Geometry optimization of metal complexes using natural internal coordinates: Representation of skeletal degrees of freedom

The geometry optimization using natural internal coordinates was applied for transition metal complexes. The original definitions were extended here for the skeletal degrees of freedom which are related to the translational and rotational displacements of the ηⁿ-bonded ligands. We suggest definitions for skeletal coordinates of ηⁿ-bonded small unsaturated rings and chains. The performance of geometry optimizations using the suggested coordinates were tested on various conformers of 14 complexes. Consideration was given to alternative representations of the skeletal internal coordinates, and the performance of optimization is compared. Using the skeletal internal coordinates presented here, most transition metal complexes were optimized between 10 and 20 geometry optimization cycles in spite of the usually poor starting geometry and crude approximation for the Hessian. We also optimized the geometry of some complexes in Cartesian coordinates using the Hessian from a parametrized redundant force field. We found that it took between two and three times as many iterations to reach convergence in Cartesian coordinates than using natural internal coordinates. © 1997 by John Wiley & Sons, Inc.     

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.     

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.     

Using redundant internal coordinates to optimize equilibrium geometries and transition states

A redundant internal coordinate system for optimizing molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between bonded atoms. Redundancies are removed by using the generalized inverse of the G matrix; constraints can be added by using an appropriate projector. For minimizations, redundant internal coordinates provide substantial improvements in optimization efficiency over Cartesian and nonredundant internal coordinates, especially for flexible and polycyclic systems. Transition structure searches are also improved when redundant coordinates are used and when the initial steps are guided by the quadratic synchronous transit approach. © 1996 by John Wiley & Sons, Inc.     

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 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.     

Green's Function Determination of Force Constants-H2O as Example

Treating isotopically substituted molecule as a perturbed system, Green's function for the perturbation are constructed and related to the force field of vibration. By spectral representation, Green's function is diagonalized in the normal coordinates. Then transforming back to the Cartesian coordinates, the Cartesian force constants are generated without solving the secular equation directly. The relations between the internal force constants and the Cartesian force constants ate given and complete internal force field can be obtained. The results for H₂O are discussed.     

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.     

Calculation of intramolecular force fields from second-derivative tensors

A practical procedure (FUERZA) to obtain internal force constants from Cartesian second derivatives (Hessians) is presented and discussed. It allows a systematic analysis of pair atomic interactions in a molecular system, and it is fully invariant to the choice of internal coordinates of the molecule. Force constants for bonds or for any pair of atoms in general are defined by means of the eigenanalysis of their pair interaction matrix. Force constants for the angles are obtained from their corresponding two-pair interaction matrices of the two bonds or distances forming the angle, and the dihedral force constants are similarly obtained using their corresponding three-pair interaction matrices. © 1996 John Wiley & Sons, Inc.     

Automatic geometry optimization for molecules with d orbitals. I. General analytic formulas for derivatives of slater orbitals

General analytic formulas for derivatives of real Slater orbitals with respect to the Cartesian coordinates x and y have been derived. The application of this result to molecular geometry optimization or to the construction of a force constant matrix is briefly discussed.     

Ab initio calculations of the ultraviolet resonance Raman spectra of uracil

An equation been derived to calculate, ab initio, the frequencies and intensities of a resonant Raman spectrum from the transform theory of resonance Raman scattering. This equation has been used to calculate the intensities of the ultraviolet resonance Raman spectra from the first π-π* excited state of uracil and 1,3-dideuterouracil. The protocol for this calculation is as follows: (1) The force constant matrix elements in Cartesian coordinate space, the vibrational frequencies, and the minimum energy ground and excited state geometries of the molecule are calculated ab initio using the molecular orbital program Gaussian 92, (2) the force constants in Cartesian coordinates are transformed into force constants in the space of a set of 3N – 6 nonredundant symmetrized internal coordinates, (3) the G matrix is constructed from the energy minimized ground state Cartesian coordinates and the GFL = LΛ eigenvalue equation is solved in internal coordinate space, (4) the elements of the L and L^?1 matrices are calculated, (5) the changes in all of the internal coordinates in going from the ground to the excited state are calculated, and (6) these results are used in combination with the transform theory of resonance Raman scattering to calculate the relative intensities of each of the 3N – 6 vibrations as a function of the exciting laser frequency. There are no adjustable parameters in this calculation, which reproduces the experimental frequencies and intensities with remarkable fidelity. This indicates that the Dushinsky rotation of the modes in the excited state of these molecules is not important and that the simplest form of the transform theory is adequate. © 1995 John Wiley & Sons, Inc.     

