Claude Aslangul

Aspects of the Localizability of Electrons in Atoms and Molecules: Loge Theory and Related Methods

Publisher Summary This chapter presents the aspects of the localizability of electrons in atoms and molecules. The focus is on the loge theory. The origin of the concept is analyzed in the first followed by mathematical definitions. Various methods are proposed to obtain an approximation of the best decomposition in loges, starting from wave functions in which the localizability of electrons is not introduced a priori. Relations with localized orbitals are discussed. On the contrary, it is shown that the loge theory can be used to build new kinds of wave functions or to improve approximate wave functions. Relations between loge functions and group functions are analyzed. The chapter also presents various applications of the loge theory for stationary states. It is shown that this theory leads to a criterion that makes it possible to distinguish from a mathematical viewpoint between covalent and dative bonds. Furthermore, a theoretical justification of empirical molecular additivity is given. A very simple procedure to establish general relations between isomerization energies is derived. The chapter also deals with the possibility of extending the localization concept to electronic excitation. It is shown that under certain conditions the excitation energy may be localized in loges. More generally, excited-state properties may be expressed as contributions from loges.

Quantum dynamics of a particle with a spin-dependent velocity

We study the dynamics of a particle in continuous time and space, the displacement of which is governed by an internal degree of freedom (spin). In one definite limit, the so-called quantum random walk is recovered but, although quite simple, the model possesses a rich variety of dynamics and goes far beyond this problem. Generally speaking, our framework can describe the motion of an electron in a magnetic sea near the Fermi level when linearization of the dispersion law is possible, coupled to a transverse magnetic field. Quite unexpected behaviours are obtained. In particular, we find that when the initial wave packet is fully localized in space, the Jz angular momentum component is frozen; this is an interesting example of an observable which, although it is not a constant of motion, has a constant expectation value. For a non-completely localized wave packet, the effect still occurs although less pronounced, and the spin keeps for ever memory of its initial state. Generally speaking, as time goes on, the spatial density profile looks rather complex, as a consequence of the competition between drift and precession, and displays various shapes according to the ratio between the Larmor period and the characteristic time of flight. The density profile gradually changes from a multimodal quickly moving distribution when the scattering rate is small, to a unimodal standing but flattening distribution in the opposite case.

Diffusion of two repulsive particles in a one-dimensional lattice

The problem of the lattice diffusion of two particles coupled by a contact-repulsive interaction is solved by finding analytical expressions of the two-body probability characteristic function. The interaction induces anomalous drift with a vanishing velocity, the average coordinate of each particle growing at large times as t1/2. The leading term of the mean-square dispersions displays normal diffusion, with a diffusion constant made smaller by the interaction by the non-trivial factor 1-1/. The space continuous limit taken from the lattice calculations allows one to establish connection with the standard problem of diffusion of a single fictitious particle constrained by a totally reflecting wall. Comparison between lattice and continuous results display marked differences for transient regimes, relevant with regards to high time resolution experiments, and in addition show that, due to slowly decreasing subdominant terms, lattice effects persist even at very large times.

Etude quantitative de la distinction entre une liaison de covalence et une liaison de coordination: les molecules de borazane et d'aminoborane

An illustration of the distinction between these two types of chemical bonds is proposed, which is based on the properties of the molecule once built. Bond energies, overlap populations and group charges vary considerably from one compound to the other; more, a recent criterion given by two of the authors seems to be particularly suitable for describing the character of the bond. Lastly, one tries to give an explicit signification to the usual chemical symbols.RésuméOn propose un exemple quantitatif de la distinction entre une liaison de covalence et une liaison de coordination; cette différenciation, fondée sur les propriétés de la molécule une fois formée, semble très nette pour les énergies de liaison, les populations de recouvrement et les charges des groupes; en particulier, un critère récent, proposé par deux des auteurs, est remarquablement vérifié. Enfin, on essaie de préciser la signification des symboles chimiques traditionnels.ZusammenfassungEs wird ein quantitatives Beispiel der Unterscheidung zwischen einer kovalenten und einer koordinativen Bindung angegeben, die auf den Eigenschaften des vorliegenden Moleküls beruht. Sie erscheint sehr günstig für die Bindungsenergien, die Überlagerungspopulationen und die Gruppenladungen. Insbesondere wird ein Kriterium, das kürzlich von zwei der Autoren angegeben wurde, gut verifiziert. Schließlich wird versucht, die üblichen chemischen Symbole zu präzisieren.

