Finding reaction paths using the potential energy as reaction coordinate

The intrinsic reaction coordinate curve (IRC), normally proposed as a representation of a reaction path, is parametrized as a function of the potential energy rather than the arc-length. This change in the parametrization of the curve implies that the values of the energy of the potential energy surface points, where the IRC curve is located, play the role of reaction coordinate. We use Caratheodory's relation to derive in a rigorous manner the proposed parametrization of the IRC path. Since this Caratheodory's relation is the basis of the theory of calculus of variations, then this fact permits to reformulate the IRC model from this mathematical theory. In this mathematical theory, the character of the variational solution (either maximum or minimum) is given through the Weierstrass E-function. As proposed by Crehuet and Bofill [J. Chem. Phys. 122, 234105 (2005)], we use the minimization of the Weierstrass E-function, as a function of the potential energy, to locate an IRC path between two minima from an arbitrary curve on the potential energy surface, and then join these two minima. We also prove, from the analysis of the Weierstrass E-function, the mathematical bases for the algorithms proposed to locate the IRC path. The proposed algorithm is applied to a set of examples. Finally, the algorithm is used to locate a discontinuous, or broken, IRC path, namely, when the path connects two first order saddle points through a valley-ridged inflection point.     

Extended hessian matrix along the reaction coordinate

We shall prove by the coordinate-free covariant treatment that the vibrational frequency along the reaction coordinate is exactly given by the extended Hessian matrix in a rigged configuration space where the reaction coordinate is identified with a straight line. The diagonalization of the extended Hessian matrix is straightforward with no recourse to the prescribed procedure of eliminating the coupling matrix elements between the promoting mode of chemical reaction and the vibrational modes.     

String model of chemical reaction coordinate

A geometrical model for estimating the effects of external fields upon a reaction path is proposed. The reaction path is defined as the intrinsic reaction coordinate (IRC) which is treated as a string. The IRC frame is introduced in order to uniquely determine the attitude of the string with respect to the reaction medium. The string is thrown in the external fields of the reaction medium, and slides or rotates, and is deformed. The cell structure attached to the string is also deformed. A perturbational approach to the reaction rate formula is presented.     

The reaction path intrinsic reaction coordinate method and the Hamilton-Jacobi theory

The definition and location of an intrinsic reaction coordinate path is of crucial importance in many areas of theoretical chemistry. Differential equations used to define the path hitherto are complemented in this study with a variational principle of Fermat type, as Fukui [Int. J. Quantum Chem., Quantum Chem. Symp. 15, 633 (1981)] reported in a more general form some time ago. This definition is more suitable for problems where initial and final points are given. The variational definition can naturally be recast into a Hamilton-Jacobi equation. The character of the variational solution is studied via the Weierstrass necessary and sufficient conditions. The characterization of the local minima character of the intrinsic reaction coordinate is proved. Such result leads to a numerical algorithm to find intrinsic reaction coordinate paths based on the successive minimizations of the Weierstrass E-function evaluated on a guess curve connecting the initial and final points of the desired path.     

An internal coordinate invariant reaction pathway

In this work we show that some properties of a potential energy surface are not independent of the choice of the coordinate frame. So the reaction pathway often described as steepest descent way does not correspond to an invariant curve under coordinate transformations. We propose an internal intrinsic reaction pathway by using some quasi-dynamical considerations (like instantaneous internal acceleration). Our work precises the intrinsic-reaction coordinates of Fukui to any set of 3N-6 internal parameters. Finally, from the equations of motion we deduce the form of the normal reaction coordinates frame anywhere along the postulated reaction pathway.     

Intrinsic reaction coordinate calculations for reaction paths possessing branching points

A 3-21+G energy surface corresponding to the proton transfer reaction in the hydroperoxyl anion solvated by one water molecule presents interesting topological features. In particular the intrinsic reaction coordinate that begins at the transition state does not lead to a minimum but to a saddle point of second order passing through two branching points. A new strategy to obtain the true reaction path in these cases is proposed.     

