Separated internal force constants for the ethylene and ethane molecules and their use for calculation of the force field for the propylene molecule

We consider the question of separation of linear combinations of force constants for ethylene and ethane. Introduction of a perturbation into the matrix of the kinematic coefficients allows us to solve the inverted vibrational problem using the matrix method of successive approximations without eliminating dependent coordinates. Such an approach makes it possible to obtain a sufficient system of equations for determining the separated internal force constants. The separated internal force constants determined for ethylene and ethane are used to calculate the force field for propylene. The calculated separated internal force constants for propylene reproduce its vibrational spectrum and the spectrum or propylene-d₆ with average deviation from experimental frequencies of 8 cm^–1. The numerical influence coefficients for stretching vibrations of the C-H bond are linearly related to the lengths of these bonds.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 22, No. 1, pp. 118–122, Janauary–February, 1986.     

ICFF: a new method to incorporate implicit flexibility into an internal coordinate force field

We introduce a new method to accurately "project" a Cartesian force field onto an internal coordinate molecular model with fixed-bond geometry. The algorithm automatically generates the Internal Coordinate Force Field (ICFF), which is a close approximation of the "source" Cartesian force field. The ICFF method reduces the number of free variables in a model by at least 10-fold and facilitates the fast convergence of geometry optimizations, an advantage that is critical for many applications such as the docking of flexible ligands or conformational modeling of macromolecules. Although covalent geometry is fixed in an ICFF model, implicit flexibility is incorporated into the force field parameters in the following two ways. First, we formulate an empirical torsion energy term in ICFF as a sixfold Fourier series and develop a procedure to calculate the Fourier coefficients from the conformational energy profiles of the fully flexible Cartesian model. The ICFF torsion parameters thus represent not only torsion component of the source force field, but also bond bending, bond stretching, and "1-4" van der Waals interactions. Second, we use a soft polynomial repulsion function for "1-5" and "1-6" interactions to mimic the flexibility of bonds, connecting these atoms. Also, we suggest a way to use a local part of the Cartesian force field to automatically generate fixed covalent geometries, compatible with the ICFF energy function. Here, we present an implementation of the ICFF algorithm, which employs the MMFF94s Cartesian force field as a "source." Extensive benchmarking of ICFF with a representative set of organic molecules demonstrates that the implicit flexibility model accurately reproduces MMFF94s equilibrium conformational energy differences (RMSD approximately 0.64 kcal) and, most importantly, detailed torsion energy profiles (RMSD approximately 0.37 kcal). This accuracy is characteristic of the method, because all the ICFF parameters (except one scaling factor in the "1-5,1-6" repulsion term) are derived directly from the source Cartesian force field and do not depend on any particular molecular set. In contrast, the rigid geometry model with the MMFF94s energy function yields highly biased estimations in this test with the RMSD exceeding 1.2 kcal for the equilibrium energy comparisons and approximately 3.4 kcal for the torsion energy profiles.     

Treatment of redundancies among internal coordinates in optimizing molecular mechanics force constants

The use of redundant coordinate bases in the construction of molecular mechanics force fields is discussed. It is shown that the intrinsic indeterminacy in a force field in redundant coordinates in general stems from the squares of the first-order redundancy relations. The necessity to use constraints in such a force field is pointed out, and a method to check whether or not a set of constraints makes the force field determinate is described. It is also explained how force fields corresponding to different sets of constraints can be transformed into one another. To facilitate the utilization of ab initio or other spectroscopic force fields, a procedure is given by which force constants pertaining to a nonredundant coordinate basis can be optimized in molecular mechanics calculations where redundant coordinates are used. © 1992 by John Wiley & Sons, Inc.     

Constrained force constants for octahedral molecules

A set of constrained force constants has been derived from experimental vibrational frequency data for eighty three octahedral molecules. Superimposing the condition that the larger value for f_d—f_dd′ be used when f_dα—f_dα″ is a maximum on the six equations relating vibrational frequencies to force constants generates a seventh. This provides a uniform set of results for all 83 molecules. The values of the force constants have a simple rationale in terms of chemical bonding theory. Some preliminary calculations for SF₆ show that these force constants are suitable for use in generating reliable molecular dynamical trajectory data.     

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.     

Use of the virial theorem to calculate molecular force constants. Harmonic force constants of the water molecule

The quantum mechanical virial theorem is used to calculate the harmonic part of the potential function for the water molecule on the basis of wave functions obtained in the self-consistent field approximation.     

Calculation of macromolecular force constants

Force constant refinement may be carried under a system of constraints imposed on well characterized modes as the refinement proceeds. Resulting stability permits the accurate calculation of many independent macromolecular force constants. A useful formalism for this technique is developed. Prediction of initially poorly characterized modes can emerge from this stability. A brief discussion of currently used refinement schemes has been included. We show that the standard least-squares procedure is a special case of our present more general method, the former being useful when the molecule is not too large.     

Ab initio MO calculation of force constants and dipole derivatives for the formamide dimer. An estimation of hydrogen-bond force constants

Ab initio SCF MO calculations (with the 4-31G basis set) have been carried out to determine the equilibrium geometry, vibrational frequencies, dipole-moment derivatives, and force constants for intermolecular modes of the formamide dimer and its d₄ and d₆ derivatives. The results are correlated with monomer calculations and experimental data for crystalline formamide.     

