Showing papers in "Advances in Quantum Chemistry in 2009"

Spectroscopy of Confined Atomic Systems: Effect of Plasma

Abstract: The effect of spatial and other external confinements on the ground and excited state energy levels of many electron atoms, ions and exotic systems is discussed. Special emphasis is given to analyzing and estimating the changes in the spectral properties of plasma embedded systems in which, apart from changes in the free particle potential due to atom plasma interaction, spatial confinement enters through the introduction of boundary conditions. Effects of weak as well as strong plasma on the dipole polarizabilities, ionization potentials, singly and doubly excited state energy levels, oscillator strengths and transition probabilities have been discussed using simple plasma models but adopting rigorous quantum chemical methods. The spectral line shifts under strongly coupled plasma have been compared with data available from laser plasma experiments. Specific attention has been given to extremely accurate estimates of the energies of different three-body systems under plasma environments. The importance of the use of finite boundary conditions originating from spatial confinement of the plasma has been demonstrated and the effect of electron correlation in estimating various confined atomic properties is shown. Attempt has been made to interpret the changes in the spectral properties of atoms trapped in cavities inside liquid helium environments, by comparing the results estimated on the basis of current quantum chemical methodologies with the data available from laser induced fluorescence experiments.

Photoionization of Atoms Encaged in Spherical Fullerenes

Abstract: Two semiempirical models for the photoionization of atoms A encaged in spherical single-walled fullerenes, both neutral C n ( n = 60 , 240 and 540) and charged C 60 ± | z | , as well as in multiwalled fullerenes, termed fullerene onions, C 60 @C 240 and C 60 @C 240 @C 540 are detailed. The models are based on the approximation of a carbon cage C n by a spherical attractive potential well of an adjustable radius R n , thickness Δ and depth U n 0 . The first model, termed Δ -potential model, accounts for the finite thickness Δ of the cage. The second model, termed δ -potential model, simulates the cage with the help of the Dirac δ -potential, thereby viewing the cage as being infinitesimally thin. A side by a side comparison of results obtained within the two models is performed. The models’ predictabilities are evaluated. Predicted trends in the modification of photoionization spectra of encaged atoms as well as electron correlation and relativistic effects in the atoms, compared to the free atoms, obtained both at the independent particle (Hartree-Fock and Dirac Hartree-Fock) and multiparticle nonrelativistic random phase approximation with exchange (RPAE) and relativistic random phase approximation (RRPA) approximation levels, are reviewed.

The Hydrogen and Helium Atoms Confined in Spherical Boxes

Abstract: The most relevant, and often utilized, methods and techniques for obtaining energy levels of confined one- and two-electron atoms are reviewed. We discuss how the electronic states of hydrogen and helium atoms are remarkably modified when subjected to extreme pressure regimes. For the hydrogen atom confined in a spherical box of impenetrable walls, we present the exact solutions and the energy eigenvalues obtained with very high accuracy. A few wave function properties calculated for the confined hydrogen atom, such as pressure, polarizability and the Fermi contact term, are discussed in some detail. Some accurate energy calculations for the hydrogen atom confined in a penetrable spherical box and when subjected to Neumann boundary conditions, are also presented. In addition, for the spherically confined helium atom we present the most accurate calculations reported to date.

Exact Relations for Confined One-Electron Systems

Abstract: Publisher Summary This chapter describes integrals of motion and boundary value problems. The chapter describes the mutual connections among the commutation relations and exponential transformations, for example, in the context of Hausdorf's relations. The coordinate transformation is a standard method for the analysis of boundary value problems. An important type of commutation relation is naturally connected with the scaling procedure. The possibility of using the virial/hypervirial relations for approximate wavefunctions depends on the form and nature of the classes of approximate functions. The chapter considers the one-parameter family of regions Ω (λ) and the corresponding boundary value problems, and also presents the main results concerning the energy changes with region modifications. Perturbation theory can be used to describe the splitting of electron energy levels for a hydrogen atom under small shifts from the center of an impenetrable spherical cavity, for example, it is demonstrated that first-order perturbation theory shows that for any s -state, the position of the atom in the center corresponds to a minimum.