Ab Initio Equilibrium Geometry and Vibrational Spectroscopic Study of 25,26,27,28-tetrahydroxycalix[4]arene

The aim of this work was to present a comprehensive vibrational spectroscopic study of 25,26,27,28-tetrahydroxycalix[4]arene. For this purpose, quantum chemical calculations were carried out at the ab initio HF/4-31G* level, as a consequence of the great size of the molecule. In the frame of these calculations, the symmetry of the molecule was investigated. Trying C 4v , C 2v , and C s symmetries as input, the geometry optimization, however, pointed to the C 2 configuration. In the latter case, all the calculated vibrational frequencies were greater than zero, and therefore the equilibrium geometry could be identified. Medium and far infrared as well as Raman spectra of the compound were measured. On the basis of the calculated force constants and geometric parameters, normal coordinate analysis was applied for the interpretation of the experimental vibrational spectra. Problems arose with the choice of the internal coordinates of the molecule, which are important from the point of view of the internal macrocycle ("lower rim") of the molecule. On the basis of the theory of redundant coordinates, a program was written for choosing the coordinates of this 16-membered cycle. Full interpretation of the vibrational fundamentals of the compound is presented. Several force constant matrix elements have surprisingly high values. As a result of the normal coordinate analysis, the relative rigidity of the lower rim was concluded. Ab initio calculations and assignment of the vibrational spectra of 25,26,27,28-tetrahydroxycalix[4]arene based on the calculations are presented.     

Formulation of the vibrational theory in terms of redundant internal coordinates

Using the generalized inverse of matrices and the method of canonical matrices, this paper develops an unambiguous formulation of the theory of small vibrations of molecules which is valid when redundant internal coordinates are used. Such treatment includes the attainment of unambiguous relationships relating the compliance matrix (the generalized inverse of the canonical force constants matrix) with experimental data (vibrational frequencies, Coriolis coupling constants, centrifugal distortion constants and mean-square amplitudes). Moreover, expressions for the elements of the Jacobian matrices of the above magnitudes with respect to the compliance constants have been also obtained as well as some sum rule relationships.     

Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3ⁿ redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θ_pqr are given by introducing Cartesian tensor rotation matrix elements X( R ). A polynomial expression and a recursion relation for X( R ) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D( R ). The expressions for X( R ) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.     

ISOEFF98. A program for studies of isotope effects using Hessian modifications

A new program for calculations of isotope effects has been developed. It requires only force constants for substrate and transition state as the external data. All other calculational steps are integrated into the program. ISOEFF98 has features of Hessian modification and scale factor optimization. The first of these allows studies of isotope effect changes upon weakening or strengthening of internal coordinates. The second feature allows fitting of the calculated isotope effect to the experimental value by scaling of molecular frequencies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Constrained optimization in delocalized internal coordinates

Using the recently introduced delocalized internal coordinates, in conjunction with the classical method of Lagrange multipliers, an algorithm for constrained optimization is presented in which the desired constraints do not have to be satisfied in the starting geometry. The method used is related to a previous algorithm by the same author for constrained optimization in Cartesian coordinates [J. Comput. Chem., 13 , 240 (1992)], but is simpler and far more efficient. Any internal (distance or angle/torsion) constraint can be imposed between any atoms in the system whether or not the atoms involved are formally bonded. Imposed constraints can be satisfied exactly. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1079–1095, 1997