The diamagnetic Kepler problem

It has turned out that the analysis ofclassical periodic orbits is the key to understanding the modulations in thequantal spectra of hydrogen Rydberg atoms in magnetic fields. Inspired by this quantum chaological connection, we analyse some fundamental periodic orbits of the diamagnetic Kepler problem, abandoning the condition of vanishing azimuthal angular momentuml _z used in the literature so far. We report the bifurcation and confluence schemes of the orbits in their dependence onl _z and discuss the structural changes in terms of catastrophe theory.     

Extended universal correlation diagrams for the quantum mechanical one-electron two-centre problem

We have recomputed exact correlation diagrams for the energy of excited states; the system consists of an electron (or a different negatively charged particle) moving in the field of two fixed point chargesZ _a ,Z _b . The ratiosq=Z _a/Z_b taken into consideration here were 1, 2, and 3. In the caseq=3, three avoided crossings of energy curves belonging to states of equal symmetry type are contained in the diagram.     

Hermitian operators for two-centre wavefunctions

Semi-analytical solutions of the Schrödinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, are derived in such a way that the resulting matrix eigenvalue equations contain real symmetric band matrices. Numerical techniques appropriate to the solution of the two simultaneous matrix eigenvalue equations are described; in particular the bisection method is used to determine precisely the significant truncation order of the matrices for a given numerical precision.     

Quantum beats and Kepler motion in fast competing photoionization and photodissociation processes

Electronic and nuclear wavepackets created by coherent excitation of an autoionized and predissociated 'complex' resonance in H2 are studied theoretically using time-dependent multichannel quantum defect theory. The calculations predict that quantum beats between the components of the complex resonance interfere with Rydberg wavepacket (Kepler) motion to yield characteristic 'mixed' flux patterns in the observable time-dependent ionization and dissociation signals.     

The two-centre shell model for the fission of metallic clusters

We calculate potential energies for charged and neutral jellium clusters which fragment in two pieces, in the framework of the liquid drop model plus Strutinsky shell corrections obtained from the two-centre harmonic oscillator. We consider the symmetric fragmentation of Na ₄ ⁺ ₂ ⁺ , Na ₁ ⁺ ₈ ⁺ , and Na₃₈. Good agreement is found with results obtained by self-consistent methods, which are much more involved.     

Binolam-AlCl: a two-centre catalyst for the synthesis of enantioenriched cyanohydrin O-phosphates

The enantioselective synthesis of cyanohydrin O‐phosphates by using in situ generated bifunctional catalysts (R)‐ or (S)‐3,3′‐bis(diethylaminomethyl)‐1,1′‐binaphthol–aluminium chloride (binolam–AlCl) is reported. The reaction, which can be described as an overall cyano‐O‐phosphorylation of aldehydes, has a wide scope and applicability. Evidence is also provided, including ab initio and DFT calculations, in support of supported by the Lewis acid/Brønsted base (LABB) dual role of the catalyst in inducing first the key enantioselective hydrocyanation, which is then followed by O‐phosphorylation. A brief screening of the synthetic usefulness of the resulting cyanohydrin O‐phosphates unveiles some interesting applications. Among them, chemoselective hydrolysis, reduction and palladium‐catalysed nucleophilic allyl substitution, thereby leading to enantiomerically enriched α‐O‐phosphorylated α‐hydroxy esters, β‐amino alcohols and γ‐cyanoallyl alcohols, respectively. Naturally occurring (?)‐tembamide and (?)‐aegeline are synthesised accordingly.     

Rate constant distribution for the target problem on lattice clusters above the percolation threshold

A diffusion-controlled reaction A+B (N _A ≫N _B (0)) has been studied by computer simulation of random walk of particles A on a simple cubic lattice with randomly closed sites above the percolation threshold. It is shown that at low concentrations of walkers the nonexponential decay of particles B derives mainly from rate constant distribution.     

Analytical force constant calculation as a minimization problem

In this letter we show that the calculation of analytic second derivatives of variational potential energy surfaces with respect to nuclear coordinates is a minimization problem.     

Finite analytical expressions for two-centre exchange integrals between having the same exponents

Closed analytical expressions are derived for some two-centre exchange integrals between Slater orbitals. Integrals involving 1s, 2s and 2p orbitals are considered with the restriction that the two orbitals have the same exponent. An expansion formula accurate for large values ofR is also derived.

---

Zusammenfasung Für eine Reihe von Zweizentren-Austauschintegralen zwischen Slaterorbitalen werden geschlossene analytische Ausdrücke mitgeteilt, wobei allerdings nur ls-, 2s- und 2p-Orbitale mit gleichen Exponenten behandelt werden. Schließlich wird noch eine asymptotische Entwicklung für großeR angegeben.

