The hyperbolic chemical bond: Fourier analysis of ground and first excited state potential energy curves of HX (X = H-Ne)

RHF/aug-cc-pVnZ, UHF/aug-cc-pVnZ, and QCISD/aug-cc-pVnZ, n = 2-5, potential energy curves of H2 X (1) summation g (+) are analyzed by Fourier transform methods after transformation to a new coordinate system via an inverse hyperbolic cosine coordinate mapping. The Fourier frequency domain spectra are interpreted in terms of underlying mathematical behavior giving rise to distinctive features. There is a clear difference between the underlying mathematical nature of the potential energy curves calculated at the HF and full-CI levels. The method is particularly suited to the analysis of potential energy curves obtained at the highest levels of theory because the Fourier spectra are observed to be of a compact nature, with the envelope of the Fourier frequency coefficients decaying in magnitude in an exponential manner. The finite number of Fourier coefficients required to describe the CI curves allows for an optimum sampling strategy to be developed, corresponding to that required for exponential and geometric convergence. The underlying random numerical noise due to the finite convergence criterion is also a clearly identifiable feature in the Fourier spectrum. The methodology is applied to the analysis of MRCI potential energy curves for the ground and first excited states of HX (X = H-Ne). All potential energy curves exhibit structure in the Fourier spectrum consistent with the existence of resonances. The compact nature of the Fourier spectra following the inverse hyperbolic cosine coordinate mapping is highly suggestive that there is some advantage in viewing the chemical bond as having an underlying hyperbolic nature.     

Solutions to linear matrix ordinary differential equations via minimal,regular, and excessive space extension based universalization

This work focuses on the solution of the linear matrix ordinary differential equations where the first derivative of the unknown matrix is equal to the same unknown matrix premultiplied by a given matrix polynomially varying with the independent variable. Work aims to get a universal form for this equation by using the space extension concept where new unknowns are defined to get more amenable form for the equation. The convergence of the series solution to this equation obtained via minimal, regular, and excessive space extension is also investigated with the aid of an appropriate norm analysis which also enables us to get error estimates for the truncated series solutions. A few illustrative examples are presented for practical convergence issues like approximation quality.     

Comparison of self-consistent field convergence acceleration techniques

The recently proposed ADIIS and LIST methods for accelerating self-consistent field (SCF) convergence are compared to the previously proposed energy-DIIS (EDIIS) + DIIS technique. We here show mathematically that the ADIIS functional is identical to EDIIS for Hartree-Fock wavefunctions. Convergence failures of EDIIS + DIIS reported in the literature are not reproduced with our codes. We also show that when correctly implemented, the EDIIS + DIIS method is generally better than the LIST methods, at least for the cases previously examined in the literature. We conclude that, among the family of DIIS methods, EDIIS + DIIS remains the method of choice for SCF convergence acceleration.     

Spectral convergence of the quadrature discretization method in the solution of the Schrodinger and Fokker-Planck equations: comparison with sinc methods

Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered.     

A Note on Mathematical Relationships Among Bond-Torsion Force Fields

A set of mathematical relationship between torsion potential functions such as trigonometric and Fourier series is presented herein. A harmonic approximation form is also introduced, and its stiffness constant is related to the parameters of trigonometric and Fourier series. Mathematical relationships between various force field parameters are presented in the form of conversion matrices.     

Diagonalization free implementation of Kohn-Sham equations with localized basis sets

A new strategy to solve the Kohn-Sham equations of density functional theory is presented which avoids diagonalization within a finite basis-set expansion. The implementation is based on an expansion of orbitals in terms of Gaussian functions and it is shown that the algorithm is competitive with more conventional approaches. The new approach is based on conjugated gradients optimization augmented by an approximate second-order update together with convergence acceleration. Computational advantages of the new algorithm are discussed under the special aspect of parallel computing. © 1997 John Wiley & Sons, Inc.     

Singularities of Møller-Plesset energy functions

The convergence behavior of M?ller-Plesset (MP) perturbation series is governed by the singularity structure of the energy, with the energy treated as a function of the perturbation parameter. Singularity locations, determined from quadratic approximant analysis of high-order series, are presented for a variety of atoms and small molecules. These results can be used as benchmarks for understanding the convergence of low-order methods such as MP4 and for developing and testing summation methods that model the singularity structure. The positions and types of singularities confirm previous qualitative predictions based on functional analysis of the Schrodinger equation.     

Numerical Evaluation of Three-Center Two-Electron Coulomb and Hybrid Integrals Over B Functions Using the HD and H\overline D Methods and Convergence Properties

The difficulties of the numerical evaluation of three-center two-electron Coulomb and hybrid integrals over B functions, arise mainly from the presence of the hypergeometric series and semi-infinite very oscillatory integrals in their analytical expressions, which are obtained using the Fourier transform method.This work presents a general approach for accelerating the convergence of these integrals by first demonstrating that the hypergeometric function, involved in the analytical expressions of the integrals of interest, can be expressed as a finite sum and by applying nonlinear transformations for accelerating the convergence of the semi-infinite oscillatory integrals after reducing the order of the differential equation satisfied by the integrand.The convergence properties of the new approach are analysed to show that from the numerical point of view the method corresponds to the most ideal situation.The numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals.     

Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.     

Semiclassical initial value series solution of the spin boson problem

A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.     

Improved semilocal convergence analysis in Banach space with applications to chemistry

We present a new semilocal convergence analysis for Secant methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies. A numerical example is also presented to illustrate the theoretical results obtained in this study.     

