Deterministic global optimization of molecular structures using interval analysis

The search for the global minimum of a molecular potential energy surface is a challenging problem. The molecular structure corresponding to the global minimum is of particular importance because it usually dictates both the physical and chemical properties of the molecule. The existence of an extremely large number of local minima, the number of which may increase exponentially with the size of the molecule, makes this global minimization problem extremely difficult. A new strategy is described here for solving such global minimization problems deterministically. The methodology is based on interval analysis, and provides a mathematical and computational guarantee that the molecular structure with the global minimum potential energy will be found. The technique is demonstrated using two sets of example problems. The first set involves a relatively simple potential model, and problems with up to 40 atoms. The second set involves a more realistic potential energy function, representative of those in current use, and problems with up to 11 atoms.     

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.     

SASS: a symmetry adapted stochastic search algorithm exploiting site symmetry

A simple symmetry adapted search algorithm (SASS) exploiting point group symmetry increases the efficiency of systematic explorations of complex quantum mechanical potential energy surfaces. In contrast to previously described stochastic approaches, which do not employ symmetry, candidate structures are generated within simple point groups, such as C2, Cs, and C2v. This facilitates efficient sampling of the 3N-6 Pople's dimensional configuration space and increases the speed and effectiveness of quantum chemical geometry optimizations. Pople's concept of framework groups [J. Am. Chem. Soc. 102, 4615 (1980)] is used to partition the configuration space into structures spanning all possible distributions of sets of symmetry equivalent atoms. This provides an efficient means of computing all structures of a given symmetry with minimum redundancy. This approach also is advantageous for generating initial structures for global optimizations via genetic algorithm and other stochastic global search techniques. Application of the SASS method is illustrated by locating 14 low-lying stationary points on the cc-pwCVDZ ROCCSD(T) potential energy surface of Li5H2. The global minimum structure is identified, along with many unique, nonintuitive, energetically favorable isomers.     

Assessment of Two Surface Monte Carlo (TSMC) method to find stationary points of (H2O)15 and (H2O)20 clusters

The Two Surface Monte Carlo (TSMC) technique reduces computational cost by using a computationally cheap biasing potential, which guides the molecular system to explore the potential energy surface of interest. It was shown earlier that the Effective Fragment Potential (EFP) can be a good choice for this biasing potential (Bandyopadhyay, J Chem Phys 122:091102, 2005) when the potential energy surface of interest is quantum mechanical. This may help in expanding the applicability of TSMC, since finding a good biasing potential is a major challenge. In the present work, the viability of TSMC method in finding stationary points of large molecular system is investigated using EFP as the biasing potential and RHF theory as the potential of interest. TSMC is applied to find the stationary points of water clusters of size 15 and 20. A semi-automated method starting from random geometries, without using any chemical intuition, found several stationary points. The simulated annealing method was used to refine the structures obtained from TSMC. Among the several low-energy structures obtained for 15 water cluster, one minimum, about 1 kcal/mol higher than the global minimum, was found. However, for 20 water cluster, no structure very close to the global minimum was obtained. Several strategies, learned from the experience of the present work, are discussed for improving the TSMC method, including the acceptance between the two energy surfaces. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Comparison of methods for finding saddle points without knowledge of the final states

Within the harmonic approximation to transition state theory, the biggest challenge involved in finding the mechanism or rate of transitions is the location of the relevant saddle points on the multidimensional potential energy surface. The saddle point search is particularly challenging when the final state of the transition is not specified. In this article we report on a comparison of several methods for locating saddle points under these conditions and compare, in particular, the well-established rational function optimization (RFO) methods using either exact or approximate Hessians with the more recently proposed minimum mode following methods where only the minimum eigenvalue mode is found, either by the dimer or the Lanczos method. A test problem involving transitions in a seven-atom Pt island on a Pt(111) surface using a simple Morse pairwise potential function is used and the number of degrees of freedom varied by varying the number of movable atoms. In the full system, 175 atoms can move so 525 degrees of freedom need to be optimized to find the saddle points. For testing purposes, we have also restricted the number of movable atoms to 7 and 1. Our results indicate that if attempting to make a map of all relevant saddle points for a large system (as would be necessary when simulating the long time scale evolution of a thermal system) the minimum mode following methods are preferred. The minimum mode following methods are also more efficient when searching for the lowest saddle points in a large system, and if the force can be obtained cheaply. However, if only the lowest saddle points are sought and the calculation of the force is expensive but a good approximation for the Hessian at the starting position of the search can be obtained at low cost, then the RFO approaches employing an approximate Hessian represent the preferred choice. For small and medium sized systems where the force is expensive to calculate, the RFO approaches employing an approximate Hessian is also the more efficient, but when the force and Hessian can be obtained cheaply and only the lowest saddle points are sought the RFO approach using an exact Hessian is the better choice. These conclusions have been reached based on a comparison of the total computational effort needed to find the saddle points and the number of saddle points found for each of the methods. The RFO methods do not perform very well with respect to the latter aspect, but starting the searches further away from the initial minimum or using the hybrid RFO version presented here improves this behavior considerably in most cases.     

