Modified two-derivative Runge–Kutta methods for the Schrödinger equation

A family of modified two-derivative Runge–Kutta (MTDRK) methods for the integration of the Schrödinger equation are obtained. Two new three-stage and fifth order TDRK methods are derived. The numerical results in the integration of the radial Schrödinger equation with the Woods–Saxon potential are reported to show the high efficiency of our new methods. The results of the error analysis are illustrated by the resonance problem.     

Using preconditioned adaptive step size Runge-Kutta methods for solving the time-dependent Schrödinger equation

If the Hamiltonian is time dependent it is common to solve the time-dependent Schr?dinger equation by dividing the propagation interval into slices and using an (e.g., split operator, Chebyshev, Lanczos) approximate matrix exponential within each slice. We show that a preconditioned adaptive step size Runge-Kutta method can be much more efficient. For a chirped laser pulse designed to favor the dissociation of HF the preconditioned adaptive step size Runge-Kutta method is about an order of magnitude more efficient than the time sliced method.     

New optimized explicit modified RKN methods for the numerical solution of the Schrödinger equation

This paper investigates a family of modified Runge-Kutta-Nyström (RKN) methods for the integration of second-order ordinary differential equations with oscillatory solutions. The order conditions for up to order five are presented. Two new optimized explicit four-stage modified RKN methods are derived by nullifying their dispersions and the dissipations in two different ways, respectively. These methods are checked to be of algebraic order five and both are dispersive of order six and dissipative of order five. The stability is examined and the error formulas are analyzed to show that advantages of the new methods compared with some highly efficient integrators from the recent literature. The high accuracy of the second new method is explained by its comparatively small dispersion and dissipation constants. In the integration of the resonance problem and the bound-states problem of the radial Schrödinger equation with the Woods-Saxon potential, the numerical results show the effectiveness and robustness of the new methods.     

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.     

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.     

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.     

Trigonometrically-fitted multi-derivative linear methods for the resonant state of the Schrödinger equation

A family of trigonometrically-fitted multi-derivative linear methods for the numerical integration of the Schrödinger equation are constructed. Numerical results show the efficiency and robustness of the new methods when applied to the radial time-independent Schrödinger equation for large energies. Error analysis is carried out and the asymptotic expressions of the local errors for large energies explain the numerical results in the case of the resonance problem.     

Embedded eighth order methods for the numerical solution of the Schrödinger equation

A new method for the approximate numerical integration of the radial Schrödinger equation is developed in this paper. Phase-lag and stability analysis of the new method is included. The new method is called the embedded method because of a simple natural error control mechanism. Numerical results obtained for the phase-shift problem of the radial Schrödinger equation show the validity of the developed theory.     

Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation

The computation of the energy eigenvalues of the one-dimensional time-independent Schrödinger equation is considered. Exponentially fitted and trigonometrically fitted symplectic integrators are obtained, by modification of the first and second order Yoshida symplectic methods. Numerical results are obtained for the one-dimensional harmonic oscillator and Morse potential.AMS subject classification: 65L15Funding by research project 71239 of Prefecture of Western Macedonia and the E.U. is gratefully acknowledged.     

Sixth algebraic order trigonometrically fitted predictor–corrector methods for the numerical solution of the radial Schrödinger equation

Our new trigonometrically fitted predictor–corrector (P–C) schemes presented here are based on the well known Adams–Bashforth–Moulton methods: the predictor is based on the fifth order Adams–Bashforth scheme and the corrector on the sixth order Adams–Moulton scheme. We tested the efficiency of our newly developed schemes against well known methods, with excellent results. The numerical experiments showed that at least one of our schemes is noticeably more efficient compared to other methods, some of which are specially designed for this type of problem. It is also worth mentioning that this is the first time that sixth algebraic order trigonometrically fitted Adams–Bashforth–Moulton P–C schemes are used to efficiently solve the radial Schrödinger equation.Active Member of the European Academy of Sciences and Arts     

An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems

In this paper we present an optimized explicit Runge-Kutta method, which is based on a method of Fehlberg with six stages and fifth algebraic order and has improved characteristics of the phase-lag error. We measure the efficiency of the new method in comparison to other numerical methods, through the integration of the Schrödinger equation and three other initial value problems.     