Chapter 5 Multiple, Localized, and Delocalized/Conjugated Bonds in the Orbital Communication Theory of Molecular Systems

Abstract: Information theory (IT) probe of the molecular electronic structure, within the communication theory of chemical bonds (CTCB), uses the standard entropy/information descriptors of the Shannon theory of communication to characterize a scattering of the electronic probabilities and their information content throughout the system chemical bonds generated by the occupied molecular orbitals (MO). These “communications” between the basis-set orbitals are determined by the two-orbital conditional probabilities: one- and two-electron in character. They define the molecular information system, in which the electron-allocation “signals” are transmitted between various orbital “inputs” and “outputs”. It is argued, using the quantum mechanical superposition principle, that the one-electron conditional probabilities are proportional to the squares of corresponding elements of the charge and bond-order (CBO) matrix of the standard LCAO MO theory. Therefore, the probability of the interorbital connections in the molecular communication system is directly related to Wiberg’s quadratic covalency indices of chemical bonds. The conditional-entropy (communication “noise”) and mutual-information (information capacity) descriptors of these molecular channels generate the IT-covalent and IT-ionic bond components, respectively. The former reflects the electron delocalization (indeterminacy) due to the orbital mixing, throughout all chemical bonds in the system under consideration. The latter characterizes the localization (determinacy) in the probability scattering in the molecule. These two IT indices, respectively, indicate a fraction of the input information lost in the channel output, due to the communication noise, and its surviving part, due to deterministic elements in probability scattering in the molecular network. Together, these two components generate the system overall bond index. By a straightforward output reduction (condensation) of the molecular channel, the IT indices of molecular fragments, for example, localized bonds, functional groups, and forward and back donations accompanying the bond formation, and so on, can be extracted. The flow of information in such molecular communication networks is investigated in several prototype molecules. These illustrative (model) applications of the orbital communication theory of chemical bonds (CTCB) deal with several classical issues in the electronic structure theory: atom hybridization/promotion, single and multiple chemical bonds, bond conjugation, and so on. The localized bonds in hydrides and delocalized π-bonds in simple hydrocarbons, as well as the multiple bonds in CO and CO2, are diagnosed using the entropy/information descriptors of CTCB. The atom promotion in hydrides and bond conjugation in π-electron systems are investigated in more detail. A major drawback of the previous two-electron approach to molecular channels, namely, two weak bond differentiation in aromatic systems, has been shown to be remedied in the one-electron approach.

Chapter 3 Exact Signal–Noise Separation by Froissart Doublets in Fast Padé Transform for Magnetic Resonance Spectroscopy

Abstract: In the present study, it is demonstrated that the fast pade transform (FPT) is capable of providing the exponential convergence rate (the spectral convergence) for the exact reconstructions of all the spectral parameters from time signals equivalent to the corresponding in vivo free induction decay curves encoded by means of magnetic resonance spectroscopy with short echo times of about 20 ms at the standard clinical magnetic field strength 1.5 T from the brain of a healthy volunteer. Further, it is shown that residual spectra (the difference between the model and input spectra) are a necessary, but not a sufficient, criterion to estimate the error invoked in quantification. Full validation of the performed quantification within the FPT is possible by monitoring stabilization of all the reconstructed spectral parameters as a function of the partial signal length for a fixed bandwidth (this is equivalent to varying the total acquisition time). Moreover, all the converged fundamental frequencies and amplitudes found in this way must further be cross-validated by checking whether they also represent the joint results of both Pade variants, the FPT (+) and the FPT (−) , inside and outside the unit circle, as done in the present study. The Froissart doublets (pole–zero cancellations) are used to unequivocally distinguish between genuine and spurious resonances in both noise-free and noise-corrupted time signals. This permits the exact reconstruction of all the genuine spectral parameters including the fundamental frequencies, the corresponding amplitudes, and the true number of physical resonances. The FPT is shown to be able to resolve and quantify tightly overlapped resonances that are abundantly seen in magnetic resonance spectra generated using encoded in vivo time signals. Most importantly, precisely such overlapping resonances are often of critical relevance for diagnostics in clinical oncology.

