Etude de la structure du cortège électronique par la méthode des états de valence

Starting with Dirac's theory, we build up a Hamiltonian for an atomic system with several electrons. The investigation of different ways of constructing the state space of the polyelectronic system leads to the definition of “electronic configuration” and “valence state”. Using these concepts a method for calculating the atomic wave functions is set forth, which allows a precise determination of spectral term energies.     

A proposed definition of resonant states

The widely cited definition of quantization in terms of square-integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many-body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non-Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest-lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy.     

Remarks on the formulations of the adiabatic theorem

By comparing the adiabatic limit of the exact solutions of the time-dependent Schrödinger equation for spin in rotating magnetic field and for harmonic oscillator with time-dependent frequency with the solutions obtained using the quantum adiabatic theorem we have demonstrated the complete agreement of the two sets of solutions and the importance of phase fixing condition for this agreement. We argue that the notions like “familiar dynamical phase” of the “usual quantum adiabatic theorem” and “an additional phase” of “geometrical origin” have been based on the unjustified neglection of the mentioned condition by applying the quantum adiabatic theorem. There is nothing to add to the quantum adiabatic theorem in which time-dependent eigenbasis satisfies phase fixing condition.     

Density functional theory for Jahn–Teller systems

The first discussion of the dynamics of Jahn–Teller systems in terms of the electronic density as the fundamental variable was given by W.J. Clinton in 1960, where the degenerate electronic configuration of a Jahn–Teller molecule was interpreted in terms of the infinite number of ways in which the charge distribution can be oriented for the same energy. The moving nuclear framework serves as the perturbation necessary to define the orientation of the charge density, with no activation energy required to put the charge cloud into motion. Recently, this notion of the electronic charge cloud in a Jahn–Teller molecule sweeping out the potential surface over which the nuclei move has found mathematical expression in our work in terms of a generalized electronic current density in nuclear-coordinate space [N. Sukumar and B.M. Deb, Int. J. Quantum Chem. 40 , 501 (1991)]. The introduction of the electronic phase as a function of both electronic and nuclear coordinates, in addition to the electronic density, is a crucial component of this formulation. In the present work, the density-based treatment is extended to the nonadiabatic situation, with the Born couplings interpreted as nonadiabatic currents in parameter space. Abelian and non-Abelian gauge transformations of these currents are discussed. © John Wiley & Sons, Inc.     

The eigenvalue problem in phase space

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c‐function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi‐representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.     

First principles view on chemical compound space: Gaining rigorous atomistic control of molecular properties

A well‐defined notion of chemical compound space (CCS) is essential for gaining rigorous control of properties through variation of elemental composition and atomic configurations. Here, we give an introduction to an atomistic first principles perspective on CCS. First, CCS is discussed in terms of variational nuclear charges in the context of conceptual density functional and molecular grand‐canonical ensemble theory. Thereafter, we revisit the notion of compound pairs, related to each other via “alchemical” interpolations involving fractional nuclear charges in the electronic Hamiltonian. We address Taylor expansions in CCS, property nonlinearity, improved predictions using reference compound pairs, and the ounce‐of‐gold prize challenge to linearize CCS. Finally, we turn to machine learning of analytical structure property relationships in CCS. These relationships correspond to inferred, rather than derived through variational principle, solutions of the electronic Schrödinger equation. © 2013 Wiley Periodicals, Inc.     

Molecular point symmetry and the phase of the electronic wave function: Tools for the prediction of critical points of potential energy surfaces

By combining a well-known theorem of Longuet-Higgins on the phase changes of electronic wave functions along loops encircling a conical intersection of potential surfaces, and a recently proven “vertical symmetry theorem” on the relative distributions of point symmetry groups and potential surface critical points in configuration space, a new result is obtained, suitable for the prediction of the presence of a critical point within a domain of configurations.     

New aspects of quantum electrodynamics on electronic structure and dynamics

Application of Alpha-oscillator theory to quantum electrodynamics (QED) solves the mystery (Feynman) of the double-slit phenomenon involved in the foundation of quantum mechanics (QM). Even if with the same initial condition given, different spots on the screen can be predicted deterministically with no introduction of hidden variables. The interference pattern is similar to, but cannot be reproduced quantitatively by, that of the QM wave function, contrary to many-years-anticipation: a new prediction, awaiting experimental test over and above the Bohr–Einstein gedanken experiment. The general proof has already been published in Ref. [3a] and the concrete numerical algorithm of the extended normal mode technique for concrete trajectory of one electron in Ref. [3b]. In this article, (1) the new “interpretation” of the QED wave function is given in section “Interpretation of Wave Function in QED”: the QED wave function used in the extended normal mode technique gives probability density distribution function of the initial values of trajectories. Moreover, (2) for the sake of demonstration of this new interpretation, the time-independent stationary state QM wave function is substituted to the QED wave function in section “Internal Self-Stress of Energetic Particles”: the QED wave function is realized by internal self-stress revealed as energy density at the initial conditions. The renewed energy density is applied to study a unified scheme for generalized chemical reactivity. This is a new kind of chemical force acting in between electrons not in between nuclei. This paves a way for more advanced time-dependent simulation of electronic structure and dynamics in chemical reaction dynamics by tracing trajectories of many electrons. © 2018 Wiley Periodicals, Inc.     

