Atomic orbitals electronic structure calculations

Once the least-squares fits to Slater functions with orbital exponents e = 1.0 are available, fits to Slater function s with oth er orbital expon cn ts can be obtained by siin ply m ii Itiplyin g th e cc s in th e above three equations by It remains to be determined what Slater orbital exponents to use in electronic structure calculation s. The two possibilities may be to use the "best atom" exponents (e = 1. f) for II. for exam pie) or to opiim i/e exponents in each calculation. The "best atom expon en ts m igh t be a rather poor ch oicc for mo lecular en viron men ts, and optirn i/.at ion of non linear exponents is not practical for large molecules, where the dimension of the space to be searched is very large.. 4 com prom isc is to use a set of standard exponents where the average values of expon en ts are optirn i/ed for a set of sin all rn olecules, fh e recom -mended STO-3G exponents are... 

There are two types of basis functions (also called Atomic Orbitals, AO, although in general they are not solutions to an atomic Schrodinger equation) commonly used in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). Slater type orbitals have die functional form... 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

A complicating factor is tlrat each spitr density matrix element is multiplied by the corresponding basis function overlap at tire nuclear positions. The orbitals having maximal amplitude at the nuclear positions are tire core s orbitals, which are usually described with less flexibility than valence orbitals itr typical electronic structure calculations. Moreover, actual atomic s orbitals are characterized by a cusp at tire nucleus, a feature accurately modeled by STOs, but only approximated by the more commonly used GTOs. As a result, tlrere are basis sets in the literature tlrat systematically improve tire description of the core orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and EPR-III (Barone 1995). 

Electronic structure calculations for transition metal carbides (Neckel 1990, Le 1990, Le et al. 1991) reveal significant contributions to cohesion by all three main types of chemical bonding. Covalent bonds are due to the formation of molecular orbitals by combining atomic d-orbitals of the metal with p-orbitals of C. Ionic bonds result from charge transfer from the metal to the non-metal. Metallic bonds are due to s electrons and also to a non-vanishing density of d-p electronic states (DOS) existing at the Fermi level (Figure 7.30). The main difference between the DOS curves calculated for stoichiometric ZrC, TiC or HfC and NbC, TaC or VC is... 

In Fig. 4.3 we plot the density dependence of the resulting exchange potentials. The relevant range of density values for electronic structure calculations is indicated by the j8-values at the origin and the expectation value of the radial coordinate of the lSl/2-orbital for the Kr and Hg atoms. One finds that relativistic effects are somewhat more pronounced for than for and are definitely relevant for inner shell features of high Z-atoms. 

With the use of the DV-Xa molecular orbital method, electronic structure calculations have been performed to investigate the impurity effect on material properties. Firstly, calculations were done for F atoms substituted for 0 (oxygen) atoms in copper oxide superconductors. It was found that the population of the atomic orbitals of F atoms is small in HOMO (highest occupied molecular orbital) and a small fraction of charge carriers enters the impurity sites. The F impurities are therefore expected to be effective for pinning magnetic flux lines in Cu oxide superconductors. 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

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