A Note on Mathematical Relationships Among Bond-Torsion Force Fields

A set of mathematical relationship between torsion potential functions such as trigonometric and Fourier series is presented herein. A harmonic approximation form is also introduced, and its stiffness constant is related to the parameters of trigonometric and Fourier series. Mathematical relationships between various force field parameters are presented in the form of conversion matrices.     

硝胺及其甲基衍生物的从头计算研究 3: 甲硝胺及其同位素衍生物的谐性力场和振动光谱

用TEXAS从头计算程序,取STO-4-21G基组,计算了甲硝胺的谐性力场和振动光谱.直接理论计算的谐性力场经由其他分子转移来的经验校正因子校正后,提供了甲硝胺振动基频的预测.预测值和甲硝胺分子在气相中的振动光谱实验值之间的平均偏差为31cm^-1.为了获得更合适的气相甲硝胺振动力场和预测它的同位素衍生物的振动光谱,我们优化了一组新的校正因子,使理论值和实验值的平均偏差减为8.9cm^-1.用这组校正因子得到的力场预测了三个同位素衍生物的振动光谱,其同位素位移的理论预测值和实验值符合良好.     

The determination of force constants from isotopic frequencies using parameter representation

The problem of fixing the molecular force field using isotopic frequencies is approached from a new angle with the help of the parameter technique. A parameter space is defined with the parameters for the molecule and its isotopic substituent as the coordinates. Making use of the constraint that the F elements are invariant under isotopic substitution in the ideal case, it is shown that unique points must exist in the parameter space enabling the determination of the force field. The method is applied to harmonic vibrations of molecules with 2 × 2 secular equations. The displays in the parameter space indicate that two such points exist for all molecules. Thus there can be two sets of force fields capable of reproducing the observed frequencies for the molecule and its isotopic substituents. Checks with mean amplitudes of vibration favour only one set. The calculated values of the force constants compare well with the values from the literature.     

Force‐field parameters of the Ψ and Φ around glycosidic bonds to oxygen and sulfur atoms

The Ψ and Φ torsion angles around glycosidic bonds in a glycoside chain are the most important determinants of the conformation of a glycoside chain. We determined force‐field parameters for Ψ and Φ torsion angles around a glycosidic bond bridged by a sulfur atom, as well as a bond bridged by an oxygen atom as a preparation for the next study, i.e., molecular dynamics free energy calculations for protein‐sugar and protein‐inhibitor complexes. First, we extracted the Ψ or Φ torsion energy component from a quantum mechanics (QM) total energy by subtracting all the molecular mechanics (MM) force‐field components except for the Ψ or Φ torsion angle. The Ψ and Φ energy components extracted (hereafter called “the remaining energy components”) were calculated for simple sugar models and plotted as functions of the Ψ and Φ angles. The remaining energy component curves of Ψ and Φ were well represented by the torsion force‐field functions consisting of four and three cosine functions, respectively. To confirm the reliability of the force‐field parameters and to confirm its compatibility with other force‐fields, we calculated adiabatic potential curves as functions of Ψ and Φ for the model glycosides by adopting the Ψ and Φ force‐field parameters obtained and by energetically optimizing other degrees of freedom. The MM potential energy curves obtained for Ψ and Φ well represented the QM adiabatic curves and also these curves' differences with regard to the glycosidic oxygen and sulfur atoms. Our Ψ and Φ force‐fields of glycosidic oxygen gave MM potential energy curves that more closely represented the respective QM curves than did those of the recently developed GLYCAM force‐field. © 2009 Wiley Periodicals, Inc., J Comput Chem, 2009     

Molecular mechanics force-field parameterization procedures

A set of procedures and guidelines are presented for the estimation of bond length, bond angle, and torsional potential constants for molecular mechanics force fields. The force field constants are ultimately derived by “subtracting” nonbonded molecular mechanics energies from corresponding molecular orbital energies using a model compound containing the chemical structure to be parameterized. Case study examples of bond length, bond angle, and torsional rotation force field parameterizations are presented. A general discussion of molecular mechanics force field parameterization strategy is included for reference and completeness. Finally, a curve-fitting program to generate force field parameters from raw data is given in Appendix I.     

