Normal mode calculation and IR band assignments of A-type zeolite

Normal mode calculation of A-type zeolite was carried out with the potential energy functions obtained from the constraint method. Mass-weighted cartesian coordinates and the pseudo-lattice method were used. The assignments of IR absorption bands were made with the calculated normal modes, by using the calculated absorption intensities of the modes and the degrees of contribution of the internal coordinates to the modes. The force constants of internal coordinate motions within the framework were also calculated and are compared with the empirical values.     

The use of internal coordinates in the variable metric method of molecular geometry optimization

A computational algorithm for the variable metric method of molecular geometry optimization using internal instead of cartesian coordinates is presented. The greater efficiency attainable using internal coordinates is shown using ethylene and methanol as examples. A high degree of accuracy in determining bond lengths and angles was achieved even when, as in the case of some ethers studied, the resulting equilibrium structures were essentially different from the initial ones constructed from experimental data.     

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.     

Force Constant Calculation in Mass-Weighted Cartesian Coordinates. Linear XYX and XYZ Type Molecules

Force constants of simple linear XYX and XYZ types of molecules have been calculated in using the method of mass-weighted cartesian coordinates. Relations and results between force constants in mass-weighted cartesian coordinates and in internal coordinates were also discussed.     

Calculation of anharmonic force constants

Non-linear transformations from internal to cartesian displacements are considered for anharmonic calculation of vibrational frequencies. Force constants in normal coordinates, up to quartic terms, are related to Christoffel symbols. The latter are tabulated for valence angles or, for torsion and wagging out of plane, evaluated through expressions which avoid differentiation of explicit functional forms for each type of internal coordinate. The same symbols are used for cartesian tensors and redundancy coefficients up to the third order.     

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.     

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.     

Finite vibrational displacements in the Eckart—Sayvetz basis

Curvilinear internal coordinates are considered in terms of cartesian displacements in a molecule-fixed basis determined by the Eckart-Sayvetz conditions. The latter are interpreted as a set of restrictions on the metrics of the space and define cartesian displacements of “pure” vibrational character expanded to any order in terms of internal coordinates. Explicit expressions for expansion coefficients are given as a function of contravariant components of the metric tensor taken from existing table. A compact notation is proposed for anharmonic force constants, expansion coefficients of redundancies and coupling terms of the rotation—vibration hamiltonian.     

A quartic force field coordinate substitution scheme using hyperbolic sine coordinates

Quartic force fields (QFF) are currently the most cost‐effective method for the approximation of potential energy surfaces for the calculation of anharmonic vibrational energies. It is known, although, that its performance can be less than satisfactory due to limitations related to slow convergence of the series. In this article, we present a coordinate substitution scheme using a combination of Morse and sinh coordinates, well adapted for its use with cartesian normal coordinates. We derive expressions for analytical integrals for use in VSCF and VCI calculations and show that the simultaneous substitution of symmetric and antisymmetric normal coordinates by Morse and sinh coordinates, respectively, significantly improves the vibrational transition frequencies for these modes in a well‐balanced fashion. The accuracy of this substitution scheme is demonstrated by comparing one and two‐dimensional sections of substituted and unsubstituted QFF with ab initio potential energy grids, as well as with vibrational energy calculations using as test cases two well‐studied benchmark molecules: water and formaldehyde. We conclude that the coordinate substitution scheme presented constitutes a very attractive alternative to simple QFFs in the context of cartesian normal coordinates.     

The quasi-independent curvilinear coordinate approximation for geometry optimization

This paper presents an efficient alternative to well established algorithms for molecular geometry optimization. This approach exploits the approximate decoupling of molecular energetics in a curvilinear internal coordinate system, allowing separation of the 3N-dimensional optimization problem into an O(N) set of quasi-independent one-dimensional problems. Each uncoupled optimization is developed by a weighted least squares fit of energy gradients in the internal coordinate system followed by extrapolation. In construction of the weights, only an implicit dependence on topologically connected internal coordinates is present. This new approach is competitive with the best internal coordinate geometry optimization algorithms in the literature and works well for large biological problems with complicated hydrogen bond networks and ligand binding motifs.     

