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A divergent perturbation series is known to yield very unreliable results for observables even at moderate coupling strengths. One of the most popular techniques in handling such series is to express them as rational functions, but it is often faithful only for small coupling. We outline here how one can gain considerable advantages in the large-coupling regime by properly embedding known asymptotic scaling relations for selected observables during construction of the aforesaid Pade approximants. Three new bypass routes are explored in this context. The first approach involves a weighted geometric mean of two neighboring PA. The second idea is to consider series for specific ratios of observables. The third strategy is to express observables as functionals of the total energy in the form of series expansions. Symanzik's scaling relation, and the virial and Hellmann–Feynman theorems, are used at appropriate places to aid each of the strategies. Pilot calculations on the ground-state perturbation series of certain observables for the quartic anharmonic oscillator problem reveal readily the benefit and novelty. © 2012 Wiley Periodicals, Inc.

Interpolation formulas, having functional forms inspired by Pade approximants, and containing at most three parameters, were derived for a number of atomic oscillator strengths. It is shown that very accurate results are obtained both for specific transitions in the helium and lithium isoelectronic sequences, and for certain series in hydrogen, helium, and lithium. Application of the methods to other systems is prevented only by the lack of reliable input and comparison data. 15 references.

We introduce an additional term to treat dispersion in NDDO-based semiempirical molecular orbital techniques. The dispersion energy is calculated using additive “atomic orbital” polarizability tensors determined with our modified version of Rivail's variational technique. Frequency-dependent polarizabilities can be obtained using Pade approximants, which proved more reliable than the Unsold approximation or a Taylor-series approach. Dispersion energies are best calculated using the Slater–Kirkwood modification of the London formula, although they can also be obtained using the Casimir–Polder approach with frequency-dependent polarizabilities. A test parameterization using MNDO as the semiempirical technique indicated that weak interactions can be reproduced satisfactorily. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

In this article, the solution of a chemical differential-algebraic equation model of general type F(y, y′, x) = 0 has been done using MAPLE computer algebra systems. The MAPLE program is given in the Appendix. First we calculate the Power series of the given equations system, then we transform it into Pade series form, which gives an arbitrary order for solving chemical differential-algebraic equation numerically. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

A new generalized expansion of the Born–Oppenheimer potential energy of diatomic molecules, combining a power series with Pade approximant, is introduced. It converges in the range of internuclear separation R∈〈0,∞) and can be used to reduce infrared and microwave spectra to parameters of a diatomic effective Hamiltonian. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002

We present a rigorous strategy, based on Stieltjes series and Pade approximants, to obtain suitable bounds for extrapolation of the quantum chemical correlation energy. Computational tests are performed for the second-order Moller-Plesset (MP2) correlation energy, and the bounds obtained are tight enough for practical calculational purposes: The associated error in most cases is much less than 1 kcal/mol. The bounds presented here are also shown to be rigorous for functional forms that represent a wide variety of methods in quantum chemistry and hence may be used in extrapolating a wide range of expressions, some of them yielding significant computational advantages compared to traditional techniques. c 2000 John Wiley & Sons, Inc. Int J Quantum Chem 79: 222-234, 2000

Polynomial and Pade representations of the kinetic energy component Tc[ρ] of the correlation energy density functional Ec[ρ] are presented in this article. Two approximate local formulas similar to the Wigner form for Ec[ρ] are investigated for Tc[ρ]. Applications of these formulas along with their two polynomial counterparts are carried out for atoms, ions, and a few simple molecules. Numerical predictions of Tc values are made for these species. Both Hartree–Fock and self-consistent-field densities are used in their evaluations. Recommended at this time is the two-parameter Pade [0, 1] formula Tc[ρ]=∫a0ρ/(1+b0ρ−1/3) dr, with a0=0.1658 and b0=6.102 (atomic units). © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 513–522, 1998

Using variational Monte Carlo, we compare the features of 118 trial wave function forms for selected ground and excited states of helium, lithium, and beryllium in order to determine which characteristics give the most rapid convergence toward the exact nonrelativistic energy. We find that fully antisymmetric functions are more accurate than are those which use determinants, that exponential functions are more accurate than are linear function, and that the Pade function is anomalously ´ accurate for the two-electron atom. We also find that the asymptotic and nodal behavior of the atomic wave function is best described by a minimal set of functions. Q 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 1001)1022, 1997

An analysis of the structure of the optimized effective Kohn-Sham exchange potential v, and its gradient approximations is presented. The potential is decomposed into the Slater potential v{sub s} and the response of v{sub s} to density variations, v{sub resp}. The latter exhibits peaks that reflect the atomic shell structure. Kohn-Sham exchange potentials derived from current gradient approaches for the exchange energy are shown to be quite reasonable for the Slater potential, but they fail to approximate the response part, which leads to poor overall potentials. Improved potentials are constructed by a direct fit of v{sub x} with a gradient-dependent Pade approximant form. The potentials obtained possess proper asymptotic and scaling properties and reproduce the shell structure of the exact v{sub x}. 44 refs., 7 figs., 4 tabs.

