Entanglement and co-tunneling of two equivalent protons in hydrogen bond pairs.

A theoretical study is reported of a system of two identical symmetric hydrogen bonds, weakly coupled such that the two mobile protons can move either separately (stepwise) or together (concerted). It is modeled by two equivalent quartic potentials interacting through dipolar and quadrupolar coupling terms. The tunneling Hamiltonian has two imaginary modes (reaction coordinates) and a potential with a single maximum that may turn into a saddle-point of second order and two sets of (inequivalent) minima. Diagonalization is achieved via a modified Jacobi-Davidson algorithm. From this Hamiltonian the mechanism of proton transfer is derived. To find out whether the two protons move stepwise or concerted, a new tool is introduced, based on the distribution of the probability flux in the dividing plane of the transfer mode. While stepwise transfer dominates for very weak coupling, it is found that concerted transfer (co-tunneling) always occurs, even when the coupling vanishes since the symmetry of the Hamiltonian imposes permanent entanglement on the motions of the two protons. We quantify this entanglement and show that, for a wide range of parameters of interest, the lowest pair of states of the Hamiltonian represents a perfect example of highly entangled quantum states in continuous variables. The method is applied to the molecule porphycene for which the observed tunneling splitting is calculated in satisfactory agreement with experiment, and the mechanism of double-proton tunneling is found to be predominantly concerted. We show that, under normal conditions, when they are in the ground state, the two porphycene protons are highly entangled, which may have interesting applications. The treatment also identifies the conditions under which such a system can be handled by conventional one-instanton techniques.

Methanol dimer formation drastically enhances hydrogen abstraction from methanol by OH at low temperature.

The kinetics of the reaction of methanol with hydroxyl radicals is revisited in light of the reported new kinetic data, measured in cold expansion beams. The rate constants exhibit an approximately 10(2)-fold increase when the temperature decreases from 200 to 50 K, a result that cannot be fully explained by tunneling, as we confirm by new calculations. These calculations also show that methanol dimers are much more reactive to hydroxyl than monomers and imply that a dimer concentration of about 30% of the equilibrium concentration can account quantitatively for the observed rates. The assumed presence of dimers is supported by the observation of cluster formation in these and other cold beams of molecules subject to hydrogen bonding. The calculations imply an important caveat with respect to the use of cold expansion beams for the study of interstellar chemistry.

Tunneling splitting in double-proton transfer: direct diagonalization results for porphycene.

Zero-point and excited level splittings due to double-proton tunneling are calculated for porphycene and the results are compared with experiment. The calculation makes use of a multidimensional imaginary-mode Hamiltonian, diagonalized directly by an effective reduction of its dimensionality. Porphycene has a complex potential energy surface with nine stationary configurations that allow a variety of tunneling paths, many of which include classically accessible regions. A symmetry-based approach is used to show that the zero-point level, although located above the cis minimum, corresponds to concerted tunneling along a direct trans - trans path; a corresponding cis - cis path is predicted at higher energy. This supports the conclusion of a previous paper [Z. Smedarchina, W. Siebrand, and A. Fernández-Ramos, J. Chem. Phys. 127, 174513 (2007)] based on the instanton approach to a model Hamiltonian of correlated double-proton transfer. A multidimensional tunneling Hamiltonian is then generated, based on a double-minimum potential along the coordinate of concerted proton motion, which is newly evaluated at the RI-CC2/cc-pVTZ level of theory. To make it suitable for diagonalization, its dimensionality is reduced by treating fast weakly coupled modes in the adiabatic approximation. This results in a coordinate-dependent mass of tunneling, which is included in a unique Hermitian form into the kinetic energy operator. The reduced Hamiltonian contains three symmetric and one antisymmetric mode coupled to the tunneling mode and is diagonalized by a modified Jacobi-Davidson algorithm implemented in the Jadamilu software for sparse matrices. The results are in satisfactory agreement with the observed splitting of the zero-point level and several vibrational fundamentals after a partial reassignment, imposed by recently derived selection rules. They also agree well with instanton calculations based on the same Hamiltonian.

