Solving the Schrödinger equation of helium and its isoelectronic ions with the exponential integral (Ei) function in the free iterative complement interaction method

We introduce here the exponential integral (Ei) function for variationally solving the Schr?dinger equation of helium and its isoelectronic ions with the free iterative complement interaction (ICI) method. In our previous study [J. Chem. Phys., 2007, 127, 224104], we could calculate very accurate energies of these atoms by using the logarithmic function as the starting function of the free ICI calculation. The Ei function has a weak singularity at the origin, similarly to the logarithmic function, which is important for accurately describing the three-particle coalescence region. The logarithmic function, however, has a node and a maximum along the radial coordinate which may be physically meaningless. In contrast, the Ei function does not have such unphysical behaviors and so would provide an improvement over the logarithmic function. Actually, using the Ei function, instead of the logarithmic function, we obtained the energy, E= -2.903 724 377 034 119 598 311 159 245 194 404 446 696 924 865 a.u. for the helium ground state with 21 035 functions, which is a slight improvement over our previous result (the bold face shows the digits that are believed to have converged). This result supports the suggestion that the Ei function is better than the logarithmic function for describing the three-particle coalescence region.     

A new method of solving the many-body Schrödinger equation

A new method of solving the many-body Schrödinger equation is proposed. It is based on the use of constant particle-particle interaction potential surfaces (IPSs) and the representation of the many-body wave function in a configuration interaction form with coefficients depending on the total interaction potential. For these coefficients the corresponding set of linear ordinary differential equations is obtained. A hierarchy of approximations is developed for IPSs. The solution of a simple exactly solvable model and He-like ions proves that this method is more accurate than the conventional configuration interaction method and demonstrates a better convergence with increasing basis set.     

An efficient six-step method for the solution of the Schrödinger equation

In this paper we develop an efficient six-step method for the solution of the Schrödinger equation and related problems. The characteristics of the new obtained scheme are:

• It is of twelfth algebraic order.
• It has three stages.
• It has vanished phase-lag.
• It has vanished its derivatives up to order two.
• All the stages of the scheme are approximations on the point \(x_{n+3}\).

This method is developed for the first time in the literature. A detailed theoretical analysis of the method is also presented. In the theoretical analysis, a comparison with the the classical scheme of the family (i.e. scheme with constant coefficients) and with recently developed algorithm of the family with eliminated phase-lag and its first derivative is also given. Finally, we study the accuracy and computational effectiveness of the new developed algorithm for the on the approximation of the solution of the Schrödinger equation. The above analysis which is described in this paper, leads to the conclusion that the new algorithm is more efficient than other known or recently obtained schemes of the literature.

     

Application of the Rayleigh—Schrödinger perturbation theory to the hydrogen atom

The Rayleigh-Schrödinger variational perturbation theory and the hyperspherical perturbation treatment of the ground state of a hydrogen-like atom are generalized to an arbitrary excited state. In both cases it is rigorously shown that the exact wavefunction is recovered by direct summation of the perturbation expansion through infinite order.     

An economical eighth-order method for the approximation of the solution of the Schrödinger equation

In this paper we introduce, for the first time in the literature, a three-stages two-step method. The new algorithm has the following characteristics: (1) it is a two-step algorithm, (2) it is a symmetric method, (3) it is an eight-algebraic order method (i.e of high algebraic order), (4) it is a three-stages method, (5) the approximation of its first layer is done on the point \(x_{n-1}\) and not on the usual point \(x_{n}\), (6) it has eliminated the phase–lag and its derivatives up to order two, (7) it has good stability properties (i.e. interval of periodicity equal to \(\left( 0, 22 \right) \). For this method we present a detailed analysis : development, errorand stability analysis. The new proposed algorithm is applied to systems of differential equations of the Schrödinger type in order to examine its efficiency.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation

A dissipative exponentially fitted method is constructed in this paper for the numerical integration of the Schr?dinger equation. We note that the present method is a nonsymmetric multistep method (dissipative method) An application to the bound-states problem and the resonance problem of the radial Schr?dinger equation indicates that the new method is more efficient (i.e. more accurate and more rapid) than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison(19) a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schr?dinger equation indicates the efficiency of the new approach.     

A variable-step Numerov method for the numerical solution of the Schrödinger equation

Numerovs method is one of the most widely used algorithms for solving second-order ordinary differential equations of the form y = f(x,y). The one-dimensional time-independent Schrödinger equation is a particular example of this type of equation. In this article we present a variable-step Numerov method for the numerical solution of the Schrödinger equation.     

An efficient and economical high order method for the numerical approximation of the Schrödinger equation

A new scheme is developed in this paper, for the first time in the literature. The new scheme: (1) is a symmetric two-step method, (2) is of three-stages scheme, (3) is a high order method (i.e of eight-algebraic order), (4) the approximations of the layers are taken place as follows: first layer on the point \(x_{n-1}\), second layer on the point \(x_{n}\), third layer on the point \(x_{n+1}\), (5) has vanished the phase-lag and its derivatives up to order four, (6) has good interval of periodicity properties [i.e. interval of periodicity equal to (0, 9.8)]. A detailed theoretical analysis is also presented. More specifically we present: (1) the development of the new method, (2) comparative error analysis (3) stability analysis. The effectiveness of the new scheme is tested via the solution of systems of coupled differential equations of the Schrödinger type.     

