Z-matrix Coordinates Research Articles

Potential Energy Surfaces and Arrangement Effects of RbNa2 Complex.

Ab initio calculations of alkaline diatomic molecule interactions with alkaline atoms provide detailed information about their electronic structure, vibrational frequencies, and spectroscopic properties, which are difficult to measure experimentally. This knowledge can aid in designing and interpreting experiments and guide the development of computational models and advanced dynamical calculation. Using the quantum chemistry ab initio methods based on multi-reference configuration interaction with Davidson correction (MCSCF/MRCI + Q), atomic effective core potentials, core-polarization potentials, and the interactions between the sodium atom and the NaRb diatomic molecule are investigated. To describe the potential energy surfaces of the RbNa2 system, we introduce two geometries described in the Z-matrix coordinates (Re, R, θ). Potential energy surfaces of the ground state 12A' and the first excited state 22A' were calculated for different approach directions of the sodium atom to the NaRb molecule and two geometries were considered. The first geometry is where the Na atom approaches the Rb atom of the RbNa dimer, and the second one is when it approaches the Na atom of the RbNa dimer. Global minima of the ground and first excited states and conical intersections between these states are determined for both geometries. The RbNa dimer in interaction with the sodium atom is found to be strongly attractive in its first excited state, which may be important for the experimenters particularly in the field of cold alkali polar dimers. Thereafter, the potential energy curves correlated to the lowest-lying dissociation limits are calculated in the linear form for the two geometrical cases (angle θ at 180°) and the atomic arrangement effect is observed.

A spur to molecular geometry optimization: Gradient-enhanced universal kriging with on-the-fly adaptive ab initio prior mean functions in curvilinear coordinates.

We present a molecular geometry optimization algorithm based on the gradient-enhanced universal kriging (GEUK) formalism with ab initio prior mean functions, which incorporates prior physical knowledge to surrogate-based optimization. In this formalism, we have demonstrated the advantage of allowing the prior mean functions to be adaptive during geometry optimization over a pre-fixed choice of prior functions. Our implementation is general and flexible in two senses. First, the optimizations on the surrogate surface can be in both Cartesian coordinates and curvilinear coordinates. We explore four representative curvilinear coordinates in this work, including the redundant Coulombic coordinates, the redundant internal coordinates, the non-redundant delocalized internal coordinates, and the non-redundant hybrid delocalized internal Z-matrix coordinates. We show that our GEUK optimizer accelerates geometry optimization as compared to conventional non-surrogate-based optimizers in internal coordinates. We further showcase the power of the GEUK with on-the-fly adaptive priors for efficient optimizations of challenging molecules (Criegee intermediates) with a high-accuracy electronic structure method (the coupled-cluster method). Second, we present the usage of internal coordinates under the complete curvilinear scheme. A complete curvilinear scheme performs both surrogate potential-energy surface (PES) fitting and structure optimization entirely in the curvilinear coordinates. Our benchmark indicates that the complete curvilinear scheme significantly reduces the cost of structure minimization on the surrogate compared to the incomplete curvilinear scheme, which fits the surrogate PES in curvilinear coordinates partially and optimizes a structure in Cartesian coordinates through curvilinear coordinates via the chain rule.

Automated handling of complex chemical structures in Z-matrix coordinates-The chemcoord library.

In this work, we present a fully automated method for the construction of chemically meaningful sets of hierarchical nonredundant internal coordinates (ICs; also commonly denoted as Z-matrices) from the Cartesian coordinates of a molecular system. Particular focus is placed on avoiding ill-definitions of angles and dihedrals due to linear arrangements of atoms, to consistently guarantee a well-defined transformation to Cartesian coordinates, even after structural changes. The representations thus obtained are particularly well suited for pathway construction in double-ended methods for transition state search and optimizations with nonlinear constraints. Analytical gradients for the transformation between the coordinate systems were derived for analytical geometry optimizations purely in Z-matrix coordinates. The geometry optimization was coupled with a Symbolic Algebra package to support arbitrary nonlinear constraints in Z-matrix coordinates, while retaining analytical energy gradient conversion. The difference to the commonly used nonhierarchical IC transformations is discussed. Sample applications are provided for a number of common chemical reactions and illustrative examples.

