparison of the TR spectra vibrational frequencies to those predicted by density functional theory calculations for the likely photochemical intermediates. The good agreement between the experimental TR vibrational... 

Figure 7.20. Top ORTEP representation from the X-ray structure of the localized radical 45, and selected bond parameters (the parenthesis show the values determined computationally at the UB3LYP/6-31G level of theory). Bottom Density functional theory calculations for the rotation of the tert-butyl group. (Taken from ref. 113.)...

Lev, D. A. Grotjahn, D. B. Amouri, H. Reversal of reactivity in diene-complexed o-quinone methide complexes insights and explanations from ab initio density functional theory calculations. Organometallics 2005, 24, 4232 -240. 

Banerjee, A., Harbola, M. K., 1999, Density-Functional-Theory Calculations of the Total Energies, Ionization Potentials and Optical Response Properties with the van Leeuwen-Baerends Potential , Phys. Rev. A, 60, 3599. 

Facelli, J. C., 1998, Density Functional Theory Calculations of the Structure and the l5N and l3C Chemical Shifts of Methyl Bacteriopheophorbide a and Bacteriochlorophyll a , J. Phys. Chem. B, 102, 2111. 

Finley, J. W., Stephens, P. J., 1995, Density Functional Theory Calculations of Molecular Structures and Harmonic Vibrational Frequencies Using Hybrid Density Functionals , J. Mol. Struct. (Theochem), 357, 225. 

Seifert, G., Kruger, K., 1995, Density Functional Theory, Calculations of Potential Energy Surfaces and Reaction Paths in The Reaction Path in Chemistry Current Approaches and Perspectives, Heidrich, D. (ed.), Kluwer, Amsterdam. 

White, C. A., Head-Gordon, M., 1996, A 7-Matrix Engine for Density Functional Theory Calculations , J. Chem. 

Wiest, O., Houk, K. N., 1996, Density Functional Theory Calculations of Pericyclic Reaction Transition Structures Top. Curr. Chem., 182, 1. 

Figure 2.14. Density functional theory calculated (BP/DNP) IR spectral profiles of the N—O bond stretching region for the i71-Af 1Cu(NO)2 12I2 complex in the attracto and repulso conformation (after [75]).

Richardson WEI, Peng C, Bashford D, Noodleman L, Case DA (1997) Incorporating Solvation Effects into Density Functional Theory Calculation of Absolute Acidities. Int J Quantum Chem 61 207-217. 

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