Theoretical study of the electronic states of calcium and calcium hydride

The lowest states of the calcium atom as a two-electron system have been studied using a non-empirical pseudopotential, valence CI, and perturbative valence—core correlation. The lowest three ²Σ⁺ states and the lowest ²Π state of calcium hydride have also been studied. The precision and the limitations of the perturbative valence—core correlation method based on the valence—core separability are illustrated.     

Toward a physical understanding of electron-sharing two-center bonds. I. General aspects

In 1916, Lewis and Kossel laid the empirical ground for the electronic theory of valence, whose quantum theoretical foundation was uncovered only slowly. We can now base the classification of the various traditional chemical bond types in a threefold manner on the one- and two-electron terms of the quantum-physical Hamiltonian (kinetic, atomic core attraction, electron repulsion). Bond formation is explained by splitting up the real process into two physical steps: (i) interaction of undeformed atoms and (ii) relaxation of this nonstationary system. We aim at a flexible bond energy partitioning scheme that can avoid cancellation of large terms of opposite sign. The driving force of covalent bonding is a lowering of the quantum kinetic energy density by sharing. The driving force of heteropolar bonding is a lowering of potential energy density by charge rearrangement in the valence shell. Although both mechanisms are quantum mechanical in nature, we can easily visualize them, since they are of one-electron type. They are however tempered by two-electron correlations. The richness of chemistry, owing to the diversity of atomic cores and valence shells, becomes intuitively understandable with the help of effective core pseudopotentials for the valence shells. Common conceptual difficulties in understanding chemical bonds arise from quantum kinematic aspects as well as from paradoxical though classical relaxation phenomena. On this conceptual basis, a dozen different bond types in diatomic molecules will be analyzed in the following article. We can therefore examine common features as well as specific differences of various bonding mechanisms.     

Relationship between the Auger parameter and the ground state valence charge at the core-ionized site

A simple semi-empirical model that correlates the Auger parameter to the ground state valence charge of the core-ionized atom with closed valence shell configuration, and which was previously applied to Cu(I) (3d¹⁰) compounds, is extended to Ba (II) (5d¹⁰), Pb (II) (5d¹⁰4s²), and Zn (II) (3d¹⁰) compounds (halides and chalcogenides). Until now, the Auger parameter was employed to separate initial and final state effects that influence the core electron binding energy. In agreement with our model, a linear relationship is found between the Auger parameter shift and the ground state Bader valence charge obtained by density functional theory (DFT) calculations.     

Solution of large configuration mixing matrices arising in partitioning technique and applications to the generalized eigenvalue problem

A method is presented that can be used (a) to determine the several lowest eigenvalues and eigenvectors of large symmetric matrices, (b) to solve the generalized eigenvalue problem associated with energy-dependent operators, that arises in computations involving energy-dependent many-body Green's functions and in the evaluation of the true parameters of the effective valence shell hamiltonian, and (c) to directly evaluate the matrices associated with resolvent operators. The applicability to large configuration mixing calculations arises when the N-electron basis functions can be easily broken down to a few dominant configurations (the primary block) and their complement. Using the partitioning technique, the effective hamiltonian within the primary block is directly evaluated. The method is extended to evaluation of the dynamical polarizability tensor, which effectively contains the contributions from all of the eigenstates of a hamiltonian matrix, without the necessity of explicitly calculating its eigenvalues and eigenvectors.     

Analysis of exact valence shell hamiltonian: Nonclassical terms and molecular based parameters

The exact valence shell effective hamiltonian is analyzed for one- and two-valence orbital systems using a second quantized formulation. The exact solutions of the exact effective hamiltonian are used, to display the meaning of each of its terms. The well-known α_υ = ?I_p and γ_υυ = I_p ? A_f relations are provided a molecular basis for certain special cases, thereby enabling molecular definitions for molecular “integrals” and allowing the determination of the molecule and bond length dependence of traditional semi-empirical “integrals”. It is shown how the presence of nonclassical terms in the effective hamiltonian destroys the pairing symmetry of alternate hydrocarbons.     

