Discrete variational methods results

Below is a brief review of the published calculations of yttrium ceramics based on the ECM approach. In studies by Goodman et al. [20] and Kaplan et al. [25,26], the embedded quantum clusters, representing the YBa2Cu307 x ceramics (with different x), were calculated by the discrete variation method in the local density approximation (EDA). Although in these studies many interesting results were obtained, it is necessary to keep in mind that the EDA approach has a restricted applicability to cuprate oxides, e.g. it does not describe correctly the magnetic properties [41] and gives an inadequate description of anisotropic effects [42,43]. Therefore, comparative ab initio calculations in the frame of the Hartree-Fock approximation are desirable. 

The theoretical results described here give only a zeroth-order description of the electronic structures of iron bearing clay minerals. These results correlate well, however, with the experimentally determined optical spectra and photochemical reactivities of these minerals. Still, we would like to go beyond the simple approach presented here and perform molecular orbital calculations (using the Xo-Scattered wave or Discrete Variational method) which address the electronic structures of much larger clusters. Clusters which accomodate several unit cells of the crystal would be of great interest since the results would be a very close approximation to the full band structure of the crystal. The results of such calculations may allow us to address several major problems ... 

Ryzhkov et al. [49] carried out a study of the electronic structure of neutral endohedral An C28 (An = Th-Md) confirming our results from fully relativistic discrete variational method. The 6d and 5f contributions to the bonding were found to be comparable for the earlier actinides. In addition, the actinides (Th-Md) series stabilize a C40 cage with a noticeable overlap between the 5f, 6d, 7s, 7p orbitals of the central actinide atom and the 2p(C) of the cage [54]. The most stable complex was found to be Pa C4o. 

To elucidate the nature of chemical bonding in metal carbides with the NaCl structure, the valence electronic states for TiC and UC have been calculated using the discrete-variational (DV) Xa method. Since relativistic effects on chemical bonding of compounds containing uranium atom become significant, the relativistic Hamiltonian, i.e., the DV-Dirac-Slater method, was used for UC. The results... 

The electronic structure of microcrystalline silicon of one-dimensional (1-D), 2-D, and 3-D clusters were calculated using the Discrete-Variational (DV)-Xa Molecular-Orbital method. The calculated results are discussed with respect to the effect of the size and the number of dimensions on the energy levels of molecular orbitals. The energy-gap (Eg) between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) decreases with the increase of cluster size amd the number of dimensions. It is found that including silicon 3d orbitals as basis sets decreases the Eg value. The results show that the components of silicon 3d orbitals in the unoccupied levels near LUMO are over 50 per cent. The calculated results predict that the Eg value will be close to the band gap of crystalline silicon when a 3-D cluster contadns more than 1000 silicon atoms with a diameter of 4nm. 

Cluster type molecular orbital calculations have proven to be powerful tools for understanding the electronic structure of molecules, clusters and solids. The Discrete Variational Xa (DV-Xa) is one of the most versatile amongst these methods in interpreting spectroscopic results and for predicting properties of polyatomic systems of great practical importance. 

In order to make a correct analysis of such an experimental spectrum, an appropriate theoretical calculation is indispensable. For this purpose, some of calculational methods based on the molecular orbital theory and band structure theory have been applied. Usually, the calculation is performed for the ground electronic state. However, such calculation sometimes leads to an incorrect result, because the spectrum corresponds to a transition process among the electronic states, and inevitably involves the effects due to the electronic excitation and creation of electronic hole at the core or/and valence levels. Discrete variational(DV) Xa molecular orbital (MO) method which utilizes flexible numerical atomic orbitals for the basis functions has several advantages to simulate the electronic transition processes. In the present paper, some details of the computational procedure of the self-consistent-field (SCF) DV-Xa method is firstly described. Applications of the DV-Xa method to the theoretical analysises of XPS, XES, XANES and ELNES spectra are... 

Cls photoemission shakeup satellites for the CO molecule were calculated with the spin-polarized discrete variational Xa method. The transition state method was applied to the estimation of multiplet peak positions for the shakeup transitions and the results are in reasonable agreement with the experimental values. 

The K0 IKa x-ray intensity ratios by photoionization and electron-capture decay have been calculated for several chemical compounds of 3d elements by the use of the discrete-variational Xo (DV-Xa) molecular orbital method. The calculated results indicate that the K/3lKa ratios depend on the excitation mode as well as the chemical effect. For the similar chemical environments the K0 /Ka ratio by photo ionization is larger than that by electron-capture decay, due to the excess 3d electron in the latter case. However, the difference is small, sometimes negligible in comparison with the chemical effect. Possible reasons for large difference in earlier experiments are discussed and future experiments are suggested. 

In the present work, we have calculated the K/3 /Koc x-ray intensity ratios for 3d transition elements excited by PI and EC, taking into account both effects described above. The calculations were made using the discrete-variational (DV) X(X molecular orbital (MO) method (19). The electronic states and wave functions in molecules were obtained for tetrahedral (Td) and octahedral (Oh) clusters. The x-ray emission rates were estimated by the DV integration method (20) with the MO wave functions in the dipole approximation. The calculated results are compared with the experimental data. 

The optimal control problem represents one of the most difficult optimization problems as it involves determination of optimal variables, which are vectors. There are three methods to solve these problems, namely, calculus of variation, which results in second-order differential equations, maximum principle, which adds adjoint variables and adjoint equations, and dynamic programming, which involves partial differential equations. For details of these methods, please refer to [23]. If we can discretize the whole system or use the model as a black box, then we can use NLP techniques. However, this results in discontinuous profiles. Since we need to manipulate the techno-socio-economic poHcy, we can consider the intermediate and integrated model for this purpose as it includes economics in the sustainabiHty models. As stated earlier, when we study the increase in per capita consumption, the system becomes unsustainable. Here we present the derivation of techno-socio-economic poHcies using optimal control appHed to the two models. 

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