A new two-step hybrid method for the numerical solution of the Schrödinger equation

With this paper, a new algorithm is developed for the numerical solution of the one-dimensional Schrödinger equation. The new method uses the minimum order of the phase-lag and its derivatives. Error analysis and the numerical results illustrate the efficiency of the new algorithm.     

Pseudospectral methods of solution of the Schrödinger equation

Several different pseudospectral methods of solution of the Schrödinger equation are applied to the calculation of the eigenvalues of the Morse potential for I₂ and the Cahill–Parsegian potential for Ar₂ [Cahill, Parsegian, J. Chem. Phys. 121, 10839 (2004)]. The calculation of the eigenvalues for the Woods–Saxon potential are also considered. The convergence of the eigenvalues with a quadrature discretization method is found to be very fast owing to the judicious choice for the weight function, basis set and quadrature points. The weight function used is either related to the exact ground state wavefunction, if known, or an approximation to it from some reference potential. We compare several different pseudospectral methods.     

Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation

In this paper we present a new method for the numerical solution of the time-independent Schrödinger equation for one spatial dimension and related problems. A technique, based on the phase-lag and its derivatives, is used, in order to calculate the parameters of the new Numerov-type algorithm. We study the relation of the local truncation error with the energy of the model of the radial Schrödinger equation and via this investigation we examine how accurate is the new method compared with other well known numerical methods in the literature. We present also the stability analysis of the new method and the relation of the interval of periodicity with the frequency of the test problem and the frequency of the new developed method. We illustrate the accuracy and computational efficiency of the new developed method via numerical examples.     

Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation

For the first time in the literature we develop in this paper a three stages symmetric six-step scheme with twelfth algebraic and eliminated phase-lag and its first derivative. An additional characteristic of the new scheme is that the first and the second layer denote the approximation of the function on the point \(x_{n+3}\). We also present a local truncation error analysis and a stability and interval of periodicity analysis and we compared the new scheme with the classical scheme (i.e. scheme with constant coefficients). Additionally, we examine in details the accuracy and computational efficiency of the new developed scheme on the numerical solution of the Schrödinger equation. The study and investigation which are presented in this paper, lead to the conclusion that the new obtained scheme is more effective than other known or recently developed methods of the literature.     

An improvement to the comparison equation method for solving the Schrödinger equation

A systematic improved comparison equation method to solve the Schrödinger equation is described. The method is useful in quantum mechanical calculations involving two or more transition or turning points and is applicable to real potentials with continuous derivatives. As a computational example of the method, a study of the bound-state problem using the Morse potential is given.     

High order four-step hybrid method with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation

In this paper, we develop a new four-step hybrid method of sixth algebraic order with vanished phase-lag and its first and second derivatives. For the obtained method we study:

– its error and

– its stability

We apply the produced method to the Schrödinger equation in order to show its efficiency.     

A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation

In this paper we introduce a new explicit hybrid Numerov-type method. This method is of fourth algebraic order and has phase-lag and its first two derivatives equal to zero. We present a stability analysis and an error analysis based on the radial Schrödinger equation. Finally we apply the new proposed method to the resonance problem of the radial Schrödinger equation and we present the final conclusion based on the theoretical analysis and numerical results.     

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrödinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrödinger equation and of coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known methods.     

A new fourteenth algebraic order finite difference method for the approximate solution of the Schrödinger equation

In the present paper we will develop and analyse a new five-stages symmetric two-step method of high algebraic order with vanished phase-lag and its first, second, third, fourth and fifth derivatives. We will construct the new method. We will compute its local local truncation error (LTE). We will produce the asymptotic form of the LTE applying the new method to the radial time independent Schrödinger equation and we will compare it with other asymptotic forms of LTE of similar methods. Applying the new method to a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, we will investigate the stability and the interval of periodicity of the new method based and we will compare the produced interval of periodicity with other intervals of similar methods. Finally, we will examine the effectiveness of the new method applying it to the coupled Schrödinger equations.     

Non-Markovian stochastic Schrödinger equations in different temperature regimes: a study of the spin-boson model

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.     

Exact solution of the Schrödinger equation with a new expansion of anharmonic potential with the use of the supersymmetric quantum mechanics and factorization method

The study involves finding exact eigenvalues of the radial Schrödinger equation for new expansion of the anharmonic potential energy function. All analytical calculations employ the mathematical formalism of the supersymmetric quantum mechanics. The novelty of this study is underlined by the fact that for the first time the recurrence formulas for rovibrational bound energy levels have been derived employing factorization method and algebraic approach. The ground state and the excited states have been determined by means of the hierarchy of the isospectral Hamiltonians. The Riccati nonlinear differential equation with superpotentials has been solved analytically. It has been shown that exact solutions exist when the potential and superpotential parameters satisfy certain supersymmetric constraints. The results obtained can be utilized both in computations of quantum chemistry and theoretical spectroscopy of diatomic molecules.