Redundant Internal Coordinates Research Articles

Multipolar electrostatics based on the Kriging machine learning method: an application to serine

A multipolar, polarizable electrostatic method for future use in a novel force field is described. Quantum Chemical Topology (QCT) is used to partition the electron density of a chemical system into atoms, then the machine learning method Kriging is used to build models that relate the multipole moments of the atoms to the positions of their surrounding nuclei. The pilot system serine is used to study both the influence of the level of theory and the set of data generator methods used. The latter consists of: (i) sampling of protein structures deposited in the Protein Data Bank (PDB), or (ii) normal mode distortion along either (a) Cartesian coordinates, or (b) redundant internal coordinates. Wavefunctions for the sampled geometries were obtained at the HF/6-31G(d,p), B3LYP/apc-1, and MP2/cc-pVDZ levels of theory, prior to calculation of the atomic multipole moments by volume integration. The average absolute error (over an independent test set of conformations) in the total atom-atom electrostatic interaction energy of serine, using Kriging models built with the three data generator methods is 11.3 kJ mol⁻¹ (PDB), 8.2 kJ mol⁻¹ (Cartesian distortion), and 10.1 kJ mol⁻¹ (redundant internal distortion) at the HF/6-31G(d,p) level. At the B3LYP/apc-1 level, the respective errors are 7.7 kJ mol⁻¹, 6.7 kJ mol⁻¹, and 4.9 kJ mol⁻¹, while at the MP2/cc-pVDZ level they are 6.5 kJ mol⁻¹, 5.3 kJ mol⁻¹, and 4.0 kJ mol⁻¹. The ranges of geometries generated by the redundant internal coordinate distortion and by extraction from the PDB are much wider than the range generated by Cartesian distortion. The atomic multipole moment and electrostatic interaction energy predictions for the B3LYP/apc-1 and MP2/cc-pVDZ levels are similar, and both are better than the corresponding predictions at the HF/6-31G(d,p) level.

Redundant internal coordinates, compliance constants and non-bonded interactions – some new insights

A long standing problem in normal mode analysis is identifying the right internal coordinates given only the cartesian coordinates, the masses of the atoms and the cartesian force constants without using any other additional chemical information. A possible solution is suggested here as drawing the normal modes obtained from the mass weighted cartesian force constant matrix and identifying the correct bonds and angles from the normal mode pictures. If chosen properly, the internal coordinates will have minimum mixing in the normal mode representation. This can in principle lead to an automation algorithm. A complete basis of internal coordinates is defined as the minimum number of valence internal coordinates that describe all the normal modes as completely as possible. It was shown in the literature that the relaxed force constants could be used as a measure of bond order in all atom–atom distance coordinates. Some of the bonded and non-bonded atom pairs can have similar values of the relaxed force constants and hence to use the relaxed force constant as a measure of bond order we need to separate the bonded pairs from the non-bonded ones. This needs extra chemical information of which pairs are bonded. The new definition of complete basis of non-redundant valence internal coordinates helps to identify the bonded pairs effectively without extra information. The hydrogen bonded water clusters (H2O)n, n = 2–6, methane dimer and methane–water complex are used as examples to verify that the relaxed force constants of bonded pairs are indeed a measure of bond order. A new definition of a complete basis of valence internal coordinates and a way of choosing them from the normal mode pictures is provided. The meaning of redundancy in redundant internal coordinates is explored. This new definition helps to identify the bonded pairs for which the relaxed force constants are a measure of bond order. Hydrogen bonded clusters are used as examples.

A study on the properties and transferability of force fields in redundant and independent coordinates

The theory of canonical and regular force fields has been used to develop a new approach to study the transferability of force constants in terms of redundant rectilinear internal coordinates. It has been found that the force field of a group of atoms is transferable when it is expressed in canonical form, and that the force field in terms of symmetry-independent coordinates is transferable when these coordinates are orthogonal to the redundancy conditions. As an illustrative example, the approach has been applied to obtain transferable harmonic force constants for the CH 3 group in methane and chloromethane molecules.

