Geometry optimization of crystals by the quasi-independent curvilinear coordinate approximation

The quasi-independent curvilinear coordinate approximation (QUICCA) method [K. Nemeth and M. Challacombe, J. Chem. Phys. 121, 2877 (2004)] is extended to the optimization of crystal structures. We demonstrate that QUICCA is valid under periodic boundary conditions, enabling simultaneous relaxation of the lattice and atomic coordinates, as illustrated by tight optimization of polyethylene, hexagonal boron nitride, a (10,0) carbon nanotube, hexagonal ice, quartz, and sulfur at the Gamma-point RPBE/STO-3G level of theory.     

The generator coordinate approximation for molecules: A review

The generator coordinate approximation is a non-adiabatic theory of molecular systems. Its fundamental outlines were developed during the 1970's. A further analysis and first applications were published during the 1980's. In this paper, we review the present status of the theory.     

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.     

An algorithm for geometry optimization without analytical gradients

A numerical algorithm for locating both minima and transition states designed for use in the ab initio program package GAUSSIAN 82 is presented. It is based on the RFO method of Simons and coworkers and is effectively the numerical version of an analytical algorithm (OPT = EF) previously published in this journal. The algorithm is designed to make maximum use of external second derivative information obtained from prior optimizations at lower levels of theory. It can be used with any wave function for which an energy can be calculated and is about two to three times faster than the default DFP algorithm (OPT = FP) supplied with GAUSSIAN 82.     

Projection operator method for geometry optimization with constraints

A new approach is presented for performing geometry optimization for stationary points on potential energy hypersurfaces with equality constraints on the internal coordinates of a polyatomic system. The working equations are the same as for unconstrained Newton–Raphson optimization in Cartesian coordinates except that projection operators are applied to the gradient and Hessian to enforce the constraints. Two reactive systems with different kinds of constraints are treated as examples: OH + H₂ → OH → H₂O + H with one constrained OH bond distance and CH₃ + H₂ → CH → CH₄ + H with one constrained H? C? H bond angle in the CH₃ group or with one constrained bond distance and one simultaneously constrained bond angle. In each case we optimized all reactants and products as well as the saddle point, all subject to the constraints.     

Chemical shift driven geometry optimization

A new method for refinement of 3D molecular structures by geometry optimization is presented. Prerequisites are a force field and a very fast procedure for the calculation of chemical shifts in every step of optimization. To the energy, provided by the force field (COSMOS force field), a pseudoenergy, depending on the difference between experimental and calculated chemical shifts, is added. In addition to the energy gradients, pseudoforces are computed. This requires the derivatives of the chemical shifts with respect to the coordinates. The pseudoforces are analytically derived from the integral expressions of the bond polarization theory. Single chemical shift values attributed to corresponding atoms are considered for structural correction. As a first example, this method is applied for proton position refinement of the D-mannitol X-ray structure. A crystal structure refinement with 13C chemical shift pseudoforces is carried out.     

Mixing parameters for geometry optimization using the Hamiltonian algorithm

We study the mixing parameters for the search of an optimal geometry using the Hamiltonian algorithm (HA) combined with ab initio molecular orbital calculations. We choose the C?CC?CC?CC dihedral angle of the butane molecule as an example. HF/3-21G level calculations are employed as the molecular orbital calculations. The distributions of the eigenvalues of mixing coefficients are fitted with the linear, quadratic, and quartic functions. Analyses of HA calculations both up to 2,000 and 60,000 iterative calculations show a possibility that the mixing process reduces the number of iterations. The low energy HF/3-21G, B3LYP/6-31G**, and PCM B3LYP/aug-cc-pVDZ optimized structures of the N-acetyl l-histidine N??-methyl amide and four water molecule supermolecule were also determined using the HA optimization method and compared to the recently determined thought to be global minimum energy structure.     

Ab-initio molecular geometry and normal coordinate analysis of pyrrolidine molecule.