Zero-point tunneling splittings in compounds with multiple hydrogen bonds calculated by the rainbow instanton method.

Zero-point tunneling splittings are calculated, and the values are compared with the experimentally observed values for four compounds in which the splittings are due to multiple-proton transfer along hydrogen bonds. These compounds are three binary complexes, namely, the formic acid and benzoic acid dimer and the 2-pyridone-2-hydroxypyridine complex, in which the protons move in pairs, and the calix[4]arene molecule, in which they move as a quartet. The calculations make use of and provide a test for the newly developed rainbow approximation for the zero-temperature instanton action which governs the tunneling splitting (as well as the transfer rate). This approximation proved to be much less drastic than the conventional adiabatic and sudden approximations, leading to a new general approach to approximate the instanton action directly. As input parameters the method requires standard electronic-structure data and the Hessians of the molecule or complex at the stationary configurations only; the same parameters also yield isotope effects. Compared to our earlier approximate instanton method, the rainbow approximation offers an improved treatment of the coupling of the tunneling mode to the other vibrations. Contrary to the conventional instanton approach based on explicit evaluation of the instanton trajectory, both methods bypass this laborious procedure, which renders them very efficient and capable of handling systems that thus far have not been handled by other theoretical methods. Past results for model systems have shown that the method should be valid for a wide range of couplings. The present results for real compounds show that it gives a satisfactory account of tunneling splittings and isotope effects in systems with strong coupling that enhances tunneling, thus demonstrating its applicability to low-temperature proton dynamics in systems with multiple hydrogen bonds.

Communication: Selection rules for tunneling splitting of vibrationally excited levels.

Five symmetry-based selection rules are formulated that relate the tunneling splitting of a vibrationally excited level to that of the ground level in molecules with a symmetric double-minimum potential. The rules, which explain why excited levels frequently have smaller splittings than zero-point levels, are used to interpret the observed and calculated splittings in malonaldehyde.

The rainbow instanton method: a new approach to tunneling splitting in polyatomics.

A new instanton approach is reported to tunneling at zero-temperature in multidimensional (MD) systems in which a "light particle" is transferred between two equivalent "heavy" sites. The method is based on two concepts. The first is that an adequate MD potential energy surface can be generated from input of the stationary configurations only, by choosing as a basis the normal modes of the transition state. It takes the form of a double-minimum potential along the mode with imaginary frequency and coupling terms to the remaining (harmonic) oscillators. Standard integrating out of the oscillators gives rise to an effective 1D instanton problem for the adiabatic potential, but requires evaluation of a nonlocal term in the Euclidean action, governed by exponential (memory) kernels. The second concept is that this nonlocal action can be treated as a "perturbation," for which a new approximate instanton solution is derived, termed the "rainbow" solution. Key to the approach is avoidance of approximations to the exponential kernels, which is made possible by a remarkable conversion property of the rainbow solution. This leads to a new approximation scheme for direct evaluation of the Euclidean action, which avoids the time-consuming search of the exact instanton trajectory. This "rainbow approximation" can handle coupling to modes that cover a wide range of frequencies and bridge the gap between the adiabatic and sudden approximations. It suffers far fewer restrictions than these conventional approximations and is proving particularly effective for systems with strong coupling, such as proton transfer in hydrogen bonds. Comparison with the known exact instanton action in two-dimensional models and application to zero-level tunneling splittings in two isotopomers of malonaldehyde are presented to show the accuracy and efficiency of the approach.

Multidimensional Hamiltonian for tunneling with position-dependent mass.

