The location of transition states: A comparison of Cartesian,Z-matrix,and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, in Z-matrix coordinates, and in natural internal coordinates for the location of transition states. In contrast to the situation with minima, where all three coordinate systems are of comparable efficiency if a reliable estimate of the Hessian matrix is available at the starting geometry, results for 25 different transition states covering a wide range of structural types demonstrate that in practice Z-matrix coordinates are generally superior. For Cartesian coordinates, the commonly used Hessian update schemes are unable to guarantee preservation of the necessary transition state eigenvalue structure, while current algorithms for generating natural internal coordinates may have difficulty handling the distorted geometries associated with transition states. The widely used Eigenvector Following (EF) algorithm is shown to be extremely efficient for optimizing transition states. © 1996 by John Wiley & Sons, Inc.     

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.     

Estimating the hessian for gradient-type geometry optimizations

Optimization methods that use gradients require initial estimates of the Hessian or second derivative matrix; the more accurate the estimate, the more rapid the convergence. For geometry optimization, an approximate Hessian or force constant matrix is constructed from a simple valence force field that takes into account the inherent connectivity and flexibility of the molecule. Empirical rules are used to estimate the diagonal force constants for a set of redundant internal coordinates consisting of all stretches, bends, torsions and out-of-plane deformations involving bonded atoms. The force constants are transformed from the redundant internal coordinates to Cartesian coordinates, and then from Cartesian coordinates to the non-redundant internal coordinates used in the specification of the geometry and optimization. This method is especially suitable for cyclic molecules. Problems associated with the choice of internal coordinates for geometry optimization are also discussed.Fellow of the Alfred P. Sloan Foundation, 1981–83     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

Geometry optimization in density functional methods

The geometry optimization in delocalized internal coordinates is discussed within the framework of the density functional theory program deMon. A new algorithm for the selection of primitive coordinates according to their contribution to the nonredundant coordinate space is presented. With this new selection algorithm the excessive increase in computational time and the deterioration of the performance of the geometry optimization for floppy molecules and systems with high average coordination numbers is avoided. A new step selection based on the Cartesian geometry change is introduced. It combines the trust radius and line search method. The structure of the new geometry optimizer is described. The influence of the SCF convergence criteria and the grid accuracy on the geometry optimization are discussed. A performance analysis of the new geometry optimizer using different start Hessian matrices, basis sets and grid accuracies is given.     

η5,η1-Coordinated cyclopentadienyl transition metal complexes featuring σ-metal–carbon ansa bridges

An overview of η⁵,η¹-coordinated transition metal ansa complexes is given. These compounds bear one or more coordinated cyclopentadienyl moieties connected with a η¹-bridging σ-bonded carbon chain featuring a length of at least two carbon atoms. Synthetic approaches, as well as characterisation and applications are described.     

Geometry optimization in cartesian coordinates: The end of the Z-matrix?

Geometry optimization directly in Cartesian coordinates using the EF and GDIIS algorithms with standard Hessian updating techniques is compared and contrasted with optimization in internal coordinates utilizing the well known Z-matrix formalism. Results on a test set of 20 molecules show that, with an appropriate initial Hessian, optimization in Cartesians is just as efficient as optimization in internals, thus rendering it unnecessary to construct a Z-matrix in situations where Cartesians are readily available, for example from structural databases or graphical model builders.     

Partial hessian fitting for determining force constant parameters in molecular mechanics

We present a new protocol for deriving force constant parameters that are used in molecular mechanics (MM) force fields to describe the bond‐stretching, angle‐bending, and dihedral terms. A 3 × 3 partial matrix is chosen from the MM Hessian matrix in Cartesian coordinates according to a simple rule and made as close as possible to the corresponding partial Hessian matrix computed using quantum mechanics (QM). This partial Hessian fitting (PHF) is done analytically and thus rapidly in a least‐squares sense, yielding force constant parameters as the output. We herein apply this approach to derive force constant parameters for the AMBER‐type energy expression. Test calculations on several different molecules show good performance of the PHF parameter sets in terms of how well they can reproduce QM‐calculated frequencies. When soft bonds are involved in the target molecule as in the case of secondary building units of metal‐organic frameworks, the MM‐optimized geometry sometimes deviates significantly from the QM‐optimized one. We show that this problem is rectified effectively by use of a simple procedure called Katachi that modifies the equilibrium bond distances and angles in bond‐stretching and angle‐bending terms. © 2016 Wiley Periodicals, Inc.     

Geometry optimization in Cartesian coordinates: Constrained optimization

An efficient algorithm for constrained geometry optimization in Cartesian coordinates is presented. It incorporates mode-following techniques within both the classical method of Lagrange multipliers and the penalty function method. Both constrained minima and transition states can be located and, unlike the standard Z-matrix using internal coordinates, the desired constraints do not have to be satisfied in the initial structure. The algorithm is as efficient as a Z-matrix optimization while presenting several additional advantages.     