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 Moller–Plesset expansion in terms of the fluctuation potentials WA and WB 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 Moller-Plesset expansion of the dispersion energy in the ring approximation is very fast. The pade approximants still accelerate this already good convergence. For all complexes studied in this paper, the sum of the corrections through the second-order in WA + WB 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.

The problem of representing an observable F(z) in the Pade scheme from its formal perturbative (Taylor) expansion in z is considered. It is demonstrated how the representation could be improved by incorporating in a simple manner, in the course of constructing such approximants, the knowledge of asymptotic (z ∞) power-law behavior of F(z). Comparison with the usual approximants is made with a thorough numerical survey on error estimates and variations of error with z, input information, and quantum number. Spectacular performance of the new strategy is exemplified. Test calculations chiefly involve various properties of the first five eigenenergy states of the quartic anharmonic oscillator system. A few consistency requirements, including the virial theorem, are also studied. © 1993 John Wiley & Sons, Inc.

In this paper a multireference constant denominator perturbation theory (CDPT) is developed to reduce incomplete basis set errors arising when solving the Schrodinger equation with a finite basis set. The advantage of this method is that very few basis functions are needed, and all calculations if carried out to high enough order in the perturbation treatment effectively use a complete basis set. As a first step the theory has been restricted to one-particle Hamiltonians and applied to the anharmonic oscillator to study the convergence properties. For perturbation calculations carried out to fifth order, results from Pade approximates show an improvement in accuracy of between one and three orders of magnitude.

Generalized Pade approximants are used to extrapolate the total energy of polymers described by alternant Hamiltonians to the bulk limit. The method provides an upper bound to the energy. The origin of (quasi) periodic oscillations of the energy per unit cell as the function of the number of unit cells is enlightened through analysis of the moments of the Hamiltonian.

Instead of the Schrodinger equation ℋΨ = EΨ subject to the boundary condition 〈φ|Ψ〉 = 1, where φ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (ℋ − ℰ)Ψℰ = aφ, where ℰ is a complex variable, with the same boundary condition, which gives a = 〈φ|ℋ|Ψℰ〉 − ℰ = ℰ1 − ℰ. Introducing the projector P = 1 − |φ〉〈φ| for the complement to O = |φ〉〈φ|, one finds easily the explicit solution Ψℰ = (1 − Pℋ/ℰ)−1φ = (1 + Tℰℋ)φ, where Tℰ = (ℰ − Pℋ)−1P = P(ℰ − PℋP)−1P is the reduced resolvent associated with the auxiliary Hamiltonian H = PℋP. The existence of these operators is discussed. It is shown that, if the parameter ℰ is real in the “discrete part” of the spectrum to ℋ, then ℰ and ℰ1 = 〈φ|ℋ|Ψℰ〉 = 〈φ|ℋ + ℋTℰℋ|Φ〉 ≡f(ℰ) bracket a true eigenvalue E satisfying the relation E = f(E). The Newton-Raphson solution to the equation F(ℰ) = ℰ − f(ℰ) = 0 is related to the variation principle. Putting ℋ = ℋ0 + V and expanding the inverse (ℰ − Pℋ0 − PV)−1 in terms of powers of V or (V − α), one gets various expansions relating to finite-order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = {h1,h2,…,hn} is a linearly independent set, the inner projection An = |h〉 〈h|A−1|h〉−1〈h| is a “rational approximation” to the operator A which converges toward A when n→∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Pade approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.

