Application of minimization methods in calculating atoms with several open shells

Basing on minimization methods, earlier suggested algorithms for the solution of a many-electron problem in Hartree-Fock-Roothaan approximation for systems with close and open shells extend over Roothaan-Hartree-Fock atomic theory (Roothaan-Bagus method). In present work the expressions for energy derivatives with respect to elements of density matrices and nonlinear parameters of atomic orbitals — orbital exponents — have been obtained to solve Hartree-Fock (HF) equations in algebraic approximation. It is possible to create an algorithm of the first-order minimization or quasi-Newton method on their basis. Calculations of atoms and ions with several open shells were carried out by minimization methods. The energy values, close to the results of numerical solution of HF equations with high accuracy of virial relation, were gained using a sufficiently narrow basic set of Slater-type AO.     

The effect of numerical error on the reproducibility of molecular geometry optimizations

Geometry optimization is one of the most often applied techniques in computational drug discovery. Although geometry optimization routines are generally deterministic, the minimization trajectories can be extremely sensitive to initial conditions, especially in case of larger systems such as proteins. Simple manipulations such as coordinate transformations (translations and rotations), file saving and retrieving, and hydrogen addition can introduce small variations ( approximately 0.001 A) in the starting coordinates which can drastically affect the minimization trajectory. With large systems, optimized geometry differences of up to 1 A RMSD and final energy differences of several kcal/mol can be observed when using many commercially available software packages. Differences in computer platforms can also lead to differences in minimization trajectories. Here we demonstrate how routine structure manipulations can introduce small variations in atomic coordinates, which upon geometry optimization, can give rise to unexpectedly large differences in optimized geometries and final energies. We also show how the same minimizations run on different computer platforms can also lead to different results. The implications of these findings on routine computational chemistry procedures are discussed.     

The molecular mechanics of valinomycin. I. Energy minimization calculations on the uncomplexed ionophore

We have used energy minimization calculations to study a number of conformations of uncomplexed valinomycin. In certain cases, x-ray diffraction atomic coordinates were used directly as input coordinates, while in other cases, conformations were found by altering the x-ray coordinates prior to minimization. Five calculated conformations are reported along with their relative energies. The conformation found theoretically to be the most stable is in agreement with earlier, cruder calculations, but does not correspond to the predominant conformation observed in nonpolar solvents. A possible rationale is presented.     

Study on the maximum accuracy of the pseudopotential density functional method with localized atomic orbitals versus plane-wave basis sets

A detailed study on the accuracy attainable with numerical atomic orbitals in the context of pseudopotential first-principles density functional theory is presented. Dimers of first- and second-row elements are analyzed: bond lengths, atomization energies, and Kohn-Sham eigenvalue spectra obtained with localized orbitals and with plane-wave basis sets are compared. For each dimer, the cutoff radius, the shape, and the number of the atomic basis orbitals are varied in order to maximize the accuracy of the calculations. Optimized atomic orbitals are obtained following two routes: (i) maximization of the projection of plane wave results into atomic orbital basis sets and (ii) minimization of the total energy with respect to a set of primitive atomic orbitals as implemented in the OPENMX software package. It is found that by optimizing the numerical basis, chemical accuracy can be obtained even with a small set of orbitals.     

An algorithm for calculating atomic D states with explicitly correlated gaussian functions

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the hamiltonian and overlap matrix elements determined with respect to the gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.     

Correlated geminal wave function for molecules: an efficient resonating valence bond approach

We show that a simple correlated wave function, obtained by applying a Jastrow correlation term to an antisymmetrized geminal power, based upon singlet pairs between electrons, is particularly suited for describing the electronic structure of molecules, yielding a large amount of the correlation energy. The remarkable feature of this approach is that, in principle, several resonating valence bonds can be dealt simultaneously with a single determinant, at a computational cost growing with the number of electrons similar to more conventional methods, such as Hartree-Fock or density functional theory. Moreover we describe an extension of the stochastic reconfiguration method, which was recently introduced for the energy minimization of simple atomic wave functions. Within this extension the atomic positions can be considered as further variational parameters, which can be optimized together with the remaining ones. The method is applied to several molecules from Li(2) to benzene by obtaining total energies, bond lengths and binding energies comparable with much more demanding multiconfiguration schemes.     

A fast SHAKE algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations

A common method for the application of distance constraints in molecular simulations employing Cartesian coordinates is the SHAKE procedure for determining the Lagrange multipliers regarding the constraints. This method relies on the linearization and decoupling of the equations governing the atomic coordinate resetting corresponding to each constraint in a molecule, and is thus iterative. In the present study, we consider an alternative method, M‐SHAKE, which solves the coupled equations simultaneously by matrix inversion. The performances of the two methods are compared in simulations of the pure solvents water, dimethyl sulfoxide, and chloroform. It is concluded that M‐SHAKE is significantly faster than SHAKE when either (1) the molecules contain few distance constraints (solvent), or (2) when a high level of accuracy is required in the application of the constraints. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 501–508, 2001     

Configuration interaction study of H2 and H3 systems

The configuration interaction method has been applied to the H₂ and H₃ systems. The effect of increasing the size of the atomic Slater-type orbital basis has been studied. A minimization procedure with respect to orbital exponents has been carried out.     