A multidimensional Hamiltonian for tunneling is formulated, based on the mode with imaginary frequency of the transition state as a reaction coordinate. To prepare it for diagonalization, it is transformed into a lower-dimension Hamiltonian by incorporating modes that move faster than the tunneling into a coordinate-dependent kinetic energy operator, for which a Hermitian form is chosen and tested for stability of the eigenvalues. After transformation to a three-dimensional form, which includes two normal modes strongly coupled to the tunneling mode, this Hamiltonian is diagonalized in terms of a basis set of harmonic oscillator functions centered at the transition state. This involves a sparse matrix which is easily partially diagonalized to yield tunneling splittings for the zero-point level and the two fundamental levels of the coupled modes. The method is tested on the well-known benchmark molecule malonaldehyde and a deuterium isotopomer, for which these splittings have been measured. Satisfactory agreement with experiment results is obtained.

Analysis of kinetic isotope effects in enzymatic carbon-hydrogen cleavage reactions.

The instanton approach, as previously applied to proton tunneling in molecular systems, is adapted to carbon-hydrogen bond cleavage catalyzed by enzymes. To compensate for the complexity of enzymatic reactions, simplifications are introduced based on the observation in numerous X-ray measurements that enzymes tend to form compact structures, which is assumed to have led toward optimization of specific parameters that govern the tunneling rate in the instanton formalism. On this basis, semiempirical equations are derived that link observed kinetic data directly to these parameters. These equations provide an analytical relation between the kinetic isotope effect and its temperature dependence for each hydrogen isotope, from which mechanistic and structural information can be extracted, including the nature of the hydrogen acceptor, the magnitude of the hydrogen transfer distance, the presence of endothermicity, and the contribution and frequency of skeletal vibrations that assist the tunneling. The method is used to analyze kinetic data reported for eight enzymatic CH-cleavage reactions; the enzymes or models thereof studied include methylmalonyl-coenzyme A mutase (coenzyme B(12)), galactose oxidase, lipoxygenase-1 with six mutants, methylamine dehydrogenase, an oxoiron(IV)porphyrin radical cation, phenylalanine hydroxylase, a bis(µ-oxo)dicopper complex, and rice α-oxygenase.

Correlated double-proton transfer. I. Theory.

The dynamics of double-proton transfer reactions is studied on a model of transfer along two identical hydrogen bonds represented by quartic double-minimum potentials. Correlation between the proton motions is introduced by a coupling term that is bilinear in the two proton coordinates; it is shown that this form properly accounts for the polarity and symmetry of the interaction and correctly reproduces the observed transfer behavior in the strong- and weak-coupling limits. The model allows a universal description of double-proton transfer mechanisms in symmetric systems in terms of the variation of a single parameter, the (dimensionless) coupling between the two hydrogen bonds. The corresponding two-dimensional (2D) transfer potential has up to nine stationary points, depending on the coupling strength. The resulting dynamics and its dependence on temperature and isotopic substitution are studied analytically by instanton techniques for the full range of the correlation parameter whereby the potential has multiple saddle points. For any coupling, the dynamics at high temperatures is dominated by classical transitions over the saddle point of lowest barrier. Strong coupling leads exclusively to synchronous transfer along a single collective coordinate, weak coupling to competition between this synchronous transfer, and stepwise transfer along local coordinates, the relative contributions of these mechanisms being governed by the temperature. Below a certain crossover temperature, transfer dynamics is dominated by the instanton, i.e., the trajectory with maximum tunneling probability. Two types of instanton are found on the 2D potential. The well-known one-dimensional instanton, corresponding to synchronous motion, exists for any coupling. It dominates at low temperatures and is responsible for any observed tunneling splittings, independent of the number of saddle points of the symmetric potential. An alternative 2D instanton, corresponding to asynchronous motion, exists for weak coupling. It is shown that under conditions where 2D tunneling dominates, it is much slower than stepwise transfer. Therefore 2D tunneling trajectories do not contribute significantly to the rate of transfer and can be ignored. The favorable quantitative aspects of the model are illustrated by an application to double-proton rate constants in porphine, which have been measured in a wide range of temperatures.

Tunneling dynamics of double proton transfer in formic acid and benzoic acid dimers.

