Direct Minimization Method Research Articles

Direct free energy minimization methods: application to grain boundaries

A critical review is given of recently developed methods for determining the atomic structures and solute concentration profiles at defects in elemental solids and substitutional alloys as a function of temperature. Exact results are given for the effective force on an atom arising from the vibrational entropy in the quasiharmonic approximation and for the occupancy of a site in the pair potential approximation. An improved, approximate formula is given for the effective force arising from the vibrational entropy. The mean field approximation that is used in the alloy problem is compared with the auto-correlation approximation. It is shown that the better statistical averaging of the auto-correlation approximation leads to effective pair interactions that are temperature and concentration dependent.

The Conjugate-Gradient Variational Analysis and Initialization Method: An Application to MONEX SOP 2 Data

Abstract A conjugate-gradient variational blending technique, based on the method of direct minimization, has been developed and applied to the problem of initialization in a limited-area model in the summer monsoon region. The aim is to blend gridded winds from a high-resolution limited-area analysis with a lower-resolution global analysis for use in a limited-area model that uses the, global analyst for boundary conditions. The ability of the variational matching approach in successfully blending meteorological analyses of varying resolutions is demonstrated. Reasonable agreement is found between the blended analyses and the imposed weak constraints, together with an adequate rate of convergence in the unconstrained minimization procedure. The technique is tested on the 1979 onset vortex vortex case using data from the FGGE Summer MONEX campaign. The results indicate that the forecasts made from the variationally matched analyses show positive impact and perform better than those from the unblended anal...

Two-configuration potential energy surface for the Ca + HF → CaF + H reaction

The three-dimensional potential energy surface for the reaction of Ca and HF has been obtained from a two-configuration direct minimization method using an extended GTO basis set. Features of the surface were examined by fitting calculated values using cubic splines. The height of the barrier to reaction was found to be quite insensitive to the direction of the impinging Ca atom for a wide range of angles of approach. The transition state occurs in the exit channel at an angle of approach of 77°, 33.6 kcal/mole (or 14.8 kcal/mole if the correlation energy error is considered) above the reactants asymptote. A further analysis of the basis set superposition error and the correlation energy error is presented, these results modify the dynamical properties of the system and give a criterion to test the validity of the previously reported two-configuration PES for BeFH and MgFH systems.

Two-configuration potential energy surface for the collinear Ca + HF → CaF + H reaction

The collinear potential energy surface and the evolution of the saddle with the collision angle for the reaction Ca + HF have been determined using a two-configuration direct minimization (TCDM) method and an extended GTO basis set. Properties of the surface are examined by fitting calculated values by means of a cubic spline procedure. Reactive mechanisms and similarities with the surface calculated for similar reactions are discussed in the light of experimental results.

A minimal multiconfigurational technique.

A direct minimization method previously presented by the authors is applied here to biconfigurational wave functions. A very moderate increasing in the time by iteration with respect to the one-determinant calculation and good convergence properties have been found. So qualitatively correct studies on singlet systems with strong biradical character can be performed with a cost similar to that required by Hartree-Fock calculations.

RHF potential energy surface for the collinear reaction of Na with HF

The potential energy surface for the reaction of Na and HF has been obtained from a RHF direct minimization method using an extended GTO basis set. Features of the surface were examined by fitting calculated values using cubic splines. The transition state occurs in the exit channel 40.9 Kcal/mole above the reactants asymptote. Possible dynamical properties of the system are also discussed.

Two-configuration mc potential energy surface for the reaction of Mg with HF

The three-dimensional potential energy surface for the reaction of Mg and HF has been obtained from a two-configuration MC direct minimization method using an extended GTO basis set. Features of the surface were examined by fitting calculated values using cubic splines. The height of the barrier to reaction was found to be quite insensitive to the direction of the impinging Mg atom for a wide range of angles of approach. The transition state occurs in the exit channel at an angle of approach of 75°, 42 kcal/mole above the reactants asymptote. Alternative paths to reaction and possible dynamical properties of the system are also discussed.

Improved calculations for anharmonic oscillators using the gradient method

We report results of energy calculations for the ground state and first excited state of anharmonic oscillators with HamiltonianH=p2+μx2+βx2q. The calculations have been performed for μ=0, ± 1,q=2, 3,..., 6 and some sets of β in the range 10−6≤β≤106. We have used as a method of direct energy minimization a simple form of the gradient method. The energy has been computed with at least 15 exact figures (in the cases μ=−1, β=10−3,q=2 and μ=0 or 1, β=104,q=6) in virtue of a new, general criterion for the optimal choice of both the dimension and the size parameter of the basis.