Simulated annealing to locate various stationary points in semiempirical methods

In this work, an algorithm was developed to study the potential energy surfaces in the coordinate spaces of molecules by a nonlocal way, in contrast to classic energy minimizers as the BFGS or the DFP method. This algorithm, based on the specificities of semiempirical methods, mixes simulated annealing and local searches to reduce computation costs. By this technique, the global energy minimum can be localized. Moreover, local minima that are close in energy to the global minimum are also obtained. If the search is not only for minima but for all stationary points (minima, saddle points…), then the energy is replaced by the gradient norm, which reaches its minimum values at stationary points. The annealing process is stopped before having accurately reached the global minimum and generates a list of geometries whose energies (respectively, whose gradients) are optimized by local minimizers. This list of geometries is shortened from the nearly equivalent geometries by a dynamic single-clustering analysis. The energy/gradient local minimizers act on the clustered list to produce a set of minima/stationary points. A targeted search of these points and reduction of the costs are reached by the way of several penalty functions. They eliminate—without energy calculation—most of the points generated by random walks on the potential energy surface. These penalty functions (on the total moment of inertia or on interatomic distances) are specific to the class of problem studied. They account for the nonrupture of either specific chemical bonds or rings in cyclic molecules, they assure that molecular systems are kept bonded, and they avoid the collapsing of atoms. © 1992 John Wiley & Sons, Inc.     

Structural variations and bonding in gold halides: a quantum chemical study of monomeric and dimeric gold monohalide and gold trihalide molecules, AuX, Au2X2, AuX3, and Au2X6 (X = F, Cl, Br, I)

The molecular structures of all gold mono- and trihalides and of their dimers have been calculated at the B3LYP, MP2, and CCSD(T) levels of theory by using relativistic pseudopotentials for all atoms except fluorine. Our computations support the experimental observation that the relative stability of the monohalides increases from the fluoride toward the iodide, while the stability trend of the trihalides is the opposite. The potential energy surface (PES) of all gold trihalides has been investigated. These molecules are typical Jahn-Teller systems; the trigonal planar D3h-symmetry geometry does not correspond to the minimum energy structure for any of them. At the same time, the amount and character of their Jahn-Teller distortion changes gradually from AuF3 to AuI3. The minimum energy geometry is a T-shaped structure for AuF3 and AuCl3, with a Y-shaped transition-state structure. For AuI3, the Y-shaped structure lies lower than the T-shaped structure on the PES. For AuBr3 and AuI3, neither of them is the global minimum but instead an L-shaped structure, which lies outside the Jahn-Teller PES. This structure can be considered to be a donor-acceptor system, or a closed-shell interaction, with I2 acting as donor and AuI as acceptor. The dimers of gold monohalides have very short gold-gold distances and demonstrate the aurophilic interaction. The dimers of the trihalides are planar molecules with two bridging halogen atoms.     

Growth and structural properties of Mg(N) (N = 10-56) clusters: density functional theory study

Using the minima hopping global geometry optimization method on density functional potential energy surface, we have studied the structural and electronic properties of magnesium clusters for a size range of Mg(N) where N = 10-56. Our exhaustive search reveals that most of our global minima are nonsymmetric in the size range above N = 20. We elucidate the evolutionary trend of the entire series and present more details about the peculiar growth of the clusters. For N > 20, it is possible to divide the cluster into two regions: the core region and the surface region. It turns out that the growth follows a peculiar cyclic pattern where the core and surface grow alternatively. The surface energy, as a function of number of atoms shows a clear signature as the number of atoms in the core increases by one. We have also carried out stability analysis and the stable sizes(magic numbers) agree very well with the experimental magic numbers reported by Diederich [J. Chem. Phys. 2011, 134, 124302]. We point out the similarities and differences between our results and sodium clusters.     

The Structures and Stability of HNOS Isomers

Potential energy surface of HNOS system is investigated by means of MP2 method with 6-311 G(d,p) basis set.The energy for each minimum and saddle point on the potential energy surface is corrected at the QCISD(T)/6-311 G(3df,2p) level of theory with zero-point vibrational energy included.As a result ,eighteen isomers are theoretically predicted and cis-HNSO is found to be global minimum on the potential energy surface,Wherein,fourteen isomers are considered as kinetically stable species,and should be experimentally observed.Comparisons are made for HNOS system with its analogues,HNO2 and NHS2.The nature of bonding and isomers‘ stability of HNOS system are similar to HNS2.The obvious similarities and discrepancies among HNOS,HNO2 and HNS2 are attributed to the hypervalent capacity of sulfur,oxygen and nitrogen atoms.     