Properties of Confined Hydrogen and Helium Atoms

Abstract: The energy spectrum and polarizabilities of hydrogen atom confined to a sphere of radius R , are analysed in terms of the numerical approach, model wave functions, and simple analytical expressions, which provide a useful description of these properties. The scaling relations are used to develop simple expressions for the energies of the confined helium atom in terms of screening effect. The considerations are extended to the hydrogen atom in an oscillator potential, and to off-centre confinement. The general results provide a clear understanding of the implications of confinement.

Modeling Pressure Effects on the Electronic Properties of Ca, Sr, and Ba by the Confined Atoms Model

Abstract: In this work three confined atoms are studied, Ca, Sr and Ba. We estimated the pressure on the system, by enclosing each atom within a sphere with rigid walls and changing the sphere radius. When the pressure is increased, these atoms undergo electronic transitions, from the s to the d orbital, as observed experimentally in the alkaline earth metals. The transition pressures obtained with this model correlate reasonably well with experimental information for Ca and Ba. For Sr, the predicted transition pressure is not close to the experimental data. Additionally, we compare the spin-potential, defined within the spin polarized Density Functional Theory version, with the singlet-triplet excitation energy, and we obtain a linear relationship between these quantities. The spin-potential towards higher multiplicities is related to the HOMO β and LUMO α gap. In this way, we show that this HOMO–LUMO gap is also related with excitation energies in confined atoms, as has been pointed out for other non-confined systems. We discuss the regions where the HOMO–LUMO gap and the excitation energies show a relationship.

Comparative Study Between the Hartree-Fock and Kohn-Sham Models for the Lowest Singlet and Triplet States of the Confined Helium Atom

Abstract: In this work, the exchange-only Kohn-Sham (KS) model is compared with the Hartree-Fock (HF) model for the lowest singlet and triplet states of the confined helium atom. The confinement on this atom is obtained by imposing Dirichlet boundary conditions. The HF equations are solved according to the Roothaan approach coupled with modified Slater Type Orbitals, where the boundary conditions are imposed. The solution of the KS equations is obtained with a numerical code adapted to work with this sort of boundary condition. For the KS exchange functional we use the local density approximation corrected by the Perdew and Zunger self-interaction approximation. It is shown that while the Perdew and Zunger proposal of incorporating the self-interaction correction is quite appropriate for the lowest singlet state of the helium atom this approach shows large discrepancies with regard to the HF method for the lowest triplet state, particularly in regions of reduced confinement radii. Thus, when electrons of the same spin are confined within small regions, the self-interaction correction scheme of Perdew and Zunger becomes inappropriate. The HF results reported in this work and those obtained with a Hylleraas expansion indicate that the correlation energy for the lowest singlet and triplet system states is almost constant with regard to the confinement radius. For the open-shell system the correlation energy is quite small and consequently the HF model can give a good description of this system.

DFT Study of Molecules Confined Inside Fullerene and Fullerene-like Cages

Abstract: The results of density functional theoretic calculations of the equilibrium geometry, electronic structure, energetic stability, and spectroscopic properties (normal vibrational frequencies, magnetic shielding constants, etc.) of a few families of model endohedral clusters with various guest molecules inside various “modestly spacious”, “tight” and “very tight” fullerene and fullerene-like inorganic cages are surveyed. Guest-cage interactions and their manifestations in changes (shifts) of these properties of the endoclusters as compared with those of the isolated (free) guests and cages are analyzed.