Direct minimization of the energy functional in the LCAO-MO density matrix formalism

A previously proposed method of energy minimization is developed for MC SCF wavefunctions formed by all-pair excitations for a closed-shell system. The orbital coefficients are optimized by a gradient approach using a suitable orthogonal transformation of the atomic basis, while optimum CI coefficients are determined solving the usual secular problem for the lowest eigenvalue, after each optimization of the orbitals. Applications to LiH and NH₃ molecules show that the method is numerically well stable, and is capable of accounting for a large part of the correlation energy giving results which compare well with those of the conventional CI method.     

Ethanol adsorption on the Si (111) surface: first principles study

Equilibrium atomic configurations and electron energy structure of ethanol adsorbed on the Si (111) surface are studied by the first principles density functional theory. Geometry optimization is performed by the total energy minimization method. Equilibrium atomic geometries of ethanol, both undissociated and dissociated, on the Si (111) surface are found and analysed. Reaction pathways and predicted transition states are discussed in comparison with available experimental data in terms of the feasibility of the reactions occurring. Analysis of atom and orbital resolved projected density of states indicates substantial modifications of the Si surface valence and conduction electron bands due to the adsorption of ethanol affecting the electronic properties of the surface.     

Torsional flexing: Conformational searching of cyclic molecules in biased internal coordinate space

A new stochastic (Monte Carlo) procedure, termed torsional flexing, has been devised for searching the conformational space of cyclic molecules. Torsional flexing causes a local, torsion angle-biased, distortion of a ring bond in a cyclic molecule. Because torsional flexing does not cause large atomic movements, even when it is applied to several bonds simultaneously, subsequent energy minimization generally proceeds rapidly. Nevertheless, the torsional flexing method is prone to generate structures that cross energy barriers so that the structure resulting after energy minimization is frequently a different conformer of the cyclic molecule. Conformational searches on cycloheptadecane, oxobrefeldin A, cyclopenta-L -alanine, and rifamycin SV based upon torsional flexing indicated that torsional flexing is among the best methods yet devised for searching the conformational space of flexible cyclic molecules. © 1993 John Wiley & Sons, Inc.     

Magnetically driven growth of anthracene thin films by organic molecular beam deposition

The possible use of a static magnetic field during organic molecular beam deposition of thin molecular films for inducing some preferential growth is discussed and the magnetic properties of diamagnetic molecules and molecular crystals are recalled. Considering prototypical materials, namely anthracene molecules and potassium phthalate substrates, which interact and may give rise to polycrystalline films with specific orientations, we show that in the presence of a magnetic field the films display a macroscopic preferential orientation as a result of minimization of the magnetic energy contribution. A very good agreement between the results of optical spectroscopy, atomic force microscopy, and predictions made on the basis of the anisotropic magnetic susceptibility of anthracene is found.     

Linear-scaling implementation of molecular electronic self-consistent field theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Lowdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introduction of a level-shift parameter in the RH Newton equations. A great advantage of the LS-TRRH method is that the optimal level shift can be determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the level-shifted Newton equations. For density averaging, the authors use the trust-region density-subspace minimization (TRDSM) method, which, unlike the traditional direct inversion in the iterative subspace (DIIS) scheme, is firmly based on the principle of energy minimization. When combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix (including a boxed fitting of the electron density), LS-TRRH and TRDSM methods constitute the linear-scaling trust-region SCF (LS-TRSCF) method. The LS-TRSCF method compares favorably with the traditional SCF/DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges to a saddle point.     

New space warping method for the simulation of large-scale macromolecular conformational changes

A space warping method, facilitating the modeling of large-scale conformational changes in mesoscopic systems, is presented. The method uses a set of "global (or collective) coordinates" that capture overall behavior, in conjunction with the set of atomic coordinates. Application of the space warping method to energy minimization is discussed. Several simulations where the method is used to determine the energy minimizing structures of simple central force systems are analyzed. Comparing the results and behavior of the space warping method to simulations involving atomic coordinates only, it is found that the space warping method scales better with system size and also finds lower minima when the potential energy surface has multiple minima. It is shown that the transformation of [Ala16]+ in vacuo from linear to globular is captured efficiently using the space warping method.     

Molecule‐specific determination of atomic polarizabilities with the polarizable atomic multipole model

Recently, many polarizable force fields have been devised to describe induction effects between molecules. In popular polarizable models based on induced dipole moments, atomic polarizabilities are the essential parameters and should be derived carefully. Here, we present a parameterization scheme for atomic polarizabilities using a minimization target function containing both molecular and atomic information. The main idea is to adopt reference data only from quantum chemical calculations, to perform atomic polarizability parameterizations even when relevant experimental data are scarce as in the case of electronically excited molecules. Specifically, our scheme assigns the atomic polarizabilities of any given molecule in such a way that its molecular polarizability tensor is well reproduced. We show that our scheme successfully works for various molecules in mimicking dipole responses not only in ground states but also in valence excited states. The electrostatic potential around a molecule with an externally perturbing nearby charge also exhibits a near‐quantitative agreement with the reference data from quantum chemical calculations. The limitation of the model with isotropic atoms is also discussed to examine the scope of its applicability. © 2012 Wiley Periodicals, Inc.     

