On the efficient path integral evaluation of thermal rate constants within the quantum instanton approximation

We present an efficient path integral approach for evaluating thermal rate constants within the quantum instanton (QI) approximation that was recently introduced to overcome the quantitative deficiencies of the earlier semiclassical instanton approach [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)]. Since the QI rate constant is determined solely by properties of the (quantum) Boltzmann operator (specifically, by the zero time properties of the flux-flux and delta-delta correlation functions), it can be evaluated by well-established techniques of imaginary time path integrals even for quite complex chemical reactions. Here we present a series of statistical estimators for relevant quantities which can be evaluated straightforwardly with any nonlinear reaction coordinates and general Hamiltonians in Cartesian space. To facilitate the search for the optimal dividing surfaces required by the QI approximation, we introduce a two-dimensional quantum free energy surface associated with the delta-delta correlation function and describe how an adaptive umbrella sampling can be used effectively to construct such a free energy surface. The overall computational procedure is illustrated by the application to a hydrogen exchange reaction in gas phase, which shows excellent agreement of the QI rates with those obtained from quantum scattering calculations.     

Proton transfer in a polar solvent from ring polymer reaction rate theory

We have used the ring polymer molecular dynamics method to study the Azzouz-Borgis model for proton transfer between phenol (AH) and trimethylamine (B) in liquid methyl chloride. When the A-H distance is used as the reaction coordinate, the ring polymer trajectories are found to exhibit multiple recrossings of the transition state dividing surface and to give a rate coefficient that is smaller than the quantum transition state theory value by an order of magnitude. This is to be expected on kinematic grounds for a heavy-light-heavy reaction when the light atom transfer coordinate is used as the reaction coordinate, and it clearly precludes the use of transition state theory with this reaction coordinate. As has been shown previously for this problem, a solvent polarization coordinate defined in terms of the expectation value of the proton transfer distance in the ground adiabatic quantum state provides a better reaction coordinate with less recrossing. These results are discussed in light of the wide body of earlier theoretical work on the Azzouz-Borgis model and the considerable range of previously reported values for its proton and deuteron transfer rate coefficients.     

Efficient estimators for quantum instanton evaluation of the kinetic isotope effects: application to the intramolecular hydrogen transfer in pentadiene

The quantum instanton approximation is used to compute kinetic isotope effects for intramolecular hydrogen transfer in cis-1,3-pentadiene. Due to the importance of skeleton motions, this system with 13 atoms is a simple prototype for hydrogen transfer in enzymatic reactions. The calculation is carried out using thermodynamic integration with respect to the mass of the isotopes and a path integral Monte Carlo evaluation of relevant thermodynamic quantities. Efficient "virial" estimators are derived for the logarithmic derivatives of the partition function and the delta-delta correlation functions. These estimators require significantly fewer Monte Carlo samples since their statistical error does not increase with the number of discrete time slices in the path integral. The calculation treats all 39 degrees of freedom quantum mechanically and uses an empirical valence bond potential based on a molecular mechanics force field.     

A new expression for the direct quantum mechanical evaluation of the thermal rate constant

Based on the formalism of Miller, Schwartz, and Tromp [J. Chem. Phys. 79, 4889(1983)], we derive a new expression for the thermal rate constant for a chemical reaction. The expression involves an unperturbed, i.e., reactant or product channel Boltzmann operator for the imaginary time propagation, making it possible to compute efficiently the rate constant for a range of temperatures. We illustrate numerical aspects with an extensive study of the one-dimensional Eckart barrier problem, as well as a study of the three-dimensional (J = 0) D + H2 problem.     

Simultaneous determination of the adsorption constant and the photoinduced electron transfer rate for a CdS quantum dot-viologen complex

Transient absorption (TA) spectroscopy of solution-phase mixtures of colloidal CdS quantum dots (QDs) with acid-derivatized viologen molecules, N-[1-heptyl],N'-[3-carboxypropyl]-4,4'-bipyridinium dihexafluorophosphate (V(2+)), indicates electron transfer occurs from the conduction band of the QD to the LUMO of V(2+) after photoexcitation of a band-edge exciton in the QD. Analysis of the magnitude of the ground state bleach of the QD as a function of the molar ratio QD:V(2+) yields the QD-ligand adsorption constant, K(a) (4.4 × 10(4) M(-1)) for V(2+) ligands adsorbed in geometries conducive to electron transfer. The value of K(a), together with the measured rates of (i) formation of the V(+?) electron transfer product and (ii) recovery of the ground state bleach of the QD, enables determination of the intrinsic rate constant for charge separation, k(CS,int) ~ 1.7 × 10(10) s(-1), the rate for a single QD-V(2+) donor-acceptor pair. This analysis confirms previous reports that the number of ligands adsorbed to each QD is well-described by a Poisson distribution. This is the first report where the QD-ligand charge transfer and binding equilibria are quantitatively investigated simultaneously with a single technique.     

