The free energy of a reaction coordinate at multiple constraints: a concise formulation

The free energy as a function of the reaction coordinate (rc) is the key quantity for the computation of equilibrium and kinetic quantities. When it is considered as the potential of mean force, the problem is the calculation of the mean force for given values of the rc. We reinvestigate the PMCF (potential of mean constraint force) method which applies a constraint to the rc to compute the mean force as the mean negative constraint force and a metric tensor correction. The latter allows for the constraint imposed to the rc and possible artefacts due to multiple constraints of other variables which for practical reasons are often used in numerical simulations. Two main results are obtained that are of theoretical and practical interest. First, the correction term is given a very concise and simple shape which facilitates its interpretation and evaluation. Secondly, a theorem describes various rcs and possible combinations with constraints that can be used without introducing any correction to the constraint force. The results facilitate the computation of free energy by molecular dynamics simulations.     

Comultiplication submodules of multiplication modules

Abstract

All rings are commutative with identity, and all modules are unital. The purpose of this work is to investigate comultiplication submodules of multiplication modules. Various properties of this class of submodules are considered. Sufficient conditions for the sum and intersection of a finite collection of comultiplication submodules to be a comultiplication submodule are also given.     

Unprecedented [V2O]6+ core of a centrosymmetric thiosemicarbazonato dimer: spontaneous deoxygenation of oxovanadium(IV)

The complex cation [{V(daptsc)(MeOH)}2(mu-O)]2+ [daptsc(2-) = 2,6-diacetylpyridine bis(thiosemicarbazonate)] is the first crystallographically elucidated dimer to possess a [V2O]6+ core, the [V(IV)-O-V(IV)]6+ structural unit, formed by cleavage of the multiple bond in the oxo-cation VO2+, is linear with the oxo group residing on a crystallographic center of inversion, and the temperature dependence of the magnetic data of the dimer is consistent with weak antiferromagnetic coupling of the d1-d1 centers.