Molecular structure of ( Z)-6-[(5-chloro-2-hydroxyphenylamino) methylene]-3-(diethylamino)cyclohexa-2,4-dienone: A combined experimental and theoretical study

The crystal structure and spectroscopic properties of ortho-hydroxy Schiff base compound ( Z)-6-[(5-chloro-2-hydroxyphenylamino)methylene]-3-(diethylamino)cyclohexa-2,4-dienone were determined by X-ray diffraction, IR and UV–Vis spectroscopy techniques. Molecules of the compound exist as NH tautomeric form in solid state. The gas phase geometry optimizations of two possible tautomeric forms of the title compound were achieved using DFT calculations at B3LYP/6-31G(d,p) level of theory. In order to describe the potential energy barrier belonging to the intramolecular proton transfer and to observe the effects of transfer on the molecular geometry, a relaxed potential energy surface (PES) scan was performed based on the optimized geometry of the NH tautomeric form by varying the redundant internal coordinate, N–H bond distance. At the same level of theory, the vibrational frequencies were calculated and compared with the experimental frequencies. The electronic absorption spectra of the tautomers were calculated using time-dependent density functional theory (TD-DFT) method started from solution phase optimized geometries and compared with the experimental ones.

Geometry Optimization with Multilayer Methods Using Least-Squares Minimization.

In this article, we introduce a least-squares minimization scheme for optimizing molecular structures with mixed quantum mechanics (QM) and molecular mechanics (MM) multilayer models. A mixed-coordinate optimization framework was developed. The QM and MM regions are modeled with redundant internal coordinates and Cartesian coordinates, respectively. Within this mixed-coordinate system, a least-squares minimization method using the quasi-Newton step as the evaluation of error is constructed. The couplings between layers are treated rigidly in accordance with the mechanical embedding approach, and the MM Hessian is approximated as a scalar constant of the root-mean-square QM Hessian eigenvalues. Both two-layer and three-layer models were tested. The performance of the method developed herein shows consistently stable and fast convergence.

An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: The use of redundant primitive coordinates

A modified effective scaling frequency factor (ESFF) method that takes advantage of the potential energy distribution (PED) coefficients calculated in the basis of redundant primitive internal coordinates is presented. This approach is simpler and more flexible than that based on the natural internal coordinates. The sets of optimal scaling factors for routine 9- and 11-parameter ESFF calculations based on B3LYP/6-311G∗∗ force fields are derived from Baker’s training set of 30 molecules (660 frequencies). The calculated root-mean-square (RMS) deviations for all frequencies are 11.42 and 11.44 cm −1 for 9- and 11-parameter scaling, respectively. They are somewhat lower than in the case of ordinary ESFF calculations. The new sets of factors seem to be particularly well suited for scaling of frequencies in the middle region of the vibrational spectra (1000–2500 cm −1) – the RMS values in this range are 8.37 for 9-, and 9.56 cm −1 for 11-parameter scaling. These values are to be compared with 9.20 and 10.29 cm −1, respectively, calculated within the natural coordinates based ESFF formalism.

Efficient computation of compliance matrices in redundant internal coordinates from Cartesian Hessians for nonstationary points

We present an extension to the theory of compliance matrices, which is valid for arbitrary nonstationary points on the potential energy hypersurface. It is shown that compliance matrices computed as the inverse of the covariant Hessian matrix obey the same invariance properties with respect to different internal coordinate systems as they do for stationary points. Furthermore, we demonstrate how the computation of compliance matrices in arbitrary sets of redundant internal coordinates starting from a Cartesian Hessian can be achieved efficiently, and we discuss their potential usefullness in geometry optimization processes

The theory of the canonical force field revisited

In a previous Letter, Kuczera’s theory of the canonical force field was generalized to the case of redundant curvilinear internal coordinates related by non-linear constraints. A central issue in this theory is the projection from the space of internal coordinates onto the pure vibrational manifold. In this Letter we present a new mathematical treatment of the projection that solves some problems of inconsistency that have arisen when applying the theory to molecules with a central atom. In addition to solving these problems, this treatment greatly simplifies the attainment of the projection and, consequently, the calculation of canonical force constants.

