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.     

Direct minimization of the energy in density functional theory

The minimization of the energy functional of the first-order density matrix γ( r , r ') is achieved using unitary transformations applied to γ. Equivalently, such transformations can be carried out also on one-electron orbitals (natural orbitals) and their occupation (integer or non-integer) numbers. The conventional local density approximation based on the electron density p( r ) is then considered as a special case. The direct minimization of the energy functional of p with respect to the parameters of the unitary transformation leads to stationary conditions that are all equivalent to the Kohn–Sham equations. Preliminary numerical tests show that the proposed algorithms for the direct minimization of the energy work in a satisfactory manner. © John Wiley & Sons, Inc.     

DFT calculation of the electronic and optical properties of Ag2PdO2 from X-ray and neutron crystallographic data

Electronic and optical properties of ternary silver palladium oxide (Ag₂PdO₂) are investigated using density functional theory. Two different possible approximations for the exchange correlation potentials were employed. The X-ray and neutron crystallographic data were optimized by minimization of the forces (1 mRy/a.u.) acting on the atoms. The electronic structure, electron space charge density, chemical bonding and optical dielectric were determined from the relaxed geometry seeking deep insight understanding of this material. Our calculated energy band gap (0.15 eV) shows a good agreement with the experimental value (0.18 eV).     

Topological analysis of electron density maps of chiral cyclodextrin-guest complexes: a steric interaction evaluation

This paper reports interaction energy values for the systems heptakis (2,3,6-tri-O-methyl)-β-cyclodextrin complexed with R- and S-flurbiprofen. An intermolecular steric interaction potential calculation method is applied from the topological analysis of electron density maps and is assessed by comparison with values obtained from the application of conventional molecular mechanics energy minimization procedures. Both topology-based and standard energy minimization methods predict that the crystalline form of the R-flurbiprofen is the most stable one.     

First Step Towards Planning of Syntheses in Solid-State Chemistry: Determination of Promising Structure Candidates by Global Optimization

A method is presented here that allows, in principle, the prediction of the existence and structure of (meta)stable solid compounds. It is based on a set of adjustable modules that are applied to the study of the energy function of the chemical system of interest. The main elements are a set of routines for global optimization and local minimization, as well as algorithms for the investigation of the phase space structure near local minima of the potential energy, and the analysis and characterization of the structure candidates. The current implementation focuses on ionic compounds, for which empirical potentials are used for the evaluation of the energy function in the first stage, and a Hartree–Fock algorithm for refinements. The global optimization is performed with a stochastic simulated annealing algorithm, and the local minimization employs stochastic quenches and gradient methods. The neighborhoods of the local minima are studied with the threshold algorithm. The results of this approach are illustrated with a number of examples: compounds of binary noble gases, and binary and ternary ionic compounds. These include several substances that have not been synthesized yet, but should stand a fair chance of being kinetically stable, for example further alkali metal nitrides besides Li₃N, as well as Ca₃SiBr₂ or SrTi₂O₅.     

A density model based on the Modified Quasichemical Model and applied to the (NaCl + KCl + ZnCl2) liquid

A theoretical model for the density of multicomponent inorganic liquids based on the Modified Quasichemical Model has been presented previously. By introducing in the Gibbs free energy of the liquid phase temperature-dependent molar volume expressions for the pure components and pressure-dependent excess parameters for the binary (and sometimes higher-order) interactions, it is possible to reproduce, and eventually predict, the molar volume and the density of the multicomponent liquid phase using standard interpolation methods. In the present article, this density model is applied to the (NaCl + KCl + ZnCl₂) ternary liquid and a Kohler–Toop-like asymmetric interpolation method is used. All available density data for the (NaCl + KCl + ZnCl₂) liquid were collected and critically evaluated, and optimized pressure-dependent model parameters have been found. This new volumetric model can be used with Gibbs free energy minimization software, to calculate the molar volume and the density of (NaCl + KCl + ZnCl₂) ternary melts.     