Unique continuation for the magnetic Schrödinger equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.     

Exponentially fitted multi-derivative linear methods for the resonant state of the Schrödinger equation

A new family of exponentially fitted P-stable one-step linear methods involving several derivatives for the numerical integration of the Schrödinger equation are obtained. Numerical results are reported to show the efficiency and robustness of the new methods specially adapted to the integration of the radial time-independent Schrödinger equation for large energies. Error analysis is carried out and the asymptotic expressions of the local errors for large energies explain the results of the numerical experiments on the resonance problem.     

A family of multiderivative methods for the numerical solution of the Schrödinger equation

A family of multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the Schrödinger equation. The methods are called multiderivative since uses derivatives of order two, four or six. Numerical application of the new obtained methods to the Schrödinger equation shows their efficiency compared with other similar well known methods of the literature.Active Member of the European Academy of Sciences and Arts     

Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation

In this paper we construct two trigonometrically fitted methods based on a classical Runge–Kutta method of England with fifth algebraic order. The methods will be used for the integration of the radial Schrödinger equation and have high efficiency as the results show. The efficiency is higher when using higher energy and this can be explained by the error analysis of the methods. More specifically the new methods have lower powers of the energy in the local truncation error and that keeps the error at lower values.PACS: 0.260, 95.10.EActive Member of the European Academy of Sciences and Arts     

New open modified trigonometrically-fitted Newton-Cotes type multilayer symplectic integrators for the numerical solution of the Schrödinger equation

In this paper we present new modified open Newton Cotes integrators and we develop a new modified trigonometrically-fitted open Newton-Cotes method. We study the connection between the new proposed schemes, the differential methods and the symplectic integrators. although The research on multistep symplectic integrators is very poor, although, much research has been done on one step symplectic integrators and several of then have obtained based on symplectic geometry. In this paper a new open modified numerical algorithm of Newton-Cotes type is produced. We present the new obtained method as symplectic multilayer integrator. The new obtained symplectic schemes are applied for the solution of the resonance problem of the radial Schrödinger Equation. The results show the efficiency of the new proposed algorithm.     

Smolyak Scheme for solving the Schrödinger equation: Application to Malonaldehyde in Full Dimensionality

In 1963 Smolyak introduced an approach to overcome the exponential scaling with respect to the number of variables of the direct product size [S. A. Smolyak Soviet Mathematics Doklady, 4, 240 (1963)]. The main idea is to replace a single large direct product by a sum of selected small direct products. It was first used in quantum dynamics in 2009 by Avila and Carrington [G. Avila and T. Carrington, J. Chem. Phys., 131, 174103 (2009)]. Since then, several calculations have been published by Avila and Carrington and by other groups. In the present study, and to push the limit to larger and more complex systems, this scheme is combined with the use of an on-the-fly calculation of the kinetic energy operator and a Block-Davidson procedure to obtain eigenstates in our home-made Fortran codes, ElVibRot and Tnum-Tana. This was applied to compute the tunneling splitting of malonaldehyde in full dimensionality (21D) using the potential of Mizukami et al. [W. Mizukami, S. Habershon, and D.P. Tew, J. Chem. Phys. 141, 1443–10 (2014)]. Our tunneling splitting calculations, 21.7±0.3 cm⁻¹ and 2.9±0.1 cm⁻¹, show excellent agreement with the experimental values, 21.6 cm⁻¹ and 2.9 cm⁻¹ for the normal isotopologue and the mono-deuterated one, respectively.     

A family of P-stable exponentially‐fitted methods for the numerical solution of the Schrödinger equation

A family of Pstable exponentiallyfitted methods for the numerical solution of the Schrödinger equation is developed in this paper. An application to the resonance problem of the radial Schrödinger equation indicates that the new method is generally more efficient than the previously developed exponentiallyfitted methods of the same kind.