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.     

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.     

A quantum generalization of intrinsic reaction coordinate using path integral centroid coordinates

We propose a generalization of the intrinsic reaction coordinate (IRC) for quantum many-body systems described in terms of the mass-weighted ring polymer centroids in the imaginary-time path integral theory. This novel kind of reaction coordinate, which may be called the "centroid IRC," corresponds to the minimum free energy path connecting reactant and product states with a least amount of reversible work applied to the center of masses of the quantum nuclei, i.e., the centroids. We provide a numerical procedure to obtain the centroid IRC based on first principles by combining ab initio path integral simulation with the string method. This approach is applied to NH(3) molecule and N(2)H(5) (-) ion as well as their deuterated isotopomers to study the importance of nuclear quantum effects in the intramolecular and intermolecular proton transfer reactions. We find that, in the intramolecular proton transfer (inversion) of NH(3), the free energy barrier for the centroid variables decreases with an amount of about 20% compared to the classical one at the room temperature. In the intermolecular proton transfer of N(2)H(5) (-), the centroid IRC is largely deviated from the "classical" IRC, and the free energy barrier is reduced by the quantum effects even more drastically.     

Multicoordinate driven method for approximating enzymatic reaction paths: automatic definition of the reaction coordinate using a subset of chemical coordinates

We present a generalization of the reaction coordinate driven method to find reaction paths and transition states for complicated chemical processes, especially enzymatic reactions. The method is based on the definition of a subset of chemical coordinates; it is simple, robust, and suitable to calculate one or more alternative pathways, intermediate minima, and transition-state geometries. Though the results are approximate and the computational cost is relatively high, the method works for large systems, where others often fail. It also works when a certain reaction path competes with others having a lower energy barrier. Accordingly, the procedure is appropriate to test hypothetical reaction mechanisms for complicated systems and provides good initial guesses for more accurate methods. We present tests on a number of simple reactions and on several complicated chemical transformations and compare the results with those obtained by other methods. Calculation of the reaction path for the enzymatic hydrolysis of the substrate by dUTPase for an active-site model with 85 atoms, including several loosely bound water molecules, indicates that the method is feasible for the study of enzyme mechanisms.     

Theoretical study of dynamical properties on reaction path in molecular internal coordinates

The dynamical properties on reaction path (IRC) in internal coordinates have been obtained, which in. clude ω_K (frequencies orthogonal to IRC), L_K (vibrational modes), B_(KF) (coupling constants between the IRC and vibra tions orthogonal to it), B_(KL) (coupling constants between every two vibrations orthogonal to IRC). A set of theory of teac. tion path in molecular internal coordinates has been also constructed. The dynamical properties, including ω_K, B_(KF) B_(KL) of the reaction H~1O~2H~3 H~4→H~1O~2 H~3H~4 have been calculated, which explicitly explain the interaction, chang ing trend and contribution of each chemical bond (including bond angle) in the reaction.     

The use of internal cartesian coordinates for describing molecular vibrations

An analysis of the influence of isotope substitution on the system of electronic-nuclear equations for an arbitrary molecular system was used as a basis for formulating invariance conditions with respect to isotope substitution of the potential energy surface written in the Cartesian coordinates rigidly bound with the center of mass of the molecule (internal Cartesian coordinates). This property of the potential function obviates the necessity of using curvilinear natural coordinates, which can be replaced by Cartesian coordinates, in theoretical studies of the vibrational spectra of molecules and their isotopomers and in solving the direct and inverse anharmonic problems. An equation for the quantum-mechanical Hamiltonian of a normal molecule in internal Cartesian coordinates was obtained.     

Molecular modelling of liquid crystal systems: An internal coordinate Monte Carlo approach

A Monte Carlo scheme is presented which is designed to provide a convenient mechanism to model accurately the internal molecular structure of liquid crystalline molecules. The technique stores atomic positions in terms of bond lengths, bond angles and dihedral angles within a Z-matrix, and the Monte Carlo scheme involves generating trial configurations from changes to the Z-matrix using the MM2 molecular mechanics potential to describe energy changes between different molecular conformations. The technique is applied to the liquid crystal molecule 4-n-pentyl-4'-cyanobiphenyl (5CB), and results are presented for the conformational populations and dihedral angle distributions of 5CB in the gas phase at 300 K. The effect of a nematic mean field on the distribution of molecular conformations is also examined via the addition of a conformation-dependent potential of mean torque to the internal energy.     