Ab Initio Calculations on Nitramide and Its Methyl Derivatives( Ⅱ )——The Harmonic Force Fields and Vibrational Spectra of Nitramide and Its Isotopic Derivatives

The harmonic force field and the vibrational spectrum of nitramide were calculated by using the ab initio gradient program TEXAS at the Hartree-Fock level with a 4-21G basis set. The directly computed theoretical harmonic force field was scaled by using empirical scale factors which are transferred from other molecules and provided an a priori prediction of fundamental frequencies and intensities. The average deviations between predicted vibrational frequencies of nitramide and experimental IR spectrum in an argon matrix are 63 cm-1 for symmetric vibrations and 41 cm-1 for antisymmetric modes. A new set of scale factors was optimized in this paper. These scale factors reduced the average deviations to 2. 3 cm-1 for symmetric modes and 0. 8 cm-1 for antisymmetric ones. The vibrational spectra of three isotopic derivatives of nitramide were predicted by using the force field resulted from the optimized set of scale factors, which are in good agreement with their experimental data in an argon matrix.     

A polarizable force field for computing the infrared spectra of the polypeptide backbone

The shapes of the amide bands in the infrared (IR) spectra of proteins and peptides are caused by electrostatically coupled vibrations within the polypeptide backbone and code the structures of these biopolymers. A structural decoding of the amide bands has to resort to simplified models because the huge size of these macromolecules prevents the application of accurate quantum mechanical methods such as density functional theory (DFT). Previous models employed transition-dipole coupling methods that are of limited accuracy. Here we propose a concept for the computation of protein IR spectra, which describes the molecular mechanics (MM) of polypeptide backbones by a polarizable force field of "type II". By extending the concepts of conventional polarizable MM force fields, such a PMM/II approach employs field-dependent parameters not only for the electrostatic signatures of the molecular components but also for the local potentials modeling the stiffness of chemical bonds with respect to elongations, angle deformations, and torsions. Using a PMM/II force field, the IR spectra of the polypeptide backbone can be efficiently calculated from the time dependence of the backbone's dipole moment during a short (e.g., 100 ps) MD simulation by Fourier transformation. PMM/II parameters are derived for harmonic bonding potentials of amide groups in polypeptides from a series of DFT calculations on the model molecule N-methylacetamide (NMA) exposed to homogeneous external electric fields. The amide force constants are shown to vary by as much as 20% for relevant field strengths. As a proof of principle, it is shown that the large solvatochromic effects observed in the IR spectra of NMA upon transfer from the gas phase into aqueous solution are not only excellently reproduced by DFT/MM simulations but are also nicely modeled by the PMM/II approach. The tasks remaining for a proof of practice are specified.     

The harmonic force field and absolute infrared intensities of butyne-2

The frequencies, harmonic force field and absolute IR intensities for butyne-2-d₀ and butyne-2-d₆ are reported. The final set of “harmonized” fundamental frequencies for butyne-2-d₀ and butyne-2-d₆ obeys the Teller—Redlich product rule very well. Starting values for the force constants were obtained from the harmonic force field of propyne, and diagonal force constants were adjusted in order to reproduce the experimental “harmonized” frequencies for the d₀ and d₆ compounds.The integrated IR intensities were measured according to the Wilson—Wells—Penner—Weber method, using nitrogen as a broadening gas. Thirteen sets of ?μ/?S values were obtained from the experimental intensities, using an iterative least-squares fitting procedure. This number could be reduced to one by use of several selection criteria. The signs of the remaining set appeared to be in complete agreement with the best set for propyne as reported both by Kondo and Koga and by Bode et al. The final ?μ/?S parameters were transformed into atomic polar tensors. Both kinds of intensity parameters are discussed and compared with corresponding parameters for related molecules.     

Nonreactive molecular dynamics force field for crystalline hexahydro-1,3,5-trinitro-1,3,5 triazine

An empirical nonreactive force field has been developed for molecular dynamics (MD)/Monte Carlo simulation of the formation, diffusion, and agglomeration of point defects in the crystal lattice of the alpha modification of hexahydro-1,3,5-trinitro-1,3,5 triazine (RDX) using flexible molecules. Bond stretching and angle bending are represented by Morse and harmonic functions, and torsion by a truncated cosine series. Nonbonded interactions, both inter- and intramolecular, are described by Buckingham potentials separately parametrized. Intermolecular electrostatic interactions are treated via a Coulomb term coupled with a smooth 15.0 A cutoff radius. Parameters were taken in part from earlier published works and were determined partly by fitting to known molecular and crystal properties of RDX. In MD simulations at constant pressure and temperature, the model was able to stabilize and maintain the correct crystal structure, symmetry, and molecular conformation of alpha-RDX. Vibrational frequencies, lattice binding energy and dimensions, coefficients of thermal expansion, and several unusually short intermolecular distances are all reproduced in satisfactory agreement with experimental data.     