The proton transfer process observed in the structure analysis and DFT calculations of (E)-2-ethoxy-6-[(2-methoxyphenylimino)methyl]phenol

The crystal and molecular structures of an o-hydroxy Schiff base derivative, (E)-2-ethoxy-6-[(2-methoxyphenylimino)methyl]phenol, have been determined by single crystal X-ray diffraction analyses at 296 and 100 K. The results from temperature-dependent structural analysis regarding the tautomeric equilibrium of the compound were interpreted with the aid of quantum chemical calculations. To clarify the tautomerization process and its effects on the molecular geometry, the gas-phase geometry optimizations of two possible tautomers of the title molecule, its OH and NH form, were achieved using DFT calculations with B3LYP method by means of 6-31 + G(d,p) basis set. In order to describe the potential barrier belonging to the phenolic proton transfer, nonadiabatic Potential Energy Surface (PES) scan was performed based on the optimized geometry of the OH tautomeric form by varying the redundant internal coordinate, O–H bond distance. The Harmonic Oscillator Model of Aromaticity (HOMA) indices were calculated in every step of the scan process so as to express the deformation in the aromaticities of principal molecular moieties of the compound. The results show that there is a dynamic equilibrium between the aromaticity level of phenol and chelate ring and furthermore π-electron coupling affecting overall molecule of the title compound. Charge transfer from phenol ring to pseudo-aromatic chelate ring increases with increasing temperature, whereas π-electron transfer from chelate ring to anisole ring is decreased as temperature increases. The most strength intramolecular H-bonds are observed for conformers close to transition state.

A relook at the compliance constants in redundant internal coordinates and some new insights

The compliance constants, the inverse of the force constant matrix elements offer the advantage that any one of them could be calculated directly without the need to calculate the full Hessian. Here we show that such a direct calculation produces accurate results. Since the diagonal G and the compliance matrix elements (G(ii),C(ii)) give unique values that are independent of the other internal coordinates, an internal frequency (nu) is defined which could be effectively used to study related molecules. The conventional coordinates used for benzene and cubane give some of the compliance constants, which differ from the values obtained by the direct method by 5%-40%, indicating that the coordinate systems used are deficient. We identify the source of this discrepancy and find a way to form the correct coordinate system in the sense that the Hessian method and the direct method will lead to the same compliance constants. This procedure takes away the simplicity of local coordinates at least in highly symmetric molecules and it is advantageous to have the potential energy distribution in redundant internal coordinates. A method is proposed for this purpose. All these imply that the most satisfactory way of describing the molecular force field is in terms of redundant internal compliance constants.

Structural and aromatic aspects for tautomerism of (Z)-6-((4-bromophenylamino)methylene)-2,3-dihydroxycyclohexa-2,4-dienone

The molecular and crystal structure of (Z)-6-((4-bromophenylamino)methylene)-2,3-dihydroxycyclohexa-2,4-dienone were determined by single crystal X-ray diffraction and spectroscopic methods. Molecules of the compound can be regarded as a resonance hybrid of cis-keto tautomer and zwitterionic form. Pairs of molecules of the compound generate pseudocyclic centrosymmetric \( R_{2}^{2} (10) \) supramolecular synthons with the aid of O–H···O type intermolecular H-bonds. Stacking of \( R_{2}^{2} (10) \) synthons along b-axis is stabilized by π···π interactions. Changes in both covalent topology and molecular geometry of the compound accompanying proton transfer were monitored by a relaxed PES scan with respect to hydroxyl bond length used as redundant internal coordinate. Quantum chemical studies at 6-311 + G(d,p) level reveal that bond lengths which are indicative to tautomerization process cannot reach their expected values even if proton transfer occurs in gas phase and pseudo-aromatic chelate ring formation has primary effect on the stabilization of NH tautomer. Resonance-assisted intramolecular H-bond affects the electronic state of its neighboring aromatic fragments.