Direct dynamics calculations based on instanton techniques are reported of tunneling splittings due to double proton transfer in formic and benzoic acid dimers. The results are used to assign the observed splittings to levels for which the authors of the high-resolution spectra could not provide a definitive assignment. In both cases the splitting is shown to be due mainly to the zero-point level rather than to the vibrationally or electronically excited level whose spectrum was investigated. This leads to zero-point splittings of 375 MHz for (DCOOH)(2) and 1107 MHz for the benzoic acid dimer. Thus, contrary to earlier calculations, it is found that the splitting is considerably larger in the benzoic than in the formic acid dimer. The calculations are extended to solid benzoic acid where the asymmetry of the proton-transfer potential induced by the crystal can be overcome by suitable doping. This has allowed direct measurement of the interactions responsible for double proton transfer, which were found to be much larger than those in the isolated dimer. To account for this observation both static and dynamic effects of the crystal forces on the intradimer hydrogen bonds are included in the calculations. The same methodology, extended to higher temperatures, is used to calculate rate constants for HH, HD, and DD transfers in neat benzoic acid crystals. The results are in good agreement with reported experimental rate constants measured by NMR relaxometry and, if allowance is made for small structural changes induced by doping, with the transfer matrix elements observed in doped crystals. Hence the method used allows a unified description of tunneling splittings in the gas phase and in doped crystals as well as of transfer rates in neat crystals.

Kinetic isotope effects for concerted multiple proton transfer: a direct dynamics study of an active-site model of carbonic anhydrase II.

The rate constant of the reaction catalyzed by the enzyme carbonic anhydrase II, which removes carbon dioxide from body fluids, is calculated for a model of the active site. The rate-determining step is proton transfer from a zinc-bound water molecule to a histidine residue via a bridge of two or more water molecules. The structure of the active site is known from X-ray studies except for the number and location of the water molecules. Model calculations are reported for a system of 58 atoms including a four-coordinated zinc ion connected to a methylimidazole molecule by a chain of two waters, constrained to reproduce the size of the active site. The structure and vibrational force field are calculated by an approximate density functional treatment of the proton-transfer step at the Self-Consistent-Charge Density Functional Tight Binding (SCC-DFTB) level. A single transition state is found indicating concerted triple proton transfer. Direct-dynamics calculations for proton and deuteron transfer and combinations thereof, based on the Approximate Instanton Method and on Variational Transition State Theory with Tunneling Corrections, are in fair agreement and yield rates that are considerably higher and kinetic isotope effects (KIEs) that are somewhat higher than experiment. Classical rate constants obtained from Transition State Theory are smaller than the quantum values but the corresponding KIEs are five times larger. For multiple proton transfer along water bridges classical KIEs are shown to be generally larger than quantum KIEs, which invalidates the standard method to distinguish tunneling and over-barrier transfer. In the present case, a three-way comparison of classical and quantum results with the observed data is necessary to conclude that proton transfer along the bridge proceeds by tunneling. The results suggest that the two-water bridge is present in low concentrations but makes a substantial contribution to proton transport because of its high efficiency. Bridging structures containing more water molecules may have lower energies but are expected to be less efficient. The observed exponential dependence of the KIEs on the deuterium concentration in H(2)O/D(2)O mixtures implies concerted transfer and thus rules out substantial contributions from structures that lead to stepwise transfer via solvated hydronium ions, which presumably dominate proton transfer in less efficient carbonic anhydrase isozymes.

Proton transfer dynamics via high resolution spectroscopy in the gas phase and instanton calculations.

Tunneling splittings have been observed in the eigenstate-resolved electronic spectrum of the 2-hydroxypyridine/2-pyridone dimer in the gas phase. Deuterium substitution experiments show that these splittings are caused by a concerted double proton transfer reaction along the O-H...O and N...H-N hydrogen bonds that hold the dimer together, substitution of the weaker and longer N...H-N bond having the larger effect. Tunneling splittings calculated by the instanton method for the zero-point level of the ground state are in good agreement with experiment for all observed isotopomers, showing that the dynamics occurs in this state, rather than in the electronically excited state.