Thomas–Fermi–Dirac–Weizsäcker Density Functional Formalism Applied to the Study of Many-electron Atom Confinement by Open and Closed Boundaries

Abstract: An assessment of the use of statistical atomic models for the study of many-electron atom confinement is presented. The Thomas–Fermi–Dirac- λ -Weizsacker TFDλW functional formalism based on known properties of the orbital electron density is shown to be an appropriate tool for the description of the ground-state energy evolution of many-electron atoms spatially limited by closed and open boundaries. A brief review of the strategy followed in the TFDλW method for the study of atoms enclosed in hard and soft spherical cavities is presented along with more refined quantitative calculations as compared with previous results. Also, detailed quantitative results are shown–for the first time–in the case of confinement by a hard prolate spheroidal box for nuclear positions located at one of the foci and for an atom located at a distance D from a hard plane. A discussion is presented on the physical consequences of different confinement geometries and the adequacy of the TFDλW formalism to explore many-electron atom confinement by open and closed boundaries.

The Energy Level Structure of Low-dimensional Multi-electron Quantum Dots

Abstract: A unified characterization of the energy-level structure of quasi-one-dimensional quantum dots is presented based on accurate computational results for the eigenenergies and wave functions, as obtained in previous studies for the case of two and three electrons, and in the present study also for four electrons. In each case the quantum chemical full configuration interaction method is adopted employing Cartesian anisotropic Gaussian basis sets. The energy-level structure is shown to be strongly dependent on the confinement strength ω and can be exemplified by the three qualitatively distinct regimes for large, medium, and small confinement strengths. To characterize the energy-level structure in the large or medium ω and the small ω regimes, the polyad quantum number, as well as its extended version, the extended polyad quantum number have been introduced. The degeneracy of energy levels for different spin states in the small ω regime is shown to be caused by the potential walls of the electron–electron interaction potential within the internal space. A systematic way of obtaining the degeneracy pattern of the energy levels in the small ω regime is also presented. Finally, qualitative differences between the energy-level structure of quasi-one-dimensional and quasi-two-dimensional quantum dots in the small ω regime are briefly discussed by referring to the different structure of their internal spaces.

The Hydrogen Atom Confined in Semi-infinite Spaces Limited by Conoidal Boundaries

Abstract: This contribution presents a sample of comments on confined atoms and molecules in the literature, a brief review of the eigenfunctions of the free hydrogen atom in different coordinate systems, an overview of the eigenfunctions of the hydrogen atom confined in spaces with closed and open conoidal boundaries, and a preview of current and future research problems on confined atoms and molecules. The first comment in the sample provides the basic idea developed in the overview; the other comments identify errors in specific articles, including their corrections, or suggesting alternative formulations of the problem incorporated in the preview. The review emphasizes the superintegrability of the free hydrogen atom, including the solutions at the ionization limit. In the overview, the confinement effect of each conoidal boundary is interpreted as a symmetry breaking effect, interpolating between the familiar degenerate energy levels of the free hydrogen atom and the vanishing energy limits at degenerate positions of the boundary for the states sharing the other two quantum numbers. The preview includes formulations of problems under investigation or about to be investigated.