Synthesis, X-Ray, spectroscopic characterization (FT-IR, NMR, UV–Vis) and quantum chemical calculations of some substituted benzoylthiourea derivatives

A new series of substituted benzoylthiourea derivatives namely N-(4-methoxybenzoyl)-N`-(3-hydroxyphenyl)thiourea (1a), N-(3-methoxybenzoyl)-N`-(3-hydroxyphenyl) thiourea (2a), N-(3-nitrobenzoyl)-N`-(3-hydroxyphenyl)thiourea (3a) and N-(4-phenylbenzoyl)-N`-(3-hydroxy phenyl)thiourea (4a) bearing phenolic groups were efficiently synthesized from the condensation reaction of benzoylisothiocyanate with 3-aminophenol. The molecular structures of 1a-4a are elucidated by FT-IR, UV–Vis, 1H and 13C NMR spectroscopic techniques. The four benzoylthiourea derivatives were obtained in crystalline forms by recrystallization in DMSO. X-ray chemical crystallographic study shows that 1a crystallizes in triclinic crystal system, while 2a-4a crystalize in monoclinic crystal system. The synthesized benzoylthiourea molecules adopt trans-cis configuration with respect to the positions of carbonoylbenzoyl and 3-hydroxyphenyl groups respectively, against the thiono C=S bonds across their C–N bonds. The corresponding predicted z-matrix coordinates and spectral data of 1a-4a were obtained using DFT and TD-DFT methods. The quantum chemical calculation have been carried out using three hybrid functionals B3LYP, B3P86 and PBE0 combined with 6–31++G(d,p) basis set. The solvent effects were taken into account implicitly using IEF-PCM model. The experimental Z-matrix coordinates and the spectral data of the tilted compounds are relatively well reproduced with good correlations coefficients.

Instanton rate constant calculations using interpolated potential energy surfaces in nonredundant, rotationally and translationally invariant coordinates.

A trivial flaw in the utilization of artificial neural networks in interpolating chemical potential energy surfaces (PES) whose descriptors are Cartesian coordinates is their dependence on simple translations and rotations of the molecule under consideration. A different set of descriptors can be chosen to circumvent this problem, internuclear distances, inverse internuclear distances or z-matrix coordinates are three such descriptors. The objective is to use an interpolated PES in instanton rate constant calculations, hence information on the energy, gradient, and Hessian is required at coordinates in the vicinity of the tunneling path. Instanton theory relies on smoothly fitted Hessians, therefore we use energy, gradients, and Hessians in the training procedure. A major challenge is presented in the proper back-transformation of the output gradients and Hessians from internal coordinates to Cartesian coordinates. We perform comparisons between our method, a previous approach and on-the-fly rate constant calcuations on the hydrogen abstraction from methanol and on the hydrogen addition to isocyanic acid. © 2018Wiley Periodicals, Inc.

On the use of different coordinate systems in mechanochemical force analyses.

Force analyses are crucial for a comprehensive understanding of mechanochemical processes. The choice of coordinate system in these kinds of analyses is a nontrivial task that determines the quality and validity of the obtained results. Here, we study the suitability of different sets of coordinates for mechanical force analyses, i.e., normal modes, delocalized internal, redundant internal, and Z-matrix coordinates. After discussing the theoretical foundations of force analyses using different coordinate systems, we investigate a number of test molecules. We show that normal modes and Z-matrix coordinates deliver useful results only if certain requirements are fulfilled and that only redundant internal coordinates yield meaningful results in all cases.

Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach: General formulation and removal of singularities

We present new techniques for an automatic computation of the kinetic energy operator in analytical form. These techniques are based on the use of the polyspherical approach and are extended to take into account Cartesian coordinates as well. An automatic procedure is developed where analytical expressions are obtained by symbolic calculations. This procedure is a full generalization of the one presented in Ndong et al., [J. Chem. Phys. 136, 034107 (2012)]. The correctness of the new implementation is analyzed by comparison with results obtained from the TNUM program. We give several illustrations that could be useful for users of the code. In particular, we discuss some cyclic compounds which are important in photochemistry. Among others, we show that choosing a well-adapted parameterization and decomposition into subsystems can allow one to avoid singularities in the kinetic energy operator. We also discuss a relation between polyspherical and Z-matrix coordinates: this comparison could be helpful for building an interface between the new code and a quantum chemistry package.

Anharmonic state counts and partition functions for molecules via classical phase space integrals in curvilinear coordinates

An algorithm is presented for calculating fully anharmonic vibrational state counts, state densities, and partition functions for molecules using Monte Carlo integration of classical phase space. The algorithm includes numerical evaluations of the elements of the Jacobian and is general enough to allow for sampling in arbitrary curvilinear or rectilinear coordinate systems. Invariance to the choice of coordinate system is demonstrated for vibrational state densities of methane, where we find comparable sampling efficiency when using curvilinear z-matrix and rectilinear Cartesian normal mode coordinates. In agreement with past work, we find that anharmonicity increases the vibrational state density of methane by a factor of ∼2 at its dissociation threshold. For the vinyl radical, we find a significant (∼10×) improvement in sampling efficiency when using curvilinear z-matrix coordinates relative to Cartesian normal mode coordinates. We attribute this improved efficiency, in part, to a more natural curvilinear coordinate description of the double well associated with the H2C-C-H wagging motion. The anharmonicity correction for the vinyl radical state density is ∼1.4 at its dissociation threshold. Finally, we demonstrate that with trivial parallelizations of the Monte Carlo step, tractable calculations can be made for the vinyl radical using direct ab initio potential energy surface evaluations and a composite QCISD(T)/MP2 method.