The molecular geometry of pyrrolidine was quantum mechanically calculated using the split valence 6-31G** basis set. Electron correlation energy has been computed employing MP2 method. The molecule showed an envelope form puckered structure with inter-plane angle of 36.4 degrees and has a total energy of -132976.80 kcal mol(-1) of which a -464.86 kcal mol(-1) electron correlation energy. The twist form of the molecule showed a twist angle of 10.2 degrees from planarity and has a total energy of -132976.05 kcal mol(-1) involving -464.097 kcal mol(-1) electron correlation energy. The normal coordinates of the molecule were theoretically analyzed on the basis of the Cs point symmetry of the envelope form. Using initial set of force constants obtained from the ab-initio calculations the fundamental vibrational frequencies were computed. The IR and laser Raman spectra of Pyrrolidine molecules were measured. All the observed vibrational bands including combination bands and overtones were assigned to normal modes with the aid of the potential energy distribution values obtained from normal coordinate calculations. The molecular force field was obtained by refining the initial set of force constants using the least square fit method. The molecular force field was determined by refining the initial set of force constants using the least square fit method instead of using the less accurate scaling factor methods. The determined molecular force field has produced simulated frequencies best match to the observed values. The low frequency molecular out-of-plane deformation modes were observed in both infrared and Raman spectra at 298 and 163 cm(-1). The barrier of ring twisting estimated from the observed ring out-of-plane vibrational mode at 163 cm(-1) was found 3.1 kcal mol(-1).     

Ab-initio molecular geometry and normal coordinate analysis of tetrahydrothiophene molecule.

The molecular geometry of tetrahydrothiophene (THT) was quantum mechanically calculated using the split valence 6-31G** basis set. Electron correlation energy has been computed employing MP2 method. The molecule showed a twist form puckered structure with a twist torsion angle of 13 degrees and has a total energy of -347,877.514 kcal/mol of which a 436.715 kcal/mol electron correlation energy. The envelope form of the molecule showed an inter-plane angle of 22 degrees and has a total energy of -347,874.430 kcal/mol involving -436.558 kcal/mol electron correlation energy. The normal coordinates of the molecule were theoretically analyzed and the fundamental vibrational frequencies were calculated. The IR and laser Raman spectra of THT molecule was measured. All the observed vibrational bands including combination bands and overtones were assigned to normal modes with the aid of the potential energy distribution values obtained from normal coordinate calculations. The molecular force field was determined by refining the initial set of force constants using the least square fit method instead of using the less accurate scaling factor methods. The determined molecular force field has produced simulated frequencies which best match the observed values. The lowest-energy modes of vibration were two molecular out-of-plane deformations, observed at 114 and 166 cm(-1). The barrier of ring twisting estimated from the observed ring out-of-plane vibrational mode at 114 cm(-1) was estimated.     

Development of an efficient geometry optimization method for water clusters

A geometry optimization method for water clusters (H(2)O)(n) was developed in the present study. The method was applied to the TIP3P and TIP4P water clusters in the range of n < or = 30, and the resulting structures were compared with the global-minimum structures in the literature (n < or = 25 for the TIP3P potential and n < or = 30 for the TIP4P potential). The method failed to reproduce the previously reported global minimum of the n = 24 TIP4P cluster. However, it was possible to find new global minima for the n = 24, 26-30 TIP3P cluster and the TIP4P clusters of 25, 28, 29, and 30 molecules.     

Quantum dynamics of the CH3 fragment: a curvilinear coordinate system and kinetic energy operators

A curvilinear coordinate system for AB(3) fragments is given. The corresponding exact kinetic energy operator is derived and a series of simpler, progressively more approximate kinetic energy operators are suggested. The operators are tailored for quantum dynamics simulations using the multiconfigurational time-dependent Hartree approach. It is outlined how these fragment coordinates can be utilized to set up coordinate systems for larger systems such as AB(3)C or AB(3)CD. Calculations of the vibrational levels of CH(3) and quantum dynamics studies investigate the accuracy of the different kinetic energy operators suggested.     

Assignment of electronic transitions by geometry optimization

Adiabatic excitation energies, excited state geometries, excited state charges, bond orders and dipole moments have been obtained for HCN, CO₂,H₂CO, HFCO, F₂CO, ethylene, trans-butadiene, furan, pyrrole and uracil using the SINDO1 semi-empirical method with configuration interaction. Our results generally agree with those ofab initio calculations and experiment satisfactorily. Geometry optimization is found to mix configurations differing in their allowedness in vertical excitation from the ground state, which in turn helps in the assignment of spectral transitions. TheV excited singlet state of trans-butadiene and various excited states of furan, pyrrole and uracil have been found to be appreciably non-planar. The single and double CC bonds are found to exchange positions due to the lowest triplet and singlet transitions of furan and pyrrole. The first triplet and first singlet transitions of uracil have been found to be of π-π* and π-σ* types respectively in agreement with recent experimental findings. On leave of absence from the Department of Physics, Banaras Hindu University, Varanasi-221005, India     