Density localization of atomic and molecular orbitals

The density localization method previously described is applied to the homonuclear diatomic molecules Li₂, Be₂, B₂, C₂, N₂, and F₂. The method is based on the minimization of the sum of the interorbital density overlap integrals. The results of the density localization method are compared with the results of the energy localization method of Edmiston and Ruedenberg and with the results of the localization procedures of Boys and of Magnasco and Perico. The agreement among the four methods is in general good. With one exception we obtain also agreement with the classical chemical concepts of electron pairs.

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Zusammenfassung Die Dichtelokalisierungsmethode, die in einer früheren Arbeit beschrieben worden ist, wird auf die homonuklearen zweiatomigen Moleküle Li₂, Be₂, B₂, C₂, N₂ und F₂ angewandt. Die Methode beruht auf der Minimierung der Summe der Dichteüberlappungsintegrale zwischen verschiedenen Orbitalen. Die Ergebnisse der Dichtelokalisierungsmethode werden mit den Resultaten der Energielokalisierungsmethode von Edmiston und Ruedenberg sowie mit denen der Lokalisierungsverfahren von Boys und von Magnasco und Perico verglichen. Die Übereinstimmung der Ergebnisse ist im allgemeinen gut. Mit einer Ausnahme entsprechen sie den klassisch-chemischen Vorstellungen.

     

Study of oxygen adsorption on β-Si3N4(0 0 0 1) by the density functional theory

The density functional theory (DFT) B3LYP method is used to theoretically investigate the interaction of O₂ with the β-Si₃N₄ surface (0 0 0 1) at 1200 °C. All the calculations have been performed at the 6-31G^∗ basis set level using H-saturated cluster. From the total energy minimization, the chemisorption on the center of the molecule lying above an Si site and the molecular axis paralleling to the surface is the most stable. After adsorption, the O–O bond is easier to dissociate compared to the free O₂. The electron transferred from the substrate to the O₂ molecule occupies the O₂ anti-bonding orbital, thus leading to a weakening off the bond strength, which is reflected by the elongated O₂ bond length. The changing trend of the O–O population and vibrational frequency is consistent with the change of the O–O bond length. The significant chemisorption energy and the short adsorption bond length indicate that the oxidation occurs on the β-Si₃N₄(0 0 0 1) surface at 1200 °C more easily.     

Molecular orbital calculations on the conformation of polypeptides and proteins

Quantum mechanical calculations using the PCILO method have been performed on the tripeptide model CH₃CO-X-Y-NHCH₃. Competition between C₅, C₇, C₁₀ rings and open structures has been investigated through mapping of the whole {φ, Ψ} conformational space and energy minimization. From these results, it appears that the C₁₀ ring simulating the folding named U-turn, involving a hydrogen bond between the i...i + 3 residues, is the most probable structure although not the most stable in energy. The results are used for predicting the frequency of U-turns in proteins, α-chymotrypsin is given as an example.     

Framework stability of nanoporous inorganic structures upon template extraction and calcination: a theoretical study of gallophosphate polymorphs

A systematic computational study of gallophosphates was undertaken. First, lattice energy minimization calculations using a formal-charge shell model potential have been carried out on a series of hypothetical gallium phosphates derived from their metallogallophosphate, aluminophosphate, or aluminosilicate analogues through atomic substitution. The minimized structures show the typical features in terms of bond angles and distances as expected in zeolitic gallophosphates. Second, the crystal structures of several gallophosphates in their calcined forms have been predicted, using for each compound lattice energy minimization and an initial model derived from its as-synthesized templated form. All the modified structures thus have the same GaPO(4) composition. The lattice energies of all the simulated gallophosphate structures were compared to that of GaPO(4)-quartz as a reference structure. Interestingly, among all predicted calcined structures, various zeolitic topologies were found. The study of the energetics of these zeotypic structures showed a linear dependence of lattice energy upon density. Strikingly, a few simulated structures showed unrealistic structural features, such as important framework distortions, often associated with the occurrence of a hexameric unit in the original as-synthesized structures. Also, those gallophosphates with structural faults were found in the upper part of the energy/density plot. To address the validity of our force field calculations in these special cases, first principles calculations were undertaken on ULM-4, chosen as a typical representative structure. Indeed, the qualitative agreement found between our results and those obtained with the nonlocal density functional theory demonstrates the robustness of our force field. Further minimization also showed that the inclusion of polarizability is crucial for yielding results comparable with those obtained using first principles methods.     