Confined Atoms Treated as Open Quantum Systems

Abstract: Every confined system, atom or molecule, if it is restrained by a physical boundary rather than by a mathematically imposed boundary condition, is an open system. This chapter is concerned with the former case, of a system bounded by the surfaces it shares with the atoms of the confining material, surfaces that enable the transfer of matter and momentum. This places the problem in the realm of the quantum mechanics of an open system. Earlier definitions of pressure, derived in analogy with the classical virial theorem for a contained system, incorrectly relate the pressure to the contribution to the system’s virial arising from external forces of constraint acting on a closed system. The quantum definition relates the pressure to the virial of the force resulting from the flux in the electronic momentum through the surface of the open system, the surface contribution to the total virial acting on an open system. A scaling procedure is used to demonstrate that the expectation value of the pressure–volume product of an open system is proportional to its surface virial. The atomic expectation values of the pressure are examined first for atoms in diatomic molecules to illustrate their general properties, followed by the determination of the pressure acting on atoms confined within a molecular cage. The final example determines the effect of very high pressures–in the range of 160 GPa–on the topological properties of the electron density and on atomic properties for a linear chain of hydrogen molecules compressed between a pair of neon atoms. The pressure exerted on an atom is very much determined by the curvature of its interatomic surface. It is found that atoms confined to an adamantane cage are bounded by highly curved surfaces and are subjected to large pressures. The Cr, Fe and Ni atoms in their carbonyl complexes are also totally confined by the interatomic surfaces they share with the carbon atoms of the carbonyl groups. Their atomic volumes are, however, larger than those of the atoms confined to an adamantane cage and their interatomic surfaces are nearly planar, unlike the curved surfaces found for the atoms confined to a cage, and the pressures exerted on the metal atoms are up to twenty times smaller. There are no ‘repulsive forces’ acting on atoms confined in an adamantane cage, but they are subject to large forces per unit area of surface, that is, to large pressures.

Perturbation Theory for a Hydrogen-like Atom Confined Within an Impenetrable Spherical Cavity

Abstract: Perturbation expansions for a hydrogen-like atom confined at the centre of an impenetrable spherical cavity, of finite radius R , are discussed in a non-relativistic approximation. Properties considered include: energy, oscillator strength, dipole polarisability and nuclear shielding factor. The appropriate form of perturbation theory to employ depends on the cavity size and three different regimes are considered: small, intermediate and large. For large cavity radii, perturbation of the unconfined atom boundary condition at r = R to satisfy a Dirichlet condition results in exponentially small deviations from the unconfined atom values which are predicted to high accuracy in first order. For small R , Rayleigh–Schrodinger perturbation theory can be used, with the electron-nucleus Coulomb interaction treated as a perturbation, to generate expansions in powers of R . These expansions, whose radii of convergence are explored, provide highly accurate results even for moderately large R (depending on the state considered). The difficult intermediate range of R values is finally investigated using Rayleigh–Schrodinger perturbation theory based on known exact solutions obtained from lobes of free-atom solutions.

Engineering Quantum Confined Silicon Nanostructures: Ab-Initio Study of the Structural, Electronic and Optical Properties

Abstract: Silicon nanostructures, in the form of nanodots, nanowires and nanoslabs have attracted a lot of interest in recent years. Quantum confinement effects play an important role with respect to both the electronic and optical properties. We review and summarize here our results concerning the properties of semiconductor nanocomplexes. Total energy calculations within the density functional theory have been carried out in order to investigate the structural, electronic and optical properties of undoped and doped Si nanosystems of different dimensionality, size and surface termination. Single-particle and Many- Body perturbation theory calculations have been carried out in order to study the optical properties, both in ground and excited state configuration, of Si nanodots. Starting from hydrogenated clusters, we have considered different Si/O bonding geometries at the interface. We provide strong evidence that not only the quantum confinement effect but also the chemistry at the interface has to be taken into account in order both, to understand the physical properties of these systems, and to provide an explanation for both the Stokes shift and the near-visible PL experimentally observed. For Si nanocrystals embedded in a SiO2 matrix, the strong interplay between the nanocrystal and the surrounding host environment and the active role of the interface region between them is pointed out, in good agreement with the experimental results. Concerning the doping, we consider B and P single- and co-doped Si nanoclusters. The neutral impurities formation energies are calculated and their dependence on the impurity position within the nanocrystal is discussed. In the case of co-doping the formation energy is strongly reduced, favoring this process with respect to the single doping. Moreover the band gap and the optical threshold are clearly red-shifted with respect to that of the pure crystals, showing the possibility of an impurity based engineering of the absorption and luminescence properties of Si nanocrystals. We also discuss here the case of multiple doping. In the case of one-dimensional systems we have calculated the structural, electronic and optical properties of hydrogenated Ge and Si nanowires of different sizes and different spatial orientations. We have analyzed how the geometrical relaxation affects the optoelectronic properties. Moreover for the smallest structures, we have calculated the electronic and optical properties overcoming the one-particle approach. Large self-energy corrections, compared to the bulk ones, have been found together with strong excitonic effects. In particular in the case of Si nanowires the calculated electronic and optical gaps compare well with the available experimental data. Indeed we show that freshly etched porous Si is better described as a distribution of interacting nanowires than as an ensemble of isolated nanoparticles. Concerning the two-dimensional nanoslab systems we show how the calculated optoelectronic properties of Si superlattices and multiple quantum wells, where CaF2 and SiO2 are the barrier mediums, are in good agreement with the experimental outcomes and we discuss the comparison between Si and Ge nanofilms.

