Quantum Tunneling Rates of Gas-Phase Reactions from On-the-Fly Instanton Calculations.

The instanton method obtains approximate tunneling rates from the minimum-action path (known as the instanton) linking reactants to the products at a given temperature. An efficient way to find the instanton is to search for saddle-points on the ring-polymer potential surface, which is obtained by expressing the quantum Boltzmann operator as a discrete path-integral. Here we report a practical implementation of this ring-polymer form of instanton theory into the Molpro electronic-structure package, which allows the rates to be computed on-the-fly, without the need for a fitted analytic potential-energy surface. As a test case, we compute tunneling rates for the benchmark H + CH4 reaction, showing how the efficiency of the instanton method allows the user systematically to converge the tunneling rate with respect to the level of electronic-structure theory.

Shallow-tunnelling correction factor for use with Wigner-Eyring transition-state theory.

We obtain a shallow-tunnelling correction factor for use with Wigner-Eyring transition-state theory (TST). Our starting point is quantum transition state theory (QTST), which approximates the accurate quantum rate as the instantaneous flux through a delocalised transition-state ensemble of ring-polymers. Expanding the ring-polymer potential to second order gives the well-known Wigner tunnelling-factor which diverges at the cross-over temperature between deep and shallow tunnelling. Here, we show how to remove this divergence by integrating numerically over the two softest ring-polymer normal modes. This results in a modified Wigner correction factor involving a one-dimensional integral evaluated along a straight line on the potential energy surface. Comparisons with accurate quantum calculations indicate that the newly derived correction factor gives realistic estimates of quantum rate coefficients in the shallow-tunnelling regime.

Reaction mechanism of the bicopper enzyme peptidylglycine α-hydroxylating monooxygenase.

Peptidylglycine α-hydroxylating monooxygenase is a noninteracting bicopper enzyme that stereospecifically hydroxylates the terminal glycine of small peptides for its later amidation. Neuroendocrine messengers, such as oxytocin, rely on the biological activity of this enzyme. Each catalytic turnover requires one oxygen molecule, two protons from the solvent, and two electrons. Despite this enzyme having been widely studied, a consensus on the reaction mechanism has not yet been found. Experiments and theoretical studies favor a pro-S abstraction of a hydrogen atom followed by the rebinding of an OH group. However, several hydrogen-abstracting species have been postulated; because two protons are consumed during the reaction, several protonation states are available. An electron transfer between the copper atoms could play a crucial role for the catalysis as well. This leads to six possible abstracting species. In this study, we compare them on equal footing. We perform quantum mechanics/molecular mechanics calculations, considering the glycine hydrogen abstraction. Our results suggest that the most likely mechanism is a protonation of the abstracting species before the hydrogen abstraction and another protonation as well as a reduction before OH rebinding.

Role of tunneling in the enzyme glutamate mutase.

The role of quantum mechanical atom tunneling during the conversion of glutamate to methylaspartate catalyzed by glutamate mutase is investigated by quantum mechanical/molecular mechanical (QM/MM) simulations based on coupled cluster and density functional calculations. The use of instanton theory allows us to calculate the tunneling contributions of up to 78 atoms in the active site. We calculate kinetic isotope effects (KIEs) and compare them to experimental data. The simulations lead to deuterium KIEs of 10 for the hydrogen abstraction from glutamate substrate and 16 for the hydrogen abstraction from methylaspartate substrate, which are consistent with the experimental results. The hydrogen abstraction from methylaspartate has higher primary deuterium and tritium (46.1) KIEs than the abstraction from glutamate. The tunneling effect increases the reaction rate by a factor of 12.3 for the hydrogen abstraction from methylaspartate at 0. Tunneling is supported by the environment by preparing the enzyme through classical motions. Consideraton of the tunneling contributions of more and more atoms around the active center shows that the motions at the ribose ring play a central role during the tunneling enhancement of the hydrogen transfers. Our simulations give new insight into the catalytic process in glutamate mutase and the way enzymes use tunneling effects for a successful catalysis.

Kinetic isotope effects calculated with the instanton method.

The ring-opening reaction of the cyclopropylcarbinyl radical proceeds via heavy-atom tunneling at low temperature. We used instanton theory to calculate tunneling rates and kinetic isotope effects with on-the-fly calculation of energies by density functional theory (B3LYP). The accuracy was verified by explicitly correlated coupled-cluster calculations (UCCSD(T)-F12). At cryogenic temperatures, we found protium/deuterium KIEs up to 13 and inverse KIEs down to 0.2. We also studied an intramolecular tautomerization reaction. A simple and computationally efficient method is proposed to calculate KIEs with the instanton method: the instanton path is assumed to be independent of the atomic masses. This results in surprisingly good estimates of the KIEs for the cyclopropylcarbinyl radical and for the secondary KIEs of the tautomerization. Challenges and capabilities of the instanton method for calculating KIEs are discussed.

The fragmentation-recombination mechanism of the enzyme glutamate mutase studied by QM/MM simulations.

The radical mechanism of the conversion of glutamate to methylaspartate catalyzed by glutamate mutase is studied with quantum mechanical/molecular mechanical (QM/MM) simulations based on density functional theory (DFT/MM). The hydrogen transfer between the substrate and the cofactor is found to be rate limiting with a barrier of 101.1 kJ mol(-1). A careful comparison to the uncatalyzed reaction in water is performed. The protein influences the reaction predominantly electrostatically and to a lesser degree sterically. Our calculations shed light on the atomistic details of the reaction mechanism. The well-known arginine claw and Glu 171 ( Clostridium cochlearium notation) are found to have the strongest influence on the reaction. However, a catalytic role of Glu 214, Lys 322, Gln 147, Glu 330, Lys 326, and Met 294 is found as well. The arginine claw keeps the intermediates in place and is probably responsible for the enantioselectivity. Glu 171 temporarily accepts a proton from the glutamyl radical intermediate and donates it back at the end of the reaction. We relate our results to experimental data when available. Our simulations lead to further understanding of how glutamate mutase catalyzes the carbon skeleton rearrangement of glutamate.

Adaptive integration grids in instanton theory improve the numerical accuracy at low temperature.

The instanton method allows to accurately calculate tunneling rates down to very low temperature. However, with lowering the temperature, the computational effort steeply increases as many more discretization points are required. This is caused in practical applications by the majority of the discretization points accumulating at a very small region in configuration space. Here, we describe a method to flexibly discretize the instanton path adapted to the temperature. Chosen appropriately, the discretization leads to a much more uniform distribution of the images (control points) along the path which reduces the number of required images by about a factor of two. Combined with a modified Newton-Raphson optimizer and successive updates of the Hessians, the proposed method provides converged reaction rates at computational costs reduced by more than an order of magnitude. We show the success of the method on analytic test potentials and on molecules with energies directly obtained from density functional theory calculations.

Locating Instantons in Many Degrees of Freedom.

We implemented and compared four algorithms to locate instantons, i.e., the most likely tunneling paths at a given temperature. These allow to calculate reaction rates, including atom tunneling, down to very low temperature. An instanton is a first-order saddle point of the Euclidean action in the space of closed Feynman paths. We compared the Newton-Raphson method to the partitioned rational function optimization (P-RFO) algorithm, the dimer method, and a newly proposed mode-following algorithm, where the unstable mode is directly estimated from the instanton path. We tested the algorithms on three chemical systems, each including a hydrogen transfer, at different temperatures. Overall, the Newton-Raphson turned out to be the most promising method, with our newly proposed mode following, being the fall-back option.