The Laplacian of the spherically averaged charge density ∇2ρ̄(r) has been computed from nonrelativistic SCF wave functions for the neutral atoms from hydrogen to uranium, and the singly positive ions, from helium to barium and lutetium to radium, in order to examine the shell structure. ∇2ρ̄(r) exhibits a number of extremal points and zeros with the absolute value of the function becoming smaller at each successive extremal point. The zeros, in particular the odd numbered zeros, are shown to exhibit good correlation with the Bohr theory of an atom while the extremal points correlate to a lesser extent. At most five shells are seen in the studied atomic cases based on the fact that the odd numbered zeros are the topological feature of ∇2ρ̄(r) most indicative of a shell.
Two-dimensional crystals of the protein streptavidin form on lipid monolayers containing biotinylated lipids. Introducing avidin as a noncrystallizable contaminant induced dendritic growth of the streptavidin crystals. Increasing the proportion of avidin to streptavidin shifted the dendritic streptavidin crystal morphology from amorphous fingers to needlelike structures. These results provide insight into the mechanism and dynamics of the two-dimensional protein crystallization process.Dendritic crystal growth is an example of spontaneous pattern generation in nonequilibrium systems.* 1 While three-dimensional dendrites have been grown and analyzed under controlled conditions,2 efforts have also been made to simplify the analysis by reduction of dimensionality, through growth in a narrow gap between parallel plates.3 * To whom correspondence should be addressed. f Department of Chemical Engineering.
Two-dimensional crystals of streptavidin formed on lipid layers at the air-water interface were cross-linked to preserve their structure during transfer to a solid support. The morphology of the crystals viewed at the air-water interface by light microscopy could then be related to the molecular structure revealed in the electron microscope. The preferred growth direction of the crystals proved to be along one of two intersecting rows of intermolecular contacts. A secondary direction arises during a transition to dendritic growth.
The radial charge distribution, D ( r ) , for the neutral atoms, hydrogen through uranium, and the singly positive ions, helium through barium and lutetium through radium, are computed from the non-relativistic SCF wavefunctions of Clementi and Roetti and McLean and McLean. The radial charge distribution is examined to see how effective it is for the determination of the shell structure of an atom or an ion with the results indicating that the maxima are the topological features of D ( r ) most indicative of shells. At most five shells are apparent based on the criterion that the maxima in D ( r ) correspond to the shells. The positions of the maxima in D ( r ) are shown to correlate well with the shell radii from the Bohr-Schrodinger theory of an atom. The small deviations are consistent with the shielding of the outer electrons from the nucleus, by the inner electrons. Furthermore, it is shown that the effects of shielding are more pronounced on the values of the radial distribution function at the extremal points. The spherically averaged charge density, p(r), is seen to be monotonically decreasing for all these ground states, a result which extends the previously known range of atomic systems for which this is true. ALFREDO M. SIMAS, ROBIN P. SAGAR, ANDREW C. T. KU et VEDENE H. SMITH, JR. Can. J. Chem. 66, 1923Chem. 66, (1988. Utilisant les fonctions d'ondes auto-cohCrentes qui ne sont pas relativistes de Clementi et Roetti ainsi que de McLean et McLean, on a calcult les distributions des charges radiales, D ( r ), des atomes neutres, de I'hydrogene a I'uranium, ainsi que des ions monovalents positifs, de I'hClium au baryum et du lutCtium au radium. On a CtudiC les distributions relatives des charges afin d'ivaluer si elles sont efficaces pour dkterminer la structure des couches d'un atome ou d'un ion; les resultats indiquent que les maxima sont les caractCristiques topologiques de D ( r ) qui donnent le plus d'indications sur les couches. On a dtmontrC qu'il existe une bonne corrClation entre les positions des maxima des D ( r ) et les rayons des couches obtenus par la thiorie de Bohr-Schrijdinger d'un atome. Les petites dkviations sont en accord avec le fait que les Clectrons des couches intkrieurs provoquent un blindage du noyau par rapport aux Clectrons extirieurs. De plus, on a dCmontrC que les effets de blindage sont plus prononcCs sur les valeurs des points extremes de la fonction de distribution radiale. On observe que la densitk de charge moyenne pour la sphere, p(r), diminue d'une faqon monotone pour tous les Ctats fondamentaux; ce rCsultat prolonge le domaine des systemes atomiques pour lesquels cette conclusion peut s'appliquer.[Traduit par la revue]
. 66, 1005 (1988).The one-electron potential, v 2 G ( r ) / 2 g P ( r ) , appearing in the Schrodinger equation for the charge density is calculated for the neutral atoms from hydrogen to uranium, and the singly positive ions, from helium to barium and lutetium to radium. These computations, utilizing the nonrelativistic SCF wavefunctions of Cle~nenti and Roetti and McLean and McLean, were performed in order to investigate the concept of shell structure as defined by this potential. v2%$(r)/2*(r) exhibits a number of zeros and extrema corresponding to classically allowed and forbidden regions, with its topology being very similar in nature to that of V2p(r). The significant difference is that v 2 V $ ( r ) / 2 f l ( r ) displays all seven shells in the heavier elements of the periodic table, whereas V 2 p ( r ) displays only five shells; i.e. compared with V 2 p ( r ) , v ' G -( r ) / 2 G ( r ) has additional zeros and extrema at large distances corresponding to the two outer shells. The positions of these additional zeros and extrema, when plotted against atomic number Z, exhibit large deviations from the Bohr model of the hydrogen atom. The outermost zeroextremum is not displayed in the transition elements; i.e. where d-orbital filling is present. The model of classically allowed and forbidden regions is shown to provide an alternative definition of shell structure to that of the simple Bohr-Schrodinger theory of an atom. The odd-numbered zeros in v 2 V $ ( r ) / 2 f l p ( r ) provide a topological feature capable of representing all shells. Furthermore, these zeros appear at precisely the atomic number as the corresponding shell predicted by the aufbau principle. Introduction Density functional theory (1) offers the possibility of reducing the N-electron problem into a more viable one-electron one. However, there are difficulties that are encountered such as N-representability, i.e. to guarantee that a computed density is representative of an ensemble of N fermions (2). In the case of the wavefunction, this N-representability is achieved by making the wavefunction antisymmetric with respect to exchange of coordinates of any two of the N indistinguishable fermions.The Schrodinger equation for the electron density as proposed by Hunter (3) is of interest since this equation is an "exact dynamical model of the motion of a single electron in a many-electron system" (3). Thus, this particular Schrodinger equation is useful for the interpretation of molecular and atomic structure based on the potential,1 {U(r) -E} = v 2 G ( r ) / 2 6 ( r ) , within which any one electron moves.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.