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.     

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.     

Force fields for metallic clusters and nanoparticles

Atomic force fields for simulating copper, silver, and gold clusters and nanoparticles are developed. Potential energy functions are obtained for both monatomic and binary metallic systems using an embedded atom method. Many cluster configurations of varying size and shape are used to constrain the parametrization for each system. Binding energies for these training clusters were computed using density functional theory (DFT) with the Perdew-Wang exchange-correlation functional in the generalized gradients approximation. Extensive testing shows that the many-body potentials are able to reproduce the DFT energies for most of the structures that were included in the training set. The force fields were used to calculate surface energies, bulk structures, and thermodynamic properties. The results are in good agreement with the DFT values and consistent with the available experimental data.     

Force field for copper clusters and nanoparticles

An atomic force field for simulating copper clusters and nanoparticles is developed. More than 2000 cluster configurations of varying size and shape are used to constrain the parametrization of the copper force field. Binding energies for these training clusters were computed using density functional theory. Extensive testing shows that the copper force field is fast and reliable for near‐equilibrium structures of clusters, ranging from only a few atoms to large nanoparticles that approach bulk structure. Nonequilibrium dissociation and compression structures that are included in the training set are also well described by the force field. Implications for molecular dynamics simulations and extensions to other metallic and covalent systems are discussed. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Some Aspects of the Implementation of Scaling Quantum Mechanical Molecular Force Fields

Some features of software implementation of the Pulay scaling procedure are considered. The advantages of the single value decomposition method for maintaining well-conditionality of the scale factor determination problem are demonstrated. The necessity of using a rational number of scale factors is shown. The possibility of obtaining transferable scale factors with the Pulay method and thus predict the vibrational spectra of related compounds is emphasized.     

Predictive Abilities of Scaled Quantum Mechanical Molecular Force Fields: Application to 1,3-Butadiene

The positions of some IR bands of the s-trans-1,3-butadiene-h ₆ and -1,1,2-d ₃ isotopomers in the gas phase have been measured using a Brucker IFS 120 HR spectrometer with a resolution of 2 cm^–1. The structural parameters of the s-trans- and s-gauche-1,3-butadiene conformers were optimized completely at the MP2/6-31G* theoretical level and their MP2/6-31G*//MP2/6-31G* quantum mechanical force fields (QMFFs) were calculated. Using only the experimental vibrational frequencies of s-trans-1,3-butadiene-h ₆ the QMFF of the s-trans conformer was corrected by Pulay's scaling method (eight scale factors were involved). The scaled QMFF was used to calculate the mean vibrational amplitudes and the Coriolis coupling constants of s-trans-1,3-butadiene-h ₆ and the vibrational frequencies of 12 of its deuterated isotopomers. The set of scale factors obtained for correction of the s-trans QMFF was transferred to the QMFF of the s-gauche conformer. Its theoretical vibrational spectrum and those of some deuterated and C¹³ isotopomers were calculated. The ability of this scaling approach (transferring of scale factors) to predict the vibrational frequencies of rotational conformers and their isotopomers, as well as other molecular characteristics, and to permit detection of perturbations of the experimental bands are discussed.     

The Dependence of Amyloid‐β Dynamics on Protein Force Fields and Water Models

We studied the dynamics of Aβ₄₀, involved in Alzheimer's disease, by using 21 methods combined from Amber03, Amber99sb‐ILDN, Charmm27, Charmm22*, OPLS‐2001, OPLS‐2006, OPLS‐2008, Gromos96‐43a1, Gromos96‐53a6, Gromos96‐54a7, and the water models SPC, TIP3P, TIP4P. Major differences in the structural ensembles were systematized: Amber03, Charmm27, and Gromos96‐54a7 stabilize the helices; Gromos96‐43a1 and Gromos53a6 favor the β‐strands (with Charmm22* and Amber99sb‐ILDN in between), and OPLS produces unstructured ensembles. The accuracy of the NMR chemical shifts was in the order: Charmm22*>Amber99sb‐ILDN>OPLS‐2008≈Gromos96‐43a1>Gromos96‐54a7≈OPLS‐2001>OPLS‐2006>Gromos96‐53a6>Charmm27>Amber03. The computed ³J_HNHα‐coupling constants were sensitive to experiment type and Karplus parameterization. Overall, the ensembles of Charmm22* and Amber99sb‐ILDN provided the best agreement with experimental NMR and circular dichroism data, providing a model for the real Aβ monomer ensemble. Also, the polar water model TIP3P significantly favored helix and compact conformations.     