Pattern of separatrices and intrinsic reaction coordinates for degenerate thermal rearrangements

A structurally stable model of the standard adiabatic gradient field of the potential energy surface for certain pericyclic reactions is derived.These reactions are not subjected to the principles of orbital isomerism or to the Woodward-Hoffmann rules.Use is made of a principle established by Ariel Fernández and Oktay Sinanolu which precludes direct meta-IRC connections between transition states.It is shown that Jahn-Teller isomers of the singlet biradicals involved in the process are not interconvertible since the biradical configuration is not a transition state but a critical point with Hessian matrix with two negative eigenvalues.The topological features of the PES obtained by combinatorial methods are in full agreement with earlier results obtained from MINDO calculations.     

Geometry optimization for peptides and proteins: comparison of Cartesian and internal coordinates

We present the results of several benchmarks comparing the relative efficiency of different coordinate systems in optimizing polypeptide geometries. Cartesian, natural internal, and primitive internal coordinates are employed in quasi-Newton minimization for a variety of biomolecules. The peptides and proteins used in these benchmarks range in size from 16 to 999 residues. They vary in complexity from polyalanine helices to a beta-barrel enzyme. We find that the relative performance of the different coordinate systems depends on the parameters of the optimization method, the starting point for the optimization, and the size of the system studied. In general, internal coordinates were found to be advantageous for small peptides. For larger structures, Cartesians appear to be more efficient for empirical potentials where the energy and gradient can be evaluated relatively quickly compared to the cost of the coordinate transformations.     

Optimal internal coordinates, vibrational spectrum, and effective Hamiltonian for ozone

In this paper the authors use the optimal internal vibrational coordinates previously determined for the electronic ground state of the ozone molecule to study the vibrational spectrum of the molecule employing the second empirical potential energy surface calculated by Tyuterev et al. [Chem. Phys. Lett. 316, 271 (2000)]. First, the authors compute variationally all the bound vibrational energy levels of the molecule up to the dissociation limit and state the usefulness of the optimal coordinates in this respect, which allows us to converge all the bound levels using relatively small anharmonic basis sets. By analyzing the expansion coefficients of the wave functions, they show then that a large portion of the vibrational spectrum of O3 can be structured in nearly separable polyadic groups characterized by the polyad quantum number N=n1+n2+n(theta) corresponding to the optimal internal coordinates. Accordingly, they determine an internal effective vibrational Hamiltonian for O3 by fitting the effective Hamiltonian parameters to the experimental vibrational frequencies, using as input parameters in the fit those extracted from an analytical second-order Van Vleck perturbation theory calculation. It is finally shown that the internal effective Hamiltonian thus obtained accurately describes the vibrational spectrum of ozone in the low and medium energy regimes.     

Quasi-Hamiltonian equations of motion for internal coordinate molecular dynamics of polymers

Conventional molecular dynamics simulations of macromolecules require long computational times because the most interesting motions are very slow compared to the fast oscillations of bond lengths and bond angles that limit the integration time step. Simulation of dynamics in the space of internal coordinates, that is, with bond lengths, bond angles, and torsions as independent variables, gives a theoretical possibility of eliminating all uninteresting fast degrees of freedom from the system. This article presents a new method for internal coordinate molecular dynamics simulations of macromolecules. Equations of motion are derived that are applicable to branched chain molecules with any number of internal degrees of freedom. Equations use the canonical variables and they are much simpler than existing analogs. In the numerical tests the internal coordinate dynamics are compared with the traditional Cartesian coordinate molecular dynamics in simulations of a 56 residue globular protein. For the first time it was possible to compare the two alternative methods on identical molecular models in conventional quality tests. It is shown that the traditional and internal coordinate dynamics require the same time step size for the same accuracy and that in the standard geometry approximation of amino acids, that is, with fixed bond lengths, bond angles, and rigid aromatic groups, the characteristic step size is 4 fs, which is 2 times higher than with fixed bond lengths only. The step size can be increased up to 11 fs when rotation of hydrogen atoms is suppressed. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18 : 1354–1364, 1997     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

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.     

Combined analytical formula for two-center Coulomb and hybrid integrals in the molecular coordinate system

Azerbaidzhan State University. Translated from Zhurnal Strukturnoi Khimii, Vol. 32, No. 1, pp. 160–161, January–February, 1991.     

