A hybrid technique of orthonormality constrained orbital optimization in SCF calculations

A recently proposed orthonormality constrained orbital optimization technique is operationally modified further by coupling it to a gradient biased method, namely the steepest descent procedure of McWeeny. The hybrid technique developed in this way is shown to have better convergence properties in closed and unrestricted open-shell calculations. The technique can be adapted to MCSCF procedures as well. The important role played by "orbital symmetries" in the operation of the method is analysed. Similarities and differences of the present method with the orthogonal gradient method are pointed out. Possible avenues of circumventing convergence difficulty that one may encounter in pathological cases, particularly in ab initio calculations involving extended basis set, are suggested.     

Convergence problems in SCF calculations: Further applications of a new technique based on the use of inverse Fock operator

A recently proposed orbital optimization technique based on the use of the inverse of the level-shifted Fock operator is successfully applied to a large number of pathologically divergent cases. The possibility of eliminating convergence problems by adopting a similar modification in the operation of the orthogonal gradient method is also successfully tested.     

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

A general theory is presented for the optimization of the coefficients of orbitals and configuration interaction expansion in the case of multiconfiguration wavefunctions containing all single excitations. The orbital coefficients are optimized by suitable orthogonal transformations of the atomic basis; the Cl coefficients are determined solving the usual secular problem. The energy minimization is performed directly by a gradient approach. The method works both for ground and excited states and no convergence difficulties are met. Computational examples are given for H₂O and H₂S molecules.     

Superlinearly converging dimer method for transition state search

Algorithmic improvements of the dimer method [G. Henkelman and H. Jonsson, J. Chem. Phys. 111, 7010 (1999)] are described in this paper. Using the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimizer for the dimer translation greatly improves the convergence compared to the previously used conjugate gradient algorithm. It also saves one energy and gradient calculation per dimer iteration. Extrapolation of the gradient during repeated dimer rotations reduces the computational cost to one gradient calculation per dimer rotation. The L-BFGS algorithm also improves convergence of the rotation. Thus, three to four energy and gradient evaluations are needed per iteration at the beginning of a transition state search, while only two are required close to convergence. Moreover, we apply the dimer method in internal coordinates to reduce coupling between the degrees of freedom. Weighting the coordinates can be used to apply chemical knowledge about the system and restrict the transition state search to only part of the system while minimizing the remainder. These improvements led to an efficient method for the location of transition states without the need to calculate the Hessian. Thus, it is especially useful in large systems with expensive gradient evaluations.     

Electrochemistry-enabled fabrication of orthogonal nanotopography and surface chemistry gradients for high-throughput screening

Gradient surfaces are emerging tools for investigating mammalian cell-surface interactions in high throughput. We demonstrate the electrochemical fabrication of an orthogonal gradient platform combining a porous silicon (pSi) pore size gradient with an orthogonal gradient of peptide ligand density. pSi gradients were fabricated via the anodic etching of a silicon wafer with pore sizes ranging from hundreds to tens of nanometers. A chemical gradient of ethyl-6-bromohexanoate was generated orthogonally to the pSi gradient via electrochemical attachment. Subsequent hydrolysis and activation of the chemical gradient allowed for the generation of a cyclic RGD gradient. Whilst mesenchymal stem cells (MSC) were shown to respond to both the topographical and chemical cues arising from the orthogonal gradient, the MSC's responded more strongly to changes in RGD density than to changes in pore size during short-term culture.     

Modified one-electron hamiltonian method in the MC SCF theory

For orbital optimization within the MC SCF theory a modification of the OEH method is proposed with the direction of descent determined according to the Fletcher–Reeves gradient method. The combined method developed on this basis ensures the convergence of the iterative process when the Hessian singularities occur. The convergence properties of the method proposed are studied by performing the ab initio water molecule calculations using two types of multiconfigurational wave functions.     

Multiparametric curve fitting XIV. Modus operandi of the least-squares algorithm MINOPT

Hybrid least-squares algorithm MINOPT for a nonlinear regression is introduced. MINOPT from CHEMSTAT package combines fast convergence of the Gauss-Newton method in a vicinity of minimum with good convergence of gradient methods for location far from a minimum. Quality of minimization and an accuracy of parameter estimates for six selected models are examined and compared with different derivative least-squares methods of five commercial regression packages.     

Daubechies wavelets as a basis set for density functional pseudopotential calculations

Daubechies wavelets are a powerful systematic basis set for electronic structure calculations because they are orthogonal and localized both in real and Fourier space. We describe in detail how this basis set can be used to obtain a highly efficient and accurate method for density functional electronic structure calculations. An implementation of this method is available in the ABINIT free software package. This code shows high systematic convergence properties, very good performances, and an excellent efficiency for parallel calculations.     

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.     

