Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Dynamic axial ligand-site exchange in facially discriminated ruthenium(II) carbonyl and rhodium(III) halide metalloporphyrins

For a series of six-coordinate Ru(II)(CO)L or Rh(III)(X–)L porphyrins which are facially differentiated by having a naphthoquinol- or hydroquinol-containing strap across one face, we show that ligand migration from one face to the other can occur under mild conditions, and that ligand site preference is dependent on the nature of L and X–. For bulky nitrogen-based ligands, the strap can be displaced sideways to accommodate the ligand on the same side as the strap. For the ligand pyrazine, we show 1 H NMR evidence for monodentate and bidentate binding modes on both faces, dependent on ligand concentration and metalloporphyrin structure, and that inter-facial migration is rapid under normal conditions. For monodentate substituted pyridine ligands there is a site dependence on structure, and we show clear evidence of dynamic ligand migration through a series of ligand exchange reactions.