Transition-state theory path dynamics

Jang S, Voth GA (2001) A relationship between centroid dynamics and path integral quantum transition state theory. J Chem Phys 112(8747-8757) Erratum 114, 1944... 

Figure 11. The solid line depicts the quantum adiabatic free energy curve for the Fe /Fe electron transfer at the water/Pt(lll) interface (obtained by using the Anderson-Newns model, path integral quantum transition state theory, and the umbrella sampling of molecular dynamics. The dashed line shows the curve from the classical calculation as given in Fig. 5. (Reprinted from Ref 14.)...

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

Let us consider first the in vacuo cases. Dynamical aspects of the reaction in vacuo may be recovered by resorting to calculations of semiclassical trajectories. A cluster of independent representative points, with accurately selected classical initial conditions, are allowed to perform trajectories according to classical mechanics. The reaction path, which is a static semiclassical concept (the best path for a representative point with infinitely slow motion), is replaced by descriptions of the density of trajectories. A widely employed approach to obtain dynamical information (reaction rate coefficients) is based on modern versions of the Transition State Theory (TST) whose original formulation dates back to 1935. Much work has been done to extend and refine the original TST. 

Quantum dynamics effects for hydride transfer in enzyme catalysis have been analyzed by Alhambra et. al., 2000. This process is simulated using canonically variational transition-states for overbarrier dynamics and optimized multidimensional paths for tunneling. A system is divided into a primary zone (substrate-enzyme-coenzyme), which is embedded in a secondary zone (substrate-enzyme-coenzyme-solvent). The potential energy surface of the first zone is treated by quantum mechanical electronic structure methods, and protein, coenzyme, and solvent atoms by molecular mechanical force fields. The theory allows the calculation of Schaad-Swain exponents for primary (aprim) and secondary (asec) KIE... 

There are alternates to probing diffusion by molecular dynamics or transition state theory. For a diffiisional path than contains a significant constriction or activation barrier, an additional force along the difiusion direction can be applied to enforce lateral difiusion pn the molecule. The hias in quantifying the difiusion that results that such an additional force injects can, in principle, be corrected-for so as to yield an estimate of the translational difiusion coefELdent in the free, unperturbed system. 

For these reasons we cannot use (7(R) as a rigid support for dynamical studies of trajectories of representative points. G(R) has to be modified, at every point of each trajectory, and these modifications depend on the nature of the system, on the microscopic properties of the solution, and on the dynamical parameters of the trajectories themselves. This rather formidable task may be simplified in severai ways we consider it convenient to treat this problem in a separate Section. It is sufficient to add here that one possible way is the introduction into G (R) of some extra coordinates, which reflect the effects of these retarding forces. These coordinates, collectively called solvent coordinates (nothing to do with the coordinates of the extra solvent molecules added to the solute ) are here indicated by S, and define a hypersurface of greater dimensionality, G(R S). To show how this approach of expanding the coordinate space may be successfully exploited, we refer here to the proposals made by Truhlar et al. (1993). Their formulation, that just lets these solvent coordinates partecipate in the reaction path, allows to extend the algorithms and concepts of the above mentioned variational transition state theory to molecules in solution. 

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

KJ is Eyring s equilibrium constant in the transition state theory rate constant [23 24]. The derivation of equation (24) here is based upon quantum dynamical arguments. Compared to eq.(9.14) in reference [24], except for the derivation paths (that are totally different), both equations are formally identical. 

Energy derivatives are essential for the computation of dynamics properties. There are several dynamics-related methods available in gamess. The intrinsic reaction coordinate (IRC) or minimum energy path (MEP) follows the infinitely damped path from a first-order saddle point (transition state) to the minima connected to that transition state. In addition to providing an analysis of the process by which a chemical reaction occurs (e.g. evolution of geometric structure and wavefunction), the IRC is a common starting point for the study of dynamics. Example are variational transition state theory (VTST [55]) and the modified Shepard interpolation method developed by Collins and co-workers... 

Rick, S.W., Lynch, D.L., Doll, J.D. The quantum dynamics of hydrogen and deuterium on the Pd(III) surface A path integral transition state theory study, J. Chem. Phys. 1993, 99, 8183. 

Mathematical and Physical Sciences, Vol. 170, D. C. Clary, Ed., D. Reidel, Boston, 1986, pp. 27—45. Reaction Path Models for Polyatomic Reaction Dynamics—From Transition State Theory to Path Integrals. 

---