Using the histogram test to quantify reaction coordinate error

Many schemes for calculating reaction rates and free energy barriers require an accurate reaction coordinate, but it is difficult to quantify reaction coordinate accuracy for complex processes like protein folding and nucleation. The histogram test, based on estimated committor probabilities, is often used as a qualitative indicator for good reaction coordinates. This paper derives the mean and variance of the intrinsic committor distribution in terms of the mean and variance of the histogram of committor estimates. These convenient formulas enable the first quantitative calculations of reaction coordinate error for complex systems. An example shows that the approximate transition state surface from Peters' and Trout's reaction coordinate for nucleation in the Ising model gives a mean committor probability of 0.495 and a standard deviation of 0.042.     

Rotation procedure in intrinsic reaction coordinate calculations

A procedure for calculating the distance between two points on an intrinsic reaction coordinate (IRC) obtained from two separate runs is proposed. The procedure mainly involves the rotation of the geometrical configuration of one point in order to obtain a configuration not rotated relative to the other point.     

Intrinsic reaction coordinate for the gas-phase pyrolysis of ethyl formate

Fukui's IRC is obtained for the pyrolysis of ethyl formate within the level of MNDO molecular orbital calculations. The activation enthalpy and entropy can be evaluated from the calculated partition functions and compared with the observed values. It is suggested that the pyrolysis would be significantly promoted when it occurs in an aprotic polar solvent.     

Time-resolved mass spectrometric study of the reaction H + Trans-2-Butene

The title reaction was studied in a discharge flow system using mass flow and modulated molecular beam sampling with phase-sensitive detection in order to obtain time-resolved mass spectrometric analysis. At total conversion exceeding 30%, the major products are methane and ethane when initially hydrogen atoms are in excess; when butene is in excess, the major products are ethane and propylene. No hydrocarbons with more than 4 carbon atoms were detected in the products. The reaction is a complicated one since the simplest reaction scheme that successfully simulates the experimental results comprises 20 elementary reactions. The simulation, coupled with sensitivity analysis, shows that with hydrogen atoms in excess, significant amounts of propylene formed in the initial decomposition of the butyl radical react further with hydrogen atoms to form methane and ethane. When butene is in excess, approximately [C₃H₆] ≈ [CH₄] + ½[C₂H₆] which means that this propylene does not react further and almost all methyl radicals end up as CH₄ or C₂H₆. At small conversion, simulation shows that the major product by far is propylene regardless of the [H]/[butene] ratio. The absence of higher hydrocarbons in the products is at variance with earlier results of Rabinovitch and coworkers; however the present work leads to a comparable value for the average rate constant ??_a = ωD/S where D and S is the amount of products arising from the decomposition and stabilization, respectively, of the butyl radical and ω is the collision frequency.     

Time-resolved velocity map imaging of methyl elimination from photoexcited anisole

To date, H-atom elimination from heteroaromatic molecules following UV excitation has been extensively studied, with the focus on key biological molecules such as chromophores of DNA bases and amino acids. Extending these studies to look at elimination of other non-hydride photoproducts is essential in creating a more complete picture of the photochemistry of these biomolecules in the gas-phase. To this effect, CH(3) elimination in anisole has been studied using time-resolved velocity map imaging (TR-VMI) for the first time, providing both time and energy information on the dynamics following photoexcitation at 200 nm. The extra dimension of energy afforded by these measurements has enabled us to address the role of πσ* states in the excited state dynamics of anisole as compared to the hydride counterpart (phenol), providing strong evidence to suggest that only CH(3) fragments eliminated with high kinetic energy are due to direct dissociation involving a (1)πσ* state. These measurements also suggest that indirect mechanisms such as statistical unimolecular decay could be contributing to the dynamics at much longer times.     

Time-resolved photoion and photoelectron imaging of NO(2)

Time-resolved photoion and photoelectron velocity mapped images from NO(2) excited close to its first dissociation limit [to NO(X(2)Pi) + O((3)P(2))] have been recorded in a two colour pump-probe experiment, using the frequency-doubled and frequency-tripled output of a regeneratively amplified titanium-sapphire laser. At least three processes are responsible for the observed transient signals; a negative pump-probe signal (corresponding to a 266 nm pump), a very short-lived transient close to the cross-correlation of the pump and probe pulses but on the 400 nm pump side, and a longer-lived positive pump-probe signal that exhibits a signature of wavepacket motion (oscillations). These transients have two main origins; multiphoton excitation of the Rydberg states of NO(2) by both 266 and 400 nm light, and electronic relaxation in the 1(2)B(2) state of NO(2), which leads to a quasi-dissociated NO(2) high in the 1(2)A(1) electronic ground state and just below the dissociation threshold. The wavepacket motion that we observe is ascribed to states exhibiting free rotation of the O atom about the NO moiety. These states, which are common for loosely bound systems such as a van der Waals complex but unusual for a chemically-bound molecule, have previously been observed in the frequency domain by optical double resonance spectroscopy but never before in the time domain.     

