Application of some methods of constrained optimization to the calculation of the molecular strain energy

The Lagrange multipliers method is applied to the minimization algorithm of the molecular potential energy proposed by Boyd in order to keep the bond lengths constant during the optimization. The results obtained for a series of molecules and the approximations supposed in the new algorithm are analysed.The first steps to include the penalty methods in the minimization of the molecular potential energy with constraints in the valence coordinates are given.     

Newton trajectories for finding stationary points on molecular potential energy surfaces

We present a new algorithm for computing Newton trajectories based on the Quadratic String Method (QSM) and explain how this can be used to find key stationary points on the molecular potential energy surface (PES). This method starts by using the intersections of Newton trajectories to locate stationary points on the PES. These points could then be used to determine the minimum energy path. The new method, called QSM-NT, is shown to be efficient and reliable for both analytical potential energy surfaces and potential energy surfaces computed from quantum chemistry calculations. The advantages and pitfalls of this method for exploring PES are discussed. In particular, the problem of discontinuous Newton trajectories is elucidated.     

铝原子Bernal多面体团簇的理论研究

将遗传算法用于铝原子团簇的构型计算.运用这种方法,从任意构型开始,较好地计算了6、8、9、10个铝原子组成的原子团簇的能量最低时的构型,发现这四种铝原子团簇的能量最低构型分别取四种Bernal多面体排列.并对得到的四种构型用密度泛函方法（DFT）进行量子化学计算,结果表明，这类构型是势能面上的极小值点,可以稳定存在.     

A reduced-restricted-quasi-Newton–Raphson method for locating and optimizing energy crossing points between two potential energy surfaces

We present a method for the location and optimization of an intersection energy point between two potential energy surfaces. The procedure directly optimizes the excited state energy using a quasi-Newton–Raphson method coupled with a restricted step algorithm. A linear transformation is also used for the solution of the quasi-Newton–Raphson equations. The efficiency of the algorithm is analyzed and demonstrated in some examples. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :992–1003, 1997     

Global energy minimization of alanine dipeptide via barrier function methods

This paper presents an interior point method to determine the minimum energy conformation of alanine dipeptide. The CHARMM energy function is minimized over the internal coordinates of the atoms involved. A barrier function algorithm to determine the minimum energy conformation of peptides is proposed. Lennard-Jones 6-12 potential which is used to model the van der Waals interactions in the CHARMM energy equation is used as the barrier function for this algorithm. The results of applying the algorithm for the alanine dipeptide structure as a function of varying number of dihedral angles are reported, and they are compared with that obtained from genetic algorithm approach. In addition, the results for polyalanine structures are also reported.     

Energy controlled insertion of polar molecules in dense fluids

We present a method to search low energy configurations of polar molecules in the complex potential energy surfaces associated with dense fluids. The search is done in the configurational space of the translational and rotational degrees of freedom of the molecule, combining steepest-descent and Newton-Raphson steps which embed information on the average sizes of the potential energy wells obtained from prior inspection of the liquid structure. We perform a molecular dynamics simulation of a liquid water shell which demonstrates that the method enables fast and energy-controlled water molecule insertion in aqueous environments. The algorithm finds low energy configurations of incoming water molecules around three orders of magnitude faster than direct random insertion. This method represents an important step towards dynamic simulations of open systems and it may also prove useful for energy-biased ensemble average calculations of the chemical potential.     

On the determination of the minimum on the crossing seam of two potential energy surfaces

An improved gradient-based algorithm is presented for the determination of the minimum energy point on the crossing seam hypersurface between two arbitrary potential energy hypersurfaces. The Hessian matrix is updated employing the gradient information. The method is demonstrated in a study of some representative cases including charge-transfer states of a typical molecular-device molecule (a rigid spiro π – σ – π molecular cation) with, as well as without, an external electric field.     

B-spline method for energy minimization in grid-based molecular mechanics calculations

A method is described for molecular mechanics calculations based on a cubic B-spline approximation of the potential energy. This method is useful when parts of the system are allowed to remain fixed in position so that a potential energy grid can be precalculated and used to approximate the interaction energy between parts of a molecule or between molecules. We adapted and modified the conventional B-spline method to provide an approximation of the Empirical Conformational Energy Program for Peptides (ECEPP) potential energy function. The advantage of the B-spline method over simpler approximations is that the resulting B-spline function is C2 continuous, which allows minimization of the potential energy by any local minimization algorithm. The standard B-spline method provides a good approximation of the electrostatic energy; but in order to reproduce the Lennard–Jones and hydrogen-bonding functional forms accurately, it was necessary to modify the standard B-spline method. This modification of the B-spline method can also be used to improve the accuracy of trilinear interpolation for simulations that do not require continuous derivatives. As an example, we apply the B-spline method to rigid-body docking energy calculations using the ECEPP potential energy function. Energies are calculated for the complex of Phe-Pro-Arg with thrombin. For this system, we compare the performance of the B-spline method to that of the standard pairwise summation in terms of speed, accuracy, and overhead costs for a variety of grid spacings. In our rigid-body docking calculations, the B-spline method provided an accurate approximation of the total energy of the system, and it resulted in an 180-fold reduction in the time required for a single energy and gradient calculation for this system. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 71–85, 1998     

