Transition State Approximation Research Articles

Bond Characterization of Chromium−Fischer Carbene Complexes: A Combined Study of Experiment and Theory

Four pentacarbonyl chromium−carbene complexes, (CO)5CrC(XR‘)R, have been studied via X-ray diffraction and molecular orbital calculations. One of the carbene complexes (XR‘ = OCH3, R = −(C⋮CPh)) has been investigated extensively at 110 K by X-ray diffraction using Mo Kα radiation. The electron density distribution of this compound as well as the asphericity in electron density around the Cr atom is clearly demonstrated by deformation density and the Laplacian of electron density. The comparison between experiment and theory is made in terms of deformation density, topological properties, and d orbital populations of Cr. Further chemical bond characterization is based on quantum mechanical molecular orbital calculation and properties associated with bond critical points. The π bond character of a metal carbene can be best represented by a Cr−C−X three-centered four-electron bond with the π electron density mainly located at either the dyz orbital of Cr or the pz orbital of X in the carbene ligand. This makes the carbene carbon an electrophilic site in the pπ direction. The resemblances and differences between amino−(XN) and alkoxy−(XO) carbene complexes are of special interest. Because the energy of π*C-N orbital is fairly close to that of the π*C⋮O (carbonyl on Cr), the π bond character is delocalized toward the M−Ccarbonyl at the trans position i.e., OCM−CN in the amino carbene case; this is in accord with the shortening of bond lengths of the trans M−Ccarbonyl for many amino−carbene complexes. The π bond-delocalization is also illustrated by the Fermi hole distribution. The bond dissociation energies of these carbene complexes are calculated at the CASSCF level and with the density functional method (DFT). The relative orbital energies are also compared with photoelectron spectroscopy (PES) data. The values based on DFT using the transition-state approximation give the best agreement with the experimental results.

A simple prediction of approximate transition states on potential energy surfaces

Given the locations, the energies, and the force constants of a reactant minimum and a product minimum and assuming that no other information is available, an analytical algorithm is formulated for determining the optimal conjecture for a surmized transition state between them. It is based on a model surface obtained by combining the two quadratic basin approximations. The method is illustrated by applications to transition states on an analytical surface and on an ab initio surface.

A Model of Coagulation Kinetics Between two Deformable Bilayers. Application to the Membrane Fusion

Starting from the classical DLVO theory describing the interaction between adjacent charged objects imbedded in an electrolyte solution, we developed a model to calculate the coagulation kinetics for large deformable bodies. We considered two flexible lamellae (e.g. lipid bilayers) interacting with each other through a double well potential. We postulated that a possible mechanism to overcome the energy barrier between the two minima requires a local deformation leading to the formation of adhering patches stabilized by short-range forces (Dispersion forces or bridged bonds formed by divalent ions) but hampered by the lamellae deformation work. These patches can either grow or re-dissolve in a fashion similar to the classical mechanism of nucleation theory. When the locally adhering zone reaches a critical diameter, its size rapidly grows causing the complete adhesion of the lamellae (transition from the secondary to the primary minimum). By combining the DLVO, ion binding and elasticity theories within the conceptual framework of the Transition State approximation, we derived a model to simulate fluctuation-induced coagulation kinetics. The model requires the knowledge of some parameters such as the bending elastic constant and the energy difference between the primary and secondary minima. A comparison with the classical DLVO results for rigid bodies and with experimental data on fusion kinetics of lipid vesicles is discussed.

Local density functional electronic structures of three stable icosahedral fullerenes

Local density functional (LDF) electronic structures of icosahedral C,, Cleo, and C240 are compared. These molecules are remarkably similar, and nothing is found to suggest that the two larger molecules are less stable than Cso. Ionization potentials are calculated by using both the transition-state approximation and differences between self-consistent-field calculations. Comparing these with one-electron eigenvalues supports the interpretation of photoelectron line shapes using theoretical cross sections calculated from LDF one-electron states of these large, highly symmetric molecules.