The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models

The multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] employs a variational principle to determine an interaction potential for a CG model from simulations of an atomically detailed model of the same system. The companion paper proved that, if no restrictions regarding the form of the CG interaction potential are introduced and if the equilibrium distribution of the atomistic model has been adequately sampled, then the MS-CG variational principle determines the exact many-body potential of mean force (PMF) governing the equilibrium distribution of CG sites generated by the atomistic model. In practice, though, CG force fields are not completely flexible, but only include particular types of interactions between CG sites, e.g., nonbonded forces between pairs of sites. If the CG force field depends linearly on the force field parameters, then the vector valued functions that relate the CG forces to these parameters determine a set of basis vectors that span a vector subspace of CG force fields. The companion paper introduced a distance metric for the vector space of CG force fields and proved that the MS-CG variational principle determines the CG force force field that is within that vector subspace and that is closest to the force field determined by the many-body PMF. The present paper applies the MS-CG variational principle for parametrizing molecular CG force fields and derives a linear least squares problem for the parameter set determining the optimal approximation to this many-body PMF. Linear systems of equations for these CG force field parameters are derived and analyzed in terms of equilibrium structural correlation functions. Numerical calculations for a one-site CG model of methanol and a molecular CG model of the EMIM(+)NO(3) (-) ionic liquid are provided to illustrate the method.     

Mathematical Connections Between Bond-Stretching Potential Functions

Mathematical connections are useful in enabling a set of parametric data from a chemical bond-stretching potential function to be applied in a computational chemistry software that adopts a different potential function. This paper establishes connections between four potential energy functions in stretching and compression of covalent bonds. The potential functions that are mathematically connected are: (i) harmonic potential, (ii) polynomial series potential, (iii) Morse potential, and (iv) Murrell–Mottram potential. Two methods are employed in obtaining the relationships between the four potential functions. The expansion approach enables the relationships to be made at large bond-stretching, whilst the differential approach allows for the connections to be made only at infinitesimal bond-stretching. For verification, parametric data from the Murrell–Mottram potential is converted to parametric data of the harmonic, polynomial series and Morse potentials. For comparison, the bond-stretching energies for these functions are plotted. Discrepancy between the Morse and the Murrell–Mottram potentials at large bond-stretching is discussed in terms of the assumed infinitesimal deformation.     

Parameterization and Application of the General Amber Force Field to Model Fluoro Substituted Furanose Moieties and Nucleosides

Molecular mechanics force field calculations have historically shown significant limitations in modeling the energetic and conformational interconversions of highly substituted furanose rings. This is primarily due to the gauche effect that is not easily captured using pairwise energy potentials. In this study, we present a refinement to the set of torsional parameters in the General Amber Force Field (gaff) used to calculate the potential energy of mono, di-, and gem-fluorinated nucleosides. The parameters were optimized to reproduce the pseudorotation phase angle and relative energies of a diverse set of mono- and difluoro substituted furanose ring systems using quantum mechanics umbrella sampling techniques available in the IpolQ engine in the Amber suite of programs. The parameters were developed to be internally consistent with the gaff force field and the TIP3P water model. The new set of angle and dihedral parameters and partial charges were validated by comparing the calculated phase angle probability to those obtained from experimental nuclear magnetic resonance experiments.     