A dimensionless reaction coordinate for quantifying the lateness of transition states

The Hammond‐Leffler postulate asserts that transition states of exothermic reactions are reactant‐like (early), whereas transition states of endothermic reactions are product‐like (late). Related postulates have been proposed to describe the sensitivity of activation barriers for reactions occurring on catalytic surfaces to the catalyst structure. To evaluate the validity of these postulates for different chemical reactions, a general method for classifying transition states as either early or late is needed. One can envision a dimensionless reaction coordinate that changes continuously and monotonically from 0 to 1 along a minimum energy reaction pathway. The value of the dimensionless reaction coordinate for the transition state (W_TS) classifies transition states as (a) early when W_TS < 0.5, (b) late when W_TS > 0.5, and (c) equidistant between reactants and products when W_TS = 0.5. In this article, we derive such a dimensionless reaction coordinate and illustrate its usefulness for several different chemical reactions. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

The reaction force: Three key points along an intrinsic reaction coordinate

The concept of the reaction force is presented and discussed in detail. For typical processes with energy barriers, it has a universal form which defines three key points along an intrinsic reaction coordinate: the force minimum, zero and maximum. We suggest that the resulting four zones be interpreted as involving preparation of reactants in the first, transition to products in the second and third, and relaxation in the fourth. This general picture is supported by the distinctive patterns of the variations in relevant electronic properties. Two important points that are brought out by the reaction force are that (a) the traditional activation energy comprises two separate contributions, and (b) the transition state corresponds to a balance between the driving and the retarding forces.     

An analytical approximation for the number of states along the reaction coordinate

A variant of a new empirical method, enables one to express a collinear triatomic potential energy surface as a family of Morse curves along “natural” bond order coordinates orthogonal to the reaction coordinate. The procedure depends on a single adjustable parameter which is related to the barrier's height. Because an analytical expression for the number of vibrational states of a Morse oscillator is available, one has an analytical approximation for the number of states along the reaction coordinate. The extrema in the number of states are utilized in various versions of classical microcanonical variational transition state theory (among which is a new version, which is in better agreement with dynamical results), to estimate the probability of a collinear reactions, as a function of the total energy. The analytical expressions are also used to analyze the origins of the maximum and minima in the number of states.     

Time-resolved electron detachment imaging of the I- channel in I2Br- photodissociation

The evolution of the I(-) channel in I(2)Br(-) photodissociation is examined using time-resolved negative-ion photoelectron imaging spectroscopy. The 388 nm photodetachment images obtained at variable delays following 388 nm excitation reveal the transformation of the excess electron from that belonging to an excited trihalide anion to that occupying an atomic orbital localized on the I(-) fragment. With increasing pump-probe delay, the corresponding photoelectron band narrows on a approximately 300 fs time scale. This trend is attributed to the localization of the excess-electron wave function on the atomic-anion fragment and the establishment of the fragment's electronic identity. The corresponding band position drifts towards larger electron kinetic energies on a significantly longer, approximately 1 ps, time scale. The gradual spectral shift is attributed to exit-channel interactions affecting the photodetachment energetics, as well as the photoelectron anisotropy. The time-resolved angular distributions are analyzed and found consistent with the formation of the asymptotic I(-) fragment.     