Detailed analysis of grid-based molecular docking: A case study of CDOCKER-A CHARMm-based MD docking algorithm

The influence of various factors on the accuracy of protein-ligand docking is examined. The factors investigated include the role of a grid representation of protein-ligand interactions, the initial ligand conformation and orientation, the sampling rate of the energy hyper-surface, and the final minimization. A representative docking method is used to study these factors, namely, CDOCKER, a molecular dynamics (MD) simulated-annealing-based algorithm. A major emphasis in these studies is to compare the relative performance and accuracy of various grid-based approximations to explicit all-atom force field calculations. In these docking studies, the protein is kept rigid while the ligands are treated as fully flexible and a final minimization step is used to refine the docked poses. A docking success rate of 74% is observed when an explicit all-atom representation of the protein (full force field) is used, while a lower accuracy of 66-76% is observed for grid-based methods. All docking experiments considered a 41-member protein-ligand validation set. A significant improvement in accuracy (76 vs. 66%) for the grid-based docking is achieved if the explicit all-atom force field is used in a final minimization step to refine the docking poses. Statistical analysis shows that even lower-accuracy grid-based energy representations can be effectively used when followed with full force field minimization. The results of these grid-based protocols are statistically indistinguishable from the detailed atomic dockings and provide up to a sixfold reduction in computation time. For the test case examined here, improving the docking accuracy did not necessarily enhance the ability to estimate binding affinities using the docked structures.     

Revised algorithms for the build-up procedure for predicting protein conformations by energy minimization

The build-up procedure for predicting low-energy conformations of polypeptides has been extended to cover the case of peptides in aqueous solutions. The revised procedure consists of five steps to be applied to each stage of the build-up. I. All low-energy minima of each of the two fragments to be joined are combined as starting points for energy minimization of the enlarged fragment, and those minima of the enlarged fragment within a certain upper bound of the lowest energy are retained. II. Whenever one of the combinations in Step I leads to an atomic overlap, the minimization is started again using a pseudoenergy function which remains finite everywhere and becomes equal to the standard energy function when no atoms overlap. III. The minima generated in Steps I and II are culled by ignoring side-chain conformations and retaining only those minima whose backbone conformations differ significantly. IV. The rotameric states of the side chains are optimized, by testing their energy of interaction with the rest of the molecule, and subjecting the whole molecule to a further round of energy minimization if the test indicates that this would reduce the energy. V. The energies of all minima are recomputed with inclusion of a term for solvation and with a smaller upper bound as the criterion for retention. The original build-up procedure consisted of Steps I and III only. Examples are presented showing the effectiveness of the new Steps II and IV in locating low-energy minima, and the problems that remain to be solved, chiefly concerning Step V, are discussed.     

Hyperfine coupling constants of the azafullerenes C19N, C59N, C69N, and C75N

The isomers of the nitrogen-substituted fullerenes (azafullerenes) C19N, C59N, C69N, and C75N are examined using all-electron Gaussian atomic orbital basis density functional theory, to determine the doublet radical geometries and hyperfine coupling constants. We find that the inaccuracy of previously calculated hyperfine coupling constants of C59N resulted from a poor treatment of the geometry optimization. We find that UB3LYP minimization of the radical geometry in the 6-31G basis, followed by single-point evaluation of the hyperfine constants in which an expanded basis is used on the atomic sites of interest, forms an efficient compromise between computational cost and accuracy with respect to experimental hyperfine constants. Using this approach, we assign the hyperfine signals observed in experiments on the C69N radical by calculating the hyperfine coupling constants for all five of the isomers and examine the electron spin density distribution. Finally, we present predicted hyperfine coupling constants for the isomers of C19N and C75N for use in the interpretation of future experiments.     

Use of simplex optimization to improve the performance of an inductively coupled plasma in atomic emission spectrometry

Simplex methods are used in the optimization of many analytical techniques. In inductively coupled plasma—atomic emission spectrometry, various performance criteria can be used, including signal-to-background ratios, signal-to-noise ratios, detection limits, and the minimization of matrix effects. The simplex method can indicate deficiencies in equipment design that restrain performance. The threshold of auto-optimization of analytical instruments has been reached.     

Geometry optimization of atomic clusters using a heuristic method with dynamic lattice searching

In this paper, a global optimization method is presented to determine the global-minimum structures of atomic clusters, where several already existing techniques are combined, such as the dynamic lattice searching method and two-phase local minimization method. The present method is applied to some selected large-sized Lennard-Jones (LJ) clusters and silver clusters described by the Gupta potential in the size range N = 13-140 and 300. Comparison with the results reported in the literature shows that the method is highly efficient and a lot of new global minima missed in previous papers are found for the silver clusters. The method may be a promising tool for the theoretical determination of ground-state structure of atomic clusters. Additionally, the stabilities of silver clusters are also analyzed and it is found that in the size range N = 13-140 there exist 12 particularly stable clusters.