On the use of isovalued surfaces to determine molecule shape and reaction pathways

Novel insights into local molecule structure and reactivity can be gained from viewing isovalued surfaces of the molecular electron density, electrostatic potential and molecular orbitals rendered as colored, 3-D objects. For example, drawing positive and negative electrostatic isopotential surfaces partitions the molecule into regions subject to nucleophilic or electrophilic attack. Similarly, coloring isodensity surfaces to indicate the magnitude of the gradient of the electron density maps the molecule surface into regions of high and low electronegativity. A basic understanding of reaction mechanisms can also come from viewing and manipulating isovalued surfaces. A theory of molecular interactions, based upon second-order perturbation theory, provides for the decomposition of the intermolecular interaction energy into steric, electrostatic and orbital interactions. Color figures illustrate the docking of reactant molecular densities, electrostatic potentials and orbitals on low-energy pathways. The figures are used to visualize the steric, electrostatic and orbital contributions to molecular interaction energy. The visualization not only identifies low-energy reaction pathways, but it frequently reveals local interactions which determine the magnitude of the total interaction energy. Similar insight is not easily obtained by simple evaluation of the total interaction energy. Approximate transition states, built from structures along low-energy approach pathways, are excellent starting points for transition state searches.

A topological electron density analysis of tricoordinate phosphorus inversion processes

This paper describes the application of topological electron density analysis to inversion processes in tricoordinate phosphorus systems. The calculations are at the Hartree-Fock self-consistent-field level using the 6-311G ∗∗ basis sets. Of particular interest are the approximate T-shaped transition states reported by Dixon and co-workers (J. Am. Chem. Soc., 108 (1986) 2461, 6821). The role that the lone pair of electrons on phosphorus plays in determining the geometry of the transition state is analyzed through the laplacian of the electron density. It is shown that the species that will invert through the T-shaped process have distances, from P to the maximum in the non-bonded charge concentration, ranging from 1.408 a.u. to 1.417 a.u., whereas those species that invert through the classical process have distances ranging from 1.432 a.u. to 1.452 a.u.

The 5d← 4fexcited states of [Ce(OH2)9]3+

A DV-Xα molecular-orbital calculation was carried out on [Ce(OH2)9]3+. The charge donation of coordinated water molecules results in a net charge of +0·354 on the central tervalent metal ion and an appreciable polarization in the coordinated waters. The coordination bond of the vertex waters is stronger than that of the equatorials. The observed transition moments and energies of the 5d ← 4f excitations as well as the observed ligand-field splitting of 2 D(5d 1) multiplet in the D 3h ligand field of [Ce(OH2)9]3+ were reasonably reproduced by use of the transition-state approximation on the base of DV-Xα molecular-orbital calculation. The model also predicts a contraction of 4f orbital while an expansion of 5d orbital in the cluster. The one-electron energies of 4f, 5d and 6s orbitals as well as the spin-orbit coupling energies in Ce3+ free ion were also calculated by the relativistic DV-Xα method. The relativistic equation reproduces the observed values only for α = 1·0 rather than the traditional value α = 0·7.

Experimental and theoretical investigation of the electronic structure and spectra of the MoN molecule

The electronic absorption spectra of the MoN molecule have been measured by intracavity laser spectroscopy, and its electronic structure has been calculated by the X..cap alpha..-SW method. The absorption spectra showed 69 quanta (of which 55 were obtained for the first time), which were assigned on the basis of their isotope shifts to the MoN molecule. The energies and oscillator strengths of the one-electron transitions to states lying below approx. 4 eV from the ground and the lowest excited electronic states were calculated in the transition-state approximation. The results are presented.

Towards a state-to-state transition state theory

We assume that, having arrived at the transition state, the branching into the different product states is independent of the initial quantum states of the reactants. This assumption plus the familiar transition state approximation (that the reaction rate is the rate of the passage across the barrier) yields an expression for the state-to-state cross section in terms of the state-to-all one, as well as microcanonical rate constants. Models, adiabatic correlations, purely statistical considerations, or collinear computations can provide the required input for the theory. Exact quantal computations on the 3D H + H2 reaction are found to satisfy the assumed factorization quite well. Furthemore, reaction probabilities derived from a line-of-centers model, with a barrier height dependent on the approach angle, account for the probabilities derived from the exact quantal computation.

