Extended electron distributions applied to the molecular mechanics of some intermolecular interactions. II. Organic complexes

Summary Extended electron distributions (XEDs) have been used to simulate the formation of complexes by intermolecular interaction via: (i) aromatic stacking; and (ii) hydrogen bonding. The results qualitatively reproduce experimental observations. In contrast, atom-centred partial charges fail to reproduce highly hydrogen-bonded systems, but make little difference in cases where interactions are driven largely by van der Waals forces. The dielectric constant used in the Coulombic term has been shown to be significant in defining the type and properties of these interactions when XEDs are employed. Some consideration has been given to solvation and entropy effects.     

Existence and regularity in the small in the calculus of variations

A local existence theorem is proved for the basic problem in the calculus of variations, that of minimizing ∝L(t, x, /.x) dt over a class of functions x assuming given boundary conditions. The Lagrangian L is only assumed to be locally Lipschitz and strictly convex in its /.x variable.     

A maximum principle for nonsmooth optimal-control problems with state constraints

Optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints. The conditions, which apply in the absence of differentiability assumptions on the data, are stated in terms of Clarke's generalized Jacobians. Although not the most general available, the conditions are derived by a novel method: this involves removal of the state constraints by introduction of a penalty term and application of Ekeland's variational principle.     

Quantitative determination of intermolecular interactions with fluorinated aromatic rings

The chemical double mutant cycle approach has been used to investigate substituent effects on intermolecular interactions between aromatic rings and pentafluorophenyl pi-systems. The complexes have been characterised using 1H and 19F NMR titrations, X-ray crystal structures of model compounds and molecular mechanics calculations. In the molecular zipper system used for these experiments, H-bonds and the geometries of the interacting surfaces favour the approach of the edge of the aromatic ring with the face of the pentafluorophenyl pi-system. The interactions are generally repulsive and this repulsion increases with more electron-withdrawing substituents up to a limit of +2.2 kJ mol(-1), when the complex distorts to minimise the unfavourable interaction. Strongly electron-donating groups cause a change in the geometry of the aromatic interaction and attractive stacking interactions are found (-1.6 kJ mol(-1) for NMe2). These results are generally consistent with an electrostatic model: the polarisation of the pentafluorophenyl ring leads to a partial positive charge located at the centre and this leads to repulsive interactions with the positive charges on the protons on the edge of the aromatic ring; when the aromatic ring has a high pi-electron density there is a large electrostatic driving force in favour of the stacked geometry which places this pi-electron density over the centre of the positive charge on the pentafluorophenyl group.     

Necessary Conditions for Free End-Time, Measurably Time Dependent Optimal Control Problems with State Constraints

Recently, necessary conditions have been derived for fixed-time optimal control problems with state constraints, formulated in terms of a differential inclusion, under very weak hypotheses on the data. These allow the multifunction describing admissible velocities to be unbounded and possibly nonconvex valued. This paper extends the earlier necessary conditions, to allow for free end-times. A notable feature of the new free end-time necessary conditions is that they cover problems with measurably time dependent data. For such problems, standard analytical techniques for deriving free-time necessary conditions, which depend on a transformation of the time variable, no longer work. Instead, we use variational methods based on the calculus of 'essential values".     

Many-body effects in the first excited subband of the n-inversion layer of Si

We have calculated the self-energies of electrons in the lowest and first excited sub-bands of Si inversion layers. The self-consistent wavefunctions calculated in the Hartree approximation were used, and dynamic screening was approximated by the Lundqvist-Overhauser model. The correlation energy of an electron in the excited band is quite large: about ?10 meV at an inversion layer density of 10¹¹ cm^?2 to about ?16 meV at 3 × 10¹² cm^?2. The calculated separation between subbands is in very good agreement with available experimental measurements. An exciton is predicted with a binding energy of 0.9 meV at N_inv = 10¹² cm^?2 calculated in the static approximation.     

A maximum principle for optimal control problems with mixed constraints

Necessary conditions in the form of maximum principles are derivedfor optimal control problems with mixed control and state constraints.Traditionally, necessary condtions for problems with mixed constraintshave been proved under hypothesis which include the requirementthat the Jacobian of the mixed constraint functional, with respectto the control variable, have full rank. We show that it canbe replaced by a weaker ‘interiority’ hypothesis.This refinement broadens the scope of the optimality conditions,to cover some optimal control problems involving differentialalgebraic constraints, with index greater than unity.