A space-saving modification of Davidson's eigenvector algorithm

A modification of Davidson's eigenvalue algorithm, based on the conjugate gradient method, is described. This method needs storage only for a few vectors (five to seven, depending on the implementation), making it practical for very large problems where disk storage is the limiting factor, without the necessity of restarting or discarding some expansion vectors. The convergence characteristics of the modified method are essentially identical with those of the original Davidson method if all expansion vectors are retained in the latter.     

New iterative scheme for a simultaneous calculation of m first eigenstates of a real symmetric matrix

A new iterative scheme for a simultaneous calculation of the m lowest eigenvalues together with their eigenvectors has been derived for a real symmetric matrix. The scheme is based on the orthogonal gradient method and is easy to use for large-scale configuration-interaction calculations of electronic wave functions. A variant of the scheme deals with nonorthogonal basis functions, which are particularly simple in the case of the bonded-function method of Boys.     

Analysis of the genetic algorithm method of molecular conformation determination

Many important problems in chemistry require knowledge of the 3-D conformation of a molecule. A commonly used computational approach is to search for a variety of low-energy conformations. Here, we study the behavior of the genetic algorithm (GA) method as a global search technique for finding these low-energy conformations. Our test molecule is cyclic hexaglycine. The goal of this study is to determine how to best utilize GAs to find low-energy populations of conformations given a fixed amount of CPU time. Two measures are presented that help monitor the improvement in the GA populations and their loss of diversity. Different hybrid methods that combine coarse GA global search with local gradient minimization are evaluated. We present several specific recommendations about trade-offs when choosing GA parameters such as population size, number of generations, rate of interaction between subpopulations, and combinations of GA and gradient minimization. In particular, our results illustrate why approaches that emphasize convergence of the GA can actually decrease its effectiveness as a global conformation search method. © John Wiley & Sons, Inc.     

An efficient method for constructing nonorthogonal localized molecular orbitals

A new method for constructing nonorthogonal localized molecular orbitals (NOLMOs) is presented. The set of highly localized NOLMOs is obtained by minimization of the spread functional starting from an initial set of canonical orthogonal molecular orbitals. To enhance the stability and efficiency, the centroids of the NOLMOs are constrained to be those of the corresponding orthogonal localized molecular orbitals (OLMOs), which are obtained with the Boys criterion in advance. In particular, these centroid constraints make the optimization for each NOLMO independent of the others, which is an attractive feature for application to large systems. The minimization with the constraints incorporated through the multiplier-penalty function method is stable and efficient in convergence. While exhibiting the classical bonding pattern in chemistry and sharing a spatial distribution similar to that of the corresponding OLMOs, the obtained NOLMOs are more compact than the corresponding OLMOs with about 10%-28% reduction in the value of the spread functional and devoid of the troublesome "orthogonalization tails."     

Direct perturbation theory in terms of energy derivatives: scalar-relativistic treatment up to sixth order

A formulation of sixth-order direct perturbation theory (DPT) to treat relativistic effects in quantum-chemical calculations is presented in the framework of derivative theory. Detailed expressions for DPT6 are given at the Hartree-Fock level in terms of the third derivative of the energy with respect to the relativistic perturbation parameter defined as λ(rel)=c(-2). They were implemented for the computation of scalar-relativistic energy corrections. The convergence of the scalar-relativistic DPT expansion is studied for energies and first-order properties such as dipole moment and electric-field gradient within the series of the hydrogen halides (HX, X = F, Cl, Br, I, and At). Comparison with spin-free Dirac-Coulomb calculations indicates that the DPT series exhibits a smooth and monotonic convergence. The rate of convergence, however, depends on the charge of the involved nuclei and significantly slows down for heavy-element compounds.     

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

The general multiconfiguration self-consistent-field method is presented along the density matrix formalism. The proposed optimization procedure for orbitals makes use of an orthogonal transformation in the space spanned by the fixed basis set. Acting on the unconstrained parameters of the transformation a direct minimization of the energy expression is performed using a gradient approach. A similar method may also be applied to the optimization of the expansion coefficients. The method works not only for the ground state of a given system, but also for any excited state, yielding an upper bound to the true energy of the considered state.     

An effective calculation procedure for two-phase equilibria in multireaction systems

In this work, we present a method for the calculation of two-phase equilibria in multireaction systems. The procedure uses a transformed composition variable in the orthogonal derivatives of the Gibbs function and the tangent plane equation to form a system of non-linear equations. We solve this system with a Newton–Raphson method and our initialization procedure uses results from the reactive stability analysis and the reactive equal area rule. With this initialization strategy, convergence occurs with only a few iterations in the numerical method. Several examples with multiple chemical reactions demonstrate the performance of our approach.     