A systematic approach to calibrate a transferable polarizable force field parameter set for primary alcohols

In this work, parameters are optimized for a charge‐on‐spring based polarizable force field for linear alcohols. We show that parameter transferability can be obtained using a systematic approach in which the effects of parameter changes on physico‐chemical properties calculated from simulation are predicted. Our previously described QM/MM calculations are used to attribute condensed‐phase polarizabilities, and starting from the non‐polarizable GROMOS 53A5/53A6 parameter set, van der Waals and Coulomb interaction parameters are optimized to reproduce pure‐liquid (thermodynamic, dielectric, and transport) properties, as well as hydration free energies. For a large set of models, which were obtained by combining small perturbations of 10 distinct parameters, values for pure‐liquid properties of the series methanol to butanol were close to experiment. From this large set of models, we selected 34 models without special repulsive van der Waals parameters to distinguish between hydrogen‐bonding and non‐hydrogen‐bonding atom pairs, to make the force field simple and transparent. © 2017 Wiley Periodicals, Inc.     

Harmonic force constants of formic acid: ab initio results and the uniqueness problem of force fields derived from vibrational data

A complete set of harmonic force constants of formic acid (HCOOH) is calculated using a 6-31G basis set. It is used to determine a modified valence force field consistent with both ab initio results and vibrational data.     

γ-吡啶甲酸及其同位素衍生物的谐性力场和振动光谱的从头计算研究

本文采用TEXAS分析梯度法从头计算程序, 以STO-4-21G基组计算了γ-吡啶甲酸的谐性力场和振动光谱。直接理论计算的谐性力场经由相关分子转移来的校正因子校正后, 得到的振动基频的预测值和固体样品红外光谱实验值之间的平均偏差为20cm^-^1(面内振动23cm^-^1, 面外振动11cm^-^1)。用这组校正因子得到的力场预测了γ-吡啶甲酸的两个同位素衍生物(-C^1^8O~2H和-C^1^6O~2D)的振动光谱, 所得同位素位移值与实验数据符合良好。对平面内振动的个别校正因子依据实验光谱进行了优化, 平面内振动的平均偏差降为15cm^-^1, 总的偏差为14cm^-^1。对预测中的偏差和某些基频的指认进行了讨论。     

Studies on nitramide and its methyl derivatives with ab initio calculations. IV. The harmonic force field and vibrational spectra of dimethylnitramine and its isotopic derivatives

The complete harmonic vibrational force field of dimethylnitramine has been calculated at the Hartree-Fock level using 4–21G basis set. The harmonic force field was then scaled with scale factors previously derived from N-methylnitramine, and the vibrational spectrum of dimethylnitramine was computed. This a priori prediction, made with no reference to observations on dimethylnitramine, agrees with the experimental IR spectrum in gas phase with a mean deviation of 8.4 cm^?1. Some of the scale factors were reoptimized by fitting of the computed force field to experimental data. The new set of scale factors reduced the mean deviation to 4.5 cm^?1, and was used to predict the vibrational spectrum of deuterated form of dimethylnitramine(-6D). Dipole moment derivatives were also calculated and used to predict infrared intensities which are comparable with experimental values.     

Application of Maclaurin Series in Relating Interatomic Potential Functions: A Review

This paper provides a short review on the application of Maclaurin series in relating potential functions within the same category of interatomic interaction. The potential functions covered are those commonly adopted in computational chemistry softwares. While various mathematical approaches have been employed in generating relationships amongst potential functions, the use of Maclaurin series has been prevalent recently due to the increasing application of polynomial series-type potential functions. In the case of covalent bond-stretching, the Maclaurin series for the exponential function is used to transform the Morse potential into the polynomial series potential, and vice versa. For covalent bond-bending, Maclaurin series for the sine and cosine functions were employed to extract polynomial angle series potential from the Fourier series and harmonic cosine potential functions, and vice versa. Finally, both the exponential and the 1/(1 – x) expressions in Maclaurin series were used in obtaining the exact relationship for the repulsive terms between two potential functions.     

Theoretical prediction of vibrational spectra. Scaled quantum mechanical (SQM) force field for fluorobenzene

The complete harmonic force field of fluorobenzene has been determined from ab initio Hartree-Fock calculations using the 4–21 Gaussian basis set. As force constants are systematically overestimated at this level of theory, the directly calculated force field was scaled by empirical factors taken over from benzene and methylfluoride. Except for a slight overestimation of the CF stretching frequency, the scaled quantum mechanical (SQM) force field obtained in this way reproduces the experimental fundamental frequencies of the parent molecule and two deuterated isotopomers within 20 cm⁻¹ (with mean deviations below 12 cm⁻¹), and experimental assignments are analyzed on this basis. Theoretical i.r. intensities reproduce the main features of the spectra fairly well.