Energy levels of paramagnetic ions: Algebra. III. The case of dN ions in cubical symmetry

The formalism developed in the first two papers of this series is applied to the investigation of a new weak-field model. This crystal-field model lies on the use of a symmetry-adapted weak-field basis and an effective Hamiltonian involving in a symmetrical way both spin- and orbit-dependent contributions. Some general properties of this Hamiltonian are studied and complete calculation of its matrix elements is conducted in a symmetry-adapted weak-field basis in the case of an arbitrary configuration nl^N in any symmetry. The case of a configuration nd^N in octahedral symmetry is fully explored. In this case, the proposed weak-field model is restricted to a 12-parameter model which accounts for isotropic and anisotropic Coulomb interactions, isotropic and anisotropic spin-orbit interactions, and crystal-field interactions. A comparison between this 12-parameter weak-field model and the 14-parameter strong-field model is established. Equivalence between the latter two models requires two constraint relations to be satisfied for some strong-field parameters. These two relations are examined with various viewpoints.     

Numerical instabilities in the computation of pseudopotential matrix elements

Steep high angular momentum Gaussian basis functions in the vicinity of a nucleus whose inner electrons are replaced by an effective core potential may lead to numerical instabilities when calculating matrix elements of the core potential. Numerical roundoff errors may be amplified to an extent that spoils any result obtained in such a calculation. Effective core potential matrix elements for a model problem are computed with high numerical accuracy using the standard algorithm used in quantum chemical codes and compared to results of the MOLPRO program. Thus, it is demonstrated how the relative and absolute errors depend an basis function angular momenta, basis function exponents and the distance between the off-center basis function and the center carrying the effective core potential. Then, the problem is analyzed and closed expressions are derived for the expected numerical error in the limit of large basis function exponents. It is briefly discussed how other algorithms would behave in the critical case, and they are found to have problems as well. The numerical stability could be increased a little bit if the type 1 matrix elements were computed without making use of a partial wave expansion.     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Unitary group approach to the many-electron problem. I. Matrix element evaluation and shift operators

This is the first paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians directly in the spin-orbit basis. In this paper we present a new and complete derivation of the matrix elements of the U(n) generators in the electronic Gel'fand basis. The approach employed differs from previous treatments in that the matrix elements of nonelementary generators are obtained directly. A general matrix element formula is derived which explicitly demonstrates the segment level formalism obtained previously by Shavitt using different methods. A simple relationship between the matrix elements of raising and lowering generators is determined which indicates that in CI calculations, only the matrix elements of raising generators need be calculated. Some results on the matrix elements of products of two generators are also presented.     

MSINDO parameterization for third‐row transition metals

The recently developed MSINDO version of the semiempirical SCF MO method SINDO1 has been parameterized for third‐row transition metals Sc to Zn. The set of reference data used for the previous parameterization of SINDO1 has been substantially increased by incorporating results of recent experiments and first‐principles calculations. A comparison of calculated heats of formation, geometries, ionization potentials, and dipole moments with literature values for more than 200 gas phase molecules is presented. The accuracy of the modified MSINDO version achieved for heats of formation and bond lengths has been considerably improved compared to SINDO1. Small clusters of transition metals and metal oxides were included in the parameterization to ensure accurate results for studies of larger systems. The application of the method to small transition metal complexes that were not included in the parameterization shows that the optimized parameters are transferable to other compounds. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 861–887, 2001     

The well-tempered model core potentials for the main-group elements Li–Rn

The well-tempered model core potential (wtMCP) parameters and valence basis sets, based on the well-tempered basis set expansion, were developed for the main-group elements Li–Rn. For the s–block elements, the valence space comprises the ns valence shell and the outermost core (n–1)p shell. For the p-block elements, the ns and np shells together with the (n–1)d shell make up the valence space. Nonrelativistic wtMCPs were developed for all atoms. Scalar-relativistic wtMCPs were prepared for all atoms heavier than Ar by using the relativistic elimination of small components to obtain the reference and core orbitals. The new potentials were tested at the restricted Hartree–Fock, second-order Mø øller–Plesset perturbation theory and density functional theory with Beckes three-parameter hybrid functional combined with Perdews 1991 gradient-corrected correlation functional levels for several diatomic molecules and the results were compared with those obtained from all-electron calculations and experimental values. Excellent agreement between the results was obtained.     