Photopolymerized diffusion-defined polyacrylamide gradient gels for on-chip protein sizing

We report on a facile diffusion-based photopatterning technique for generating linear and non-linear decreasing pore-size gradients in cross-linked polyacrylamide gels. Diffusion of low viscosity polymer precursor solutions and a two-step photopatterning process were used to define the decreasing pore-size gradient gels in a microfluidic format, thus eliminating the need for controlled mixing and delivery of polymer precursor solutions. We present an analytical model of the non-steady state diffusion process and numerically evaluate that model for direct comparison with empirical characterizations of the gradient gels. We show that the analytical model provides an effective means to predict the steepness and linearity of a desired gradient gel prior to fabrication. To assess electrophoretic assay performance in the microfluidic gradient gels, on-chip sizing of protein samples (20-116 kDa) was investigated. Baseline resolution of six proteins was demonstrated in 4 s using 3.5% to 10% polyacrylamide gradient gels. The demonstrated ability to conduct efficient protein sizing in ultra-short separation lengths (0.3 cm) means low applied electric potentials are needed to achieve the electric field strengths required for protein separations. The low required electric potentials relax operating constraints on electrical components, as is especially important for translation of the assay into pre-clinical and clinical settings. The gradient gel fabrication method reported is amenable to adaptation to non-sizing protein assays, as well as integration with upstream sample preparation steps and subsequent orthogonal downstream assays.     

Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm

In this article, we study the numerical solution of singularly perturbed non-linear autonomous initial-value problems by a non-standard algorithm on layer-resolving nonuniform meshes. Here, we use the piecewise-uniform Shishkin meshes, and two other nonuniform meshes which resolve the difficulties arising from the steep gradient of the solution in the initial layer. The present method is intended for solving the nonlinear problem without linearization and provides third-order convergence results. Linear stability of this method is studied. Numerical experiments are carried out to verify the efficiency and accuracy of the method.     

Concentration and fractionation by isoelectric trapping in a micropreparative-scale multicompartmental electrolyzer having orthogonal pH gradients. Part 1

A multicompartmental electrolyzer called ConFrac has been developed and tested for micropreparative-scale isoelectric trapping separations. ConFrac contains n separate, minimalistic isoelectric trapping core units, each with a separate anode compartment, anodic flow-through compartment, collection compartment, cathodic flow-through compartment and a shared cathode compartment. The collection compartments are all isolated from each other and have volumes of 100 μL each. The liquid held in the collection compartments is stagnant. The respective anodic and cathodic flow-through compartments are hydraulically serially connected to each other by flexible, minimum-length, narrow internal diameter tubes. The respective feed solutions whose volumes are larger and variable are recirculated through the serially connected flow-through compartments. Poly(vinyl alcohol)-based buffering membranes are placed between the anode compartments, anodic flow-through compartments, collection compartments, cathodic flow-through compartments and cathode compartment. The membranes establish two orthogonal pH gradients in ConFrac. The primary pH gradient is parallel with the direction of the recirculating flows and orthogonal to that of the electric field. The secondary pH gradient is parallel with the direction of the electric field and orthogonal to that of the recirculating flows. Since the recirculating liquids are kept in thermostated reservoirs and the residence times in the flow-through compartments are shorter than 2 s, ConFrac can tolerate power loads as high as 2 W without overheating the solutions. The operation and performance of ConFrac has been quantitatively characterized: four 25 μM ampholytic components were isolated from 5 mL of feed solution in 20 min and their concentration increased approximately 50-fold.     

A new Lanczos-based algorithm for simulating high-frequency two-dimensional electron spin resonance spectra

The Lanczos algorithm (LA) is a useful iterative method for the reduction of a large matrix to tridiagonal form. It is a storage efficient procedure requiring only the preceding two Lanczos vectors to compute the next. The quasi-minimal residual (QMR) method is a powerful method for the solution of linear equation systems, Ax = b. In this report we provide another application of the QMR method: we incorporate QMR into the LA to monitor the convergence of the Lanczos projections in the reduction of large sparse matrices. We demonstrate that the combined approach of the LA and QMR can be utilized efficiently for the orthogonal transformation of large, but sparse, complex, symmetric matrices, such as are encountered in the simulation of slow-motional 1D- and 2D-electron spin resonance (ESR) spectra. Especially in the 2D-ESR simulations, it is essential that we store all of the Lanczos vectors obtained in the course of the LA recursions and maintain their orthogonality. In the LA-QMR application, the QMR weight matrix mitigates the problem that the Lanczos vectors lose orthogonality after many LA projections. This enables substantially more Lanczos projections, as required to achieve convergence for the more challenging ESR simulations. It, therefore, provides better accuracy for the eigenvectors and the eigenvalues of the large sparse matrices originating in 2D-ESR simulations than does the previously employed method, which is a combined approach of the LA and the conjugate-gradient (CG) methods, as evidenced by the quality and convergence of the 2D-ESR simulations. Our results show that very slow-motional 2D-ESR spectra at W-band (95 GHz) can be reliably simulated using the LA-QMR method, whereas the LA-CG consistently fails. The improvements due to the LA-QMR are of critical importance in enabling the simulation of high-frequency 2D-ESR spectra, which are characterized by their very high resolution to molecular orientation.