There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Publisher Summary Many interesting phenomena can be described in terms of a resonance structure. This chapter discusses biorthogonal basis sets and the characteristics of Gamow–Siegert states. These two concepts form a convenient basis on which to formulate the rest of the material. The chapter provides a discussion of the complex coordinate theorems; the properties of the bound, scattering, and resonance states; and analytical examples to illustrate these properties. A variational principle is derived in the chapter along with a discussion on various types of configuration interaction and self-consistent calculational approaches that have used this variational principle is presented. The generalization of the stabilization techniques for configuration integration, self-consistent field, and many-body calculations to what the complex stabilization method for the real Hamiltonian is discussed in the chapter.

Recent Computational Developments in the use of Complex Scaling in Resonance Phenomena

A benchmark theoretical study of the electronic ground state and of the vertical and adiabatic singlet-triplet (ST) excitation energies of benzene (n=1) and n-acenes (C4n+2H2n+4) ranging from naphthalene (n=2) to heptacene (n=7) is presented, on the ground of single- and multireference calculations based on restricted or unrestricted zero-order wave functions. High-level and large scale treatments of electronic correlation in the ground state are found to be necessary for compensating giant but unphysical symmetry-breaking effects in unrestricted single-reference treatments. The composition of multiconfigurational wave functions, the topologies of natural orbitals in symmetry-unrestricted CASSCF calculations, the T1 diagnostics of coupled cluster theory, and further energy-based criteria demonstrate that all investigated systems exhibit a A1g singlet closed-shell electronic ground state. Singlet-triplet (S0-T1) energy gaps can therefore be very accurately determined by applying the principles of a focal p...

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A benchmark theoretical study of the electronic ground state and of the singlet-triplet split of benzene and linear acenes

Time dependent variation perturbation calculations have been performed for estimating the transition energies, os- cillator strengths and transition probability values for a few dipole allowed states of compressed hydrogen atom confined in a weakly coupled plasma. The compression is obtained by embedding the atom at the centre of an impenetrable spherical box. The dipole polarizability of the atom is evaluated at each confinement radius with respect to different plasma screening param- eters. The effect of pressure due to spatial confinement on the dipole polarizability and other atomic properties is analyzed. Results obtained are useful for the diagnostic determination of astrophysical and laboratory plasmas and for the calculation of collision rate coefficients needed for computing opacity of stellar envelopes - a quantity of importance in the context of stellar structure and pulsations.

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Energy levels and structural properties of compressed hydrogen atom under Debye screening

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.

Spectroscopy of Confined Atomic Systems: Effect of Plasma

Very accurate variational calculations have been performed to determine the ground state energy of the negative hydrogen ion when it is embedded in a variety of Debye plasmas. The results predict a high degree of stability even under strong plasma conditions. {copyright} {ital 1996 The American Physical Society.}

Detachment energies for a negative hydrogen ion embedded in a variety of Debye plasmas

The purpose of the present study is twofold. First, the techniques of correlated wave functions for two-electron systems have been extended to obtain results for P and D states in a screening environment, and in particular for Debye screening. In these calculations, the satisfaction of both the quantum virial theorem and a related sum rule has been enforced and found to provide a high degree of stability of the solutions. Second, in order to facilitate the general use of correlated wave functions in combination with sum rule stability criteria, a rather systematic computational approach to this notoriously cumbersome method has been developed and thoroughly discussed here. Accurate calculations for few-electron systems are of interest to plasma diagnostics; in particular, when inaccuracies in binding energies are drastically magnified as they occur in exponents of Boltzmann factors.

Calculations of properties of screened He-like systems using correlated wave functions.

The pair-function formalism has been extended to the calculation of atomic and ionic properties in plasma environments that are modeled in terms of analytic screening potentials of the usual Debye form and the more general Debye-Laughton type. The theory has been applied to two-electron systems. As particular applications, the stability of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ and the energy splitting of several 1sns and 1snp states of the He atom have been studied in model plasma environments as well as the energies of allowed and forbidden transitions.

Pair-function calculations for two-electron systems in model plasma environments

Aspects of the influence of a strong Debye plasma environment on the negative hydrogen ion and the neutral helium atom have been studied. Contrary to earlier work, in the present calculation all interactions have been screened. This increases the stability of the systems to the extent that neither the H- ion nor the ground state of helium will lose an electron by pressure ionization. It has been found that the charge distribution of H- remains remarkably constant over a vast range of values of the Debye parameter D. © 1996 John Wiley & Sons, Inc.

Negative hydrogen and helium in a variety of debye plasmas

Computational studies are presented for atoms in screening environments. Numerical solution of atomic Hartree–Fock equations in the Slater type orbitals basis are presented for screened Debye electron–electron as well as electron–nucleus interaction. Slater integrals are evaluated using a Gauss–Laguerre quadrature. Detailed numerical results are presented for various atoms and Debye lengths establishing the method as well as showing the opposing effects of electron–nucleus and electron–electron screening that may induce ambiguity in physical properties extracted from experimental plasma data. A detailed discussion of convergence properties is given. The relevant outcome of the present study revealed the fact that the present expansion method and the previously published Legendre expansion method (Winkler, Int. J. Quantum Chem. 2010, 110, 3129) have their best convergence properties in different regions of the Debye screening length. This is important for applications. In addition, the new expansion method can be implemented for other screening potentials, where other approaches fail. © 2013 Wiley Periodicals, Inc.

Peter Winkler

Papers

Detachment energies for a negative hydrogen ion embedded in a variety of Debye plasmas

Calculations of properties of screened He-like systems using correlated wave functions.

Pair-function calculations for two-electron systems in model plasma environments

Negative hydrogen and helium in a variety of debye plasmas

Computation of screened two-electron matrix elements