A dynamic lattice searching method for fast optimization of Lennard-Jones clusters

A highly efficient unbiased global optimization method called dynamic lattice searching (DLS) was proposed. The method starts with a randomly generated local minimum, and finds better solution by a circulation of construction and searching of the dynamic lattice (DL) until the better solution approaches the best solution. The DL is constructed adaptively based on the starting local minimum by searching the possible location sites for an added atom, and the DL searching is implemented by iteratively moving the atom located at the occupied lattice site with the highest energy to the vacant lattice site with the lowest energy. Because the DL can greatly reduce the searching space and the number of the time-consuming local minimization procedures, the proposed DLS method runs at a very high efficiency, especially for the clusters of larger size. The performance of the DLS is investigated in the optimization of Lennard-Jones (LJ) clusters up to 309 atoms, and the structure of the LJ(500) is also predicted. Furthermore, the idea of dynamic lattice can be easily adopted in the optimization of other molecular or atomic clusters. It may be a promising approach to be universally used for structural optimizations in the chemistry field.     

A dynamic lattice searching method with interior operation for unbiased optimization of large Lennard-Jones clusters

For improving the efficiency of dynamic lattice searching (DLS) method for unbiased optimization of large Lennard-Jones (LJ) clusters, a variant of the interior operation (IO) proposed by Takeuchi was combined with DLS. The method is named as DLS-IO. In the method, the IO moves outer atoms with higher energy toward the coordinates center, i.e., (0, 0, 0), of a cluster and a local minimization (LM) follows each IO. This makes the interior atoms more compact and the outer atoms more uniformly distributed with lower potential energy. Therefore, the starting structure for DLS operations is closer to the global optimum compared with the randomly generated structures. On the other hand, a method to identify the central atom is proposed for the central vacancy problem. Optimizations of LJ(500), LJ(561), LJ(660), LJ(665), and LJ(670) were investigated with the DLS-IO, and the structural transition during the optimization was analyzed. It was found that the method is efficient and unbiased for optimization of large LJ clusters, and it may be a promising approach to be universally used for structural optimizations.     

Nonplanarity at tri-coordinated aluminum and gallium: cyclic structures for X3Hn(m) (X = B, Al, Ga)

Structures and energies of X3H3(2-), X3H4-, X3H5, and X3H6+ (X = B, Al and Ga) were investigated theoretically at B3LYP/6-311G(d) level. The global minimum structures of B are not found to be global minima for Al and Ga. The hydrides of the heavier elements Al and Ga have shown a total of seven, six and eight minima for X3H3(2-), X3H(4-), and X3H5, respectively. However, X3H(6+) has three and four minima for Al and Ga, respectively. The nonplanar arrangements of hydrogens with respect to X3 ring is found to be very common for Al and Ga species. Similarly, species with lone pairs on heavy atoms dominate the potential energy surfaces of Al and Ga three-ring systems. The first example of a structure with tri-coordinate pyramidal arrangement at Al and Ga is found in X3H(4-) (2g), contrary to the conventional wisdom of C3H3+, B3H3, etc. The influence of pi-delocalization in stabilizing the structures decreases from X3H3(2-) to X3H6+ for heavier elements Al and Ga. In general, minimum energy structures of X3H4-, X3H5, and X3H6+ may be arrived at by protonating the minimum energy structures sequentially starting from X3H3(2-). The resonance stabilization energy (RSE) for the global minimum structures (or nearest structures to global minimum which contains pi-delocalization) is computed using isodesmic equations.     

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.     

On the investigation of the low-energy conformational space of molecules

A new perspective on traditional energy minimization problems is provided by a connection between statistical thermodynamics and combinatorial optimization (finding the minimum of a function depending on many variables). The joint use of a new method for uncovering the global minimum of intramolecular potential energy functions, based on following the asymptotic behavior of a system of stochastic differential equations, and an iterative-improvement technique, whereby a search for relative minima is made by carrying out local quasi-Newton minimizations starting from many distinct points of the energy hypersurface, proved most effective for investigating the low-energy conformational space of molecules.     