Internal‐to‐Cartesian back transformation of molecular geometry steps using high‐order geometric derivatives

In geometry optimizations and molecular dynamics calculations, it is often necessary to transform a geometry step that has been determined in internal coordinates to Cartesian coordinates. A new method for performing such transformations, the high‐order path‐expansion (HOPE) method, is here presented. The new method treats the nonlinear relation between internal and Cartesian coordinates by means of automatic differentiation. The method is reliable, applicable to any system of internal coordinates, and computationally more efficient than the traditional method of iterative back transformations. As a bonus, the HOPE method determines not just the Cartesian step vector but also a continuous step path expressed in the form of a polynomial, which is useful for determining reaction coordinates, for integrating trajectories, and for visualization. © 2013 Wiley Periodicals, Inc.     

Analytical hessian fitting schemes for efficient determination of force‐constant parameters in molecular mechanics

Building upon our recently developed partial Hessian fitting (PHF) method (Wang et al., J. Comput. Chem. 2016 , 37, 2349), we formulated and implemented two other rapid force‐field parameterization schemes called full Hessian fitting (FHF) and internal Hessian fitting (IHF), and comparisons were made among these three parameterization schemes to assess their performance. FHF minimizes deviation between the Hessian matrices in Cartesian coordinates computed by quantum mechanics (QM) and molecular mechanics (MM), to determine the best possible MM force‐constant parameters. While PHF requires step‐by‐step fittings of 3 × 3 partial Hessian matrices, FHF compares the lower triangular part of the QM and MM Hessian matrices, which allows simultaneous determination of all force‐constant parameters. In addition to this simple FHF scheme, IHF was developed such that it considers the Hessian matrices in redundant internal coordinates, where all possible internal coordinates that arise from the user‐defined interatomic connectivity are utilized. The results show that IHF performs best overall, followed by PHF and then FHF. Python‐based programing codes were developed to automate various tedious steps involved in the parameterization processes. © 2017 Wiley Periodicals, Inc.     

A stable (amino)(phosphino)carbene as bidentate ligand for palladium and nickel complexes: Synthesis,structure, and catalytic activity

Two original complexes featuring an (amino)(phosphino)carbene η²-bonded to the metal have been obtained in 60% and 80% yields, by addition of the corresponding stable carbene to PdCl₂(cod) and NiCl₂(PPh₃)₂, respectively. Both complexes have been fully characterized including X-ray diffraction studies. The catalytic activity of the palladium complex has been evaluated for aryl amination reactions.     

Exohedral Complexation of B40, C60 and Arenes with Transition Metals: A Comparative DFT Study

Holes are inevitable in borospherenes. The surface topography of B₄₀ and its π MOs isolobal to benzene allow for better η⁷‐, η⁶‐ and η³‐ exohedral complexation with transition metal fragments than it is possible with C₆₀ and arenes. η⁷‐complexes of B₄₀ is lower in energy than the η⁶‐complexes for metal fragments such as C₅H₅Mn, C₄H₄Fe, and C₃H₃Co that have relatively diffuse frontier orbitals. The fragment C₆H₆Cr prefers η⁶‐coordination. Near‐isodesmic equations based on density functional theory computations of the transition metal complexes of B₄₀, C₆₀ and C₆H₆ support these anticipations. Transition metal complexation increases the stability of B₄₀.     

Using internal coordinates to describe photoinduced geometry changes in MLCT excited states

A resonance Raman intensity analysis of the metal-to-ligand charge-transfer (MLCT) transition for the rhenium compound Re(2-(2'-pyridyl)quinoxaline)(CO)(3)Cl (RePQX) is presented. Photoinduced geometry changes are calculated, and the results are presented using the vibrational normal modes and the redundant internal coordinates. A density functional theory calculation is used to determine the ground-state nonresonant Raman spectrum and a transformation matrix that transforms the redundant internal coordinates into the normal modes. The normal modes nu(37) (rhenium coordination sphere distortion) and nu(75) (ligand skeletal stretch) show the largest photoinduced geometry change (Delta = 1.0 and 0.7, respectively). A single carbonyl mode is enhanced in the resonance Raman spectra. Time-dependent density functional theory is used to calculate excited-state geometry changes, which are subsequently used to determine the signs of the photoinduced normal mode displacements. Transforming to internal coordinates reveals that all the CO bond lengths are displaced in the excited state. The Re-C and C-C ligand bond lengths are also displaced in the excited state. The results are discussed in terms of a simple one-electron picture for the electronic transition. Many bond angles and torsional coordinates are also displaced by the metal-to-ligand charge transfer, and most of these are associated with the rhenium coordination sphere. It is demonstrated that using internal coordinates presents a clear picture of the geometry changes associated with photoinduced electron transfer in metal polypyridyl systems.     