Tucker core consistency for validation of restricted Tucker3 models

In Tucker3 analysis of three-way data array obtained from a chemical or biological system, it is sometimes possible to use a priori knowledge about the system to specify what is called a restricted Tucker3 model. Often, the restricted Tucker3 model is characterized by having some elements of the core forced to zero. As a simple example, an F-component PARAFAC model can be seen as a restricted (F, F, F) Tucker3 model in which only superdiagonal elements of the core are allowed to be nonzero. The core consistency diagnostic was previously introduced by Bro and Kiers for determining the proper number of components in PARAFAC analysis. In the current study, this diagnostic is extended to other restricted Tucker3 models to validate the appropriateness of the applied constraints. The new diagnostic is named Tucker core consistency (TuckCorCon). When the dimensionality and the pattern of the restricted core is valid, the simple core of restricted Tucker3 model and a corresponding unrestricted core will be similar and in this case the TuckCorCon will be close to maximum (100%). A simulated chemical equilibrium data set and two experimental data sets were used to evaluate the applicability of the TuckCorCon to decide about the appropriateness of dimensionality and pattern of the core nonzero elements in the restricted Tucker3 models.     

Computational modeling of the adsorption and photodegradation of 4-chlorophenol on anatase TiO2 particles

In the present work, the adsorption and photodegradation of 4-chlorophenol (4-CP) on the (100) surface of TiO₂ anatase with semiempirical SCF MO method MSINDO has been investigated. The (100) surface is modeled with free clusters (TiO₂)_n, where n = 20–80. The surface lattice titanium atoms, which are Lewis acid sites, are considered as adsorption sites. Molecular dynamics (MD) simulations have been used for the investigation of 4-CP adsorption conformations and the surface reaction mechanism studies. The 4-CP molecule has revealed parallel adsorption upon optimization, whereas under excitation conditions the perpendicular configuration is dominant. The aromatic ring cleavage by atomic oxygen has been studied computationally and accordingly, the relevant mechanism was suggested. By comparison with experimental and other theoretical calculations, it is shown that MSINDO can reproduce literature data with acceptable accuracy.     

Asymmetric rotor-like probes to polarized fluorescence study of the macroscopically oriented uniaxial media: Model parameters recognition

In this paper the possibility of the numerical data modelling in the case of angle- and time-resolved fluorescence spectroscopy is investigated. The asymmetric fluorescence probes are assumed to undergo the restricted rotational diffusion in a hosting medium. This process is described quantitatively by the diffusion tensor and the aligning potential. The evolution of the system is expressed in terms of the Smoluchowski equation with an appropriate time-developing operator. A matrix representation of this operator is calculated, then symmetrized and diagonalized. The resulting propagator is used to generate the synthetic noisy data set that imitates results of experimental measurements. The data set serves as a groundwork to the χ² optimization, performed by the genetic algorithm followed by the gradient search, in order to recover model parameters, which are diagonal elements of the diffusion tensor, aligning potential expansion coefficients and directions of the electronic dipole moments. This whole procedure properly identifies model parameters, showing that the outlined formalism should be taken in the account in the case of analysing real experimental data.     

On the use of the lichen Ramalina celastri (Spreng.) Krog. & Swinsc. as an indicator of atmospheric pollution in the province of Córdoba, Argentina, considering both lichen physiological parameters and element concentrations

The lichen Ramalina celastri (Spreng.) Krog. & Swinsc. was used to study trace-element atmospheric pollution in Córdoba (Argentina). 38 samples from a 1999 campaign were analyzed by NAA and a number of physiological parameters was determined. In first-ever comparisons, no correlations were found between altitude, physiological parameters and determined elements, which allowed the full comparison of element data in lichen samples throughout the whole survey area. After the application of Monte Carlo assisted factor analysis to the elemental matrix, five source profiles were found and mapped. The attribution of the sources is discussed.     