多原子分子力场的模型势函法与MNDO,MINDO／3方法的研究

用量子力学方法获得分子的模型力场现已占有很重要的位置.然而由于理论上很难获得体系高精度解析势能面,使力场的理论计算遇到了困难.直到Pulay提出梯度法后,量子化学方法才被广泛用于多原子分子的振动分析.但由于该方法采用数值微分通过能量的第二阶导数所获得的力场矩阵对角元误差很大,需将其伸缩、变角对角力常数分别扣除10％、20％,才能获得与观测值相近的结果;且ab initio计算花机时太多,应用于大分子计算存在一定的困难.为此本文把前文提出的确定力常数的模型势函法与MNDO、MINDO/3等半经验方法结合起来,构造多原子分子的力场,并用于振动分析,获得了满意的结果.     

Calculation of Second- and Higher-Order Force Constants Including Electron Correlation Effects and Hartree-Fock Force Constant Scaling

A formalism suitable for practical implementation is suggested for computation of quadratic, cubic, and quartic force constants using the configuration interaction (CI) method. Expressions are compared which involve the Hartree-Fock (HF) harmonic and anharmonic force constants calculated by the HF and CI methods. Also, physical assumptions are formulated to perform scaling of the diagonal harmonic and, which is more important, of anharmonic HF force constants with a common scaling factor. This approach is consistent with the data of ab initio quantum-chemical calculations. A table is presented which compares the results of valence force constant calculations for a series of simple organic molecules.     

分子力场发展的新趋势

分子模拟中的力场方法是用来精确计算分子结构和能量的计算方法,它通过原子核的位置来计算体系能量。最初的分子力场都是针对某一特定体系的,它们的许多参数要由观测数据拟合得到。当时要建立新的分子力场是十分困难的,因为实验归属振动谱带需要花费大量的时间。所以此后大多数工作者都致力于发展涵盖尽可能多体系的“求全”型分子力场,这种趋势一直延续至今。但是随着各个学科研究的不断深入,所需要研究的体系越来越复杂,要求的精度也越来越高。在保证相当精度的条件下,“求全”型的分子力场要想涵盖所有需要研究的体系常常是十分困难的事情。2003年问世的Direct Force Field软件包能够便捷的建立针对某一特定分子体系,并且有相当精度的分子力场。它的出现为分子力场从“求全”转为向“求精”发展提供了可能。     

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.     

ForceFit: A code to fit classical force fields to quantum mechanical potential energy surfaces

The ForceFit program package has been developed for fitting classical force field parameters based upon a force matching algorithm to quantum mechanical gradients of configurations that span the potential energy surface of the system. The program, which runs under UNIX and is written in C++, is an easy‐to‐use, nonproprietary platform that enables gradient fitting of a wide variety of functional force field forms to quantum mechanical information obtained from an array of common electronic structure codes. All aspects of the fitting process are run from a graphical user interface, from the parsing of quantum mechanical data, assembling of a potential energy surface database, setting the force field, and variables to be optimized, choosing a molecular mechanics code for comparison to the reference data, and finally, the initiation of a least squares minimization algorithm. Furthermore, the code is based on a modular templated code design that enables the facile addition of new functionality to the program. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Force field modeling of conformational energies: Importance of multipole moments and intramolecular polarization

The ability of force fields to reproduce relative conformational energies for seven molecules is probed. The correlation against LMP2/cc‐pVTZ results for standard force field employing fixed partial charges deteriorates as the molecules become more polar. Inclusion of multipole moments and intramolecular polarization can improve the correlation, and both contributions are of similar importance. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Development and testing of a general amber force field

We describe here a general Amber force field (GAFF) for organic molecules. GAFF is designed to be compatible with existing Amber force fields for proteins and nucleic acids, and has parameters for most organic and pharmaceutical molecules that are composed of H, C, N, O, S, P, and halogens. It uses a simple functional form and a limited number of atom types, but incorporates both empirical and heuristic models to estimate force constants and partial atomic charges. The performance of GAFF in test cases is encouraging. In test I, 74 crystallographic structures were compared to GAFF minimized structures, with a root-mean-square displacement of 0.26 A, which is comparable to that of the Tripos 5.2 force field (0.25 A) and better than those of MMFF 94 and CHARMm (0.47 and 0.44 A, respectively). In test II, gas phase minimizations were performed on 22 nucleic acid base pairs, and the minimized structures and intermolecular energies were compared to MP2/6-31G* results. The RMS of displacements and relative energies were 0.25 A and 1.2 kcal/mol, respectively. These data are comparable to results from Parm99/RESP (0.16 A and 1.18 kcal/mol, respectively), which were parameterized to these base pairs. Test III looked at the relative energies of 71 conformational pairs that were used in development of the Parm99 force field. The RMS error in relative energies (compared to experiment) is about 0.5 kcal/mol. GAFF can be applied to wide range of molecules in an automatic fashion, making it suitable for rational drug design and database searching.     