Constrained variations for diatomic forces and force constants

The results of earlier studies of diatomic forces and force constants are used to formulate force constraints. It is proposed that such constrained variations should improve force constant calculations.

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Zusammenfassung Mittels früherer Ergebnisse für Bindungskräfte bei zweiatomigen Molekülen werden Neben-bedingungen für die Wellenfunktion formuliert. Es ist anzunehmen, da\ dies zu einer Verbesserung der Berechnung für die entsprechenden Kraftkonstanten führt.

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Résumé Les résultats d'études antérieures des forces et des constantes de force diatomiques sont utilisés pour formuler des contraintes de force. De telles variations contraintes devraient améliorer les calculs de constante de force.

     

Ab initio computation of force constants: Part VI. Applications of the force relaxation method for geometry optimization

Techniques to improve the computational efficiency of the force relaxation method are discussed. Force constants for fragments in previously computed smaller molecules can be transferred to construct a guess force constant matrix. Additional force constants that may be needed can be computed by a procedure which uses only one additional force calculation per diagonal force constant required. A scaling technique to improve convergence on the optimized geometry is discussed.     

Calculation of molecular vibrations: Selective scaling factors for semiempirical force constants

The complete force constant matrices of a set of 50 aliphatic and aromatic hydrocarbons are calculated at the density functional theory B3LYP/6–31+G(d, p) and semiempirical PM3 levels of theory. After transformation from Cartesian to nonredundant internal coordinates, the errors in the semiempirical force constants are systematically analyzed. The force constants of the C(SINGLE BOND)C stretching coordinates can be easily corrected by a second-order fit. Thus, only two parameters are needed to reduce the mean error from 21.2 to 1.23%. The errors of other internal coordinates, particulary those including torsional modes, exhibit a larger diversity. The performance of the correction scheme in predicting vibrational spectra is shown for several examples including buckminsterfullerene (C₆₀). © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 2050–2059, 1997     

Determination of the molecular force constants of colloid particles

The possibility of analysis of the shape of colloid particles in solutions of latex and French and Moldovan wines with a flow-type ultramicroscope and digital photo camera was studied. The micrographs taken can be used to determine distances between particles by comparison with particle radii found by an independent method. The concepts of Deryagin-Landau-Vervey-Overbeek’s physical theory of stability of colloid systems were used to calculate the molecular force constants for wine particles.     

Ab initio SCF computation of force constants for CO2

Force constants for CO₂ have been evaluated using SCF wave functions. The effect ofd basis functions and geometry are investigated. Comparison with experimental values shows that a large error, due to neglect of electron correlation, occurs for theK ₁₂ interaction stretch force constant.     

Quasi-Hamiltonian equations of motion for internal coordinate molecular dynamics of polymers

Conventional molecular dynamics simulations of macromolecules require long computational times because the most interesting motions are very slow compared to the fast oscillations of bond lengths and bond angles that limit the integration time step. Simulation of dynamics in the space of internal coordinates, that is, with bond lengths, bond angles, and torsions as independent variables, gives a theoretical possibility of eliminating all uninteresting fast degrees of freedom from the system. This article presents a new method for internal coordinate molecular dynamics simulations of macromolecules. Equations of motion are derived that are applicable to branched chain molecules with any number of internal degrees of freedom. Equations use the canonical variables and they are much simpler than existing analogs. In the numerical tests the internal coordinate dynamics are compared with the traditional Cartesian coordinate molecular dynamics in simulations of a 56 residue globular protein. For the first time it was possible to compare the two alternative methods on identical molecular models in conventional quality tests. It is shown that the traditional and internal coordinate dynamics require the same time step size for the same accuracy and that in the standard geometry approximation of amino acids, that is, with fixed bond lengths, bond angles, and rigid aromatic groups, the characteristic step size is 4 fs, which is 2 times higher than with fixed bond lengths only. The step size can be increased up to 11 fs when rotation of hydrogen atoms is suppressed. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18 : 1354–1364, 1997     

Intermolecular force constants of hcn in the condensed phase

A simple theory of linear lattice is applied to the hydrogen bonded linear chain system of HCN to calculate the intermolecular force constants at different temperatures in the condensed phase. The strong CN bond is assumed to remain unperturbed in the hydrogen bond formation. The sharp change in intermolecular force constant while passing from the crystalline to the liquid phase is interpreted as a characteristic of this phase transition (fusion).     

八乙基金属卟啉的轴向配位反应研究及稳定常数测定

测定了八乙基金属卟啉 (OEP) M中当M=Zn, Cu, FeCl 或 MnCl时在CH2Cl2, N,N-二甲基甲酰胺(DMF), 二甲亚砜(DMSO), 吡啶(Py)等4种不同非水溶剂中的紫外-可见光谱, 探讨了溶剂的性质对八乙基金属卟啉光谱的影响. 结果表明八乙基锌、铁、锰卟啉在CH2Cl2中能够与配位溶剂如DMF, DMSO或pyridine 等发生轴向配位反应生成五配位或六配位的金属卟啉配合物. 用微量光谱滴定法测定了Zn、 Fe、 Mn等金属卟啉和配位溶剂发生轴向配位反应的稳定常数(logK). 讨论了中心金属离子的电负性以及配位溶剂的给电子能力对稳定常数的影响.