A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.     

Density functional theory of born couplings: Consequences for electron flow in Jahn–Teller molecules and superconductors

The dynamics of Jahn–Teller systems has recently been discussed in terms of generalized electronic charge and current densities in nuclear-coordinate space. The introduction of the electronic phase as a function of both electronic and nuclear coordinates, in addition to the electronic density, was a crucial component of this formulation. Here, a densitybased treatment of Born couplings is derived from first-principles quantum mechanics beyond the Born–Oppenheimer approximation. Because of the degenerate electronic configuration of a Jahn–Teller molecule, there are an infinite number of ways in which the charge distribution can be oriented for the same energy, leading to a vanishing bond hardness for the molecule in the symmetric nuclear configuration. Further, the moving nuclear framework serves as the perturbation necessary to define the orientation of the charge density, leading to unhindered rotation of the charge cloud. This leads to the dynamical Jahn–Teller problem, namely, the coupling of electronic and nuclear motions through the Born coupling terms. Applications to superconductivity theory are discussed. © 1995 John Wiley & Sons, Inc.     

Competition between excitation and electronic decay of short-lived molecular states

The nuclear dynamics accompanying the excitation to and the subsequent decay of an electronic state is discussed. Particular attention is paid to cases, in which the whole process cannot be divided into two steps (excitation and decay) since the excitation and the decay times are of the same order of magnitude. The recently introduced time-dependent formulation of the theory describing the wave packets’ dynamics is extended to include the excitation process. The wave packets can be related to the intensity of the emitted particles. Most of the resulting integrals can actually be performed by employing eigenstates of the Hamiltonians corresponding to the involved potential energy surfaces. This leads to the so called “timeindependent” formulation of the theory. Computational details of the implementation of the corresponding “timedependent” and “time-independent” methods are presented. Illustrative applications are given to illuminate both the influence of the excitation process and the lifetime of the decaying state. It emerges that the intuitive interpretation of the spectra (within the above two step model) may fail. Insight into the process is gained by studying the evolution of the spectra as a function of time. The appearance of “atomic lines” due to dissociative decaying and final states is investigated in some detail.     

Constituent units and framework conformations in zeolite networks

The building units of known zeolite networks have been reinvestigated and related to the framework density. Only a relatively small number of four-connected nets, which can be generated readily from observed constituent units, are “permissible” zeolite networks. Criteria for gauging the “permissibility” of zeolite-type networks in terms of geometrical constraints caused by conformational requirements have been investigated.     

Electronic stress tensor analysis of molecules in gas phase of CVD process for gesbte alloy

We analyze the electronic structure of molecules which may exist in gas phase of chemical vapor deposition process for GeSbTe alloy using the electronic stress tensor, with special focus on the chemical bonds between Ge, Sb, and Te atoms. We find that, from the viewpoint of the electronic stress tensor, they have intermediate properties between alkali metals and hydrocarbon molecules. We also study the correlation between the bond order which is defined based on the electronic stress tensor, and energy‐related quantities. We find that the correlation with the bond dissociation energy is not so strong while one with the force constant is very strong. We interpret these results in terms of the energy density on the “Lagrange surface,” which is considered to define the boundary surface of atoms in a molecule in the framework of the electronic stress tensor analysis. © 2015 Wiley Periodicals, Inc.     

Electronic structure of some mesoscopic systems: II. Electronic composites

We study the electronic properties of a mesoscopic system composed of an array of straight, infinite rods made of an isotropic medium and embedded in a regular way in an isotropic background. Such a composite system has two-dimensional periodicity in the plane perpendicular to the rods. Using a Fourier series expansion, the corresponding Schrödinger equation is solved within the effective-mass approximation. The electronic band structure is computed for the wave vector in the transverse plane, and is illustrated by dispersion curves along the principal directions of the two-dimensional Brillouin zone as well as by the histograms of the density of states. The main result is the appearance of absolute energy gaps in the two-dimensional band structure.