MCDF calculation of argon Auger process

The Auger transitions in argon are computed with a multiconfiguration Dirac-Fock (MCDF) program using Slater's transition state approximation. All the transitions are investigated with the smallest possible basis of configurations; then some of them are improved using a larger basis set. Fairly good agreement is obtained with other theoretical calculations and with experimental data.

Effect of rotational excitation on chemical reaction cross sections

Cross section calculations are reported for the 3D chemical reactions of H, Cl and O(3 P) atoms with H2(v=0, j) molecules in selected initial rotational states j. A detailed quantum transition state approximation is used. Good agreement is obtained with accurate quantum results for the H + H2 reaction. The agreement with quasiclassical trajectory results is also quite good for all three reactions. In most cases, it is found that the reactive cross sections decrease as j is increased.

Short-time behavior of quantum correlation functions in rate theory

The quantum autocorrelation function of number fluctuations, important in reaction rate theory, decays as ¦ t¦ 3 2 near t = 0. This is in contrast to the classical case, where the decay is proportional to ¦ t¦with coefficient determined by the transition state approximation to the reaction rate.

Brownian dynamics simulation of a chemical reaction in solution

We present a study, using the brownian dynamics simulation technique, of a simple model of a chemical reaction in solution. The model consists of the transfer of a particle between two substrate species in a reaction complex which interacts with its surroundings through frictional effects and random force terms. We pay particular attention to the regime in which both the barrier to transfer and the degree of interaction between the transferred particle and its environment are low. Correlation functions associated with the kinetic rate constant are investigated, and the dependence upon temperature, barrier height, mass and coupling coefficient are studied. Even at low barrier height it is possible to define the rate constant and to obtain near-Arrhenius behaviour. The classical isotope effect is found to hold in the high mass limit. The transition state approximation is investigated and the approach to the diffusion controlled limit is observed as the mass and coupling increase. Comparison is also made wit...

Electronic structure and electronic absorption spectra of molybdenum and tungsten oxotetrachlorides

Electronic absorption spectra of the molecules MoOCl4 and WOCl4 have been measured and their electronic structure has been calculated on the basis of the SCF-Xα-SW theory in the overlapping atomic sphere model. Ionisation potentials and allowed optical transition energies have been found in the transition state approximation. The interpretation of the electronic absorption spectra of gaseous MoOCl4 and WOCl4 is given.

Molecular dynamics simulation of a chemical reaction in solution

We have carried out a study, using the molecular dynamics simulation technique, of a simple model describing a chemical reaction in solution. The model consists of a three body reaction complex, in which a light particle transfers between two heavier species, surrounded by a ‘bath’ of 106 spherical liquid particles. Within the reaction complex we employ a three body potential with an effective barrier to transfer which depends strongly upon the separation of the heavy particles. We study the behaviour of a correlation function related to the kinetic rate constant for the transfer reaction and examine the validity of the transition state approximation to this quantity. It is found that transition state theory improves as the mass of the transferred particle increases. Arrhenius plots yield an effective activation energy, although the transfer is not best described as a barrier hopping process. We also compute the predicted incoherent neutron scattering spectrum due to light particle motion and compare with a recent study of proton transfer in the system trifluoracetic acid/water using this experimental technique. It is found that the transfer contributes to the quasielastic line in the predicted spectrum and also produces a broad, low intensity, feature extending into the inelastic region. Both effects would be expected to be difficult to observe experimentally.

Brownian dynamics study of a polymer chain of linked rigid bodies

A Brownian dynamics model for the backbone chain of a macromolecule is developed as a system of linked rigid bodies so that constraints on valence angles and bond lengths are satisfied exactly. For comparison, a corresponding flexible model is developed in which bond lengths and valence angles are held nearly constant by strong harmonic potentials. Equilibrium properties and barrier crossing rates are examined theoretically and by computer simulation of both models, with differences arising due to the presence of constraints in the rigid case. A compensating potential based on the metric determinant of unconstrained coordinates in the rigid model is found to eliminate the effect of constraints. Barrier crossing rates in the transition state approximation are studied when a force fixed in space is applied to the end atoms of the three-bond chain. An exact transition state rate formula developed for this case predicts curved Arrhenius plots of barrier crossing rates; this result is confirmed by computer simulation.