Approximate molecular orbital theory: the ese mo formalism.: III. Parametrization for all- or valence-electron calculation using accurate STF bases for molecules containing 1st, 2nd,and 3rd row atoms. SO2 results

The key components of a completely theoretical parametrization of the essential-structural-elements molecular orbital (ESE MO) formalism using Slater-type AO basis in the LCAO SCF procedure are discussed. Special attention is paid to the problem of separability into core and valence parts of the total molecular wavefunction, including the case where valence functions strongly overlap neighbouring core orbitals. The use of Huzinaga and Cantu effective hamiltonian is proposed. The parametrization is tested in relation to the SO₂ molecules. The role of sulphur 3d functions in bonding as predicted by the present ESE MO calculations and ab initio calculations are compared. The present parametrization appears to adequately handle both the core/valence separation, and the diffuse higher valence sulphur 3d functions in this system.     

Generalized perturbation theory of effective valence shell hamiltonians

The second quantized effective valence shell hamiltonian of Iwata and Freed is generalized to incorporate valence orbital energies into the perturbative (matrix) energy denominators, eliminating convergence difficulties in calculations of atomic valence shell hamiltonians. When the matrix energy denominators are taken to be the simplest form our generalized effective hamiltonian reduces to Brandow quasi-degenerate theory.     

A computationally efficient exact pseudopotential method. I. Analytic reformulation of the Phillips-Kleinman theory

Even with modern computers, it is still not possible to solve the Schrodinger equation exactly for systems with more than a handful of electrons. For many systems, the deeply bound core electrons serve merely as placeholders and only a few valence electrons participate in the chemical process of interest. Pseudopotential theory takes advantage of this fact to reduce the dimensionality of a multielectron chemical problem: the Schrodinger equation is solved only for the valence electrons, and the effects of the core electrons are included implicitly via an extra term in the Hamiltonian known as the pseudopotential. Phillips and Kleinman (PK) [Phys. Rev. 116, 287 (1959)]. demonstrated that it is possible to derive a pseudopotential that guarantees that the valence electron wave function is orthogonal to the (implicitly included) core electron wave functions. The PK theory, however, is expensive to implement since the pseudopotential is nonlocal and its computation involves iterative evaluation of the full Hamiltonian. In this paper, we present an analytically exact reformulation of the PK pseudopotential theory. Our reformulation has the advantage that it greatly simplifies the expressions that need to be evaluated during the iterative determination of the pseudopotential, greatly increasing the computational efficiency. We demonstrate our new formalism by calculating the pseudopotential for the 3s valence electron of the Na atom, and in the subsequent paper, we show that pseudopotentials for molecules as complex as tetrahydrofuran can be calculated with our formalism in only a few seconds. Our reformulation also provides a clear geometric interpretation of how the constraint equations in the PK theory, which are required to obtain a unique solution, are themselves sufficient to calculate the pseudopotential.     

Non local potential model for atomic barium

We calculate the barium atomic states using a two-electron effective hamiltonian via a configuration interaction method. We propose a non-local potential with a minimum number of three core states to separate the electromagnetic contributions from Pauli's. We give the first twenty monoexcited states and twenty diexcited states below the first ionization limit.     

Negativity of ground-state interaction energies of atom-excess-electron systems

Any atom—excess-electron system confined to an arbitrary finite region of space that is free of external fields and which can be characterized by a non-relativistic spin-free hamiltonian has, in the limit of infinite nuclear mass of the atom, an exact ground-state expectation value of the atom—electron interaction hamiltonian that is negative-definite.     

One-center two electron repulsion parameters in the π and all-valence semi-empirical theories

To examine the applicability of the γ = I_p — EA approximation to all-valence semi-empirical CI calculations, two simple models are studied. One is the model for π theory and the other is for all-valence theory. It is demonstrated that we cannot use the same two electron repulsion integrals in the γ only CI and in the all valence CI. The true effective parameters for π orbitals are evaluated within the second model from first principles by using a method recently proposed by Iwata and Freed.     