Test of the quantum instanton approximation for thermal rate constants for some collinear reactions

Two variants of the recently developed quantum instanton (QI) model for calculating thermal rate constants of chemical reactions are applied to several collinear atom-diatom reactions with various skew angles. The results show that the original QI version of the model is consistently more accurate than the "simplest" quantum instanton version (both being applied here with one "dividing surface") and thus to be preferred. Also, for these examples (as with other earlier applications) the QI results agree well with the correct quantum rates (to within approximately 20% or better) for all temperatures >200 K, except for situations where dynamical corrections to transition state theory (i.e., "re-crossing" dynamics) are evident. (Since re-crossing effects are substantially reduced in higher dimensionality, this is not a cause for serious concern.) A procedure is also described which facilitates use of the METROPOLIS algorithm for evaluating all quantities that appear in the QI rate expression by Monte Carlo path integral methods.     

Components for integral evaluation in quantum chemistry

Sharing low-level functionality between software packages enables more rapid development of new capabilities and reduces the duplication of work among development groups. Using the component approach advocated by the Common Component Architecture Forum, we have designed a flexible interface for sharing integrals between quantum chemistry codes. Implementation of these interfaces has been undertaken within the Massively Parallel Quantum Chemistry package, exposing both the IntV3 and Cints/Libint integrals packages to component applications. Benchmark timings for Hartree-Fock calculations demonstrate that the overhead due to the added interface code varies significantly, from less than 1% for small molecules with large basis sets to nearly 10% for larger molecules with smaller basis sets. Correlated calculations and density functional approaches encounter less severe performance overheads of less than 5%. While these overheads are acceptable, additional performance losses occur when arbitrary implementation details, such as integral ordering within buffers, must be handled. Integral reordering is observed to add an additional overhead as large as 12%; hence, a common standard for such implementation details is desired for optimal performance.     

Dual fluorescence of ellipticine: excited state proton transfer from solvent versus solvent mediated intramolecular proton transfer

Photophysical properties of a natural plant alkaloid, ellipticine (5,11-dimethyl-6H-pyrido[4,3-b]carbazole), which comprises both proton donating and accepting sites, have been studied in different solvents using steady state and time-resolved fluorescence techniques primarily to understand the origin of dual fluorescence that this molecule exhibits in some specific alcoholic solvents. Ground and excited state calculations based on density functional theory have also been carried out to help interpretation of the experimental data. It is shown that the long-wavelength emission of the molecule is dependent on the hydrogen bond donating ability of the solvent, and in methanol, this emission band arises solely from an excited state reaction. However, in ethylene glycol, both ground and excited state reactions contribute to the long wavelength emission. The time-resolved fluorescence data of the system in methanol and ethylene glycol indicates the presence of two different hydrogen bonded species of ellipticine of which only one participates in the excited state reaction. The rate constant of the excited state reaction in these solvents is estimated to be around 4.2-8.0 × 10(8) s(-1). It appears that the present results are better understood in terms of solvent-mediated excited state intramolecular proton transfer reaction from the pyrrole nitrogen to the pyridine nitrogen leading to the formation of the tautomeric form of the molecule rather than excited state proton transfer from the solvents leading to the formation of the protonated form of ellipticine.     

Using a family of dividing surfaces normal to the minimum energy path for quantum instanton rate constants

One of the outstanding issues in the quantum instanton (QI) theory (or any transition-state-type theory) for thermal rate constants of chemical reactions is the choice of an appropriate "dividing surface" (DS) that separates reactants and products. (In the general version of the QI theory, there are actually two dividing surfaces involved.) This paper shows one simple and general way for choosing DSs for use in QI theory, namely, using the family of (hyper) planes normal to the minimum energy path on the potential energy surface at various distances s along it. Here the reaction coordinate is not one of the dynamical coordinates of the system (which will in general be the Cartesian coordinates of the atoms), but rather simply a parameter which specifies the DS. It is also shown how this idea can be implemented for an N atom system in three-dimensional space in a way that preserves overall translational and rotational invariance. Numerical application to a simple system (the collinear H+H(2) reaction) is presented to illustrate the procedure.     

Effect of pressure on the proton transfer rate from a photoacid to a solvent. 4. Photoacids in methanol

The pressure dependence of the excited-state proton dissociation rate constant of four photoacids, 2-naphthol-6,8-disulfonate (2N68DS), 10-hydroxycamptothecin (10-CPT), 5-cyano-2-naphthol (5CN2), and 5,8-dicyano-2-naphthol (DCN2), are studied in methanol. The results are compared with the results of the pressure dependence study we recently conducted for several photoacids in water, ethanol, and propanol. The pressure dependence is explained using an approximate stepwise two-coordinate proton transfer model. The increase in rate, as a function of pressure, manifests a strong dependence of proton tunneling on the distance which decreases with an increase of pressure between the two oxygen atoms involved in the process. The decrease in the proton transfer rate with increasing pressure reflects the dependence of the reaction on the solvent relaxation rate. We found that, for the relatively weak photoacids 2N68DS, 10-CPT, and 5CN2, the proton transfer rate constant increases by a factor of about 5-8 at a pressure of about 1.5 GPa. For a strong photoacid like DCN2, the rate increase was only by a factor of 2.     