Ordinary Rayleigh-Schroudinger perturbation theory with Moller-Plesset (RSMP) partitioning is used to calculate second- and third-order correlation corrections to the CHF polarizability and dipole moment of the water molecule by a finite-field procedure. Pade approximants are found to be useful in accelerating the convergence of the property perturbation expansions. Field-induced polarization functions suitable for polarizability calculations are determined. The average polarizability calculated, neglecting vibrational averaging, with Dunning's (9s5p/4s-4s2p/2s) contracted GTO basis set augmented by field-induced lslp2d/lp polarization functions is within 3 per cent of the experimental result. Correlation corrections to the dipole moment and polarizability of the water molecule calculated by the finite-field RSMP and single + double excitation CI(SDCI) methods for the same basis set are found to be in close agreement. The RSMP approach has the advantages of being size-consistent and of being capable of greater efficiency than the SCDI method. Comparative calculations carried out using Epstein-Nesbet partitioning show that through third order RSEN correlation perturbation expansions for the dipole moment and polarizability are less rapidly convergent than RSMP expansions. However, reasonable accord with RSMP results can be achieved by using Pade approximants to accelerate the convergence of RSEN energy perturbation expansions. The convergence of RSMP property correlation expansions based on the zeroth-order uncoupled-Hartree-Fock (UCHF) and coupled-Hartree-Fock (CHF) approximations are compared through third order. Whereas the CHF + RSMP expansions are for practical purposes fully converged, the UCHF + RSMP expansions are not adequately converged.

We examine various perturbation–variational approximations for molecular electronic energy when the fifth-order perturbational energies are available. Such theories require very little additional computation once the sequence of perturbation energies are known yet supply a bound even when the peturbation sequence is poorly convergent. We choose for computational examples results obtained very rapidly from a zeroth order wave function consisting of doubly occupied localized bonds and examine polarization within these bonds, delocalization, and bond breaking. In general, we find that the fifth-order energy itself, and especially the [2, 1] Pade approximant on this sequence, are especially accurate in estimating the total energy and more accurate than any variational scheme when the zeroth order localized wave function is a good description of the electronic structure. The variational results, however, are nearly as accurate, and a [1, 0] Pade on the sequence of variational results is remarkably robust, even in those cases where the perturbation sequence is poorly defined. We also examine several scaling techniques, or partitionings of the Hamiltonian. Although these scaling techniques do accelerate convergence of the perturbation sequence, none that we examine give better results, than the [2, 1] pade, which is independent of any scaling.

The contributions of the triple and quadruple excitations to the fifth-order perturbation energy for the perturbation configuration interaction using localized orbitals (PCILO) method are derived. This completes the development of a fifth-order constant denominator perturbation theory initiated in a previous paper [5] with the single and double excitations. This theory is tested on molecules containing strained ring geometries, stretched bonds, strongly polarized bonds, and delocalized pi systems: cases where the starting zero order reference wave function poorly describes the system. Although the perturbation expansions turn out to be slowly convergent, the Pade approximant taken from an energy series which itself is constructed from Pade approximants provides results accurate to within a few kilocalories/mole of benchmark calculations. Computational times as in the original PCILO procedure remain proportional to N3, where N is the number of bonds.

This paper serves a twofold purpose. First, Lowdin's inner projection in both nonperturbative and perturbative forms is applied to the quartic anharmonic oscillator. Inner projection with perturbation theory yields rational approximations to Brillouin–Wigner-type perturbation expansions. These lower bounds are compared with [N − 1, N] Pade approximants to the Rayleigh–Schrodinger perturbation series for this problem. These Pades are also expressible as the even convergents, w2N, of a Stieltjes-type continued fraction. The latter representation has certain advantages with respect to its Pade counterpart. Inner projection without perturbation theory provides significantly better results than the perturbative version. The application of inner projection techniques to a perturbed hydrogen atom is not straightforward. The usual problems associated with the continuum spectrum of hydrogen are present. By means of a nonunitary “tilting” transformation associated with the Lie group SO(4, 2), these problems may be bypassed. In the SO(4, 2)-reformulated eigenvalue problem, a reinterpretation of the basic variables, as developed by Silverstone and Moats, yields a new Hamiltonian that permits direct use of the inner projection method. This method has been applied to the ground state of the hydrogen atom in a magnetic field, using both four- and eight-dimensional basis manifolds. This represents the first application of inner projection to this problem.

It is shown that trial functions involving Pade approximants yield satisfactory results for the ground state of the helium atom. In particular, the five-parameter form reproduces the best variational function of the type Ψ = e–zsϕ(u) obtained numerically, to a remarkable extent.

The behavior of a rigid rotor perturbed by an electric field is studied. An hypervirial calculation, to obtain the perturbation series without calculation of the perturbed wavefunctions, is used to determine the Stark-shifted eigenvalues. A numerical estimation of the m equals; 0 m = 1 transition for the lowest vibrational state (v = 0) and the first rotational state (J = 1) for the HF and DF molecules using Pade approximants is made.