Arene transition metal chemistry : III. Arene exchange phenomena

Arene exchange between free arene and an arene—transition metal complex is reviewed with respect to structure, thermodynamics, kinetics and reaction mechanism. Catalysis of such arene exchange is also examined. Experimental results are presented for arene exchange between arene and the arene—metal complexes: η⁶-arene-1,5-cyclooctadieneruthenium, the cationic η⁶-arene-1,5-cyclooctadieneiridium and η⁶-arenetricarbonylmolybdenum. Mechanistic features of these exchange reactions are discussed.     

An analytical computation of Christoffel symbols for reaction coordinate and trajectory treatments under internal coordinates

We examine the Hessian matrix of the potential energy under internal coordinates. We report all Christoffel symbols which exist for molecules if we use the known coordinates such as bond distances, bond angles, torsion angles, and out-of-plane angles. We use as an example triatomic HCN in an extended geometry.     

The choice of internal coordinates in complex chemical systems

This article presents several considerations for the appropriate choice of internal coordinates in various complex chemical systems. The appropriate and black box recognition of internal coordinates is of fundamental importance for the extension of internal coordinate algorithms to all fields where previously Cartesian coordinates were the preferred means of geometry manipulations. Such fields range from local and global geometry optimizations to molecular dynamics as applied to a wide variety of chemical systems. We present a robust algorithm that is capable to quickly determine the appropriate choice of internal coordinates in a wide range of atomic arrangements. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

1H NMR structural increments for alkyl substituted η3-allyl transition metal complexes

The ¹H NMR spectra of various alkyl substituted η³-allyl transition metal complexes (M?Ni, Ru) have been analysed. The chemical shifts of the η³-allyl protons can be calculated using additive increments; the values of the syn and anti vicinal proton-proton coupling constants approach each other on alkyl substitution of the η³-allyl group.     

Molecular geometry and vibrational spectroscopy of potassium O,O-diethyldithiophosphate

The aim of this work was to build a good theoretical and experimental basis for the further study of changes in structure and spectra of the O,O-diethyldithiophosphate anion upon its adsorption on the surfaces of transition metal sulfides.Infrared and Raman spectra of potassium O,O-diethyldithiophosphate were recorded. High level quantum chemical calculations were carried out to optimize the molecular geometry of both the potassium salt and its anion. Vibrational force constants were calculated from the second derivative of the molecular energy function with respect to the Cartesian coordinates of the atoms. With the aid of the optimized geometry and the calculated vibrational force constants a normal coordinate analysis was carried out to characterize the molecular vibrational modes and to assign the vibrational frequencies.     

Theoretical DFT studies of chromium tricarbonyl complexes with polycyclic aromatic ligands

This paper presents a theoretical analysis of the structures of tricarbonyl chromium complexes of carbo- and heterocyclic polyaromatic ligands (PAL) and the mechanisms of interring haptotropic rearrangement in such complexes performed by using density functional theory (DFT) with the nonempirically constructed PBE functional and extended split basis sets. The reaction paths were calculated for interring haptotropic rearrangements and rotations of the metalcarbonyl fragment in the regioisomeric complexes. The structures and energy characteristics of stationary points of the systems were determined. The migration of the Cr(CO)₃ group was shown to occur at the periphery of the ligand via transition states with the structure of η³-allylic or η⁴-trimethylmethane complexes. Calculated geometries of the complexes and the activation barriers were in a close agreement with the experimental data.The reaction of η⁶-tricarbonylchromium complexes of PAL with n-BuLi (lithiation) was also studied by the DFT. The kinetic and thermodynamic factors that control the direction and selectivity of metallation were calculated for the model η⁶-biphenylenetricarbonylchromium complex. Both approaches indicate that lithiation occurs exclusively at the aromatic ring bonded to the transition metal, which agrees with the experimental data. The selectivity inside this ring is governed by a thermodynamic factor. The solvation effects were simulated for the lithium salt of the model η⁶-naphthalenechromium tricarbonyl complex in which lithium is localized at the α(1)-position of coordinated ring. The simulation showed the most stable coordination of the lithium atom with two THF molecules. Addition of extra THF molecules is thermodynamically unfavorable. The tricarbonylchromium complexes of naphthalene, biphenyl, biphenylene and dibenzothiophene calculated relative energies for all solvated by two THF molecules lithium salts indicate that the difference in energies ΔE ? 1 kcal mol^?1 corresponds to the experimentally observed absence of selectivity, while the difference more than 2.5  kcal mol^?1 corresponds to the selectivity of the reaction. No additional coordination of lithium to heteroatom was observed for the sulfur-containing dibenzothiophene complex. Similar calculation shows that double metallation in the dibenzothiophene complex occurs at positions 1 and 4. The developed approach enables one to predict the direction and selectivity of metallation reactions of transition metal complexes with different arenes and thus to synthesize labeled complexes for the investigation of degenerate IRHR.