DL-FIND: An Open-Source Geometry Optimizer for Atomistic Simulations

Geometry optimization, including searching for transition states, accounts for most of the CPU time spent in quantum chemistry, computational surface science, and solid-state physics, and also plays an important role in simulations employing classical force fields. We have implemented a geometry optimizer, called DL-FIND, to be included in atomistic simulation codes. It can optimize structures in Cartesian coordinates, redundant internal coordinates, hybrid-delocalized internal coordinates, and also functions of more variables independent of atomic structures. The implementation of the optimization algorithms is independent of the coordinate transformation used. Steepest descent, conjugate gradient, quasi-Newton, and L-BFGS algorithms as well as damped molecular dynamics are available as minimization methods. The partitioned rational function optimization algorithm, a modified version of the dimer method and the nudged elastic band approach provide capabilities for transition-state search. Penalty function, gradient projection, and Lagrange-Newton methods are implemented for conical intersection optimizations. Various stochastic search methods, including a genetic algorithm, are available for global or local minimization and can be run as parallel algorithms. The code is released under the open-source GNU LGPL license. Some selected applications of DL-FIND are surveyed.

Systematic First Principles Parameterization of Force Fields for Metal−Organic Frameworks using a Genetic Algorithm Approach

A systematic strategy is proposed to derive the necessary force field parameters directly from first principles calculations of nonperiodic model systems to reproduce both the structure and curvature of the reference potential energy surface. The parameters are determined using a genetic algorithm combined with a novel fitness criterion based on a representation of structure and curvature in a set of redundant internal coordinates. Due to the efficiency of this approach it is possible to abandon the need for transferability of the parameters. The method is targeted for the application on metal-organic frameworks (MOFs), where parameters for molecular mechanics force fields are often not available, because of the wide range of possible inorganic fragments involved. The scheme is illustrated for Zn4O-based IRMOF materials on the example of MOF-5. In a "building block" approach parameters are derived for the two model systems basic zinc formate (Zn4O(O2CH)6), and dilithium terephthalate with reference data obtained from density functional theory. The resulting potential gives excellent agreement with the structure, vibrational frequencies, thermal behavior and elastic constants of the periodic MOF-5.

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.

The Gradient Curves Method: An Improved Strategy for the Derivation of Molecular Mechanics Valence Force Fields from ab Initio Data.

A novel force-field development strategy is proposed that tackles the well-known difficulty of parameter correlations arising in a conventional least-squares optimization. In the first step of the new gradient curves method (GCM), continuity criteria are imposed to transform the raw multidimensional ab initio training data to distinct sets of one-dimensional data, each associated with an individual energy term. In the second step, the transformed data suggest suitable analytical expressions, and the parameters in these expressions are fitted to the transformed data; that is, one does not have to postulate a priori analytical expressions for the force-field energy terms. This approach facilitates the derivation of valence terms. Benchmarks have been performed on a set of small molecules. The results show that the new method yields physically acceptable energy terms exactly when a conventional parametrization would suffer from parameter correlations, that is, when an increasing number of redundant internal coordinates is used in the force-field model. The generic treatment of parameter correlations in the proposed method facilitates an intuitive physical interpretation of the individual terms in the force-field expression, which is a prerequisite for the transferability of force-field models.

Empirical Valence-Bond Models for Reactive Potential Energy Surfaces Using Distributed Gaussians.

A new method for constructing empirical valence bond potential energy surfaces for reactions is presented. Building on the generalized Gaussian approach of Chang-Miller, V12(2)(q) is represented by a Gaussian times a polynomial at the transition state and generalized to handle any number of data points on the potential energy surface. The method is applied to two model surfaces and the HCN isomerization reaction. The applications demonstrate that the present method overcomes the divergence problems encountered in some other approaches. The use of Cartesian versus internal or redundant internal coordinates is discussed.