Energy-barrier models for membrane transport

Energy-barrier models are analyzed to find hidden assumptions and establish ranges of validity. The analysis proceeds by comparison with integrated results for model continuum membranes. The main conclusions are that a simple energy-barrier model has a wide range of validity, is remarkably accurate even when its conditions of validity are not strictly met, and is almost always superior to the analogous equations of irreversible thermodynamics. Its major limitations are a possible nonphysical divergence at high electric fields or volume flows caused by breakdown of the transition-state approximation, and the inability to treat multicomponent mixtures except in a pseudobinary (Nernst-Planck) approximation.

Dynamical treatment of unimolecular decomposition reactions. II. Short-range interfragment coupling and incomplete randomization

We investigate the dynamics of an isolated polyatomic molecule undergoing unimolecular dissociation. The intramolecular vibrational energy transfer step is treated by applying a theory of vibrational relaxation previously developed for permanently bound molecules. In contrast to an earlier study, the decomposition step is treated here subject to the assumption that, when sufficient energy is concentrated in the reaction coordinate, transitions between internal states of the molecule occur only while the incipient fragments are near their minimum classically allowed separation. Under such circumstances, it is generally not possible to divide configuration space cleanly into a randomized and a nonrandomized region, as required by the RRKM theory of unimolecular reactions. In spite of this, it is shown that, under certain conditions, the RRKM specific rate constant expression may remain valid. More generally, it is shown that the accurate specific rate constant assumes a form predicted by the transition state treatment of unimolecular reactions. The explicit rate expression derived here reveals factors which determine the accuracy of the transition state approximation for unimolecular reactions—the basic, simplifying assumption that a transition state exists which coincides with a ’’configuration of no return’’ for both decomposition and association reactions. Estimates based on our rate formulas suggest that the transition state approximation may often be sufficiently accurate to justify use of the RRKM rate expression. As part of our analysis of randomization and decomposition dynamics, we examine the time-dependent behavior of a molecule which has fragment–fragment interactions as described above but which is constrained to remain bound by the presence of an artificial ’’wall’’ which prevents the fragments from separating completely. It is found that statistical equilibrium may be achieved in this system—a significant result because the system violates some rather stringent assumptions previously invoked to prove randomization in bound molecules. This motivates a future search for a more general theory of intramolecular relaxation.

Statistical mechanics of isomerization dynamics in liquids and the transition state approximation

In this article, time correlation function methods are used to discuss classical isomerization reactions of small nonrigid molecules in liquid solvents. Molecular expressions are derived for a macroscopic phenomenological rate constant. The form of several of these equations depend upon what ensemble is used when performing averages over initial conditions. All of these formulas, however, reduce to one final physical expression whose value is manifestly independent of ensemble. The validity of the physical expression hinges on a separation of time scales and the plateau value problem. The approximations needed to obtain transition state theory are described and the errors involved are estimated. The coupling of the reaction coordinate to the liquid medium provides the dissipation necessary for the existence of a plateau value for the rate constant, but it also leads to failures of Wigner’s fundamental assumption for transition state theory. We predict that for many isomerization reactions, the transmission coefficient will differ significantly from unity and that the difference will be a strong function of the thermodynamic state of the liquid solvent.

Photoisomerization of some bicyclooctanones in solution

The ketene derived products and unsaturated aldehydes produced upon solution photolysis of a series of bicyclooctanones (bicyclo[2.2.2]octan-2-one (1), bicyclo[3.2.1]octan-2-one (6), l,8,8-trimethylbicyclo[3.2.1]octan-2-one (12)) are reported. Estimated strain energies of the approximate transition states involved and conformational analysis are utilized as guides to rationalize the results. This approach should assist more accurate predictions of photolysis products from related systems in the same solvent(s). The product ratios are solvent sensitive and preliminary experiments indicate that a primary amine (cyclohexylamine) in the photolysis solution enhances the production of ketene derived product.