Theoretical study on the potential energy surface and vibrational energy levels of HXeI

The potential energy surface for the electronic ground state of the HXeI molecule is constructed by using the internally contracted multi-reference configuration interaction with the Davidson correction（icMRCI＋Q）method and large basis sets. The stabilities and dissociation barriers are identified from the potential energy surfaces.The three-body dissociation channel is found to be the dominate dissociation channel for HXeI.Based on the obtained potentials,vibrational energy levels of HXeI are calculated using the Lanczos algorithm.Our theoretical results are in excellent agreement with the available observed values.     

Force reversed method for locating transition states

To identify the transition state accurately and efficiently on a high-dimensional potential energy surface is one of the most important topics in kinetic studies on chemical reactions. We present here an algorithm to search the transition state by so-called force reversed method, which only requires a rough reaction direction instead of knowing the initial state and final state. Compared to the nudged elastic band method and the dimer method that require multiple images, the present algorithm with only single image required saves significantly the computational cost. The algorithm was implemented in the first-principle periodic total energy calculation package and applied successfully to several prototype surface processes such as the adsorbate diffusion and dissociation on metal surfaces. The results indicate that the force reversed method is efficient, robust to identify the transition state of various surface processes.     

Reaction path following by quadratic steepest descent

An efficient steepest descent algorithm for the integration of minimum energy paths, based on local quadratic approximations of the potential energy surface, is presented. The algorithm incorporates a selection procedure for the points at which the second derivatives of the energy are calculated fully or partially, thus minimizing the computational effort while maintaining high accuracy. This makes the method especially well suited for application in variational transition state theory calculations with tunnelling corrections, which have very high accuracy requirements. The performance of the algorithm is illustrated by ab initio calculations for four chemical reactions of differing complexity. The overall computational cost is less than for, or comparable to that of, first- or second-order algorithms published previously. Received: 28 July 1998 / Accepted: 10 August 1998 / Published online: 28 October 1998     

Global energy minimization by rotational energy embedding

Given a sufficiently good empirical potential function for the internal energy of molecules, prediction of the preferred conformations is nearly impossible for large molecules because of the enormous number of local energy minima. Energy embedding has been a promising method for locating extremely good local minima, if not always the global minimum. The algorithm starts by locating a very good local minimum when the molecule is in a high-dimensional Euclidean space, and then it gradually projects down to three dimensions while allowing the molecule to relax its energy throughout the process. Now we present a variation on the method, called rotational energy embedding, where the descent into three dimensions is carried out by a sequence of internal rotations that are the multidimensional generalization of varying torsion angles in three dimensions. The new method avoids certain kinds of difficulties experienced by ordinary energy embedding and enables us to locate conformations very near the native for avian pancreatic polypeptide and apamin, given only their amino acid sequences and a suitable potential function.     

Theoretical Studies on the Potential Energy Surface and Vibrational Energy Levels of HXeBr

The potential energy surface for the electronic ground state of the HXeBr molecule is constructed from more than 4200 ab initio points calculated using the internally contracted multi-reference configuration interaction method with the Davidson correction （icMRCI ＋ Q）. The stabilities and dissociation barriers are identified from the potential energy surface. The three-body dissociation channel is found to be the dominant dissociation channel for HXeBr. Low-lying vibrational energy levels of HXeBr calculated using the Lanczos algorithm are found to be in good agreement with the available experimental band origins.     

Algorithm for preliminary evaluation of the correlation time of local dynamics in some polymeric materials

A method for preliminary evaluation of the correlation time of local dynamics of the polymeric chains, using the experimental nuclear magnetic resonance spin-lattice relaxation data, is proposed. The algorithm is based on the model of the passage of a particle over a potential energy barrier. This method was utilized for some molten polymers and concentrated polymeric solutions.     