Density localization of atomic and molecular orbitals

Density localized molecular Orbitals are computed for the molecules LiH, LiF, BF, BN, CO, C₂H₂, CH₄, NH₃, H₂O, and HF. The density localization method is based on the minimization of the sum of the interorbital density overlap integrals. The results of this method are compared to the results of the energy localization method of Edmiston and Ruedenberg and the localization procedures of Boys and of Magnasco and Perico. The agreement among the results obtained by these four methods is in general good. With a few exceptions the localized molecular orbitals agree with the classical chemical concepts.

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Zusammenfassung Dichtelokalisierte Molekülorbitale sind berechnet worden für die Moleküle LiH, LiF, BF, BN, CO, C₂H₂, CH₄, NH₃, H₂O und HF. Die Dichtelokalisierungsmethode beruht auf der Minimisierung der Summe der Dichteüberlappungsintegrale zwischen verschiedenen Orbitalen. Die Ergebnisse dieses Verfahrens werden verglichen mit den Resultaten der Energielokalisierungsmethode von Edmiston und Ruedenberg und den Verfahren von Boys und von Magnasco und Perico. Die übereinstimmung zwischen den verschiedenen Methoden ist im allgemeinen gut. Mit einigen Ausnahmen werden die klassischen chemischen Vorstellungen von Elektronenpaaren reproduziert.

     

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.     

Molecular geometry and excited electronic states. Theoretic study on the molecular geometry and the fluorescence lifetime difference of isometric phenylnapkthalenes1

The equilibrium molecular geometries of 1-phenyl- and 2-phenyl-naphthalenes in the electronic S_o and S₁ states have been calculated by minimization of the total energy with respect to all molecular coordinates. The singlet term systems of both isomers have been determined using these results. Although the fluorescence transition energy is nearly the same in both cases it was found that the corresponding electronic transitions were allowed in 1-phenylnaphthalene but were forbidden in 2-phenylnaphthalene. The result explains the different fluorescence lifetimes observed.     

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.     

Flexible docking simulations: Scaled collective variable Monte Carlo minimization approach using Bezier splines,and comparison with a standard Monte Carlo algorithm

An algorithm for docking a flexible ligand onto a flexible or rigid receptor, using the scaled‐collective‐variables Monte Carlo with energy minimization approach, is presented. Energy minimization is shown to be one of the best techniques for distinguishing between native‐ and nonnative‐generated conformations. Incorporation of this technique into a Monte Carlo procedure enables one to distinguish the native conformation directly during the conformational search. It avoids the generation of a large number of ligand conformers for which more sophisticated energy evaluation tools would have had to be applied to identify the nativelike conformations. The efficiency of the Monte Carlo minimization was greatly improved by incorporating a new grid‐based energy evaluation technique using Bezier splines for which the energy function, as well as all of its derivatives, can be deduced from the values at the grid points. Comparison between our ECEPP/3‐based algorithm and the Monte Carlo algorithm presented elsewhere (Hart, T. N.; Read, R. J. Prot Struct Funct Genet 1992, 13, 206–222) has been made for docking NH₂ D Phe Pro Arg COOH, the noncovalent analog of NH₂ D Phe Pro Arg chloromethylketone (PPACK), onto the active site of human α‐thrombin. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 244–252, 1999     

Structure as a function of relaxation in glassy polymers

Previously, amorphous glasses were simulated by carrying out energy minimization on initial conformations generated by growing polymer chains in a periodic cube. It was not known what degree of relaxation these simulated glasses possessed. The degree relaxation is determined by the thermal history of the bulk polymer and in turn determines numerous important properties. Constant stress molecular dynamics (CSMD) followed by energy minimization was used to simulate different thermal histories of an isotactic poly(propylene) glass. This simulation approach produced glasses in which the degree of energetic relaxation was a function of the thermal history. Based on the simulated cohesive energy density, the simulation using minimization alone produced a glass that was energetically annealed. However, the local geometry and x-ray structure factor indicate that it has a different structure than those obtained using CSMD followed by energy minimization and may not be structurally relaxed.     

