Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.     

Variable step molecular dynamics: An exploratory technique for peptides with fixed geometry

A new algorithm is presented for performing molecular dynamics simulations of peptides with fixed geometry, with the aim of simulating conformational changes and of exploring conformational space. The principle of the method is to expand the potential energy as a Taylor's series in the coordinates around the current point, retaining the force and its first two derivatives, and obtain a series solution of the resulting differential equations using a method due to Lyapunov. By choosing the time step so that the second term in the series is small compared to the first, the true solution can in principle be approximated to any desired degree of accuracy. The algorithm has been used to solve numerically Lagrange's equations of motion for N-acetyl alanine amide and N-acetyl methionide amide, regarded as fixed at their C-termini, under the influence of the ECEPP/2 potential energy function, and time steps of 15–30 fsec have been achieved with little variation in the total energy. Possible directions for future development are discussed.     

Dilute solution properties of a polyurethane. I. Linear polymers

A linear polyurethane of high molecular weight was prepared in solution by the polyaddition of equimolar amounts of ethylene glycol and methylene bis(4-phenyl isocyanate). The polymer was fractionated by using a direct sequential extraction procedure, with a solvent–nonsolvent system consisting of N,N′-dimethylformamide (DMF) and acetone (A). The resulting fractions were characterized by viscosity and lightscattering measurements. The relationship between the intrinsic viscosity and molecular weight was found in DMF at 25°C. to be [η] = 3.64 × 10^?4M^0.71. The unperturbed polymer chain dimensions were determined from intrinsic viscosity measurements carried out under experimentally determined theta conditions.     

Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals

In the bulk scaling limit of the Gaussian Unitary Ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the Fredholm determinant of the sine kernel over this interval. A formal asymptotic expansion for the determinant ass tends to infinity was obtained by Dyson. In this paper we replace a single interval of lengths bysJ, whereJ is a union ofm intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect tos of the determinant equals a constant (expressible in terms of hyperelliptic integrals) timess, plus a bounded oscillatory function ofs (zero ifm=1, periodic ifm=2, and in general expressible in terms of the solution of a Jacobi inversion problem), pluso(1). Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.Research supported by National Science Foundation grant DMS-9216203.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

Quantum-chaotic scattering effects in semiconductor microstructures

We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations-a sensitivity of the conductance to either Fermi energy or magnetic field-and weak-localization-a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak-localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak-localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal-approximation theory in describing the magnitude of these quantum transport effects.     

Second-order nonlinear optical properties of the Ag(111)/electrolyte interface in the presence of self-assembled monolayers containing conjugated pi systems. I. Alpha-functionalized terthiophene films on Ag(111)

The nonlinear optical properties of self-assembled monolayers obtained from bonding two different alpha-functionalized terthiophenes (alpha-T3) to (111) silver electrode surfaces have been investigated using second harmonic generation (SHG). The two (alpha-T3) compounds used were functionalized with alkane chains of different lengths (C8 and C4), and each was terminated with a thiol anchoring group. A nitrile group was attached to the terminal thiophene ring of the (alpha-T3) compound with the C4 chain. The orientation of the polarization of the incident beam was changed systematically and gradually between "p" and "s" orientation and the SH signal (isotropic and anisotropic contributions) analyzed in both directions ("P" and "S"). The symmetry of the system was reduced by the presence of the adlayers from C3v to C3. The dependence on the applied potential and the incident wavelength has also been studied. The relative magnitudes and phases of the various second-order tensor elements have been estimated and compared with the values for a bare surface. A resonance process in the (alpha-T3) pi moiety has been investigated, and from this, the effective "band-gap" energies of the organic semiconductor SAMs (i.e., the energy difference between the pi-pi bands) have been estimated.     

Semiclassical approach to collision-induced emission in the presence of intense laser radiation: An aspect in the study of cooperative chemical and optical pumping

Collion-induced emission in molecular systems in an intense laser field is studied using the semiclassical approach, with a view towards cooperative chemical and optical pumping in laser production. The formalism is developed with the electronic-field representation, which treats collision and radiative interaction on the same footing. Electronic-field surfaces can be regarded as forming spectra for spontaneous emission; and particular emission events can be accounted for by propagating classical trajectories on emission electronic-field surfaces. Pre-emission loss from the excited state is dealt with by propagating classical trajectories on a loss surface along a complex contour of emission branch points. This loss surface is derived on the basis of localized radiative couplings between electronic-field states and provides a framework to treat the general problem of discrete state-continuum interactions. The formalism is applied to a two-state, collinear exponential model to compute S-matrix elements and transition probabilities between asymptotic states.     

Acceleration of convergence in Monte Carlo simulations of aqueous solutions using the metropolis algorithm. Hydrophobic hydration of methane

Convergence problems encountered in the computer simulations of aqueous solutions are discussed. Solute–solvent radial distribution functions are shown to converge very poorly when the standard Metropolis Monte Carlo procedure is applied. To overcome this difficulty, several modifications are made in the Metropolis method. Optimum maximum step sizes are determined for simulations of liquid water. A scheme is employed for preferential sampling of both the solvent and the solute molecules. To test these modifications, a simulation is carried out for pure liquid water, treating a single water molecule as a “solute.” The convergence of the radial distribution functions is found to be accelerated significantly. A further test is made by simulating an aqueous solution of methane, consisting of one methane molecule (using the EPEN /2 potential for methane–water interactions) and 124 water molecules (using the MCY potential for water–water interactions). Again, the convergence of solute–solvent radial distribution functions is found to be accelerated. The computation of partial molar thermodynamic quantities, however, still suffers from convergence difficulties. This problem is discussed in detail. The EPEN /2 potential is found to yield structural and thermodynamic features of hydrophobic hydration that are consistent with available experimental and theoretical results for aqueous solutions of methane.