Classical Wigner model based on a Feynman path integral open polymer

Abstract

When using mathematical models to computationally investigate a chemical system it is important that the methods used are accurate enough to account for the relevant properties of the system and at the same time simple enough to be computationally affordable. This thesis presents research that so far has resulted in three published papers and one unpublished manuscript. It concerns the application and development of computational methods for chemistry, with some extra emphasis on the calculation of reaction rate constants. In astrochemistry radiative association is a relevant reaction mechanism for the formation of molecules. The rate constants for such reactions are often difficult to obtain though experiments. In the first published paper of the thesis a rate constant for the formation of the hydroxyl radical, through the radiative association of atomic oxygen and hydrogen, is presented. This rate constant was calculated by a combination of different methods and should be an improvement over previously available rate constants. In the the second published paper of this thesis two kinds of basis functions, for use with a variational principle for the dynamics of quantum distributions in phase space, i.e. Wigner functions, is presented. These are tested on model systems and found to have some appealing properties. The classical Wigner method is an approximate method of simulation, where an initial quantum distribution is propagated in time with classical mechanics. In the third published paper of this thesis a new method of sampling the initial quantum distribution, with an imaginary time Feynman path integral, is derived and tested on model systems. In the unpublished manuscript, this new method is applied to reaction rate constants and tested on two model sysv Abstract tems. The new sampling method shows some promise for future applications. vi Preface You have guessed right; I have lately been so deeply engaged in one occupation that I have not allowed myself sufficient rest, as you see: but I hope, I sincerely hope, that all these employments are now at an end, and that I am at length free.

Calculation of reaction rate constants from a classical Wigner model based on a Feynman path integral open polymer

“…In this section expressions for calculating reaction rate constants with the OPCW method will be derived. These derivations follow the ones in the original OPCW article [13] quite closely, but use the symmetrised Boltzmann operator and will be specialised to the flux-side trace, C fs (t) in Equation 2.…”

“…x j +x (j mod N)+1 2 and H(y j , p j ) is the classical Hamiltonian. If V(y j ) is independent of momentum, all momenta except p N 2 and p N can be integrated out analytically as done in the original derivation of the OPCW method [13]. This gives…”

“…where δ(x) is the Dirac delta function. Using the momentum integrals of Appendix A in our previous article [13] a quantity…”

“…The integrals in equations 16 and 17 were evaluated through the Metropolis Monte Carlo method [18]. The maximum stepsize for the Monte Carlo was chosen and updated according to the procedure in the original OPCW work [13]. A Maxwell-Boltzmann distribution of the inverse temperature β/N was used to sample the momentum, p N .…”

“…the instanton reaction rate theory [8][9][10], the statistical adiabatic channel model [11] and Ring Polymer Molecular Dynamics (RPMD) [12]. This article presents reaction rate constant calculations using a recent implementation of the classical Wigner method by the same authors [13]. Section 2 is an introduction to a general formulation of chemical reaction rates, then section 3 combines this theory with the Open Polymer Classical Wigner (OPCW) method.…”