A simple method for derivation of the rovibrational Hamiltonian: application to orthogonal radau coordinate systems as special cases

The molecular Hamiltonian of polyatomic molecules has been obtained. A general choice of internal coordinates depending on external parameters was considered. The rovibrational Hamiltonian for this set of coordinate system was derived in general terms as a function of the external parameters a and b. This procedure is also applicable to various kinds of internal coordinates in a straightforward way. The rovibrational Hamiltonian of triatomic molecules is considered as an application of this general formulation. In addition, orthogonal Radau coordinates are considered as cases of this new approach     

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     

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.     

First-principle calculation of reduced masses in vibrational analysis using generalized internal coordinates: some crucial aspects and examples

In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.     

Generalized approximation to the reaction path: the formic acid dimer case

A set of mass-weighted internal coordinates was derived and applied to the double proton transfer reaction in the formic acid dimer (FAD). The coordinate set was obtained starting from the Hirschfelder "mobile" by an optimization procedure consisting of a sequence of kinematic rotations. In FAD, the optimization procedure leads to three coordinates that do change significantly along the reaction path. These coordinates span the reaction space, whereas the remaining modes are treated in a harmonic approximation. The effect that the dimer dissociative motion has on the ground and excited vibrational states dynamics was explored. In the frequency region corresponding to the symmetric OH-stretch vibration four doublets have been identified with splittings of 2.76, 0.07, 0.60, and 4.03 cm(-1).     

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.     

A modular strategy for generating starting conformations and data structures of polynucleotide helices for potential energy calculations

We describe a simple and rapid algorithm for generating data structures and starting coordinates of polynucleotides for potential energy calculations. The algorithm is tailored to investigations in cartesian coordinate, rather than dihedral angle, space. First, instead of a tree structure for molecular design, we set up a helix from a simple list of bonds for the basic DNA subunits (sugar, phosphate, and bases). Second, instead of using successive transformations to obtain a set of coordinates in one reference frame, we apply a simple “matching” routine to patch DNA subunits. Third, we avoid ring closure and geometry optimization by allowing deviations from equilibrium values only for P? O3′ bond lengths and O5′? P? O3′ bond angles at the residue connection sites. A double-stranded helix is constructed from duplex building blocks (2 hydrogen-bonded nucleotides) which are in turn built from the basic structural units. Every building block is constructed from two sets of geometric variables: {α, β, γ, χ, P, τ_max}, one for each strand. The building blocks are then assembled into a helix by using the 6 rigid body transformations {Δx, Δy, Δz, Θ_ROLL , Θ_TILT , Θ_TWIST }. For cartesian space programs, generating starting coordinates by this procedure is particularly useful as an alternative to using actual crystal structure coordinates. After describing the algorithm in detail, we illustrate how it was used to generate model A, B, and Z DNA helices. We conclude by suggesting how the algorithm can be used to pursue a build-up technique and to set up a wide range of starting conformations in the goal of locating novel helical structures.     

Group theory and normal coordinate calculations

A method is proposed to obtain a set of computer generated symmetry adapted basis vectors which completely factorize a matrix associated with any operator which commutes with all the symmetry operations of a molecule. The cartesian coordinates of all atoms are used to derive the irreducible representations of the appropriate molecular point group and the transformation properties of any starting set of coordinates. Symmetry coordinates are then projected out each irreducible representation.     

Global optimization of binary Lennard-Jones clusters using three perturbation operators

Global optimization of binary Lennard-Jones clusters is a challenging problem in computational chemistry. The difficulty lies in not only that there are enormous local minima on the potential energy surface but also that we must determine both the coordinate position and the atom type for each atom and thus have to deal with both continuous and combinatorial optimization. This paper presents a heuristic algorithm (denoted by 3OP) which makes extensive use of three perturbation operators. With these operators, the proposed 3OP algorithm can efficiently move from a poor local minimum to another better local minimum and detect the global minimum through a sequence of local minima with decreasing energy. The proposed 3OP algorithm has been evaluated on a set of 96 × 6 instances with up to 100 atoms. We have found most putative global minima listed in the Cambridge Cluster Database as well as discovering 12 new global minima missed in previous research.