Solvation and proton transfer in polar molecule nanoclusters

Proton transfer in a phenol-amine complex dissolved in polar molecule nanoclusters is investigated. The proton transfer rates and mechanisms, as well as the solvation of the complex in the cluster, are studied using both adiabatic and nonadiabatic dynamics. The phenol-amine complex exists in ionic and covalent forms and as the size of the cluster increases the ionic form gains stability at the expense of the covalent form. Both the adiabatic and nonadiabatic transfer reaction rates increase with cluster size. Given a fixed cluster size, the stability of the covalent state increases with increasing temperature. The proton transfer rates do not change monotonously with an increase in temperature. A strong correlation between the solvent polarization reaction coordinate and the location of the phenol-amine complex in the cluster is found. The ionic form of the complex strongly prefers the interior of the cluster while the covalent form prefers to lie on the cluster surface.     

Path integral calculation of thermal rate constants within the quantum instanton approximation: application to the H + CH4 --> H2 + CH3 hydrogen abstraction reaction in full Cartesian space

The quantum instanton approximation for thermal rate constants of chemical reactions [Miller, Zhao, Ceotto, and Yang, J. Chem. Phys. 119, 1329 (2003)], which is modeled after the earlier semiclassical instanton approach, is applied to the hydrogen abstraction reaction from methane by a hydrogen atom, H + CH4 --> H2 + CH3, using a modified and recalibrated version of the Jordan-Gilbert potential surface. The quantum instanton rate is evaluated using path integral Monte Carlo approaches based on the recently proposed implementation schemes [Yamamoto and Miller, J. Chem. Phys. 120, 3086 (2004)]. The calculations were carried out using the Cartesian coordinates of all the atoms (thus involving 18 degrees of freedom), thereby taking explicit account of rotational effects of the whole system and also allowing the equivalent treatment of the four methane hydrogens. To achieve such a treatment, we present extended forms of the path integral estimators for relevant quantities that may be used for general N-atom systems with any generalized reaction coordinates. The quantum instanton rates thus obtained for the temperature range T = 200-2000 K show good agreement with available experimental data, which gives support to the accuracy of the underlying potential surface used.     

Monte Carlo simulation of the diabatic free energy curves for a dissociative electron transfer reaction in a polar solvent

In the present work we have carried out a Monte Carlo simulation of a dissociative electron transfer reaction in a polar solvent. In particular, we have chosen as a very simple model the electrochemical reduction of hydrogen fluoride to give a hydrogen atom and a fluoride anion in a dipolar solvent. From a classical point of view, the electron transfer occurs at the intersection region S* of the diabatic potential hypersurfaces H_pp and H_ss, corresponding to the precursor and successor complexes, respectively. We have evaluated both diabatic surfaces using potential functions that have been built up with ab initio methods by us. For each of the obtained configurations the parameter ΔE = H_ss – H_pp has been calculated. This parameter is then used as the reaction coordinate for obtaining the diabatic free energy curves of the reaction. Because the activation energy is high, a suitable mapping potential along with the statistical perturbation theory is employed to force the system to evolve toward the intersection region S*. A total of 68,340,000 configurations have been generated. The main conclusion of this article is that Marcus' relationship seems to fail for this kind of inner-sphere processes. © 1992 by John Wiley & Sons, Inc.     

Proton transfer in nanoconfined polar solvents. II. Adiabatic proton transfer dynamics

The reaction dynamics for a model phenol-amine proton transfer system in a confined methyl chloride solvent have been simulated by mixed quantum-classical molecular dynamics. In this approach, the proton vibration is treated quantum mechanically (and adiabatically), while the rest of the system is described classically. Nonequilibrium trajectories are used to determine the proton transfer reaction rate constant. The reaction complex and methyl chloride solvent are confined in a smooth, hydrophobic spherical cavity, and radii of 10, 12, and 15 A have been considered. The effects of the cavity radius and the heavy atom (hydrogen bond) distance on the reaction dynamics are considered, and the mechanism of the proton transfer is examined in detail by analysis of the trajectories.     

Following the solvent directly during ultrafast excited state proton transfer

Excited state intramolecular proton transfer in 1-chloroacetylaminoanthraquinone is investigated from the perspective of the solvent. Using a new two-dimensional nonlinear optical spectroscopy the solvent response is probed directly as the proton transfer takes place. The measurements indicate that solvent reorganization controls the proton transfer in acetonitrile by dynamically shifting the position of equilibrium in the excited state, even on subpicosecond time scales.