Extrapolation and interpolation of asymptotic series by self-similar approximants

The problem of extrapolation and interpolation of asymptotic series is considered. Several new variants of improving the accuracy of the self-similar approximants are suggested. The methods are illustrated by examples typical of chemical physics, when one is interested in finding the equation of state for a strongly interacting system. A special attention is payed to the study of the basic properties of fluctuating fluid membranes. It is shown that these properties can be well described by means of the method of self-similar approximants. For this purpose, the method has been generalized in order to give accurate predictions at infinity for a function, whose behavior is known only at the region of its variable close to zero. The obtained results for fluctuating fluid membranes are in good agreement with the known numerical data.     

Construction of approximate entropic forces for finitely extensible nonlinear elastic (FENE) polymers

When the stress applied to a Rouse-like polymer chain is large enough, one must use anharmonic entropic spring forces in order to keep the chain contour length from increasing to unphysical values. Although one can derive “exact” equations relating the spring extension to the entropic force produced by a finitely extensible non-linear elastic (FENE) random-walk polymer, such expressions are usually of little interest because their complexity would entail large evaluation times in numerical studies by computer. Moreover, these expressions can rarely be used directly in analytical studies. In this article, we describe a systematic method to construct analytically simple yet numerically accurate expressions to relate the entropic force to the extension of an entropic spring for a random-walk polymer chain in arbitrary dimension d ≥ 2. These expressions are modified Pade approximants which yield the correct asymptotic behaviours in both the small and large extension limits. It is shown that the well-known Warner empirical approximation is but a limiting case (for infinite dimensions).     

Thermogenesis: Identification by means of pade approximants

The paper describes how to obtain an analytic approximation to the transfer function of a conduction calorimeter, namely, a procedure to identify the calorimetric system. In this case Pade approximants are used on the Laplace transform of the thermogram. The feasibility of the method is tested on two models which span the frequency range usually attained by actual calorimeters. The influence of random noise and baseline drift have also been analyzed. The results show that three or four time constants are correctly obtained.     

Non-Born-Oppenheimer approximation for very weakly bound states of molecular anions

The influence of nuclear rotation on weak electron binding in the long range field of a linear polar molecule is treated in a way that leads ultimately, with suitable approximation, to the familiar equations for close coupling of electron-nuclear-rotational motions. Subsequently, a conventional pseudopotential approximation is invoked to examine the rotational spectra of HCN and DCN anions. It is shown that the number of rotationally excited anion states cannot be reliably predicted by assuming that zero binding occurs when the rotational energy equals the electron affinity obtained in the Born-Oppenheimer approximation. A method is suggested for combining accurate molecular orbital and parameterized pseudopotential methods to provide accurate electron affinities for very weakly bound anionic states.     

An improved version of Junmeng–Fang–Weiming–Fusheng approximation for the temperature integral

An improved version of Junmeng–Fang–Weiming–Fusheng approximation for the temperature integral has been developed. The accuracy of the improved approximation for the temperature integral has been tested by some numerical analyses. The systematic analysis of the relative errors involved in the kinetic parameters obtained from Junmeng–Fang–Weiming–Fusheng integral method and its improved version has been also carried out. The results have shown that the improved approximation is more accurate than Junmeng–Fang–Weiming–Fusheng approximation as the solution of the temperature integral, and that more accurate kinetic parameters can be determined from the integral method based on the improved temperature integral approximation.     

On Padé approximants to virial series

Padé approximants have long been used to predict virial series coefficients and to provide equations of state for low and high density materials. However, some justified criticism has appeared about this procedure. Although we agree to impose several restrictions on the use of Padé approximants, we indicate that the Padé approximant is still an excellent way to predict the first unknown virial series coefficients. As an example, we report a calculation of the B11=128.6 and B12=155 virial coefficients of the three dimensional hard sphere model that are in excellent agreement with the two most recent estimates. We also consider that the commonly used method to choose among Padé approximants is not completely reliable for this specific application and suggest an alternative new method.     

Third-order many-body perturbation theory for the ground state of the carbon monoxide molecule

Many-body perturbation calculations have been performed for the ground state of the carbon monoxide molecule at its equilibrium internuclear separation. The calculations are complete through third order within the algebraic approximation; i.e., the state functions are parameterized by expansion in a finite basis set. All two-, three-, and four-body terms are rigorously determined, and many-body effects are found to be very important. A detailed comparison is made with a previously reported configuration interaction study. Padé approximants to the energy expansion are constructed. The many-body perturbative wave function is used in the Rayleigh quotient to produce upper bounds to the electronic energy.     