Two‐component calculations of spin–orbit effects for a van der Waals molecule Rn2

Electronic structures of the weakly bound Rn₂ were calculated by the two‐component Møller–Plesset second‐order perturbation and coupled‐cluster methods with relativistic effective core potentials including spin–orbit operators. The calculated spin–orbit effects are small, but depend strongly on the size of basis sets and the amount of electron correlations. Magnitudes of spin–orbit effects on D_e (0.7–3.0 meV) and R_e (−0.4∼−2.2 Å) of Rn₂ are comparable to previously reported values based on configuration interaction calculations. A two‐component approach seems to be a promising tool to investigate spin–orbit effects for the weak‐bonded systems containing heavy elements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 139–143, 1999     

Ab initio quantum chemical calculations of enolized and deprotonated forms of Β-diketones and their derivatives

The data of ab initio quantum chemical calculations of Β-diketones and their anions performed in split valence-shell bases are presented. These compounds are of great interest as ligands in numerous chelate complexes. The electronic structures of substituted enolized Β-diketones containing the period III elements S and Cl are compared. It is shown that using the Huzinaga pseudopotential for describing core electronic shells makes it possible to adequately reproduce the main peculiarities of the structure of the outer electronic layers at a qualitative level. The basis used substantial affects on the calculated values of atomic charges, bond orders, and dipole moments, whereas the one-electron energies are less sensitive to the choice of the basis. Translated fromZhurnal Struktumoi Khimii, Vol.40, No. 2, pp. 234–241, March–April, 1999.     

Quasiclassical trajectory study of the O(3P) + CH4 --> OH + CH3 reaction with a specific reaction parameters semiempirical Hamiltonian

We present a theoretical study of the O(3P) + CH4 --> OH + CH3 reaction using electronic structure, kinetics, and dynamics calculations. We calculate a grid of ab initio points at the PMP2/AUG-cc-pVDZ level to characterize the potential energy surface in regions of up to 1.3 eV above reagents. This grid of ab initio points is used to derive a set of specific reaction parameters (SRP) for the MSINDO semiempirical Hamiltonian. The resulting SRP-MSINDO Hamiltonian improves the quality of the standard Hamiltonian, particularly in regions of the potential energy surface beyond the minimum-energy reaction path. Quasiclassical-trajectory calculations are used to study the reaction dynamics with the original and the improved MSINDO semiempirical Hamiltonians, and a prior surface. The SRP-MSINDO semiempirical Hamiltonian yields OH rotational distributions in agreement with experimental results, improving over the results of the other surfaces. Thermal rate constants estimated with Variational Transition State Theory using the SRP-MSINDO Hamiltonian are also in agreement with experiments. Our results indicate that reparametrized semiempirical Hamiltonians are a good alternative to generating potential energy surfaces for accurate dynamics studies of polyatomic reactions.     

MSINDO parameterization for third‐row main group elements

A new parameterization of the SINDO model is introduced in the recently developed MSINDO version for third‐row main group elements Ga, Ge, As, Se, and Br. It is shown that the accuracy achieved for structure and stability as well as ionization potential and dipole moment of compounds containing these elements is good enough to make the extension a suitable tool for the study of larger systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 974–987, 2000     

Unitary group approach to the many-electron problem. II. Adjoint tensor operators for U(n)

In this paper we present a derivation of the U(n) adjoint coupling coefficients for the representations appropriate to many-electron systems. Since the states of a many-fermion system are to comprise the totally antisymmetric Nth rank tensor representation of U(2n), the work of this paper enables the matrix elements of the U(2n) generators to be evaluated directly in the U(n) × U(2) (i.e., spin orbit) basis using their transformation properties as adjoint tensor operators. A connection between the adjoint coupling coefficients, as derived in this paper, and the matrix elements of certain (spin independent) two-body operators is also presented. This indicates that in CI calculations, one may obtain the matrix elements of spin-dependent operators from the known matrix elements of certain spin-independent two-body operators. In particular this implies a segment-level formula for the matrix elements of the U(2n) generators in the spin-orbit basis.