Classification of vibrational resonances in the energy spectrum of the formaldehyde molecule and Katz’s branch points

The Rayleigh–Schrödinger perturbation theory of high orders and the algebraic Padé–Hermite approximants are used to determine the singular points of a vibrational energy function of the formaldehyde molecule dependent on a complex perturbation parameter as on the argument. It is shown that the Fermi, Darling–Dennison, and other higher-order vibrational resonances are related to Katz’s points—common branch points on the complex plane of the energy of two vibrational states. Analysis of Katz’s points that connect different vibrational states allows one to reveal essential resonance perturbations, to introduce an additional classification for them, and to determine the polyad structure of an energy spectrum.     

Pade approximants and the anharmonic oscillator

The diagonal Padé approximants of the perturbation series for the eigenvalues of the anharmonic oscillator (a βκ¹ perturbation of p² + κ²) converge to the eigenvalues.     

Perturbation theory and Padé approximants in realistic large-matrix models of the nuclear effective interaction

Realistic extended shell-model calculations are used to construct exact effective Hamiltonians, the perturbation series for the effective Hamiltonian to any order, and the [N+1, N] Padé approximants to the series. It is found that the Padé approximants give reliable results even when the series diverge, but that for both convergent and divergent series reasonably accurate results can be obtained only in fifth, or even seventh order. In addition, the poles of the low-order Padé approximants are not always reliable indicators of singularities of the perturbation series. The perturbation series and Padé approximants for the Q-box (energy-dependent effective Hamiltonian) are no more accurate in low orders than those for the usual effective Hamiltonian. Explicit formulas for the matrix Padé approximants are given in an appendix.     

A method for summation of perturbation series

A method for summation of perturbation series is developed. It consists of a proper rearrangement of the power series through an appropriate redefinition of the perturbation parameter. Well-known divergent power series appearing in a φ⁴-scalar field theory or in the treatment of the linear confining potential model and the Stark effect in hydrogen are used as illustrative examples. A comparison between present results and accurate, numerical ones shows that our series converge even faster than sequences of Padé or Borel-Padé approximants.     

Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.     

Eigenvalues of structures with uncertain elastic boundary restraints

The eigenvalue problems of structures with random elastic boundary supports are studied in this paper. Using the perturbation method, the differential equations including stochastic distributed parameters and random boundary conditions that govern the eigenproblems are transformed to a series of deterministic differential equations and boundary conditions. Then the differential equations and boundary conditions are discretized utilizing the finite element method (FEM). This process is easy to be implemented since the resulting perturbation equations with different orders own the same FEM meshes. The first-order and second-order sensitivities of eigenvalues are derived through solving the deterministic algebraic equations. Upon determining these sensitivities of eigenvalues, the approximate statistic expressions of random eigenvalues considering uncertain elastic boundary supports are established. At the end, several numerical examples are given to illustrate the application and effectiveness of the present method. Comparison of the present numerical results with those from the Monte-Carlo simulation method verifies the accuracy of the developed method.     

Padé approximants and the effective interaction

Analytic properties of the effective interaction allow us to indicate the positions of the poles of low-order Padé approximants and the domain of convergence of the series of Padé approximants. All evidence favors the conjecture that the Padé approximants will converge to that branch of the effective interaction which reproduces the model space states, if the series converges.     

Calculation of vibrational HDO energy levels: Analysis of perturbation theory series

Series of the Rayleigh-Schrödinger perturbation theory are analyzed and summated by the example of the HD¹⁶O molecule for vibrational energy levels. Particular attention is given to determining the location of singularities-branching points corresponding to the intersection of levels in the complex plane. Numerical analysis demonstrates the divergence of the series for states involved in the Fermi resonance; however, summation by the method of Padé-Hermite approximants makes it possible to reconstruct the levels by coefficients of the series with sufficient accuracy. It is found that resonance-coupled states have common branching points, which leads to the coincidence of series’ coefficients in high orders. Branching points’ characteristics permitting one to obtain a comparatively simple representation of high order corrections are determined.     

Optical and magnetic studies on the phosphorescent state of phthalazine in polar and non-polar hosts

The ground state potential energy curve for the beryllium dimer is calculated using non-degenerate many-body perturbation theory and the multi-configuration self-consistent-field/configuration interaction method. Quasi-degeneracy in this system makes it useful in exploring the limitation of the applicability of the non-degenerate formulation of diagrammatic many-body perturbation theory. Both methods are applied within the algebraic approximation defined by a contracted gaussian basis set of triple zeta quality. It is shown that non-degenerate perturbation theory can lead to a potential energy curve which is in close agreement with the configuration interaction curve when taken to third order in the energy and [2/1] Padé approximants constructed.     