---

Resumé Obtention d'expressions analytiques compactes pour certaines intégrales d'échange bicentriques entre orbitales de Slater. On considère des intégrales impliquant des orbitales ls, 2s et 2p avec comme restriction l'égalité des exposants orbitaux. Un développement valable pour les grandes valeurs deR est aussi obtenu.

---

---

On leave from the Institute Ruder Boskovi, Zagreb, Jugoslavia     

On the Dirac–Kepler problem: The Johnson–Lippmann operator,supersymmetry, and normal-mode representations

The relativistic Kepler problem is discussed, with emphasis on the exact supersymmetry of the problem. It is shown that the supersymmetry is generated by the Johnson–Lippmann operator. Two related operators are found to generate new supersymmetries in an extended function space. Each of these supersymmetries may be disguised as radial supersymmetries. The radial supersymmetries are discussed and it is shown that each of them defines a normal-mode representation of the hydrogen-atom radial functions. Thus, one obtains two different, but equivalent, analytical expressions for these functions. The expressions are well known, but are rederived here in the light of the new understanding. Finally, the nonrelativistic image of the relativistic supersymmetry is constructed and its generators shown to be identical with those recently presented in the literature. © 1995 John Wiley & Sons, Inc.     

Diffusion constant of a nonspecifically bound protein undergoing curvilinear motion along DNA

The mechanism of a protein's diffusion along a DNA segment is a subject of much current interest because of the involvement of this diffusion in numerous biological processes, including the recognition of DNA sequences and chemical modifications of DNA. In this work we present a theoretical derivation of the diffusion coefficient of a nonspecifically bound protein, assuming that the protein follows a helical track along the DNA. It is shown that, for protein-sized molecules, the principal contribution to the total translational friction comes from the curvilinear motion along the helix, and this contribution is given by 6pietaRR(oc)(2) + 8pietaR(3), where R is the protein radius, ROC is the distance of separation between the center of mass of the protein and the helical axis of DNA, and eta is the viscosity of the medium. The translational diffusion of the protein along the helical track of DNA is thus predicted to have a nearly R(-3) size dependence, not the R(-1) dependence characterizing simple translational diffusion. It is shown that this expression gives rather good estimates of the translational diffusion coefficient measured in single molecule experiments.     

Matrix elements for dipole transitions in the statement of the quantum problem for molecules with restrictions on nuclear motion

Based on the previously proposed solution for the Schrödinger equation, which corresponds to the representation of a molecule as a dynamically stable geometric figure [1–3] and explicitly reflects the continuous correlation between electronic and nuclear motions with retaining the separation of variables, expressions for the matrix elements of the dipole transition are obtained. It is shown that the equation common for all states has the matrix form. The appearance of parameters having simple meanings and simultaneously affecting both vibrational and electronic components of the general wave function and electron vibrational energy levels is noted. This enables the statement and solution of the corresponding inverse spectral problems.     

A constant voltage constant current wheatstone bridge configuration

For many measurement applications a Wheatstone bridge configuration can be used to measure physical quantities, such as pressure, force or velocity. Offset behaviour plays a vital role in the overall accuracy. Often, ambient temperature changes cause undesired changes in parameters that are supposed to be constant.In this paper, an integrated silicon double bridge configuration is presented using a sophisticated sensor biasing principle in one bridge, whilst simultaneously maintaining the supply current and the supply voltage constant.Sensor biasing is achieved by using thermal feedback between the two bridges. Applying this principle will reduce drift to a large extent and may even make laser or any other trimming procedure obsolete. Moreover, full integration of the sensor and the signal processing parts, consisting of complicated highly accurate analogue circuits, is possible due to the constant parameters provided. To demonstrate the utility of this concept, an integrated silicon anemometer is described.     

A constant current coulometer for the high precision analysis of uranium

 A constant current coulometry device was designed, built and evaluated. It is capable of controling the coulometric titration process and to deliver the necessary current and time supplies/measurements with extremely high accuracy. The device was applied to the coulometric titration of uranium. The chemical part of the system was adapted and improved and a reliable procedure was developed. The method is highly accurate, standard deviations lie around 0.06% relative. It is an absolute method and directly traceable to the SI units. Received: 4 March 1996/Revised: 23 December 1996/Accepted: 7 January 1997     

A method for enlarging the Kerr constant of polymer-stabilised blue phases

Polymer-stabilised blue phase (PSBP) is one of the most promising materials for display devices because of its superior electro-optical properties compared with conventional nematics. However, the application of the PSBP has a serious practical issue in that the driving voltage required is too high to drive with thin film transitors, that is, the magnitude of the Kerr constant of PSBPs is insufficient. We present a useful method for increasing the Kerr constant based on the control of polymer aggregation structure using a cross-linker with a chiral structure. The director distortion arising from polymer networks in the PSBP seems to be responsible for the resulting improvement.