Derivation of class II force fields. I. Methodology and quantum force field for the alkyl functional group and alkane molecules

A new method for deriving force fields for molecular simulations has been developed. It is based on the derivation and parameterization of analytic representations of the ab initio potential energy surfaces. The general method is presented here and used to derive a quantum mechanical force field (QMFF) for alkanes. It is based on sampling the energy surfaces of 16 representative alkane species. For hydrocarbons, this force field contains 66 force constants and reference values. These were fit to 128,376 quantum mechanical energies and energy derivatives describing the energy surface. The detailed form of the analytic force field expression and the values of all resulting parameters are given. A series of computations is then performed to test the ability of this force field to reproduce the features of the ab initio energy surface in terms of energies as well as the first and second derivatives of the energies with respect to molecular deformations. The fit is shown to be good, with rms energy deviations of less than 7% for all molecules. Also, although only two atom types are employed, the force field accounts for the properties of both highly strained species, such as cyclopropane and methylcyclopropanes, as well as unstrained systems. The information contained in the quantum energy surface indicates that it is significantly anharmonic and that important intramolecular coupling interactions exist between internals. The representation of the nature of these interactions, not present in diagonal, quadratic force fields (Class I force fields), is shown to be important in accounting accurately for molecular energy surfaces. The Class II force field derived from the quantum energy surface is characterized by accounting for these important intramolecular forces. The importance of each 4.2 to 18.2%. This fourfold increase in the second derivative error dramatically demonstrates the importance of bond anharmonicity in the ab initio potential energy surface. The Class II force field derived from the quantum energy surface is characterized by accounting for these important intramolecular forces. The importance of each of the interaction terms of the potential energy function has also been assessed. Bond anharmonicity, angle anharmonicity, and bond/angle, bond/torsion, and angle/angle/ torsion cross-term interactions result in the most significant overall improvement in distorted structure energies and energy derivatives. The implications of each energy term for the development of advanced force fields is discussed. Finally, it is shown that the techniques introduced here for exploring the quantum energy surface can be used to determine the extent of transferability and range of validity of the force field. The latter is of crucial importance in meeting the objective of deriving a force field for use in molecular mechanics and dynamics calculations of a wide range of molecules often containing functional groups in novel environments. © 1994 by John Wiley & Sons, Inc.     

Analysis of Single-Walled Carbon Nanotubes Using a Chemical Bond Element Model

A three dimensional nano-scale finite element model (FEM), called the chemical bond element model, is proposed for the simulation of mechanical properties of single-walled carbon nanotubes (SWCNTs) based upon molecular mechanics method. Chemical bonds between carbon atoms are modeled by chemical bond elements. The constants of a sub-stiffness matrix are determined by using a linkage between molecular mechanics and continuum mechanics. In order to evaluate the correctness and performance of the proposed model, simulation was done to determine the influence of nanotube wall thickness, radius and length on the elastic modulus (Young's modulus and shear modulus) of SWCNTs. The simulation results show that the choice of wall thickness significantly affects the Young's modulus and shear modulus. The force field constants is also very important, because the elastic modulus is sensitive to force field constants and the elastic properties of SWCNT are related to the radii of the tubes. The contribution of length to elastic modulus is insignificant and can be ignored. In comparison with the Young's modulus and shear modulus reported in the literature, the presented results agree very well with the corresponding theoretical results and many experimental measurements. Furthermore, if the force constants are properly chosen, the present method could be conveniently used to predict the mechanical behavior of other single-walled nanotubes such as boron nitride nanotubes. The results demonstrate the value of the proposed model as a valuable tool in the study of mechanical behaviors of carbon nanotubes and in the analysis of nanotube-based equipments.     

Derivation of class II force fields. VIII. Derivation of a general quantum mechanical force field for organic compounds

A class II valence force field covering a broad range of organic molecules has been derived employing ab initio quantum mechanical "observables." The procedure includes selecting representative molecules and molecular structures, and systematically sampling their energy surfaces as described by energies and energy first and second derivatives with respect to molecular deformations. In this article the procedure for fitting the force field parameters to these energies and energy derivatives is briefly reviewed. The application of the methodology to the derivation of a class II quantum mechanical force field (QMFF) for 32 organic functional groups is then described. A training set of 400 molecules spanning the 32 functional groups was used to parameterize the force field. The molecular families comprising the functional groups and, within each family, the torsional angles used to sample different conformers, are described. The number of stationary points (equilibria and transition states) for these molecules is given for each functional group. This set contains 1324 stationary structures, with 718 minimum energy structures and 606 transition states. The quality of the fit to the quantum data is gauged based on the deviations between the ab initio and force field energies and energy derivatives. The accuracy with which the QMFF reproduces the ab initio molecular bond lengths, bond angles, torsional angles, vibrational frequencies, and conformational energies is then given for each functional group. Consistently good accuracy is found for these computed properties for the various types of molecules. This demonstrates that the methodology is broadly applicable for the derivation of force field parameters across widely differing types of molecular structures. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1782-1800, 2001     