Surprises in the suddenly-expanded infinite well

I study the time evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |?(x, t)|2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |?(x, t)|2 is constant on finite intervals for x. Elements of theoretical explanation are given by analysing the singular component of the second derivative ?xx?(x, t). Analytical closed expressions are obtained for some specific times, which easily allow us to show that, at these times, the density organizes itself into regular patterns provided the size of the box is large enough; more, above some critical size depending on the specific time, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.

Effect of a forbidden site on a d-dimensional lattice random walk

We study the effect of a single excluded site on the diffusion of a particle undergoing a random walk in a d-dimensional lattice. The determination of the characteristic function allows us to find explicitly the asymptotical behaviour of physical quantities such as the particle average position (drift) �� x� (t) and the mean square deviation �� x 2 � (t) −� �x� 2 (t). In contrast to the one-dimensional case, where �� x� (t) diverges at infinite times (�� x� (t) ∼ t 1/2 ) and where the diffusion constant D is changed due to the impurity, the effects of the latter are shown to be much less important in higher dimensions: for d 2, �� x� (t) is simply shifted by a constant and the diffusion constant remains unaltered although dynamical corrections (logarithmic for d = 2) still occur. Finally, the continuum space version of the model is analysed; it is shown that d = 1i s the lower dimensionality above which all the effects of the forbidden site are irrelevant.

Dynamical properties of a random walk in a one-dimensional random medium

Abstract The random motion of a particle in a one-dimensional continuous random medium with random forces is investigated by making use of the large-scale equivalence of the properties of the walk with those of a directed random walk on a discrete lattice. When the disorder is weak enough, a normal drift-diffusion regime takes place, whereas for strong disorder anomalous behaviours occur. It is shown that these results may be obtained by introducing a renormalized lattice with a finite spacing. In the normal phase this spacing appears explicitly in the result, while it turns out to be irrelevant in the anomalous ones. The dynamical exponents as well as the prefactors of the power-laws of the particle position and of its average dispresion are explicitly calculated in the anomalous phases.

Random-random walk on an asymmetric chain with a trapping attractive center

The random walk of a particle on an asymmetric chain in the presence of an attractive center, possibly trapping, is examined by means of the equivalent transfer rates technique. Both the situations of ordered and disordered hopping rates are studied. It is assumed that initially the particle is located on the attractive center. The (average) probability of presence of the particle at its initial point is computed as a function of time. In the ordered case this quantity decreases exponentially toward its limiting value (with in certain cases an inverse power-law prefactor), while in the presence of disorder it decreases according to a power law, with an exponent depending both on disorder and on asymmetry. When the possibility of trapping is taken into account, this model is relevant for the description of the transfer of energy in a photosynthetic system. The amount of energy conserved within the chain, as a function of time, and the average lifetime of the particle before it is captured by the trap are examined in both ordered and disordered situations.

One-dimensional motion in a biased random medium: Random potential versus random force

The one-dimensional motion of a particle in a biased random medium is studied in both cases of a random potential and of a random force. The correlations are not the same in both cases, and this is at the source of entirely different drift and diffusion behaviours. As a result, drift and diffusion are always normal in the presence of a random potential while anomalous motion and dynamical phase transitions may appear in the presence of a random force.

Random walk on a one-dimensional random lattice: Equivalence between bond disorder and site disorder models

Abstract The master equation describing the one-dimensional diffusive motion of a particle on an asymmetric random lattice is examined in the two cases of bond and site disorder. Both models are shown to possess the same long-time properties, thus falling into the same university class.