Efficient geometry optimization of molecular clusters

By combining valence coordinates (stretches, bends, and torsions) to describe intramolecular degrees of freedom with inverse distance coordinates for intermolecular degrees of freedom, we derive an efficient set of coordinates for the geometry optimization of molecular clusters. We illustrate the efficacy of our new coordinates by considering randomly generated clusters of dihydrogen and water molecules. Compared to optimizations in Cartesian coordinates, the number of cycles required for convergence is reduced by up to a factor of 30. In addition, for the dihydrogen clusters, optimizations using our new cluster coordinates consistently converge to lower energy structures than the corresponding Cartesian optimizations. Our method is far more efficient than optimizations using Z-matrix coordinates, and it avoids all problems with near-linear bond angles that are endemic with a Z-matrix description of the cluster geometry. Additionally, by constraining all the intramolecular degrees of freedom in a completely automated manner, we are able to carry out full rigid-body optimizations. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 69–76, 2000

Efficient geometry optimization of molecular clusters

By combining valence coordinates (stretches, bends, and torsions) to describe intramolecular degrees of freedom with inverse distance coordinates for intermolecular degrees of freedom, we derive an efficient set of coordinates for the geometry optimization of molecular clusters. We illustrate the efficacy of our new coordinates by considering randomly generated clusters of dihydrogen and water molecules. Compared to optimizations in Cartesian coordinates, the number of cycles required for convergence is reduced by up to a factor of 30. In addition, for the dihydrogen clusters, optimizations using our new cluster coordinates consistently converge to lower energy structures than the corresponding Cartesian optimizations. Our method is far more efficient than optimizations using Z-matrix coordinates, and it avoids all problems with near-linear bond angles that are endemic with a Z-matrix description of the cluster geometry. Additionally, by constraining all the intramolecular degrees of freedom in a completely automated manner, we are able to carry out full rigid-body optimizations. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 69–76, 2000

Graphical approach for defining natural internal coordinates

A simple, customizable connectivity scheme is rigorously defined in which pairs of atoms are classified into three categories. The tools of graph theory are used to analyze the molecular graph and to efficiently find rings and ring assemblies through a combination of pruning and homeomorphic reduction. The definition of natural internal coordinates is extended in a nonredundant fashion for the various cases of weakly interacting components and for fused ring systems. The ring system coordinates were tested and found to be superior to Z-matrix coordinates. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 504–511, 1998

Ab initio geometry optimization for large molecules

Various geometry optimization techniques are systematically investigated. The rational function (RF) and direct inversion in the iterative subspace (DIIS) methods are compared and optimized for the purpose of geometry optimization. Various step restriction and line search procedures are tested. The model Hessian recently proposed by Lindh et al. has been used in conjunction with different Hessian update procedures. Optimization for over 30 molecules have been performed in Z-matrix coordinates, local normal coordinates, and curvilinear natural internal coordinates, using the same approximations for the Hessian in all cases. The most effective and stable procedure for optimization of equilibrium structures was found to be the DIIS minimization in natural internal coordinates using the BFGS update of the model Hessian. Our method shows faster overall convergence than all previously published methods for the same test suite of molecules. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1473–1483, 1997

Ab initio geometry optimization for large molecules

Various geometry optimization techniques are systematically investigated. The rational function (RF) and direct inversion in the iterative subspace (DIIS) methods are compared and optimized for the purpose of geometry optimization. Various step restriction and line search procedures are tested. The model Hessian recently proposed by Lindh et al. has been used in conjunction with different Hessian update procedures. Optimization for over 30 molecules have been performed in Z-matrix coordinates, local normal coordinates, and curvilinear natural internal coordinates, using the same approximations for the Hessian in all cases. The most effective and stable procedure for optimization of equilibrium structures was found to be the DIIS minimization in natural internal coordinates using the BFGS update of the model Hessian. Our method shows faster overall convergence than all previously published methods for the same test suite of molecules. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1473–1483, 1997

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.

A comparison of geometry optimization with internal, cartesian, and mixed coordinates

Improvements in MO programs and computer speeds have dramatically increased the size of molecules that can be optimized by MO methods. This has put a considerable strain on existing geometry optimization techniques. For a given molecular system, the performance of a geometry optimization method can vary significantly depending on the coordinates used. Nonredundant internal coordinates (e.g., Z-matrix coordinates) are easily constructed and behave well for acyclic molecules but can be quite troublesome for cyclic systems. Cartesian coordinates behave well for rigid, cyclic molecules but can be quite poor for flexible, acyclic systems. Mixed cartesian and internal coordinates have been suggested as a means of combining the best of both approaches and are particularly convenient for cyclic systems with flexible substituents. The efficiency of these different approaches is compared for a number of molecules. © 1992 John Wiley & Sons, Inc.