Robert J. Low

Time reversible and symplectic integrators for molecular dynamics simulations of rigid molecules

Molecular dynamics integrators are presented for translational and rotational motion of rigid molecules in microcanonical, canonical, and isothermal-isobaric ensembles. The integrators are all time reversible and are also, in some approaches, symplectic for the microcanonical ensembles. They are developed utilizing the quaternion representation on the basis of the Trotter factorization scheme using a Hamiltonian formalism. The structure is similar to that of the velocity Verlet algorithm. Comparison is made with standard integrators in terms of stability and it is found that a larger time step is stable with the new integrators. The canonical and isothermal-isobaric molecular dynamics simulations are defined by using a chain thermostat approach according to generalized Nosé-Hoover and Andersen methods.

A generalized scaling of a chiral index for molecules

We propose a scaling of an intrinsic molecular chirality index calculated from atomic positions. Its application to the design of chiral molecules through a consideration of atomic or group chiral indices is discussed for a range of molecules from small molecules to liquid crystal molecules. It is found to indicate the trend in change of chirality of a molecule caused by an atom or group substitution. Comparison is made between the method of scaling and an analytic calculation of the index for an orthogonal tetrahedron. We investigate the effect of substitution of atoms and groups in real liquid crystal molecules by application of the generalized chiral index.

Measuring order and biaxiality

When one considers systems with orientational order, such as liquid crystals, there are various aspects of the ordering that one wishes to quantify. Two properties of the system that one wishes to identify are its overall director, and the extent to which it exhibits biaxiality. We give an exposition of some standard approaches to these issues suitable for advanced undergraduates or beginning postgraduates.

The geometry of the space of null geodesics

The topology and geometry of the space of null geodesics N of a space‐time M are used to study the causal structure of the space‐time itself. In particular, the question of whether the topology of N is Hausdorff or admits a compatible manifold structure carries information on the global structure of M, and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic.

Twistor linking and causal relations

The space of null geodesics, N, of Minkowski space, M, is a five-dimensional CR manifold diffeomorphic to R3 x S2, and the celestial sphere of a point in M is a smooth S2 in N. The dimensionality is therefore appropriate for the consideration of linking properties. An appropriate definition of linking is given, both in terms of the homology of N and in terms of intersection theory. The relationship between this concept of linking and the causal structure of M is explored.

The Space of Null Geodesics (and a New Causal Boundary)

The space of null geodesics, G, of a space-time, M, carries information on various aspects of the causal structure M. In this contribution, we will review the space of null geodesics, G, and some natural structures which it carries, and see how aspects of the causal structure of M are encoded there. If M is strongly causal, then G has a natural contact manifold structure, points are represented in G by smooth Legendrian S 2 s, and the relationships between these S 2 s reflect causal relationships between the points of M. One can also attempt to pass in the opposite direction with the intention of constructing a space-time from a family of S 2 si nG; this process suggests a means of attaching end-points to null geodesics of M, and thereby constructing a causal boundary. We close by summarizing some open questions in this general area.

Helical twisting power and chirality indices

We review the statistical theory of helical twisting power and propose an alternative derivation of the results obtained by Nordio and co-workers [1999, J. chem. Soc. Perkin Trans., 2, 411]. We examine chiral indices which include an interaction between the host and dopant, then compare these with chiral indices dependent only on the dopant structure. We discuss the need for the inclusion of the interaction between dopant and host in the description of chiral induction, and discuss alternative measures of chirality and their shortcomings. We show that both the previously suggested surface indices and the newly proposed one vanish for at least one class of non-trivial chiral structures. We compare the results obtained from these surface indices with a more general 3D chiral index and evaluate these indices for four helicene (M)-1–4 molecules.

Correlation between molecular chirality and helical twisting power: a computer simulation study

The helical structure of the chiral nematic phases induced by chiral dopants in nematic solvents provides a macroscopic information of the molecular chirality of the dopant. Chiral dopant molecules considered in this paper are challenging systems because their twisting ability has a relatively strong dependence on the molecular structure. Several classes of the flexible chiral dopants such as the binaphthyl derivatives characterized by a conformational chirality are investigated. In addition, we will investigate the odd–even effect on the helical twisting power of two azobenzene derivatives and the twist inversion with temperature and solvent of the (1S)-1-naphthalen-1-ylethyl amide and trans-4,5-diaryl-1,3-dioxolanes derivatives, respectively. In this work two different theoretical approaches, the surface model and the scaled chiral index, are studied which are combined with molecular dynamics simulations of the single molecule in the gas phase. The results are compared with the experimental values of the helical twisting power.

Stable singularities of wave-fronts in general relativity

A wave-front in a space-time M is a family of null geodesics orthogonal to a smooth spacelike two-surface in M; it is of some interest to know how a wave-front can fail to be a smoothly immersed surface in M. In this paper we see that the space of null geodesics N of M, considered as a contact manifold, provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave-fronts.

Twistor linking and causal relations in exterior Schwarzschild space

We investigate the causal structure of , the spacetime consisting of that portion of Schwarzschild space outside the event horizon. Two points, p and q in which do not lie on a common null geodesic are chronologically related if and only if their celestial spheres, P and Q have non-zero linking number in , the space of null geodesics of .