A hybrid elastic band string algorithm for studies of enzymatic reactions

A common challenge in theoretical biophysics is the identification of a minimum energy path (MEP) for the rearrangement of a group of atoms from one stable configuration to another. The structure with maximum energy along the MEP approximates the transition state for the process and the energy profile itself permits estimation of the transition rates. In this work we describe a computationally efficient algorithm for the identification of minimum energy paths in complicated biosystems. The algorithm is a hybrid of the nudged elastic band (NEB) and string methods. It has been implemented in the pDynamo simulation program and tested by examining elementary steps in the reaction mechanisms of three enzymes: citrate synthase, RasGAP, and lactate dehydrogenase. Good agreement is found for the energies and geometries of the species along the reaction profiles calculated using the new algorithm and previous versions of the NEB and string techniques, and also those obtained by the common method of adiabatic exploration of the potential energy surface as a function of predefined reaction coordinates. Precisely refined structures of the saddle points along the paths may be subsequently obtained with the climbing image variant of the NEB algorithm. Directions in which the utility of the methods that we have implemented can be further improved are discussed.     

Identification of ligand binding sites on proteins using a multi-scale approach

Identification of a ligand binding site on a protein is pivotal to drug discovery. To date, no reliable and computationally feasible general approach to this problem has been published. Here we present an automated efficient method for determining binding sites on proteins for potential ligands without any a priori knowledge. Our method is based upon the multiscale concept where we deal with a hierarchy of models generated using a k-means clustering algorithm for the potential ligand. This is done in a simple approach whereby a potential ligand is represented by a growing number of feature points. At each increasing level of detail, a pruning of potential binding site is performed. A nonbonding energy function is used to score the interactions between molecules at each step. The technique was successfully employed to seven protein-ligand complexes. In the current paper we show that the algorithm considerably reduces the computational effort required to solve this problem. This approach offers real opportunities for exploiting the large number of structures that will evolve from structural genomics.     

Using a pruned basis, a non-product quadrature grid, and the exact Watson normal-coordinate kinetic energy operator to solve the vibrational Schrödinger equation for C2H4

In this paper we propose and test a method for computing numerically exact vibrational energy levels of a molecule with six atoms. We use a pruned product basis, a non-product quadrature, the Lanczos algorithm, and the exact normal-coordinate kinetic energy operator (KEO) with the π(t)μπ term. The Lanczos algorithm is applied to a Hamiltonian with a KEO for which μ is evaluated at equilibrium. Eigenvalues and eigenvectors obtained from this calculation are used as a basis to obtain the final energy levels. The quadrature scheme is designed, so that integrals for the most important terms in the potential will be exact. The procedure is tested on C(2)H(4). All 12 coordinates are treated explicitly. We need only ~1.52 × 10(8) quadrature points. A product Gauss grid with which one could calculate the same energy levels has at least 5.67 × 10(13) points.     

Automatic generation of potential energy and property surfaces of polyatomic molecules in normal coordinates

A procedure for the automatic construction of Born-Oppenheimer (BO) potential energy and molecular property surfaces in rectilinear normal coordinates is presented and its suitability and accuracy when combined with vibrational structure calculations are assessed. The procedure relies on a hierarchical n-mode representation of the BO potential energy or molecular property surface, where the n-mode term of the sequence of potentials/molecular properties includes only the couplings between n or less vibrational degrees of freedom. Each n-mode cut of the energy/molecular property surface is first evaluated in a grid of points with ab initio electronic structure methods. The ab initio data are then spline interpolated and a subsequent polynomial fitting provides an analytical semiglobal representation for use in vibrational structure programs. The implementation of the procedure is outlined and the accuracy of the method is tested on water and difluoromethane. Strategies for improving the proposed algorithm are also discussed.     

Bridging coarse-grained models by jump-in-sample simulations

We present an efficient method to construct coarse-grained (CG) models from models of finer resolution. The method estimates the free energies in a generated sample of the CG conformational space and then fits the entire effective potential surface in the high-dimensional CG conformational space. A jump-in-sample algorithm that uses a random jumping walk in the CG sample is used to iteratively estimate the free energies. We test the method in a tetrahedral molecular fluid where we construct the intermolecular effective potential and evaluate the CG molecular model. Our algorithm for calculating the free energy involves an improved Wang-Landau (WL) algorithm, which not only works more efficiently than the standard WL algorithm, but also can work in high-dimensional spaces.     

用于真实蛋白质结构预测的一种新的优化方法

用"相对熵"作为优化函数,提出了一个有效快速的折叠预测优化算法.使用了非格点模型,预测只关心蛋白质主链的走向.其中只用到了蛋白质主链上的两两连续的Cα原子间的距离信息以及20种氨基酸的接触势的一个扩展形式.对几个真实蛋白质做了算法测试,预测的初始结构都为比较大的去折叠态,预测构象相对于它们天然结构的均方根偏差(RMSD)为5～7 A.从原理上讲,该方法是对能量优化的改进.