An intrinsic reaction coordinate calculation of the torsion-internal rotation potential of hydrogen peroxide and its isotopomers

Intrinsic reaction coordinate (IRC) calculations of the internal rotation (torsional) potentials for H(2)O(2) and its isotopomers HDO(2) and D(2)O(2) were carried out at the CCSD(T)/CBS//aug-cc-pVDZ level. Two extrapolation methods were used to obtain energies in the complete basis set (CBS) limit. The full IRC potential was constructed from scans from the C(2v) (cis) and C(2h) (trans) transition states to the equilibrium C(2) (gauche) structure. The IRC potential for H(2)O(2) was fit to a five-term Fourier function; coefficients were compared with values obtained from spectroscopic data. The twofold IRC torsional potentials were used to obtain torsional eigenvalues, which yielded values of the transitions between various ntau states. These results compare favorably with Raman and near-infrared data. Our calculations provide values of the cis and trans barriers of 2495 and 364 cm(-1), respectively, which are in good agreement with both previously calculated and experimentally derived values. It appears that coupling between torsional motion and other degrees of freedom is not significant in these molecules.     

Geometry optimization of molecular clusters and complexes using scaled internal coordinates

Scaled internal coordinates are introduced for use in the geometry optimization of systems composed of multiple fragments, such as solvated molecules, clusters, and biomolecular complexes. The new coordinates are related to bond lengths, bond angles and torsion angles by geometry-dependent scaling factors. The scaling factors serve to expedite the optimization of complexes containing outlying fragments, without hindering the optimization of the intramolecular degrees of freedom. Trial calculations indicate that, at asymptotic separations, the scaling factors improve the rate of convergence by a factor of 4 to 5.     

An improved algorithm for the analytical computation of solvent-excluded volume. The treatment of singularities in solvent-accessible surface area and volume functions

An algorithm for the analytical computation of solvent-excluded volume is presented as part of our efforts to develop an improved computational model for a solvent effect term, in which the work required to create a cavity in the solvent is expressed as a function of the solvent-excluded volume. In this article we describe mathematical developments in the analytical integration of solvent-accessible surface (SAS) area, the singularities in SAS area and volume functions, and the procedures required to detect and treat singularities. Techniques to increase algorithm performance are presented, which improve computational speed by about five times, on the average. The accuracy of the analytical method for volume computation is compared with the accuracy of two numerical methods: the numerical integration of SAS area and the point-by-point scanning method. This algorithm calculates the volume of the spheres confined among their intersection planes and resembles a numerical integration of surface area by summing up volume layers. These characteristics make the algorithm useful in analytically calculating the work required to create a convex cavity in a solvent and the work (pΔV) associated with a change in the solvent-excluded volume of the solute due to solvent pressure. © 1995 by John Wiley & Sons, Inc.     

A new analytical reaction for the determination of perrhenate

Perrhenate is quantitatively extracted into methyl isobutyl ketone from aqueous solutions containing copper(II), azide and an excess of 2,2'-bipyridine. Measurement of the extracted copper either by spectrophotometry or by atomic absorption spectrometry, allows the determination of perrhenate in the ranges 16–40 μg ml^?1 or 3–16 μg ml^?1 in the final dilution, respectively. The procedure is highly selective, being applicable in the presence of a large concentration of molybdate and a considerable number of foreign ions. The extracted species corresponds to the formula CuN₃(bipy)₂ ReO₄.     

On the use of optimal internal vibrational coordinates for symmetrical bent triatomic molecules

The use of generalized internal coordinates for the variational calculation of excited vibrational states of symmetrical bent triatomic molecules is considered with applications to the SO2, O3, NO2, and H2O molecules. These coordinates depend on two external parameters which can be properly optimized. We propose a simple analytical method to determine the optimal internal coordinates for this kind of molecules based on the minimization with respect to the external parameters of the zero-point energy, assuming only quadratic terms in the Hamiltonian and no quadratic coupling between the optimal coordinates. The optimal values of the parameters thus obtained are shown to agree quite well with those that minimize the sum of a number of unconverged energies of the lowest vibrational states, computed variationally using a small basis function set. The unconverged variational calculation uses a basis set consisting of the eigenfunctions of the uncoupled anharmonic internal coordinate Hamiltonian. Variational calculations of the excited vibrational states for the four molecules considered carried out with an increasing number of basis functions, also evidence the excellent convergence properties of the optimal internal coordinates versus those provided by other normal and local coordinate systems.     

Rhodium catalyzed reaction of internal alkynes with organoborons under CO atmosphere: a product tunable reaction

Alkynes react with organoborons under a CO atmosphere in the presence of a rhodium(I) catalyst to afford mainly 5-aryl-2(5H)-furanones, α,β-unsaturated ketones, and indanones. The product selectivity can be tuned by modifying the reaction conditions.