Solution of constrained hartree-fock and hartree-fock-bogolubov equations by a direct energy minimization method

A new formulation of the energy minimization method to solve constrained Hartree-Fock and Hartree-Fock-Bogolubov equations is proposed, in which attention is focused on the determination of Lagrange multipliers. Some preliminary results about pairing correlations obtained with this method within the HFB approximation are reported.

Direct Minimization Methods for Adaptive Control of Non-Minimum Phase Systems

Adaptive control of non-minimum phase systems is not straightforward. Direct pole-placement methods fail, since they involve cancellation of an unstable model. Indirect methods, based on explicit identification, have the problem that the model may occasionally be non-stabilizable.In this paper another approach is studied. The control objective is formulated as an optimization criterion. It thus contains minimum-variance and dead-beat control as special cases. The criterion is minimized on-line with respect to the regulator parameters. Auxiliary parameters, necessary to determine the gradients, are estimated using an Instrumental Variable Technique. The results are compared with other approaches in simulations.

Self-consistent molecular orbital methods. XVI. Numerically stable direct energy minimization procedures for solution of Hartree–Fock equations

Direct energy minimization methods are examined for solution of the Hartree–Fock equations in a basis expansion. Given a set of trial spin– orbitals, an initial direction for a univariate search of spin–orbital space is specified using general energy-weighted coordinates. The first energy minimum in such a search then provides an initial point for the next iterative step. Techniques are proposed for the maintenance of spin–orbital continuity during this process. Comparison of various energy-weighted coordinates indicates that most effective convergence is achieved if the search path is chosen to pass through the point corresponding to the classical procedure involving the construction of successive approximations to the Fock matrix. The utility of the new method is illustrated by the completion of a previous systematic study of Hartree–Fock energies for small organic species.

Point defects in molybdenum

Abstract The displacement fields around an isolated vacancy, an interstitial and a number of divacancy configurations in molybdenum have been calculated using a method of direct minimization of the potential energy of the real-space atomic configuration. The atomic interaction is described by a short-range empirical pair-wise potential which is matched to the elastic constants and the vacancy formation energy (2·2 eV). The interstitial formation energy is found to be 8 eV and the most stable divacancy configuration occurs at a separation of [2, 0, 0]α with a binding energy of 0·33 eV.

The convergence properties of direct methods of energy minimization with respect to linear coefficients in the LCAO-MO-SCF approach

It is shown that in the LCAO-MO-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner first suggested by Fletcher, then the Hessian of the problem is singular. It is suggested that this singularity may well account for the slow convergence observed using direct energy minimization methods to solve the SCF problem. Ways of avoiding the consequences of this singularity are discussed.

Some Consequences of the Nonlinearity of the Hartree-Fock Approach, Demonstrated on the Example of the PPP Model for Closed Shell Alternant Hydrocarbons

It is shown that the nonuniqueness of the H–F solutions of the PPP model for alternant hydrocarbons is generally a consequence of the fact that the part E2 of the energy expectation value E = E1 + E2, connected with two electron operators in the Hamiltonian, can exhibit various minima. Existence conditions for minima of E2 are given. These minima have bond orders corresponding to distribution of pure single and pure double bonds as in classical chemical structural formulas (Kekulé and Dewar formulas). There are cases for which the superposition of the part E1 does not erase the minima corresponding to Kekulé and Dewar formulas. By the direct minimization method, two distinct electronic H–F distributions are obtained for realistic or nearly realistic parametrizations and geometries in the case of annulenes and long polyenes. Cases of different H–F solutions with the same symmetry as the symmetry of the molecule itself are found. A conjecture is made that there may be two stable geometric configurations for very long polyenes. The possible connection is discussed between the strange features of the Hartree-Fock approach for long linear polyenes and their instability.

An algorithm for real-time computer control of a tracking system with a nonlinearity

A tracking antenna control system with a nonlinearity is considered. The system is uniformly sampled and the control signal is computed at each sampling interval to achieve optimal position alignment with minimum energy expenditure. A direct minimization method permitting efficient real-time implementation is adopted.