A general procedure, based on the two-point Pade approximant technique, is formulated, enabling an interpolation of the low-lying energy levels of the PPP (Pariser–Parr–Pople) Hamiltonian over the divergency gap separating the asymptotic perturbation expansions in both strongly and weakly correlated limits. The methods for the determination of the first few terms of the perturbation expansions in both limits are also briefly outlined. The procedure is applied to the five lowest singlet energy levels of the PPP model of the benzene molecule and the results are compared with the exact solutions for this model. The advantages as well as the shortcoming of this approach are pointed out and illustrated on the model calculation for benzene. The applicability of this technique to larger π-electronic systems is discussed and a feasible procedure for its implementation is formulated.

Euler transformation for accelerating convergence of a series is considered in the context of handling divergent (asymptotically convergent) perturbation series. A generalized (parametrized) version of this transformation is developed, based on the conjecture of Dalgarno and Stewart, which works better. Viewed from this standpoint, the Pade approximants follow as a special case of the parametrized Euler transformation (PET), as is the case with the μ transformation procedure of Feenberg in a perturbative context. The PET is shown to serve as a more general method of handling a divergent series and is able to appreciate the construction and convergence behavior of specific sequences of Pade approximants. The role of parametrization in the context of the Z−1 perturbation theory of atoms is also noted and the workability of the adopted strategy is demonstrated by choosing some specific test cases.

Diagrammatic many-body perturbation theory is used to calculate the potential energy function for the X1 σ+ state of the CO molecule near the equilibrium nuclear configuration. Spectroscopic constants are derived from a number of curves which are obtained from calculations taken through third order in the energy. By forming [2/1] Pade approximants to the constants we obtain: re = 1.125 A (1.128 A), Be = 1.943 cm−1 (1.9312 cm−1), aBe = 0.0156 cm−1 (0.0175 cm−1), We = 2247 cm−1 (2170 cm−1), WeXe = 12.16 cm−1 (13.29 cm−1), where the experimental values are given in parenthesis.

The problem of direct Pade summation of Rayleigh–Schrodinger perturbation coefficients for the H atom Stark effect is discussed in light of the location of the Pade representation of the expected Bender-Wu branch cuts in the complex F (field) plane. The resulting conclusion that such direct summation of the divergent perturbation expansion [for the real part of E(F)] might give suggestive, but certainly not convergent, results is documented. The relationship between the Pade problem of representation of cuts and corresponding problems in L2-discretized scattering theory is pointed out. An alternative Pade expansion with newly placed cuts is proposed which is expected to, and indeed does, have excellent convergence properties for real F. The real and imaginary parts of the Stark broadened and shifted hydrogenic ground state are thus obtained directly from the (real) Rayleigh–Schrodinger perturbation coefficients for a large range of fields indicating the power and simplicity of the method.

Our procedure for employing analytical gradients of ab initio potential energy hypersurfaces in the description of centrifugally distorted molecules is applied to an asymmetric top, namely water. Both single determinantal (HF/6-31G**) and configuration-interaction (CISD/6-31G**) surfaces were utilized. Quartic centrifugal spectroscopic coefficients are obtained in both cases and are in reasonable agreement with experiment. It is shown that the calculated dependence of the energy upon the rotational angular momentum is better represented by a Pade approximant than by the conventional power series.

The study of large orders of perturbation theory in various problems is reviewed: the anharmonic oscillator, the Zeeman and Stark problems, double wells, and the like. Pade and Borel summability and path integral ideas are discussed. The rigorous results on the subject are summarized.

We report in this paper the results of outer and inner valence IP calculations for the HF molecule using two different many-body methods for the direct evaluation of energy differences. The first is the nonperturbative coupled-cluster based linear response theory (LRT) and the second is the hermitian open-shell many-body perturbation theory (MBPT). A Huzinaga-Dunning (9s5p→ 5s3p/3s) basis has been used. LRT uses an "ionization operator" S as in the equation of motion method (EOM) to generate the ionized states from a coupled-cluster type of ground state. S is chosen to consist of single ionization and ionization-cum-shake-up operators, thus treating the Koopmans as well as the shake-up states on equal footing. LRT would thus be capable of computing both the outer and the inner valence regions with equal facility. This is borne out by the results. For the open-shell MBPT, the model space is chosen to be spanned by the singly ionized determinants. The convergence of the results for the inner valence region is slow, and the results obtained from the [2, 1] Pade' approximants are presented. Unlike the LRT, the inner valence region is not reproduced with full complexity in MBPT, indicating that it is essential to modify the theory by way of expanding the model space to contain the shake-up determinants also.