Chapter 6 Quantum Mechanical Methods for Loss-Excitation and Loss-Ionization in Fast Ion–Atom Collisions

Abstract: Inelastic collisions between bare nuclei and hydrogen-like atomic systems are characterized by three main channels: electron capture, excitation, and ionization. Capture dominates at lower energies, whereas excitation and ionization prevail at higher impact energies. At intermediate energies and in the region of resonant scattering near the Massey peak, all three channels become competitive. For dressed or clothed nuclei possessing electrons, such as hydrogen-like ions, several additional channels open up, including electron loss (projectile ionization or stripping). The most important aspect of electron loss is the competition between one- and two-electron processes. Here, in a typical one-electron process, the projectile emits an electron, whereas the target final and initial states are the same. A prototype of double-electron transitions in loss processes is projectile ionization accompanied with an alteration of the target state. In such a two-electron process, the target could be excited or ionized. The relative importance of these loss channels with single- and double-electron transitions involving collisions of dressed projectiles with atomic systems is also strongly dependent on the value of the impact energy. Moreover, impact energies determine which theoretical method is likely to be more appropriate to use for predictions of cross sections. At low energies, an expansion of total scattering wave functions in terms of molecular orbitals is adequate. This is because the projectile spends considerable time in the vicinity of the target, and as a result, a compound system comprised of the projectile and the target can be formed in a metastable molecular state which is prone to decay. At high energies, a perturbation series expansion is more appropriate in terms of powers of interaction potentials. In the intermediate energy region, atomic orbitals are often used with success while expanding the total scattering wave functions. The present work is focused on quantum mechanical perturbation theories applied to electron loss collisions involving two hydrogen-like atoms. Both the one- and two-electron transitions (target unaffected by collision, as well as loss-ionization) are thoroughly examined in various intervals of impact energies varying from the threshold via the Massey peak to the Bethe asymptotic region. Systematics are established for the fast, simple, and accurate computations of cross sections for loss-excitation and loss-ionization accounting for the entire spectra of all four particles, including two free electrons and two free protons. The expounded algorithmic strategy of quantum mechanical methodologies is of great importance for wide applications to particle transport physics, especially in fusion research and hadron radiotherapy. This should advantageously replace the current overwhelming tendency in these fields for using phenomenological modeling with artificial functions extracted from fitting the existing experimental/theoretical data bases for cross sections.

Chapter 2 Quantum Linear Superposition Theory for Chemical Processes: A Generalized Electronic Diabatic Approach

