Evaluation Of First Derivatives Research Articles

Boundary element approach to mass and charge transport in electrochemical cells

This paper presents a boundary element scheme for solving a steady-state non-linear problem in electrochemistry involving diffusion, convection and migration of several ions in a dilute solution. In the boundary element formulation, the velocity field is divided into an average and a perturbation, and the fundamental solution of the diffusion-convection equation for constant velocity is applied. The perturbation velocity and migration terms are included through a domain discretization. Singular integrations are encountered in the evaluation of first and second derivatives, necessary in the iteration process. Transformations are performed to avoid directly evaluating Bessel functions for Cauchy and hyper-singular integrations. The development of an implicit-explicit iteration process is shown to reduce the computing time of the solution. Several examples with Dirichlet, Neumann and Robin (mixed) types of boundary conditions are tested to assess the numerical scheme.

Parametric correction of self-adjoint finite element models in the presence of multiple eigenvalues

The subject of this article concerns the calculation of the direct and inverse sensitivities of linear elastodynamic systems. This study is limited to the case of conservative mechanical structures which can be represented by discrete models having symmetric positive definite matrices. The first part reviews the equations for the first derivatives of the eigensolutions with respect to the design variables in the presence of multiple eigenvalues (direct sensitivity). These expressions are derived using the modal method. The second part concerns the parametric correction of a model from the observed data. The correction method envisaged is based on the minimization of a residual which is a function of the distance between the eigensolutions of the model and the system (inverse sensitivity). The fundamental difficulties in the evaluation of the first derivatives of the eigensolutions, when the model possesses multiple eigenvalues, are avoided by the introduction of a generalized parametrization constructed o...

Efficient finite difference derivatives for algebraic modelling systems

A procedure is described which determines Jacobian incidence structure and the constant/nonconstant nature of each Jacobian element via examination of the text of function expression strings. This procedure may be used to minimize the effort required to evaluate by finite differences the Jacobian of a set of functions. Target applications involve algebraic modelling systems and other systems with interpreted functions which require evaluation of first derivatives. Computational experience is presented and discussed.

Compact finite difference schemes with spectral-like resolution

Finite difference schemes providing an improved representation of a range of scales (spectral-like resolution) in the evaluation of first, second, and higher order derivatives are presented and compared with well-known schemes. The schemes may be used on non-uniform meshes and a variety of boundary conditions may be imposed. Schemes are also presented for derivatives at mid-cell locations, for accurate interpolation and for spectral-like filtering. Applications to fluid mechanics problems are discussed.

Trust region methods for structural optimization using exact second order sensitivity

AbstractThe performance of multiplier algorithms for structural optimization has been significantly improved by using trust regions. The trust regions are constructed using analytical second order sensitivity, and within this region, the augmented Lagrangian ϕ is minimized subject to bounds. Evaluation of first and second derivatives of ϕ by the adjoint method does not require derivations of individual (implicit) constraint functions, which makes the method economical. Eight test problems are considered and a vast improvement over previously used multiplier algorithms has been noted. Also, the algorithm is robust with respect to scaling, input parameters and starting designs.

Evaluation of Coulombic lattice sums for vibrational calculations in crystals: an extension of Bertaut's method

The well known general method due to Bertaut for calculating the Coulombic lattice sums in crystals has been extended to evaluating the Coulombic contribution to the dynamical matrices in harmonic lattice-dynamical calculations. The procedure is particularly simple and close to usual crystallographic routines, in line with Bertaut's treatment of the general problem. It involves the evaluation of first derivatives of structure factors for two satellite reflexions of indices h0 + q and h0-q for each reflexion of indices h0, with respect to the primitive lattice, where q is the wave vector; the 'self' contribution can be directly evaluated from the second derivatives of F(h0)'s. A particularly interesting feature is that the contribution of the macroscopic field is already included, and it derives from the two satellites of F(O00).

Analytical differentiation of the energy contribution due to triple excitations in quadratic configuration interaction theory

Formulae for the analytical differentiation of the energy contribution due to triple excitations (T) within quadratic configuration interaction (QCI) theory are derived. Combining these formulae with previously derived formulae for the evaluation of analytical first derivatives for QCI theory with single (S) and double excitations (D), an algorithm is developed to calculate analytical QCISD(T) energy gradients. The applicability of this algorithm is demonstrated by calculating the equilibrium geometry of CH 2OO at the QCISD(T)/6-31G(d, p) level of theory.

Analytical differentiation of the energy contribution due to triple excitations in fourth-order Møller-Plesset perturbation theory

Formulae for the analytical differentiation of the energy contribution due to triple (T) excitations within fourth-order Møller-Plesset (MP4) perturbation theory are derived. Combining these formulae with previously derived formulae for the evaluation of analytical first derivatives of the energy contributions due to single (S), double (D), and quadruple (Q) excitations at MP4 (Chem. Phys. Letters 138 ( 1987 ) 131 ), an algorithm is developed to calculate analytical MP4 (SDTQ) energy gradients. Various ways of implementing this algorithm on a computer are discussed and the applicability of the corresponding computer programs is demonstrated by calculating equilibrium geometries, dipole moments, harmonic vibrational frequencies, and infrared intensities for some test molecules at the MP4 (SDTQ) level. The importance of triple excitations for an accurate description of multiple bonds is emphasized.