Method of local increments for the calculation of adsorption energies of atoms and small molecules on solid surfaces. 2. CO/MgO(001)

The method of local increments is used in connection with an embedded cluster approach and wave function based quantum chemical ab initio methods to describe the adsorption of a single CO molecule on the MgO(001) surface. The first step in this approach is a conventional Hartree-Fock calculation. The occupied orbitals are then localized by means of the Foster-Boys localization procedure, and the full system is decomposed into several "subunits" that consist of the orbitals localized at the CO molecule and at the Mg and O atoms of the MgO cluster. The correlation energy is expanded into a series of local n-body increments that are evaluated separately and independently. In this way, big savings in computer time can be achieved because (a) the treatment of a large system is replaced with a series of much faster calculations for small subsystems and (b) the big basis sets necessary for describing dispersion effects are only needed for the atoms in the respective subsystem while all other atoms can be treated by medium size Hartree-Fock type basis sets. The coupled electron pair approach, CEPA, an approximate coupled cluster method, is used to calculate the correlation energies of the various subsystems. For the vertical adsorption of CO on top a Mg atom of the MgO(001) surface with the C atom toward Mg, the individual one- and two-body increments are calculated as functions of the CO-MgO separation and a full potential energy curve is constructed from them. A very shallow minimum with an adsorption energy of 0.016 eV at a Mg-C distance of 3.04 ? is found at the Hartree-Fock level, while inclusion of correlation (dispersion) effects shortens the Mg-C distance to 2.59 ? and yields a much larger adsorption energy of 0.124 eV. This is in very good agreement with the best experimental value of 0.14 eV. The basis set superposition error, BSSE, was fully corrected for by the counterpoise method and the bonding mechanism was analyzed at the Hartree-Fock level by means of the constrained space orbital variation, CSOV, analysis.     

Global optimization of binary Lennard-Jones clusters using three perturbation operators

Global optimization of binary Lennard-Jones clusters is a challenging problem in computational chemistry. The difficulty lies in not only that there are enormous local minima on the potential energy surface but also that we must determine both the coordinate position and the atom type for each atom and thus have to deal with both continuous and combinatorial optimization. This paper presents a heuristic algorithm (denoted by 3OP) which makes extensive use of three perturbation operators. With these operators, the proposed 3OP algorithm can efficiently move from a poor local minimum to another better local minimum and detect the global minimum through a sequence of local minima with decreasing energy. The proposed 3OP algorithm has been evaluated on a set of 96 × 6 instances with up to 100 atoms. We have found most putative global minima listed in the Cambridge Cluster Database as well as discovering 12 new global minima missed in previous research.     

Can a physics‐based,all‐atom potential find a protein's native structure among misfolded structures? I. Large scale AMBER benchmarking

Recent work has shown that physics-based, all-atom energy functions (AMBER, CHARMM, OPLS-AA) and local minimization, when used in scoring, are able to discriminate among native and decoy structures. Yet, there have been only few instances reported of the successful use of physics based potentials in the actual refinement of protein models from a starting conformation to one that ends in structures, which are closer to the native state. An energy function that has a global minimum energy in the protein's native state and a good correlation between energy and native-likeness should be able to drive model structures closer to their native structure during a conformational search. Here, the possible reasons for the discrepancy between the scoring and refinement results for the case of AMBER potential are examined. When the conformational search via molecular dynamics is driven by the AMBER potential for a large set of 150 nonhomologous proteins and their associated decoys, often the native minimum does not appear to be the lowest free energy state. Ways of correcting the potential function in order to make it more suitable for protein model refinement are proposed.     

Clever and efficient method for searching optimal geometries of lennard-jones clusters

An unbiased algorithm for determining global minima of Lennard-Jones (LJ) clusters is proposed in the present study. In the algorithm, a global minimum is searched by using two operators: one modifies a cluster configuration by moving atoms to the most stable positions on the surface of a cluster and the other gives a perturbation on a cluster configuration by moving atoms near the center of mass of a cluster. The moved atoms are selected by employing contribution of the atoms to the potential energy of a cluster. It was possible to find new global minima for LJ506, LJ521, LJ536, LJ537, LJ538, and LJ541 together with putative global minima of LJ clusters of 10-561 atoms reported in the literature. This indicates that the present method is clever and efficient for cluster geometry optimization.     

Exploring the free energy surfaces of clusters using reconnaissance metadynamics

A new approach is proposed for exploring the low-energy structures of small to medium-sized aggregates of atoms and molecules. This approach uses the recently proposed reconnaissance metadynamics method [G. A. Tribello, M. Ceriotti, and M. Parrinello. Proc. Natl. Acad. Sci. U.S.A. 107(41), 17509 (2010)] in tandem with collective variables that describe the average structure of the coordination sphere around the atoms/molecules. We demonstrate this method on both Lennard-Jones and water clusters and show how it is able to quickly find the global minimum in the potential energy surface, while exploring the finite temperature free energy surface.     

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.