氮、氮双中心三电子键的分子力学力参数研究

1，n－氮、氮双环烷烃（n＝4－6）第一电离能很低（＜7eV）[1]．动力学和光谱学研究表明，当电离（氧化）后形成的阳离子游离基（I）是中间体，且其中某些分子存在三电子σ键[2]．成键轨道σ（N，N）有两个电子，反键轨道有一个电子（如），这些中间体的寿命短则几毫秒、几秒，长到几小时甚至几个月，存在干晶体或对质子有惰性的溶剂中．有人［1，3］还推测出该三电子σ键键能约本文尝试用分子力学uuz方法对阳离子游离基（！）进行计算，采用nontoger方法问得到了一些力场参数、用这些力场参数，选择文献已作报导的1，5一氮、氮双环（3，3…     

Computation of pressure components due to Class II force fields

It has been shown that purely angle dependent terms in a force field do not contribute to the total pressure in a molecular simulation. However, this is not the case for the individual components of the pressure tensor, and is also untrue for crossterms in the force field including bond stretch. In this article, we show that virial contributions to the pressure tensor are easily computed in terms of bond distance vectors and atom forces for the bond topologies present in a Class II force field. Results from a simulation of a phospholipid biomembrane using the cff97 force field show that angle and torsional crossterms make a significant contribution to the pressure tensor.     

Alternative expressions for energies and forces due to angle bending and torsional energy

We have derived alternative expressions for computing the energies and forces associated with angle bending and torsional energy terms commonly used in molecular mechanics and molecular dynamics computer programs. Our expressions address the problems of singularities that are intrinsic in popular angle energy functions and that occur from other chain rule derivations of force expressions. Most chain rule derivations of expressions for Cartesian forces due to angle energies make use of relations such as where ? is a bond or torsion angle, E(?) is energy, and ?/?x represents a derivative with respect to some Cartesian coordinate. This expression leads to singularities from the middle term, ?1/sin ?, when ? is 0 or π. This is a problem that prevents the use of torsional energy expressions that have phase angles, ?°, other than 0 or π, such as in E(?) = κ[1 + cos(n? ? phsi;°)]. Our derivations make use of a different, but equivalent, form of the chain rule: This form still possesses singularities for the bond angle forces since the last factor is undefined when ? is 0 or π. However, the alternate form may be used to great advantage for the torsional angle forces where no such problem arises. The new expressions are necessary if one desires the use of torsional energy expressions with general phase angles. Even for energy expressions in common use, i.e., with phase angles of 0 or π, our force expressions are as computationally efficient as the standard ones. The new expressions are applicable to all molecular simulations that employ restrained, or phase-shifted, torsional angle energy expressions.     

Calculation of alcohol conformations by molecular mechanics

The geometries and energies of simple alcohols were calculated with a molecular mechanics force field. The force field requires the application of the charge interaction model with charges calculated by the CNDO/2 method, the importance of electrostatic interactions for the equilibrium of rotamers about the C-O bond exceeds that of van der Waals interactions. The calculated rotamer populations are discussed with regard to the value of ¹H NMR coupling constants ³JHCOH and other experimental data.     

1,4-Interactions in molecular mechanics calculations of Ethers

Van der Waals and electrostatic interactions are found to be insufficient for the calculation of conformational energies of ethers by molecular mechanics. Low order torsional potential functions must be added for the potential about C-O bonds. A onefold term necessary for the CCOC-fragment is interpreted to be a substitute for gauche interactions present in CCCC-, but missing in CCOC-fragments. For the COCO fragment the anomeric effect must be included explicitly as another torsional energy term, but no such term is required to stabilize the gauche conformation for OCCO. With the resulting ether force field the geometries and energies of model compounds, many of them 1,3-dioxanes, are calculated with good accuracy.     

Alcohols,ethers, carbohydrates,and related compounds. II. The anomeric effect

The anomeric effect has been studied for a variety of compounds using the MM4 force field, and also using MP2/6-311++G(2d,2p) ab initio calculations and experimental data for reference purposes. Geometries and energies, including conformational, rotational barriers, and heats of formation were examined. Overall, the agreement of MM4 with the experimental and ab initio data is good, and significantly better than the agreement obtained with the MM3 force field. The anomeric effect is represented in MM4 by various explicit terms in the force constant matrix. The bond length changes are accounted for with torsion-stretch elements. The angle changes are accounted for with torsion-bend elements. The energies are taken into account with a number of torsional terms in the usual way. A torsion-torsion interaction is also of some importance. With all of these elements included in the calculation, the MM4 results now appear to be adequately accurate. The heats of formation were examined for a total of 12 anomeric compounds, and the experimental values were fit by MM4 with an RMS error of 0.42 kcal/mol.     

