Graph transformation method for calculating waiting times in Markov chains.

We describe an exact approach for calculating transition probabilities and waiting times in finite-state discrete-time Markov processes. All the states and the rules for transitions between them must be known in advance. We can then calculate averages over a given ensemble of paths for both additive and multiplicative properties in a nonstochastic and noniterative fashion. In particular, we can calculate the mean first-passage time between arbitrary groups of stationary points for discrete path sampling databases, and hence extract phenomenological rate constants. We present a number of examples to demonstrate the efficiency and robustness of this approach.

Finding pathways between distant local minima.

We report a new algorithm for constructing pathways between local minima that involve a large number of intervening transition states on the potential energy surface. A significant improvement in efficiency has been achieved by changing the strategy for choosing successive pairs of local minima that serve as endpoints for the next search. We employ Dijkstra's algorithm [E. W. Dijkstra, Numer. Math. 1, 269 (1959)] to identify the "shortest" path corresponding to missing connections within an evolving database of local minima and the transition states that connect them. The metric employed to determine the shortest missing connection is a function of the minimized Euclidean distance. We present applications to the formation of buckminsterfullerene and to the folding of various biomolecules: the B1 domain of protein G, tryptophan zippers, and the villin headpiece subdomain. The corresponding pathways contain up to 163 transition states and will be used in future discrete path sampling calculations.

Analysis of cooperativity and localization for atomic rearrangements.

We propose measures of localization and cooperativity for the analysis of atomic rearrangements. We show that for both clusters and bulk material cooperative rearrangements usually have significantly lower barriers than uncooperative ones, irrespective of the degree of localization. We also find that previous methods used to sample stationary points are biased towards rearrangements of particular types. Linear interpolation between local minima in double-ended transition state searches tends to produce cooperative rearrangements, while random perturbations of all the coordinates, as sometimes used in single-ended searches, have the opposite effect.

A doubly nudged elastic band method for finding transition states.

A modification of the nudged elastic band (NEB) method is presented that enables stable optimizations to be run using both the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton and slow-response quenched velocity Verlet minimizers. The performance of this new "doubly nudged" DNEB method is analyzed in conjunction with both minimizers and compared with previous NEB formulations. We find that the fastest DNEB approach (DNEB/L-BFGS) can be quicker by up to 2 orders of magnitude. Applications to permutational rearrangements of the seven-atom Lennard-Jones cluster (LJ7) and highly cooperative rearrangements of LJ38 and LJ75 are presented. We also outline an updated algorithm for constructing complicated multi-step pathways using successive DNEB runs.

Correlated ab initio study of nucleic acid bases and their tautomers in the gas phase, in a microhydrated environment and in aqueous solution. guanine: surprising stabilization of rare tautomers in aqueous solution.

Altogether eight keto and enol tautomers of guanine were studied theoretically in the gas phase, in a microhydrated environment (1 and 2 water molecules) and in bulk water. The structures of isolated, as well as mono- and dihydrated tautomers were determined by means of the RI-MP2 method using the extended TZVPP (5s3p2d1f/3s2p1d) basis set. The relative energies of isolated tautomers included the correction to higher correlation energy terms evaluated at the CCSD(T)/aug-cc-pVDZ level. The relative enthalpies at 0 K and relative free energies at 298 K were based on the above-mentioned relative energies and zero-point vibration energies, temperature-dependent enthalpy terms and entropies evaluated at the MP2/6-31G level. The keto form having hydrogen atom at N7 is the global minimum while the canonical form having hydrogen atom at N9 represents the first local minimum at all theoretical levels in vacuo and in the presence of 1 and 2 water molecules. All three unusual rare tautomers having hydrogens at N3 and N7, at N3 and N9, and also at N9 and N7 are systematically considerably less stable and can be hardly detected in the gas phase. The theoretical predictions fully agree with existing theoretical as well as experimental results. The effect of bulk solvent on the relative stability of guanine tautomers was studied by self-consistent reaction field and molecular dynamics free energy calculations using the thermodynamic integration method. Bulk solvent, surprisingly, strongly favored these three rare tautomers over all remaining low-energy tautomers and probably only these forms can exist in water phase. The global minimum (tautomer with hydrogens at N3 and N7) is by 13 kcal/mol more stable than the canonical form (3rd local minimum). Addition of one or two water molecules does not change the relative stability order of isolated guanine tautomers but the respective trend clearly supports the surprising stabilization of three rare forms.