Time-resolved fluorescence measurements using microlens array and area imaging devices

Time-resolved fluorescence (TRF) assay formats are frequently used technologies in high-throughput screening. In this article, we have characterised the novel Plate::Vision(2) 96-microlens array reader (Carl Zeiss Jena GmbH, Germany) and compared it to the novel LEADseeker Generation IV multimodality imaging system (LEADseeker Gen IV; Amersham Biosciences UK Ltd., UK) for applications in the TRF mode. In europium measurements using the TRF mode, the Plate::Vision displayed a limit of detection for europium of approximately 3 pM, which was comparable to two established TRF readers, the Discovery and the Victor V (both PerkinElmer Life Sciences Inc., USA). The LEADseeker's limit of detection only extended down to europium concentrations of approximately 10 pM in these experiments. For TRF resonance energy transfer (TR-FRET) experiments, a europium-biotin (Eu-biotin) conjugate was titrated with a streptavidin-allophycocyanin (SA-APC) conjugate. The Plate::Vision produced Z' values larger than 0.5 for the acceptor fluorophor emission with concentrations of Eu-biotin as low as 3 nM combined with 175 pM SA-APC. To achieve Z' values of at least 0.5 with the LEADseeker, concentrations of 10 nM Eu-biotin combined with SA-APC of at least 0.8 nM were required. In a drug screening application using TR-FRET, the energy transfer from a europium-labelled protein X (Eu-protein X) to a complex of biotinylated peptide Y with SA-APC was measured. Using the Plate::Vision, a Z' factor larger than 0.5 for the acceptor fluorophor emission was only obtained for a Eu-protein X concentration of at least 10 nM in combination with biotinylated peptide Y/SA-APC at saturating concentrations. Both the Plate::Vision and the LEADseeker show good quality results for applications in the TRF mode and enable an increased throughput based on their shortened measurement time in comparison to classic photomultiplier tube-based readers.     

Time-resolved gas-phase kinetic and quantum chemical studies of the reaction of silylene with oxygen

Time-resolved kinetic studies of the reaction of silylene, SiH2, generated by laser flash photolysis of phenylsilane, have been carried out to obtain rate constants for its bimolecular reaction with O(2). The reaction was studied in the gas phase over the pressure range 1-100 Torr in SF(6) bath gas, at five temperatures in the range 297-600 K. The second order rate constants at 10 Torr were fitted to the Arrhenius equation: [see text] The decrease in rate constant values with increasing temperature, although systematic is very small. The rate constants showed slight increases in value with pressure at each temperature, but this was scarcely beyond experimental uncertainty. From estimates of Lennard-Jones collision rates, this reaction is occurring at ca. 1 in 20 collisions, almost independent of pressure and temperature. Ab initio calculations at the G3 level backed further by multi-configurational (MC) SCF calculations, augmented by second order perturbation theory (MRMP2), support a mechanism in which the initial adduct, H(2)SiOO, formed in the triplet state (T), undergoes intersystem crossing to the more stable singlet state (S) prior to further low energy isomerisation processes leading, via a sequence of steps, ultimately to dissociation products of which the lowest energy pair are H2O+SiO. The decomposition of the intermediate cyclo-siladioxirane, via O-O bond fission, plays an important role in the overall process. The bottleneck for the overall process appears to be the T-->S process in H2SiOO. This process has a small spin-orbit coupling matrix element, consistent with an estimate of its rate constant of 1x10(9) s-1 obtained with the aid of RRKM theory. This interpretation preserves the idea that, as in its reactions in general, SiH2 initially reacts at the encounter rate with O2. The low values for the secondary reaction barriers on the potential energy surface account for the lack of an observed pressure dependence. Some comparisons are drawn with the reactions of CH2+O2 and SiCl2+O2.     

Time-resolved resonance Raman study of the reaction of the 2-fluorenylnitrenium ion with 2-fluorenylazide

A time-resolved resonance Raman investigation of the reaction of the 2-fluorenylnitrenium ion with 2-fluorenylazide in a mixed aqueous solvent is presented. The reaction of the 2-fluorenylnitrenium ion with 2-fluorenylazide in the mixed aqueous solution generates two new species on the microsecond time scale. One of these species is identified as 2,2'-azobisfluorene, and the other species is tentatively assigned to a 1,4-bis-(2,2'-fluorenyl)-tetrazadiene cation intermediate. The structure and properties of these two species are briefly discussed. The reaction of the 2-fluorenylnitrenium ion with 2-fluorenylazide is also briefly compared to that of the 2-fluorenylnitrenium ion reactions with guanosine and water.