First-principles calculation of local atomic polarizabilities

Common methods of determining atomic polarizabilities suffer from the inclusion of nonlocal effects such as charge polarization. A new method is described for determining fully ab initio atomic polarizabilities based on calculating the response of atomic multipoles to the local electrostatic potential. The localized atomic polarizabilities are then used to calculate induction energies that are compared to ab initio induction energies to test their usefulness in practical applications. These polarizabilities are shown to be an improvement over the corresponding molecular polarizabilities, in terms of both absolute accuracy and the convergence of the multipolar induction series. The transferability of localized polarizabilities for the alkane series is also discussed.     

An efficient second-order MC SCF method for long configuration expansions

A new second-order optimisation procedure for general MC SCF wavefunctions is described. The method shows greatly improved convergence as compared to previous methods. Using a determinant-based direct CI procedure which avoids the construction of a formula tape, very long complete active space (CAS SCF) wavefunctions can be handled. Energy averages of several states can also be optimised. Sample calculations for CH₂, FeO, and the vinoxy radical CH₂CHO with up to 178916 configurations are presented.     

Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems

A new method for Ewald summation in planar/slablike geometry, i.e., systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast particle mesh Ewald (PME)-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.     

Ewald summations in systems with two-dimensional periodicity

This study presents formulas for the electrostatic energy of lattices with two-dimensional periodicity, based on Fourier representations and alternatively on the Ewald procedure for convergence acceleration. The work extends the contributions of previous investigators by taking full advantage of plane-group symmetry and by providing analytical formulas for all derivatives of the energy through second order. The derivatives considered include those with respect to the positions of all charges within the unit cell, those with respect to the lattice vectors (cell deformations), and those involving both types of variables. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 385–404, 1998     

Graph—theoretical approach to the calculation of physico-chemical properties of polymers

A topological extrapolation method for the calculation of various properties (melting points, refractive indices, specific rotation, etc.) of infinite linear polymers is developed. The rapid convergence of the oligomer extrapolation series results in very good agreement between calculated and experimental values. Comparison between the proposed method and other empirical or semi-empirical methods (the group addition method and the Pade-approximation) is also presented.     

The Fourier space restricted Hartree-Fock method for the electronic structure calculation of linear poly(tetrafluoroethylene)

Building on the pioneering work of Jean-Marie André and working in the laboratory he founded, the authors have developed a code called FT-1D to make Hartree-Fock electronic structure computations for stereoregular polymers using Ewald-type convergence acceleration methods. That code also takes full advantage of all line-group symmetries to calculate only the minimal set of two-electron integrals and to optimize the computation of the Fock matrix. The present communication reports a benchmark study of the FT-1D code using polytetrafluoroethylene(PTFE) as a test case. Our results not only confirm the algorithmic correctness of the code through agreement with other studies where they are applicable, but also show that the use of convergence acceleration enables accurate results to be obtained in situations where other widely-used codes(e.g., PLH and Crystal) fail. It is also found that full attention to the line-group symmetry of the PTFE polymer leads to an increase of between one and two orders of magnitude in the speed of computation. The new code can therefore be viewed as extending the range of electronic-structure computations for stereoregular polymers beyond the present scope of the successful and valuable code Crystal.     

Fourier-Transform-Spektroskopie im Infraroten

Fourier transform spectroscopy has gained widespread interest in recent years, also in chemical laboratories, because Fourier spectrometers are now commercially available for the whole infrared region. It is the purpose of this article to give an introduction to the principles of this new technique and its general advantages over conventional spectroscopic methods. Some of the shortcomings of Fourier spectroscopy are also critically reviewed.     

Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations

The fast multipole method proposed by Greengard and Rokhlin (GR) is applied to large biomacromolecular systems. In this method, the system is divided into a hierarchy of cells, and electric field exerted on a particle is decomposed into two parts. The first part is a rapidly varying field due to nearby cells, so that it needs rigorous pairwise calculations. The second part is a slowly varying local field due to distant cells; hence, it allows rapid calculations through a multipole expansion technique. In this work, two additional possibilities for improving the performance are numerically examined. The first is an improvement of the convergence of the expansion by increasing the number of nearby cells, without including higher-order multipole moments. The second is an acceleration of the calculations by the particle–particle and particle–mesh/multipole expansion (PPPM/MPE) method, which uses fast Fourier transform instead of the hierarchy. For this purpose, the PPPM/MPE method originally developed by the authors for a periodic system is extended to a nonperiodic isolated system. The advantages and disadvantages of the GR and PPPM/MPE methods are discussed for both periodic and isolated systems. It is numerically shown that these methods with reasonable costs can reduce the error in potential felt by each particle to 0.1–1 kcal/mol, much smaller than the 30-kcal/mol error involved in conventional simple truncations. © 1994 by John Wiley & Sons, Inc.     

Ewald-type formulas for Gaussian-basis studies of one-dimensionally periodic systems

The history of computations at Namur and elsewhere on the electronic structures of stereoregular polymers is briefly reviewed to place the work reported here in the context of related efforts. Our earlier publications described methods for the formal inclusion of Ewald-type convergence acceleration in band-structure computations based on Gaussian-type orbitals, and that work is here extended to include a discussion of the calculation of total energies. It is noted that the continuous nature of the electronic density leads to different functional forms than are encountered for point-charge lattice sums. Examples are provided to document the correctness and convergence properties of the formulation.