Logarithmic energy functions in z−1 perturbation theory

We exploit some known relationships between the nuclear attraction, interelectronic repulsion, and total non-relativistic energies of atoms as functions of the nuclear charge. By approximation the component energies by means of padé approximants, we obtain compact and reliable total energy functions, which contain logarithmic terms, for several isoelectronic sequences.     

Self-similar power transforms in extrapolation problems

A method is suggested allowing for the improvement of accuracy of self-similar factor and root approximants, constructed from asymptotic series. The method is based on performing a power transforms of the given asymptotic series, with the power of this transformation being a control function. The latter is defined by a fixed-point condition, which improves the convergence of the sequence of the resulting approximants. The method makes it possible to extrapolate the behaviour of a function, given as an expansion over a small variable, to the region of the large values of this variable. Several examples illustrate the effectiveness of the method     

Padé approximants to the evolution operator through the Lippmann Schwinger variational principle

Using the connection between the evolution operator and the stationary value of the Lippmann–Schwinger functional, approximations to this operator are obtained using diagonal Padé approximants. A harmonic oscillator with a non-hermitean perturbation proportional to powers of the bosonic creation operator is considered and its evolution operator is evaluated. The poles of the spectral representation obtained by this method are compared to both: the ones of the usual perturbative expansion and those of the exact solution. Extensions to Hermitian Hamiltonians are discussed, involving the necessity of inverting more complex operators in the calculation of the Fourier transform. However, the approximation obtained by this procedure becomes exactly unitary. © 1995 John Wiley & Sons, Inc.     

Accurate eigenvalues of three- through ten-electron atomic isoelectronic sequences from low-order Z-dependent perturbation theory combined with variational constraints and screening

A method is developed, based on Rayleigh-Schrödinger perturbation theory combined with variational constraints and screening, for obtaining accurate atomic eigenvalues from third-order 1/Z expansions. Application of the procedure to the ground states of the 3NV10 electron atomic sequences yields energies of 99.95–100.05% or greater accuracy, a marked improvement over those obtained from other third-order summations including Padé approximants. In the important test cases of the Be and Ne atoms, our results are found to exceed in accuracy all but the most elaborate ab initio calculations.     

On the Padé method for sequence acceleration and calculation of Madelung constants

An iterative variant of Padé approximants (PA) is presented and employed to accelerate convergence of sequences. Pilot calculations on partial lattice sums for NaCl and CsCl crystals have been explicitly shown to offer accurate estimates of their Madelung constants.     

Towards the Development and Applications of Manifestly Spin-free Multi-reference Coupled Electron-pair Approximation-like Methods: A State Specific Approach

As a practical tool of being applicable to bigger molecules, a full-blown state-specific multi-reference coupled cluster formalism developed by us (Mahapatra et al. in J Chem Phys 110:6171, 1999) would be rather demanding computationally, and it is worthwhile to look for physically motivated approximation schemes which capture a substantial portion of the correlation of the full-blown theory. In this spirit, we have recently proposed coupled electron-pair approximation (CEPA)-like various approximants to the parent spin-adapted state-specific multi-reference coupled cluster (SS-MRCC) theory which depend on the inclusion of EPV terms to various degree. Here, the space of excitations is confined to the first order interactive virtual space generated by the cluster operator, but the EPV terms are included exactly. We call them spin-free state specific multi-reference CERA (SS-MRCEPA) theories. They work within the complete active space (CAS) and have been found to be very effective in bypassing the intruders, similar in performance to that of the parent SS-MRCC theory. The spin-adaptation of the working equations of both the SS-MRCC and the CEPA-like approximants is a non-trivial exercise. In this paper, we delineate briefly the essentials of a spin-free formulation of the SS-MRCC and SS-MRCEPA theories. This allows us to include open-shell configuration state functions (CSF) in the CAS. We consider three variants of SS-MRCEPA method. Two are explicitly orbital invariant: (1) SS-MRCEPA(0), a purely lineralized version of the SS-MRCC theory, (2) SS-MRCEPA(I), which includes all the EPV terms explicitly and exactly in an orbital invariant manner and (3) the SS-MRCEPA(D), which emerges when we keep only the diagonal terms of a set of dressed operators in the working equations. Unlike the first two, the third version is not invariant under the orbital transformation within the set of doubly occupied core, valence and virtual orbitals. The SS-MRCEPA methods produce very encouraging results as was evidenced in the applications on the computation of potential energy surfaces for the ground states of LiH and HF molecules.     