CH-stretching Overtone Spectra of a Fast Rotating Methyl Group: 2-CH3 and 2-CHD2 Pyridines

The CH-stretching overtone spectra of the methyl group in gaseous 2-CH(3) and 2-CHD(2) methylpyridines are recorded with conventional Fourier transform near-infrared spectroscopy in the Deltav(CH) = 1-4 regions and by intracavity laser photoacoustic spectroscopy in the Deltav(CH) = 5 and 6 regions. All spectra exhibit a complex structure. They are analyzed with a theoretical model that incorporates, within the adiabatic approximation, the coupling of the anharmonic CH-stretch vibrations described by Morse potentials with the quasifree internal rotation of the methyl group and with isoenergetic combination states involving the six angle deformation modes of the methyl group. The molecular vibrations are calculated in terms of redundant internal coordinates in an unambiguous canonical form. A simultaneous analysis of different isotopic derivatives is thus achieved. The Fermi resonance coupling parameters are those previously determined for toluene. The technique of diabatic rotations is used to disentangle the multiple avoided crossings occurring along the internal rotation coordinate theta in the calculated spectra, which become rapidly very dense owing to the low symmetry of the system. This simulation is successful in reproducing the experimental spectra. In addition, the transferrability of the Fermi resonance coupling parameters between two parent molecules is demonstrated.

Convergence Properties of the Normal Mode Optimization and Its Combination with Molecular Geometry Constraints

Optimization in the normal mode coordinates has been established as a useful tool for modeling of vibrational spectra (J. Chem. Phys. 2002, 117, 4126). In this work the algorithm is extended with the aid of harmonic penalty functions to allow for multiple restraints of geometry parameters, such as bond lengths, bond and dihedral angles, and for simultaneous optimization of more molecules. Additionally, geometry optimization when atomic nuclei are maintained on the constant electrostatic potential surface was implemented and its applications for solvent models are discussed. Model systems include small molecules, water cluster, antiparallel β-sheet peptide containing intermolecular hydrogen bonds, periodic α-helix and a parallel β-sheet segments. The normal mode method provided better numerical stability than the conventional redundant internal coordinates, especially for weakly hydrogen-bonded systems, while the speed of the optimization was found similar as for the Cartesian coordinates.

Geometry optimization methods for modeling large molecules

Geometry optimization is an essential part of quantum chemical applications. The diversity of the scaling of different methods from linear to exponential implies that there are different requirements for a chosen optimization method. The proposed method aims to meet two requirements, good scaling with size and reliability, which would be a good match for redundant internal coordinate system-based optimization techniques with linear scaling coordinate transformation. The new optimization algorithm uses screened Cholesky decomposition for coordinate transformations and an iterative subspace optimization method. The iterative subspace appears in the course of any optimization. However, few methods are available for using such information efficiently. The Geometry Optimization using Direct Inversion in the Iterative Subspace method is known to have good scaling and efficiency, but poor reliability. Building a Hessian-like matrix in the iterative subspace allows one to take advantage of the reliability offered by Rational Function Optimization, Eigenvector Following and Trust Radius Method (TRM), but still avoid a consequent computational penalty. Also, the new approach steps away from the regular quadratic approximation related to the Newton methods by assuming a simple linear connection between gradient and coordinate changes.

Transferability and Physicochemical Interpretation of Canonical Force Fields in Redundant Internal Coordinates: Pyridazine and 3,6-Dichloropyridazine

The behavior of various force fields for pyridazine and 3,6-dichloropyridazine coming from different origins (i.e., HF/6-31G*, MP2/6-311G**, BLYP/6-31G*, B3LYP/6-31G* previously scaled) and empirical force fields transferred from the benzene molecule is investigated. These potentials are subjected to a linear least-squares refinement with the available experimental information considered as observed data. Moreover, the transferability properties of the force fields for these related molecules are analyzed. From the refined force fields which gave the most satisfactory results in the fitting procedure, the physicochemical meaning of the force constants in terms of redundant simple internal coordinates in the unambiguous canonical form is studied. Although the results are not completely decisive for the pyridazine series, significant conclusions are drawn pertaining to their force fields, some of which are calculated for the first time. First, the utility of canonical force fields is shown through the force...