Development of glycyl radical parameters for the OPLS-AA/L force field

On the basis of quantum chemical calculations C(alpha)-glycyl radical parameters have been developed for the OPLS-AA/L force field. The molecular mechanics hypersurface was fitted to the calculated quantum chemical surface by minimizing their molecular mechanics parameter dependent sum-of-squares deviations. To do this, a computer program in which the molecular mechanics energy derivatives with respect to the parameters were calculated analytically was developed, implementing the general method of Lifson and Warshel (J Chem Phys 1968, 49, 5116) for force field parameter optimization. This program, in principle, can determine the optimal parameter set in one calculation if enough representative value points on the quantum chemical potential energy surface are available and there is no linear dependency between the parameters. Some of the parameters in quantum calculations, including several new torsion types around a bond as well as angle parameters at a new central atom type, are not completely separable. Consequently, some restrictions and/or presumptions were necessary during parameter optimization. The relative OPLS-AA energies reproduced those calculated quantum chemically almost perfectly.     

On the geometry and internal rotation of XSCF3 compounds (X = F,Cl and CF3)

The molecular structures (r_α⁰ values) for XSCF₃ with X = F, Cl and CF₃ have been determined by electron diffraction of gases. While the geometry (C-F bond length and FCF angle) of the CF₃ groups and the bond angle at the sulfur atom depend very little on the substituent X, the S-C bond length increases with decreasing electronegativity of X from 1.805 (3) Å for X = F to 1.824 (6) Å for X = Cl. Torsional force constants for the CF₃ groups were derived from vibrational amplitudes. A strong increase of this force constant is observed between FSCF₃ (f_τ = 0.09 (2) mdyn Å) and CISCF₃ (f_τ = 0.18 (5) mdyn Å). The torsional frequencies derived from the electron diffraction experiment agree very well with the values observed in the far IR spectra for CISCF₃, and CF₃SCF₃. A force field for CF₃SCF₃ has been derived from IR and Raman data.     

Force-field parametrization of retro-inverso modified residues: Development of torsional and electrostatic parameters

Summary Torsional and the electrostatic parameters for molecular mechanics studies of retro-inverso modified peptides have been developed using quantum mechanical calculations. The resulting parameters have been compared with those calculated for conventional peptides. Rotational profiles, which were obtained spanning the corresponding dihedral angle, were corrected by removing the energy contributions associated to changes in interactions different from torsion under study. For this purpose, the torsional energy associated to each point of the profiles was estimated as the corresponding quantum mechanical energy minus the bonding and nonbonding energy contributions produced by the perturbations that the variation of the spanned dihedral angle causes in the bond distances, bond angles and the other dihedral angles. These energies were calculated using force-field expressions. The corrected profiles were fitted to a three-term Fourier expansion to derive the torsional parameters. Atomic charges for retro-inverso modified residues were derived from the rigorously calculated quantum mechanical electrostatic potential. Furthermore, the reliability of electrostatic models based on geometry-dependent charges and fixed charges has been examined.     

COSMIC(90): An improved molecular mechanics treatment of hydrocarbons and conjugated systems

Summary Four modifications to the COSMIC molecular mechanics force field are described, which greatly increase both its versatility and the accuracy of calculated conformational energies. The Hill non-bonded van der Waals potential function has been replaced by a two-parameter Morse curve and a new H-H potential, similar to that in MM3, incorporated. Hydrocarbon energies in particular are much improved.A simple iterative Hückel pi-electron molecular orbital calculation allows modelling of conjugated systems. Calculated bond lengths and rotational barriers for a series of conjugated hydrocarbons and nitrogen heterocycles are shown to be as accurate as those determined by the MM2 SCF method.Explicit hydrogen-bonding potentials for H-bond acceptor-donor atom pairs have been included to give better hydrogen bond energies and lengths. The van der Waals radii of protonic hydrogens are reduced to 0.5 Å and the energy well depth is increased to 1.0 kcal mol^-1.Two new general atom types, N⁺ _sp ² and O^- _sp ³ , have been introduced which allow a wide variety of charged conjugated systems to be studied. A minimum of parameterisation is required, as the new types are easily included in the Hückel scheme which automatically adjusts bond and torsional parameters according to the defined bond-order relationships.     