Abstract: Complete basis states (BSs), in abstract configuration space-projected quantum mechanics (QM), permit representations of any physical and chemical process elicited by quantum states changes. For a material 1-system, defined by n -electrons and m -nuclei, BSs including relevant fragments cover a representation of chemical species identifiable by spectral response toward electromagnetic (EM) radiations. Reactants, products, and intermediate species are expressed as specific linear superpositions where the amplitude in square modulus at a given BS controls the relative intensity to the spectrum rooted at the corresponding energy eigenstate. Moreover, there is no trace that quantum numbers characterizing BSs would be changed as a function of particular regions of nuclear or electronic configuration space. The exact Coulomb Hamiltonian generates BSs. However, in this basis set, this operator does not generate evolution measured by changes of amplitudes in time, only time phases change. This operator in semiclassical models cannot drive effective time evolution via changes of amplitudes for the electronic quantum states either. The presence of a driving external field, for example, EM fields, is a sufficient condition to produce evolution standing for the physical process. It is a matter of logics that if the exact operator and the semiclassical one do not generate time evolution, then approximate models – such as computational Born–Oppenheimer (BO) – should not do it as well. However, standard (s-)BO scheme does change basis quantum numbers as a function of nuclear configuration space leading to chemical reaction representation. This apparent contradiction and possible solutions are examined here. By introducing the concept of abstract generalized electronic diabatic (a-GED) and a-BO models, electronuclear separability models are examined. Sets of noninteracting many-I-frame fragments leading to asymptotic states descriptions are included together with sets of quantum states for the one-I-frame system providing BSs to describe dissociation/association processes in chemistry. The theory takes on a clear semiclassical flavor. This approach permits introducing nuclear fixed configuration concept and relate theoretical states to laboratory ones in a natural manner. The approach leads to a generalization of the many-state reactivity models. General semiclassic schemes are introduced in Section 6 , which permit integration of one-I-frame to many-I-frames states. Planting one-electron functions at nuclear positions is the origin of the parametric dependence of s-BO wave functions, and it explains why the method displays chemical behavior. This atomic-orbital algorithm permits connecting one-I-frame semiclassic electronic states to asymptotic ones in a continuous way. A ghost atomic-orbital model is introduced to facilitate diabatic studies and reinstate the linear superposition model. As indicated in the “Contents,” some other subjects are examined from the present perspective. This includes the Jahn–Teller effect defined in this new diabatic framework and the nature of the BO scheme.

Exact Solutions for Confined Model Systems Using Kummer Functions

Abstract: We treat model systems where an electron is confined in a region of space. The particular models considered have solutions which may be expressed in terms of the Kummer functions. Both standard and non-standard Kummer functions are used in these models and a comprehensive summary of the usual and exceptional Kummer functions is given. The definition of confinement is widened to treat radial confinement in any spherical shell, including the asymptotic region and cases where the electron is confined to a lower dimension. Initially we consider the theory in K dimensional space and then give particular examples in 1, 2, and 3 dimensions. A commonly treated model is the radially confined hydrogen atom in 3 dimensions with an infinite barrier on a confining sphere so that the wavefunction is identically zero on this sphere. We have extended this model to treat a more general model of spherical confinement where the derivative of the charge density is zero on the confining sphere. It is shown that the analogous models for the radial harmonic oscillator and radial constant potentials may be treated using a generic technique.

Chapter 1 Multireference and Spin–Orbit Calculations on Photodissociations of Hydrocarbon Halides

Abstract: The mechanistic photodissociations of several series of aliphatic and aryl halides have been theoretically investigated using multireference ab initio calculations. The photodissociations of halomethanes were also investigated by spin–orbit ab initio calculations. Based on the calculated results of complete active space self-consistent field (CASSCF), CASSCF with second-order perturbation (CASPT2), multistate CASPT2 (MS-CASPT2), or MS-CASPT2 with spin–orbit interaction through complete active space state (MS-CASPT2/CASSI-SO), the experimentally observed photodissociation channels were clearly assigned, the photochemical and photophysical processes were quantitatively described. The photodissociation mechanisms of the targeted molecules were deeply discussed by high-quality potential energy curves (PECs) and critical points of the potential energy surfaces (PESs).

Chapter 4 Reflections on Formal Density Functional Theory

Abstract: It is discussed that many variations on the Hohenberg–Kohn construction of in principle exact density functionals exist and that the treatment of the kinetic energy in Kohn–Sham theory can be generalized to many other types of orbital-dependent energy components, all yielding formally exact density functionals. These generalizations present opportunities for alternative density functional approximations, but they also raise questions regarding the foundations and physical implications of density functional theory.