Analysis of the equation-of-motion theory of electron affinities and ionization potentials

An analysis of the equation-of-motion (EOM) method for computing molecular electron affinities and ionization potentials is presented. The method is compared with the Dyson equation approach of Green function theory. Particular emphasis is devoted to clarifying the similarities between these two theories when carried out to second and to third order. The Epstein—Nesbet hamiltonian and the notion of diagonal scattering renormalization have been used to systematize this comparison.     

Core–Valence correlations and relativistic effects in heavy atoms for molecular applications

Theoretical core effective potential methods are widely used in valence-only electron molecular calculations. These methods, which imply the frozen-core approximation, work well for the elements of the righthand side of the periodic table but are often unrealistic for metallic elements with highly polarizable cores. For these atoms one has to consider the polarization of the cores under the influence of the electric field created by the valence electrons. Moreover, relativistic corrections must be added for heavy atoms. Various theoretical approaches of core–valence interactions (polarization and core–valence correlations) will be reviewed, with a special emphasis on practical methods of calculation. The problem of handling the relativistic effects will mainly be discussed within the two-component Pauli formalism. It will be shown that the Foldy–Wouthuysen transformation is not the unique way for deriving relativistic corrections and that the second-order Dirac equation also provides a good starting point for obtaining relativistic corrections. Analytical exact results are given for the hydrogen atom. The accuracy of this approach is tested on many-electron atoms and molecules. It is finally shown that the problem of the core-valence separation is relevant to the general methodology of effective Hamiltonians that seems to provide the best promising way for filling the gap between the semiempirical and purely theoretical ab initio methods.     

Electron affinity of the sodium atom within the coupled-channel hyperspherical approach

We present a nonadiabatic calculation, within the hyperspherical adiabatic approach, for the ground state energy of the alkali-metal negative ions. An application to the sodium negative ion (Na-) is considered. This system is treated as a two-electron problem in which a model potential is used for the interaction between the Na+ core and the valence electrons. Potential curves and nonadiabatic couplings are obtained by a direct numerical calculation, as well as the channel functions. An analysis of convergence is made and comparisons of the electron affinity with results of prior work of other authors are given.     

Spectroscopy on Rydberg states of sodium atoms on the surface of helium nanodroplets

One- and two-photon excitation spectra of sodium atoms on the surface of helium droplets are reported. The spectra are recorded by monitoring the photoionization yield of desorbed atoms as function of excitation frequency. The excitation spectra involving states with principal quantum number up to n = 6 can be reproduced by a pseudodiatomic model where the helium droplet is treated as a single atom. For the lowest excited states of sodium, the effective interaction potentials for this system can be approximated by the sum of NaHe pair potentials. For the higher excited states, the interaction of the sodium valence electron with the helium induces significant configuration mixing, leading to a failure of this approach. For these states, effective interaction potentials based on a perturbative treatment of the interactions between the valence electron, the alkali positive core, and the helium, as described in detail in the accompanying publication, yield excellent agreement with experiment.     

Superfine and hyperfine structures in the v3 band of 32SF6

We present a first detailed account of our theoretical approach to reproduce observed superfine and hyperfine structures in the ν₃ band of SF₆ and we display various observed and calculated patterns of superfine clusters exhibiting hyperfine effects. The main operators of the hamiltonian are derived and the associated constants are related to molecular parameters. We show that, owing to the off-diagonal terms in the hyperfine hamiltonian, a mixing occurs between vibration—rotation states with different point-group symmetry species. As a consequence, superfine and hyperfine structures have to be considered simultaneously and hyperfine hamiltonian matrices connecting several vibration—rotation states need to be diagonalized to reproduce the spectra. We analyse in greater detail a few typical examples from which several molecular constants have been determined (e.g. t₀₄₄, c_d). For the first time, the sign c_d is obtained. Also an effective change, Δc_d, is found between upper and lower levels which can be readily interpreted as a manifestation of the tensor spin—vibration interaction.