A new force field (ECEPP-05) for peptides, proteins, and organic molecules

Parametrization and testing of a new all-atom force field for organic molecules and peptides with fixed bond lengths and bond angles are described. The van der Waals parameters for both the organic molecules and the peptides were taken from J. Phys. Chem. B 2003, 107, 7143 and J. Phys. Chem. B 2004, 108, 12181. First, the values of the 1-4 nonbonded and electrostatic scale factors appropriate to the new force field were determined by computing the conformational energies of six model molecules, namely, ethanol, ethylamine, propanol, propylamine, 1,2-ethanediol, and 1,3-propanediol with different values of these factors. The partial atomic charges of these molecules were obtained by fitting to the electrostatic potentials calculated with the HF/6-31G quantum-mechanical method. Two different charge models (single- and multiple-conformation-derived) were also considered. We demonstrated that the charge model has a stronger effect on the conformational energies than the 1-4 scaling. The choice of a charge model affected the conformational energies of even the smallest molecules considered, whereas the effect of the 1-4 electrostatic or nonbonded scaling was apparent only for 1,3-propanediol. The best agreement with high-level ab initio data was obtained with the multiple-conformation-derived charges and with no scaling of the 1-4 nonbonded or electrostatic interactions (scale factors of 1.0). Next, the torsional parameters of a large number of neutral and charged organic molecules, assumed to be models of the side chains of the 20 naturally occurring amino acids, were computed by fitting to rotational energy profiles obtained from ab initio MP2/6-31G calculations. The quality of the fits was high with average errors for torsional profiles of less than 0.2 kcal/mol. To derive the torsional parameters for the peptide backbone, the partial atomic charges of the 20 neutral and charged amino acids were obtained by fitting to the electrostatic potentials of terminally blocked amino acids using the HF/6-31G quantum-mechanical method. Then, the phi-psi energy maps of Ac-Ala-NMe and Ac-Gly-NMe were computed using MP2/6-31G//HF/6-31G quantum-mechanical methods. The phi-psi energy map of Ac-Ala-NMe was used for refinement of the nonbonded parameters for the backbone nitrogen and hydrogen bonded to it. Subsequently, the main-chain torsional parameters were obtained by fitting the molecular mechanics energies to the phi-psi energy maps of Ac-Ala-NMe and Ac-Gly-NMe. The transferability of the entire force field was demonstrated by reproducing the main energy minima of terminally blocked Ala3 from the literature. The performance of the force field was also evaluated by simulating crystal structures of small peptides. By comparison of simulated and experimental data, examination of the torsional-angle and atom-positional root-mean-square deviations of the energy-minimized crystal structures from the corresponding X-ray model structures demonstrated high accuracy of the force field.     

Validation of the general purpose tripos 5.2 force field

A molecular mechanics force field implemented in the Sybyl program is described along with a statistical evaluation of its efficiency on a variety of compounds by analysis of internal coordinates and thermodynamic barriers. The goal of the force field is to provide good quality geometries and relative energies for a large variety of organic molecules by energy minimization. Performance in protein modeling was tested by minimizations starting from crystallographic coordinates for three cyclic hexapeptides in the crystal lattice with rms movements of 0.019 angstroms, 2.06 degrees, and 6.82 degrees for bond lengths, angles, and torsions, respectively, and an rms movement of 0.16 angstroms for heavy atoms. Isolated crambin was also analyzed with rms movements of 0.025 angstroms, 2.97 degrees, and 13.0 degrees for bond lengths, angles, and torsions respectively, and an rms movement of 0.42 angstroms for heavy atoms. Accuracy in calculating thermodynamic barriers was tested for 17 energy differences between conformers, 12 stereoisomers, and 15 torsional barriers. The rms errors were 0.8, 1.7, and 1.13 kcal/mol, respectively, for the three tests. Performance in general purpose applications was assessed by minimizing 76 diverse complex organic crystal structures, with and without randomization by coordinate truncation, with rms movements of 0.025 angstroms, 2.50 degrees, and 9.54 degrees for bond lengths, angles and torsions respectively, and an average rms movement of 0.192 angstroms for heavy atoms.     

A molecular mechanics (MM3) study of fluorinated hydrocarbons

A molecular mechanics study of small saturated hydrocarbons (up to C-6), substituted by up to six fluorines was carried out with the MM3 force field. Perfluorobutane and Teflon were also studied. A parameter set was developed for use in the calculation of bond lengths, bond angles, torsion angles, conformational energies, barriers to rotation, dipole moments, moments of inertia and vibrational frequencies for these compounds. The results are in good agreement with experiment when only one or two fluorines are present, but some rather large discrepancies were noted when the F/H ratio becomes high. These can be taken into account only by using a force field more complicated than MM3. Some of the requirements of such a force field are delineated. Some pertinent ab initio results are also reported in this article.