Singularity analysis of fourth-order Møller–Plesset perturbation theory

The usefulness of Møller–Plesset perturbation theory, a standard technique of quantum chemistry, is determined by singularities in the corresponding energy function in the complex plane of the perturbation parameter. A method is developed that locates singularities from fourth-order perturbation series, using quadratic approximants with bilinear conformal mappings.     

Nonperturbative approximation schemes for the effective interaction in nuclei

Several methods are investigated to deal with the singularities of the effective interaction which cause the perturbation series to diverge. Some of these methods, which are found to be quite useful, are based upon a semiphenomenological treatment of the “intruder” states. One method, based upon the use of Padé approximants, seems to be free of such semiphenomenological input data and appears to yield a particular effective interaction which has as eigenvalues those levels of the system for which the corresponding eigenfunctions have maximum overlap with the model space. The methods are investigated with the help of simple examples. Those methods which are found to be useful are applied to the 0⁺ and 2⁺ levels of ¹⁸O.     

Double wells: Perturbation series summable to the eigenvalues and directly computable approximations

We give a rigorous proof of the analyticity of the eigenvalues of the double-well Schrödinger operators and of the associated resonances. We specialize the Rayleigh-Schrödinger perturbation theory to such problems, obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method. By an approximation we obtain new series expansions directly computable, still summable, which, in the case of the Herbst-Simon model, can be given in an explicit form.Partially supported by Ministero della Pubblica Istruzione     

多项式势阱中粒子能级的微扰计算

利用超位力定理(HVT)和Hellmann—Feynman定理(HFT)．可把势阱中粒子能级的微扰计算简化为代数公式．从而方便计算机编程．对于可展开为多项式的势阱,综合考虑势能多项展开式中截断近似和有限级次的递推计算二者引入的误差．我们重新定义了近似计算的阶次．作为例子,我们给出了高斯势的能级近似计算．     

Padé approximants on a lattice

The linear harmonic oscillator on a lattice is solved analytically. Wave functions and energy eigenvalues are expressed in terms of Mathieu functions and characteristic values of the Mathieu equation respectively. The Padé-approximant method for calculating the continuous limit of energy eigenvalues is tested. It is found that the values of approximants at infinity do not converge. A modification of the Padé method is proposed which leads to convergent series. Implications for more complicated systems are discussed.     

Method of self-similar factor approximants

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.     

Eigenwerte des Lippmann-Schwinger-Kerns für Yukawa- und Nukleon-Nukleon-Potentiale

In several papers we recently obtained simple high-energy asymptotic expansions for the solutions and eigenvalues of wave equations containing generalized superpositions of Yukawa potentials. In the present article we extend these investigations to the calculation of phase shifts and eigenvalues of the Lippmann-Schwinger kernel. We also calculate corresponding Padé approximants and illustrate, by means examples, their usefulness, even in regions of low energies.     

Anharmonic Parameters of Vibrationally Excited Molecules

An algebraic method of perturbation theory for eigenvalues and eigenvectors is used to examine the intramolecular parameters of excited vibrational states. Based on this method, formulas for higher-order coefficients of the matrix of vibrational modes, changes in the Cartesian coordinates of vibrating atoms, and anharmonic elements of the tensor of inertia moment are derived.     

Algebraic approach to the study of excited vibrational states

The algebraic method of perturbation theory for eigenvalues and eigenvectors is used for the theoretical study of molecular parameters of excited vibrational states. Within the framework of this method, formulas for higher-order elements of the vibration form tensors are obtained. They can be used for the study of changes in intramolecular parameters of vibrations and the anharmonic coefficients of kinematic interaction.     

Calculation of vibrational energy levels of triatomic molecules with the C2v and Cs symmetries by summing divergent series of the Rayleigh-Schr?dinger perturbation theory

The Rayleigh-Schr?dinger perturbation theory is applied to calculation of vibrational energy levels of triatomic molecules with the C _2v and C _s symmetries: SO₂, H₂S, F₂O, HOF, HOCl, and DOCl. Particular attention is given to the states coupled by anharmonic resonances; for such states, the perturbation theory series diverge. To sum these series, the known methods of Padé, Padé-Borel, and Padé-Hermite and the method of power moments are used. For low-lying levels, all the summation methods give satisfactory results, while the method of quadratic Padé-Hermite approximants appears to be more efficient for high-excited states. Using these approximants, the structure of singularities of the vibrational energy, as a function in the complex plane, is studied.