Two-sheeted Surface Research Articles

Vibronic eigenstates and the geometric phase effect in the 2E″ state of NO3.

The 2E″ state of NO3, a prototype for the Jahn-Teller effect, has been an enigma and a challenge for a long time for both experiment and theory. We present a detailed theoretical study of the vibronic quantum dynamics in this electronic state, uncovering the effects of tunnelling, geometric phase, and symmetry. To this end, 45 vibronic levels of NO3 in the 2E″ state are determined accurately and analyzed thoroughly. The computation is based on a high quality diabatic potential representation of the two-sheeted surface of the 2E″ state developed by us [W. Eisfeld et al., J. Chem. Phys. 140, 224109 (2014)] and on the multi-configuration time dependent Hartree approach. The vibrational eigenstates of the NO3- anion are determined and analyzed as well to gain a deeper understanding of the symmetry properties of such D3h symmetric systems. To this end, 61 eigenstates of the NO3- anion ground state are computed using the single sheeted potential surface of the 1A1 state published in the same reference quoted above. The assignments of both the vibrational and vibronic levels are discussed. A simple model is proposed to rationalize the computed NO3 spectrum strongly influenced by the Jahn-Teller couplings, the associated geometric phase effect, and the tunnelling. Comparison with the available spectroscopic data is also presented.

Full-dimensional diabatic potential energy surfaces including dissociation: The 2E″ state of NO3

A scheme to produce accurate full-dimensional coupled diabatic potential energy surfaces including dissociative regions and suitable for dynamical calculations is proposed. The scheme is successfully applied to model the two-sheeted surface of the (2)E″ state of the NO3 radical. An accurate potential energy surface for the NO₃⁻ anion ground state is developed as well. Both surfaces are based on high-level ab initio calculations. The model consists of a diabatic potential matrix, which is expanded to higher order in terms of symmetry polynomials of symmetry coordinates. The choice of coordinates is key for the accuracy of the obtained potential energy surfaces and is discussed in detail. A second central aspect is the generation of reference data to fit the expansion coefficients of the model for which a stochastic approach is proposed. A third ingredient is a new and simple scheme to handle problematic regions of the potential energy surfaces, resulting from the massive undersampling by the reference data unavoidable for high-dimensional problems. The final analytical diabatic surfaces are used to compute the lowest vibrational levels of NO₃⁻ and the photo-electron detachment spectrum of NO₃⁻ leading to the neutral radical in the (2)E″ state by full dimensional multi-surface wave-packet propagation for NO3 performed using the Multi-Configuration Time Dependent Hartree method. The achieved agreement of the simulations with available experimental data demonstrates the power of the proposed scheme and the high quality of the obtained potential energy surfaces.

Some properties of a solution of the Ernst equation

Using Ehlers and unitary transformations, from Bonanos solution of the Ernst equation, we build a new vacuum stationary axisymmetric solution of Einstein equations depending on three parameters. The parameters are associated with the total mass of the source and its angular momentum. The third parameter produces a topological deformation of the ergosphere making it a two-sheet surface, and for some of its values forbids the Penrose process.

Variations in molecular compactness and chain entanglement during the compression of grafted polymers

Characterizing the effect of geometrical confinement on mean polymer shape is an important step towards understanding and controlling molecular behaviour at interfaces. In this work, we study the configurational transitions and molecular shape changes that take place when a grafted polymer (or mushroom) is compressed by a hard plane. The polymer is modelled as a single. permanently-grafted chain with a Lennard-Jones interaction between monomer beads. For this model, we have monitored molecular size, asphericity, and chain entanglements as a function of compression, from the regime of self-avoiding walks to the regime of collapsed polymers. With these tools. we show that strong confinement can produce chain compactization and disentanglement even in the presence of a mild attractive interaction. Our results provide limit values to the degree of compression and monomer attraction that is necessary to deform strongly collapsed polymer mushrooms. Schematic representation of chain entanglements as a function of molecular size and monomer-monomer interaction. The surface has a bifurcation point (denoted by B), marking the shape transition from a soft to a hard polymer. At weak interactions, we find a two-sheet surface associated with the compressing behaviour of soft mushrooms, In this case, the compactization of free mushrooms corresponds to paths on the lower surface, whereas the transition to flat polymers takes place on the upper surface. (The direction of compression is indicated by an arrow in the drawing.) For strong interactions, we find only a one-sheet surface associated with the behaviour of hard mushrooms, On this surface, polymer shape is not affected by weak confinement, but strong confinement leads directly into flat polymers. In order to compare with Figs. 7-10, the letters a and b denote the limits of low and high compression, L = 100 A and L = 10 A, respectively.

Vertical two-phase flow in porous media

New concepts are introduced to describe single-component two-phase flow under gravity. The phases can flow simultaneously in opposite directions (counterflow), but information travels either up or down, depending on the sign of the wavespeedC. Wavespeed, saturation and other quantities are defined on a two-sheeted surface over the mass-energy flow plane, the sheets overlapping in the counterflow region. A saturation shock is represented as an instantaneous displacement along a line of constant volume fluxJ Q in the flow plane. Most shocks are of the wetting type, that is, they leave the environment more saturated after their passage. When flow is horizontal all shocks are wetting, but it is a feature of vertical two-phase flow that for sufficiently small mass and energy flows there also exist drying shocks associated with lower final saturations.

On the stress-strain state near a three-dimensional crack in a two-sheeted surface

A limit scheme of a two-sheeted Riemannian surface is used to illustrate special features encountered in the course of the study of the asymptotic form of the stresses and displacements near the edge of a three-dimensional crack. The fundamental first, second and mixed boundary-value problems are formulated on this surface by analogy with the case of a single plane, and are solved explicitly by quadratures by reducing them to a Riemann boundary-value matrix problem with a constant coefficient. The sheets of the surface are in a generalized plane stress state and have, generally speaking, different stress constants and different thicknesses. A scheme for investigating the stress-strain state of another two-sheeted construction different from the Riemannian surface is elucidated briefly.

On the propagation of elastic vibrations from a point source within an anisotropic half-space

There is investigated the solution, obtained in /1/, for the problem of elastic vibrations propation in an anisotropic half-space with four elastic constants, from a point source of an instantaneous pulse type. By using the results from /2–4/, the features of wave propagation as a function of the elastic constants relationships are studied. The case is examined, when the elastic constants satisfy conditions for which the wave fronts have no acute-angled edges. A Riemann surface is constructed for a single-valued determination of the solution and the correspondence between points of this surface and the wave fields is studied. To determine the incident and reflected waves on one two-sheeted surface, there is introduced the concept of a two-layer domain of definition of the reflected waves. Expressions are obtained for the primary, propagated from the source and secondary perturbation fronts reflected from the half-space boundary.

On the ambiguity in the specification of a two-sheeted surface by its branch curve

It is familiar that a “double plane”, i.e. an algebraic surface conceived as consisting of each point of a plane counted twice, is unambiguously determined when its “branch curve”, or locus of points in the plane which count as two coincident points of the surface (the general point of the plane counting as two distinct points), is assigned; the only condition which must be satisfied by this curve being that it is of even order. In fact, if f(x, y) = 0 is the Cartesian equation of the curve, the double plane can be regarded as the limit of the surfacewhere A is a large constant, or as the projection of this surface, for finite A, from the point at infinity on the z-axis. If, however, f = 0 is of odd order, it is easily seen that the line at infinity also forms part of the branch curve of the double plane. A repeated portion of the branch curve is ineffective, for if φ (x, y) = 0 is any other curve, the surfaceis birationally identical with the former merely by putting