Direct energy functional minimization under orthogonality constraints

The direct energy functional minimization problem in electronic structure theory, where the single-particle orbitals are optimized under the constraint of orthogonality, is explored. We present an orbital transformation based on an efficient expansion of the inverse factorization of the overlap matrix that keeps orbitals orthonormal. The orbital transformation maps the orthogonality constrained energy functional to an approximate unconstrained functional, which is correct to some order in a neighborhood of an orthogonal but approximate solution. A conjugate gradient scheme can then be used to find the ground state orbitals from the minimization of a sequence of transformed unconstrained electronic energy functionals. The technique provides an efficient, robust, and numerically stable approach to direct total energy minimization in first principles electronic structure theory based on tight-binding, Hartree-Fock, or density functional theory. For sparse problems, where both the orbitals and the effective single-particle Hamiltonians have sparse matrix representations, the effort scales linearly with the number of basis functions N in each iteration. For problems where only the overlap and Hamiltonian matrices are sparse the computational cost scales as O(M2N), where M is the number of occupied orbitals. We report a single point density functional energy calculation of a DNA decamer hydrated with 4003 water molecules under periodic boundary conditions. The DNA fragment containing a cis-syn thymine dimer is composed of 634 atoms and the whole system contains a total of 12,661 atoms and 103,333 spherical Gaussian basis functions.     

Momentum density and molecular geometry. Bent BH 2 − and linear BH 2 +

Based on a special form of the molecular virial theorem, the recently proposed method of momentum density for interatomic interactions is here applied to the problem of molecular geometry. Two molecules BH ₂ ^– and BH ₂ ⁺ , which have the same nuclear framework but favor respectively bent and linear conformations, are comparatively studied. Using an approximate Hartree-Fock momentum density, the total molecular energy (including the nuclear repulsion) is partitioned into orbital components, and a geometry correlation diagram is derived. An atom-bond partitioning of the total energy is also examined based on the one- and two-center decomposition of the momentum density.     

高硅沸石骨架结构及其稳定性的模拟计算(I)*

The lattice energy of a series of high-silica zeolites was determined using the lattice energy minimization method. The results were compared to the lattice energy of dense polymorphs of SiO2. All high-silica zeolites frameworks are only 30～67kJ•mol-1 less stable than α-quartz This may imply that there is little energy barrier to the formation of high-silica zeolites frame-works and explain the structural diversity observed for high-silica zeolites. The relationships of calculated lattice energies and framework Structures was disscussed. The results revealed a good linear relationship between framework density of these molecular sieves and all-silica framework lattice energies.     

An analysis of the structural and energetic properties of deoxyribose by potential energy methods

We discuss the three fundamental issues of a computational approach in structure prediction by potential energy minimization, and analyze them for the nucleic acid component deoxyribose. Predicting the conformation of deoxyribose is important not only because of the molecule's central conformational role in the nucleotide backbone, but also because energetic and geometric discrepancies from experimental data have exposed some underlying uncertainties in potential energy calculations. The three fundamental issues examined here are: (i) choice of coordinate system to represent the molecular conformation; (ii) construction of the potential energy function; and (iii) choice of the minimization technique. For our study, we use the following combination. First, the molecular conformation is represented in cartesian coordinate space with the full set of degrees of freedom. This provides an opportunity for comparison with the pseudorotation approximation. Second, the potential energy function is constructed so that all the interactions other than the nonbonded terms are represented by polynomials of the coordinate variables. Third, two powerful Newton methods that are globally and quadratically convergent are implemented: Gill and Murray's Modified Newton method and a Truncated Newton method, specifically developed for potential energy minimization. These strategies have produced the two experimentally-observed structures of deoxyribose with geometric data (bond angles and dihedral angles) in very good agreement with experiment. More generally, the application of these modeling and minimization techniques to potential energy investigations is promising. The use of cartesian variables and polynomial representation of bond length, bond angle and torsional potentials promotes efficient second-derivative computation and, hence, application of Newton methods. The truncated Newton, in particular, is ideally suited for potential energy minimization not only because the storage and computational requirements of Newton methods are made manageable, but also because it contains an important algorithmic adaptive feature: the minimization search is diverted from regions where the function is nonconvex and is directed quickly toward physically interesting regions.