Isomerization of stilbene using enforced geometry optimization

Using our recently proposed quantum chemical model to simulate the effect of external forces acting on a molecule (Wolinski and Baker, Mol Phys 2009, 107, 2403), which we subsequently termed enforced geometry optimization (EGO), we investigate structural isomerism in C₁₄H₁₂, starting from cis‐stilbene. By applying an external force to pairs of carbon atoms, one from each “half” of the molecule, we have generated 10 different structural isomers. Each was characterized as a minimum by vibrational analysis. Not only can EGO generate potentially new, metastable isomers it can also provide good initial guesses for transition states connecting the starting and final structures, thus giving an estimate of the stability of the new isomers to rearrangement back to the starting material. In addition to the new isomers, we provide a full set of vibrational fundamentals for cis‐ and trans‐stilbene and 4a,4b‐dihydrophenanthrene. The agreement with experimental assignments is excellent, with mean average deviations for the stilbenes of 5.0 cm^?1 or less. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Shape optimization of a cruciform geometry for biaxial testing of polymers

The presented literature review of cruciform shapes used for biaxial characterization of materials indicates that the majority of shapes can be divided into two large groups when the following selection criteria are taken into consideration: (i) the shape of the outer boundaries and (ii) the load capacity needed to achieve failure in the biaxial region. Manipulation of the outer shape boundaries appears to be essential to bundle the applied loads to the central zone where failure is intended to be built up. For each group, one particular cruciform design is reported whereby the outer boundaries are based on a single curved shape. Although the use of discontinuous double radii edges should be avoided according to earlier reports [1,2], it is shown here through the construction of an optimization algorithm, that the use of a single curve for the outer boundaries leads to strains in the arms that are strongly dependent on these single curved edges. Numerical simulations based on the finite element method as well as experiments performed on polymeric test pieces in combination with DIC measurements, show good agreement on this matter and demonstrate this sensitivity very clearly.     

On analytical derivatives for geometry optimization in the polarizable continuum model

The present paper is dedicated to the analytical computation of shape derivatives in the polarizable continuum model. We derive expressions for the interaction energy’s sensitivity with respect to variations of the cavity’s shape by means of the Hadamard representation of the shape gradient. In particular, by using the adjoint approach, the shape gradient depends only on two solutions of the underlying electrostatic problem. We further formulate boundary integral equations to compute the involved quantities.     

Square planar vs tetrahedral geometry in four coordinate iron(II) complexes

The geometric preferences of a family of four coordinate, iron(II) d6 complexes of the general form L2FeX2 have been systematically evaluated. Treatment of Fe2(Mes)4 (Mes = 2,4,6-Me3C6H2) with monodentate phosphine and phosphite ligands furnished square planar trans-P2Fe(Mes)2 derivatives. Identification of the geometry has been accomplished by a combination of solution and solid-state magnetometry and, in two cases (P = PMe3, PEt2Ph), X-ray diffraction. In contrast, both tetrahedral and square planar coordination has been observed upon complexation of chelating phosphine ligands. A combination of crystallographic and magnetic susceptibility data for (depe)Fe(Mes)2 (depe = 1,2-bis(diethylphosphino)ethane) established a tetrahedral molecular geometry whereas SQUID magnetometry and M?ssbauer spectroscopy on samples of (dppe)Fe(Mes)2 (dppe = 1,2-bis(diphenylphosphino)ethane) indicated a planar molecule. When dissolved in chlorinated solvents, the latter compound promotes chlorine atom abstraction, forming tetrahedral (dppe)Fe(Mes)Cl and (dppe)FeCl2. Ligand substitution reactions have been studied for both structural types and are rapid on the NMR time scale at ambient temperature.     

Berry exchange coordinate geometry in 3-methyl-2-hydroxycyclopenten-1-one tin esters

Three new penta- and hexacoordinated tin compounds (1-3) were prepared from PhSnCl₃, Ph₂SnCl₂ and Ph₃SnOH and 3-methyl-2-hydroxy-2-cyclopenten-1-one (L). Compounds 1-3 were characterized by IR, mass spectra, elemental analysis, ¹H, ¹³C, and ¹¹⁹Sn NMR. The ligand acts as a bidentate giving the tin esters and coordinating the tin by the carbonyl group. Compound 1 (PhSnCl₂L · EtOH) has an hexacoordinated tin atom, with an octahedral distorted geometry, which is a stereogenic center. Compounds 2 (Ph₂SnClL) and 3 (Ph₃SnL) have pentacoordinated tin atoms. The structures were determined by X-ray diffraction analyses. In the solid state 1 presents a racemic pair, linked by strong hydrogen bonds and 2 and 3 “Berry exchange coordinate” geometry.