The expectation value coupled-cluster method and analytical energy derivatives

A new method for correlation energies and properties based upon the coupled-cluster expectation value (XCC) expression is developed. The size-extensive XCC method is shown to satisfy the generalized Hellmann-Feynman theorem, aiding the evaluation of first and higher derivatives to define molecular properties or to search energy surfaces and predict vibrational spectra. Though truncated at order n, XCC( n) methods include effects of infinite-order correlation, containing MBPT( n) as a special case. In XCC(5), the contributions from connected quadruple excitations arise, but these may be evaluated with only an ∼ n 7 algorithm instead of the ∼ n 10 that might have been expected.

The analytic evaluation of energy first derivatives for two-configuration self-consistent-field configuration interaction (TCSCF-CI) wave functions. Application to ozone and ethylene

An efficient formulation for the analytic evaluation of two-configuration self-consistent-field configuration interaction (TCSCF-CI) energy first derivatives is presented. Use is made of the Z-vector approach of Handy and Schaefer. The procedure outlined does not require the transformation of derivative integrals. We have applied the new procedure to the ground states of ozone and ethylene. For ozone, the (DZP) TCSCF-CI procedure yields a structure which is much improved relative to that obtained using a single-reference description. However, the incorrect ordering of the harmonic vibrational frequencies ω1(a1) and ω3(b2), which was first found using a TCSCF description of O3, remains at the TCSCF-CI level of theory. Anharmonic vibrational frequencies (utilizing TCSCF-CI harmonic frequencies and SCF anharmonic corrections) are reported for ethylene. With a DZP basis set, 11 of the 12 TCSCF-CI fundamental frequencies of C2H4 agree very well with experiment, the mean absolute error being 2.4%. However, the infrared inactive ν8=799 cm−1 is 15.0% below the accepted experimental value. The utilization of a TZ2P basis set resolves this discrepancy. The TZ2P TCSCF-CI ν8=922 cm−1 is 1.9% below experiment, and the average absolute error of all of the TZ2P fundamentals is only 2.6%.

A New dimension to quantum chemistry: Theoretical methods for the analytic evaluation of first, second, and third derivatives of the molecular electronic energy with respect to nuclear coordinates

Perhaps even more important to quantum chemistry than enhancements in computer technology has been the theoretical development of analytical derivative methods. It may be realistically stated that such analytic derivative methods provide a new dimension to molecular electronic structure theory. The combination of the two effects (computational advances and theoretical progress) has led to explosive growth in the field. This account primarily describes theoretical developments at Berkeley over the past five years. Analytic first, second and third derivative methods have been developed for single-configuration self-consistent-field (SCF) wavefunctions. Analytic first and second derivative techniques now exist for multiconfiguration (MC) SCF and configuration interaction (CI) wavefunctions. These methods have been used to address a variety of chemical problems, some of which are described here.

First derivatives of correlated wave functions by a matrix‐oriented method: Preliminary application to molecular dipole and quadrupole moments

AbstractA direct, matrix‐oriented procedure in the context of the self‐consistent electron pairs (SCEP) method has been implemented for the evaluation of first derivatives of the energy of a variational configuration interaction wavefunction. This has been applied in calculating dipole and quadrupole moments of the correlated charge distributions of several small molecules. As already established, the analytical differentiation being carried out yields properties slightly different than those calculated by integration with the one‐electron density. The matrix‐oriented analytical differentiation is computationally competitive with construction of the one‐electron density.

On the Convergence of Some Interval-Arithmetic Modifications of Newton’s Method

In this paper we consider the meanwhile well-known interval-arithmetic modifications of the Newton's method and of the so-called simplified Newton’s method for solving systems of nonlinear equations. Starting with an interval-vector which contains a zero, we give for the first time sufficient conditions for the convergence of these methods to this solution. If the starting interval-vector contains no solution, then under the very same conditions the methods under consideration will break down after a finite number of steps. The interval-arithmetic evaluation of the first derivative is only involved in these conditions.

Some unexpected relationships between first, second and third derivative electron repulsion integrals for diatomic and triatomic molecules

It is demonstrated that for diatomic molecules, second and third derivative electron repulsion integrals need not be explicitly evaluated. Similarly for triatomic molecules an analogous simplification applies to all second derivative and 94% of all third derivative integrals. Such higher integral derivatives may be determined from intermediate quantities required in the evaluation of integral first derivatives, in conjunction with translational and rotational invariance properties. This result also has implications for theoretical studies of tetra-atomic and larger molecules. Specifically, the evaluation of most two- and three-center integral derivatives becomes very efficient.

A Newton-type curvilinear search method for optimization

A new search method is presented for unconstrained optimization. The method requires the evaluation of first and second derivatives and defines a curve along which a undimensional step takes place. For large step-size, the method performs as Newton's method, but it does not fail where the latter fails. For small step-size, the method behaves as the gradient method.