Scaling Function Between the Exponential-6 and the Generalized Lennard-Jones Potential Functions

The van der Waals forces for non-bonded interaction can be expressed either by the Exponential-6 or by the Lennard-Jones(m-n) potential functions, whereby m > n. Hitherto a relationship exists between the Exponential-6 and the Lennard-Jones(12-6) potential functions, with a scaling factor = 13.772 at or near the equilibrium and = 12.0 for long range interaction. This paper attempts to develop relationships between Exponential-6 and a more generalized Lennard-Jones(m-n). Analysis reveals that the relationship exists only when n = 6 and that two sets of scaling factors (as functions of index m) applies for the relationship between Exponential-6 and the Lennard-Jones(m-6), whereby m > 6.     

Relationship Between Morse and Murrell-Mottram Potentials at Long Range

This paper develops a relationship between the Morse potential and the Murrell-Mottram's 2-body potential. By expressing both potentials in terms of a repulsive term and an attractive term, and approximating Maclaurin's expansion, comparison of coefficients and indices of the repulsive and attractive terms leads to parametric connections between these two potential functions. Non-dimensional curves of these potentials at long range show good agreement as compared to those obtained previously. A set of parametric relationships obtained recently, together with the presently proposed connections, are useful in paving a way for the development of a potential function converter.     

The quest for the best nonpolarizable water model from the adaptive force matching method

The recently introduced adaptive force matching (AFM) method is used to develop a significantly improved pair‐wise nonpolarizable potential for water. A rigid version of the potential is also presented to enable larger time steps for biological simulations. In this work, it is demonstrated that the AFM method can be used to systematically assess the importance of each functional term during the construction of a force field. For a water potential, it is established that a single off‐atom charge center (M) in the plane of water outperforms two out‐of‐plane charge sites for reproducing intermolecular forces. The four‐site pair‐wise nonpolarizable force field developed in this work rivals some of the most sophisticated polarizable models in terms of reproducing accurate ab initio forces. The force fields are parameterized to perform best in the temperature range from 0 to 40°C. Equilibrium and dynamical properties calculated with the flexible and rigid force fields are in good agreement with experimental results. For the flexible model, the agreement improves when path integral simulation is performed. These force fields provide high‐quality results at a very low computational cost and are thus well suited to atomistic scale biological simulations. The AFM method provides a mechanism for selecting important terms in force field expressions and is a very promising tool for producing accurate force fields in condensed phases. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Contact Potential Difference of Au and GaInAs by Electrostatic Force Microscopy

 For a metallic surface (Au) and highly doped (N+) and (P+) semiconductor surfaces (GaInAs) and for localised zones (2 × 2 μm) we have measured using an electrostatic force microscope the variation of the gradient of the electrostatic force by the signal (phase of the oscillating movement of the metallised tip) as a function of the sample-tip potential difference (− 4 V to + 4 V). In both cases the signal shows a quadratic variation with the sample-tip potential difference. The variation of the signal is of the order of magnitude of the theoretical predictions obtained by modelling the shape of the tip by a truncated cone + a portion of a sphere. Using the parabolic curve that fits the experimental results, the value of the contact potential difference, corresponding to a zero value of the electrostatic force gradient, can be determined with an accuracy of 50 mV. The contact potential difference, measured between the metallised tip and the metal (Au), taken as a reference, allows the work function of the metal tip to be determined (5.25 eV). The values of the contact potential difference for the GaInAs (N+) and (P+) surfaces can be explained by the Fermi level pinning due to surface charges.     

Searching for globally optimal functional forms for interatomic potentials using genetic programming with parallel tempering

Molecular dynamics and other molecular simulation methods rely on a potential energy function, based only on the relative coordinates of the atomic nuclei. Such a function, called a force field, approximately represents the electronic structure interactions of a condensed matter system. Developing such approximate functions and fitting their parameters remains an arduous, time-consuming process, relying on expert physical intuition. To address this problem, a functional programming methodology was developed that may enable automated discovery of entirely new force-field functional forms, while simultaneously fitting parameter values. The method uses a combination of genetic programming, Metropolis Monte Carlo importance sampling and parallel tempering, to efficiently search a large space of candidate functional forms and parameters. The methodology was tested using a nontrivial problem with a well-defined globally optimal solution: a small set of atomic configurations was generated and the energy of each configuration was calculated using the Lennard-Jones pair potential. Starting with a population of random functions, our fully automated, massively parallel implementation of the method reproducibly discovered the original Lennard-Jones pair potential by searching for several hours on 100 processors, sampling only a minuscule portion of the total search space. This result indicates that, with further improvement, the method may be suitable for unsupervised development of more accurate force fields with completely new functional forms.     

分子力场进展

分子力学（简称ＭＭ）是近年来化学家常用的一种计算方法。与量子力学从头计算和半经验方法相比，用分子力学处理大分子可以大大节省计算时间，而且，在大多数情况下，用分子力学方法计算得到的分子几何构型参数与实验值之间的差值可在实验误差范围之内。所以，分子力学是研究生物化学体系的有效和可行的手段。分子力学的核心是分子力场。本文介绍了分子力场的量子力学背景、分子力场和光谱力场之间的关系。分子力场的一般形式、分力