A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra

This paper shows that using the Padé–Laplace (PL) method for deconvolution of multi-exponential functions (stress relaxation of polymers) can produce ill-conditioned systems of equations. Analysis of different sets of generated data points from known multi-exponential functions indicates that by increasing the level of Padé approximants, the condition number of a matrix whose entries are coefficients of a Taylor series in the Laplace space grows rapidly. When higher levels of Padé approximants need to be computed to achieve stable modes for separation of exponentials, the problem of generating matrices with large condition numbers becomes more pronounced. The analysis in this paper discusses the origin of ill-posedness of the PL method and it was shown that ill-posedness may be regularized by reconstructing the system of equations and using singular value decomposition (SVD) for computation of the Padé table. Moreover, it is shown that after regularization, the PL method can deconvolute the exponential decays even when the input parameter of the method is out of its optimal range.     

Simulations of cyclic voltammetric and chronoamperometric electrode responses at a disk electrode using combinations of spherical and cylindrical electrode geometries

Using geometric models based on one-dimensional transport at spheres and cylinders, three methods for improving the simulation of voltammetric behavior of a disk electrode have been explored. One method is based on the common assumption of equivalency between the limiting currents for a disk and a hemisphere under steady-state diffusion conditions. The second method involves the use of a partial-sphere geometry which is a better approximation that is suitable at the extreme diffusional limits achievable at a disk electrode of fully planar and steady-state transport. The third method, which is generally applicable, is a further refinement that uses a combination of appropriate one-dimensional spherical and cylindrical geometries. The results demonstrate that reasonably accurate approximations of disk behavior for several reaction mechanisms can be achieved in a fraction of the time required to compute the more rigorous two-dimensional model. We propose that the approximation serve primarily as a fast way to explore system behavior and establish approximate values of the relevant parameters. More accurate computations can then be performed using the two-dimensional model.     

Note: The weak-correlation limit of the three-electron harmonium atom

Asymptotic energy expressions for the weak-correlation limits of the two lowest energy states of the three-electron harmonium atom are obtained in closed forms. When combined with the known results for the strong-correlation limit, these expressions, which are correct through the second order of perturbation theory, yield robust Pade? approximants that allow accurate estimation of energies in question for all magnitudes of the confinement strength.     

Simple approximants for natural orbitals of harmonium

Simple approximations to the natural orbitals (NOs) of harmonium with enforced correct short- and long-range asymptotics yield accurate bounds for the NO occupancies. In particular, expressions involving Pade approximants with just one variational parameter are capable of producing the largest NO occupancies with accuracy better than 10(-4). The comparison of two cases with different coupling strengths omega (1.948 51相似文献   

Møller–Plesset expansion of the dispersion energy in the ring approximation

Coupled-cluster equations for the calculation of the nonexpanded (fully damped) dispersion energy are derived. These equations are solved in the ring approximation using the Møller–Plesset expansion in terms of the fluctuation potentials W_A and W_B for the individual molecules. Numerical results of high-order perturbative calculations for the He, H₂, LiH, H₂O, and HF dimers are presented and compared with the converged results computed using the same basis sets. It is found that the convergence of the Møller-Plesset expansion of the dispersion energy in the ring approximation is very fast. The padé approximants still accelerate this already good convergence. For all complexes studied in this paper, the sum of the corrections through the second-order in W_A + W_B reproduces over 99% of the converged value. The sum of third- and higher-order corrections in the ring approximation is found to be one or two orders of magnitude smaller than the sum of second-order terms not included in the ring approximation and, therefore, may be safely neglected. Thus, it appears that a second-order calculation, which does not require iterating coupled-cluster equations or solving random phase approximation equations, offers the best compromise between accuracy and computational requirements. © 1993 John Wiley & Sons, Inc.     

Efficient step size selection for the tau-leaping simulation method

The tau-leaping method of simulating the stochastic time evolution of a well-stirred chemically reacting system uses a Poisson approximation to take time steps that leap over many reaction events. Theory implies that tau leaping should be accurate so long as no propensity function changes its value "significantly" during any time step tau. Presented here is an improved procedure for estimating the largest value for tau that is consistent with this condition. This new tau-selection procedure is more accurate, easier to code, and faster to execute than the currently used procedure. The speedup in execution will be especially pronounced in systems that have many reaction channels.     

Use of asymptotic analysis of the large activation- energy limit to compare graphical methods of treating thermogravimetry data

An earlier numerical analysis showed that the second approximate method of Horowitz and Metzger can be rendered exceedingly accurate for reduction of thermogravimetry data. It is demonstrated here that this result can be justified on the basis of an asymptotic expansion with a nondimensional activation energy as the large parameter. The order of magnitude of the error is ascertained for this and two other approximate methods. Higher-order terms in the approximation are developed.