The second-order dispersion energy between interacting hydrogen pairs has been evaluated by the Borel-integral method. The results are in complete harmony with those obtained by the method of Pade approximants.

A perturbation approach based on resolvent technique and Pade approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin–Wigner perturbation theory. The truncated power series is extrapolated by Pade approximants of type II. The method is tested on a CI calculation of the energy of the lowest 1Σ state of the B2 molecule.

In reply to Kaldor's [Int. J. Quantum Chem. XX, XXX (1985)] criticism of our study of simple four-electron models, in which the degree of quasi-degeneracy can be continuously varied, by the finite-order nondegenerate many-body perturbation theory, we examine in more detail a simple two-state model that the used to substantiate his claim that “the low order sum of the perturbation series is not very meaningful” in view of its divergence. It is shown that in contrast to Kaldor's claim, the partitioning used increases the radius of convergence of the considered perturbation series and is in principle capable to make it convergent. It is also shown that the convergence of the series is not very essential and that even divergent series can provide useful estimate of the exact result, particularly when the resummation techniques, such as Pade approximants or continued fractions, are employed. Finally, the shortcomings of the existing multi-reference perturbation approaches, which Kaldor advocates, are pointed out.

The degenerate Rayleigh-Schrodinger (RS), Murrell–Shaw–Musher–Amos (MSMA), and Eisenschitz–London—Hirschfelder–van der Avoird (EL–HAV) perturbation theories are formulated and applied through high order to the interaction of a hydrogen atom with a proton for the excited 2sσg, 3pσu, 3dσg, and 4fσu states of the system. In contrast to the 1sσg and 2pσu states all the formalisms fail to provide a convergent series for any of the above states. However, numerical evidence is given that for sufficiently large internuclear separations the RS series is able to reproduce the Coulomb energy with an accuracy better than an arbitrary fraction of the exchange energy. Similarly the exchange energy can be reproduced by the sum of a few exchange corrections of the MSMA method. It has also been shown that both the MSMA and EL–HAV series can be continued analytically beyond the convergence radius by means of Pade approximants.

The theory of the Baker–Gammel approximants is applied to the spectral density function associated with the adjacency matrix. As a result, various upper and lower bounds to topological invariants are obtained in a systematic manner. In particular, the (N, M)-type lower bound to the total π-electron energy of benzenoid hydrocarbons is derived for the first time.

Several approximate analytical formulas for the multidimensional isotropic bounded oscillators and the bounded hydrogen atom are compared. Numerical results show that the coth z method is in both cases better than the Pade approximants method. Perturbational polynomials, necessary in order to build the approximate eigenvalues, are obtained through the hypervirial perturbative method.

Pade approximants of type II have been applied to solve equations of the type F(z) = 0. The method is compared with the first-order iteration procedure, the Aitken-Samuelson formula, and the Newton-Raphson tangential method. As a test example the partitioning technique in its most simple form is applied to the Hamiltonian of a rigid symmetric-top molecule in a static electric field. The proposed algorithm is found to be superior to the first-order procedure and the Aitken-Samuleson formula, and at least as effective as the Newton-Raphson method.

The [N, M] Pade approximants to functions formally associated to power series expanssions are expressed in terms of expectation values of inverse matrices. These formulae, which can be derived by the inner-projection technique, lead to a simple analysis of the properties of serveral approximation methods and their inter-relationships, in particular Gaussian integration, continued factorization and Pade approximations, which are of current interest in the calculation of physical properties. A relation with Fredholm integral equations and expansions of the resolvent is also discussed. The use of operator inequalities in a systematic fashion is particularly convenient when both the function being approximated and the coefficients of the power series have physically meaningful expressions as moments of operators.

The Z-expansion of two-electron systems is analyzed with the Pade technique with emphasis on establishing analytical properties of the function E(Z) formally associated with the power series expansion. The concept of critical point in this connection is stressed. For this sequence it occurs at Zc = 0.911246 with E(Zc) = −0.415184. The structure of E(Z) for Z < Zc is investigated. The use of Pade approximants to extrapolate values of electron affinities is emphasized.

In the course of work on the ground state of the helium isoelectronic series using nonlinear trial wave functions of the Pade type, a minimization procedure was employed in which all the linear parameters were obtained by matrix diagonalization, while the nonlinear parameters were determined by a subroutine based on nonlinear optimization techniques. This procedure was tested on the Hylleraas six-term function for helium, and yielded a lower value for the energy, namely, E = -2.9033294 a.u. It was found that the values quoted in the literature correspond to the first step in the iterative cycle; as such, they do not correspond to a fully optimized function. 1 table. (RWR)