Numerov Method Research Articles

Theoretical calculations on 12 Λ-S and 23 Ω states of CBr+ cation in the gas phase: Potential energy curves, spectroscopic parameters and spin–orbit coupling

The potential energy curves (PECs) of X1Σ+, a3Π, 13Σ+, 13Δ, 11Δ, 11Σ−, 13Σ−, 11Π, 21Σ+, 23Π, 21Π and 23Σ+ Λ-S states of CBr+ cation and corresponding 23 Ω states are calculated for the first time using the CASSCF method, which is followed by the internally contracted MRCI approach with the aug-cc-pVQZ basis set. All the Λ-S states involved are found to be bound and dissociate into the first dissociation limit of CBr+ cation. Of these Λ-S states, only the 13Σ+ and 13Σ− are inverted ones. The spin–orbit (SO) coupling is accounted for by the state interaction approach with the Breit–Pauli Hamiltonian. Core–valence correlation is included by a cc-pCVTZ basis set. Relativistic correction is calculated with the third-order Douglas–Kroll Hamiltonian approximation at the level of cc-pVQZ basis set. To obtain more reliable results, the PECs obtained by the MRCI calculations are corrected for size-extensivity errors by means of the Davidson modification. The PEC crossings of different Λ-S states are studied. With these PECs, the spectroscopic parameters of all the Λ-S and Ω states involved are obtained by fitting the first ten vibrational levels whenever available, which are calculated by solving the rovibrational Schrödinger equation using the Numerov's method. The spectroscopic parameters are compared with those reported in the literature. Excellent agreement is found between the present results and available measurements. In particular, the energy separation of 352.26cm−1 between the a3Π0+ and the a3Π1 Ω states agrees well with the measurements of 369±8cm−1, and the ωe results of 907.45 and 907.08cm−1 for the a3Π0+ and a3Π1 Ω states are in excellent agreement with the measurements of 906±2 and 903±6cm−1, respectively. These show that the spectroscopic parameters obtained in the present paper can be expected to be reliable predicted ones.

New explicit adapted Numerov methods for second-order oscillatory differential equations

This paper is concerned with a novel class of explicit adapted Numerov (AN) methods for solving initial value problems of second-order oscillatory differential equations of the form y″+ω2y=f(x,y). The scheme of the methods incorporate the special structure of the problem carried by the term ω2y, so that they can integrate the linear equation y″+ω2y=0 without truncation error. A new type of simplifying assumption is proposed on the linear operators related to the quadratures that lead to the AN scheme. A practical AN method of order five and three AN methods of order six are derived. The phase and stability properties of the new methods are analyzed. The results of numerical experiments show the robustness and competence of the new methods compared with some highly efficient codes in the recent literature.

Study of the vibrational characteristics of the homonuclear diatomic nuclear schrödinger equation with a numerov method using a number of empirical potential functions

Many empirical potential energy functions have been modeled to represent the potential energy function of a diatomic molecule along whole range of internuclear distance coordinate, whereby one can determine certain molecular constants. Here we employ various potential functions such as Morse, Rydberg, Varshni(II), Varshni(III), Varshni(VI), Poschl-Teller, Hulburt-Hirschfelder, Lippincott, Frost-Musulin, Linnet, and Rosen-Morse, and the Numerov method to solve the nuclear Schrodinger equation for F2, as an example of a homonuclear diatomic molecule. Herewith, the vibrational and vibration-rotation energy levels are obtained and excellent accuracy is achieved. The potential of employing the Numerov method in engineering physics computations is emphasized.

Quark Model Three Body Calculations for the Hypertriton Bound State

Hyperspherical three body calculations are performed to study and review the various properties of the hypertriton bound state nucleus  3 H in the quark model using -N potentials. In these calculations we study the different effects of the -N potentials on the hypertriton bound states as well as the separation energy B. A combination of realistic two body N-N potentials with various - N potentials are considered. Complete symmetric and mixed symmetric wave functions are introduced. using the renormalized Numerov method. The agreement between the calculated  3 H binding energies and the available experimental data basically depends on the type of the -N interactions used in the calculations. It was found that the -N potentials are the most effective part in the hypertriton binding energy as well as the separation energy B where the -N potentials is very effective to bound or unbound the  3 H hyper nucleus

Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force

The paper addresses the issue of effectively using the direct numerical method for static analysis of the flexible curved uniform cantilever beam under a tip-concentrated follower force. The angle of inclination of the follower force with respect to the deformed axis of the beam remains unchanged during deformation. After changing the variables, the original non-linear boundary value problem transforms into the initial-value problem for pendulum equation. The resulting initial value problem is solved numerically using a modified Numerov's method. In contrast to the usually used iteration methods (e.g. shooting technique), the problem is solved without iterations by direct numerical method. Some qualitative conclusions were made using Kirchhoff's kinetic analogy. It is shown that there are no critical loads in the Euler sense (divergence) for any values of the initial curvature and angle of inclination of the follower force. An extension of direct numerical method to curved spring-hinged cantilever subjected to follower force is also proposed. The paper presents some equilibrium configurations of the uniform curved fixed and spring-hinged cantilevers under normal and tangential follower force obtained by direct method.

Matrix Numerov method for solving Schrödinger’s equation

We recast the well-known Numerov method for solving Schrödinger’s equation into a representation of the kinetic energy operator on a discrete lattice. With just a few lines of code in a high-level programming environment such as mathematica, it is simple to calculate and plot accurate eigenvalues and eigenvectors for a variety of potential problems. We illustrate the method by calculating high-accuracy solutions for the |x| potential.

Measurements of Relative Transition Oscillator Strengths of Rydberg States in Cesium Magneto-Optical Trap

The oscillator strengths of 6 P 3/2 → n S / n D transitions are measured by fitting ultracold Cs Rydberg atom field-ionized spectra. The oscillator strength ratios η( n ) = f D 5/2 ( n )/ f D 3/2 ( n ) and ρ( n ) = f S 1/2 ( n + 2)/ f D 3/2 ( n ), which agree with the calculations based on the wave functions obtained by Numerov's method, are determined. The obtained results indicate that the ratios are almost independent of the principal quantum number n .

Asymptotic - Numerical Method for Third-Order Singular Perturbation Problems

ABSTRACT In this paper, a class of third order singularly perturbed boundary value problems with suitable boundary conditions is considered. The third order boundary value problem is transformed to asymptotically equivalent second order boundary value problem. This problem is solved efficiently by using fitted Numerov method. Linear and non-linear examples are solved to illustrate the method and relative errors with L2-norms are presented to support the method.

Coupled-channel theory of photoionization microscopy

We develop a quantum-mechanical coupled-channel theory to simulate spatial distributions of electron current densities, produced in photoionization for nonhydrogenic atoms in the presence of a uniform external electric field. The coupled Schr\"odinger equations are numerically solved using the renormalized Numerov method. The expression for the outgoing wave function for photoelectrons ejected from the nonhydrogenic atomic source is derived. The theory is applied to investigations of photoionization for ground-state Li atoms. The distributions of electron current densities are computed and compared to the corresponding experimental images. Excellent agreement is obtained. It is, furthermore, found that the presence of the nonhydrogenic residual ion significantly changes the differential cross sections and/or electron current densities with respect to the hydrogenic case. Finally, the implications of the presence of the atomic core for quantum resonance tunneling are also analyzed.

Trigonometrically fitted block Numerov type method for y′′ = f(x, y, y′)

A trigonometrically fitted block Numerov type method (TBNM), is proposed for solving y???=?f(x, y, y?) directly without reducing it to an equivalent first order system. This is achieved by constructing a continuous representation of the trigonometrically fitted Numerov method (CTNM) and using it to generate the well known trigonometrically fitted Numerov method (TNUM) and three new additional methods, which are combined and applied in block form as simultaneous numerical integrators. The stability property of the TBNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

Electronic spectrum of 17 electronic states of BN molecule: A theoretical study

The potential energy curves (PECs) of the X3Π, a1Σ+, b1Π, A3Σ+, B3Σ−, c1Δ, D3Π, 15Π, 31Σ+, 33Π, 21Π, 23Σ+, 13Δ, 15Σ+, 43Π, 23Σ− and 15Σ− electronic states of the BN molecule are calculated using an ab initio quantum chemical method. The PEC calculations have been made for internuclear separations from 0.06 to 1.20nm using the complete active space self-consistent field (CASSCF) method, which is followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with a correlation-consistent aug-cc-pV5Z basis set. To improve the quality of PECs, core-valence correlation and relativistic corrections are included. Relativistic correction calculations are carried out using the third-order Douglas–Kroll Hamiltonian (DKH3) approximation. Core–valence correlation corrections are included using a cc-pCVQZ basis set. Relativistic corrections are calculated at the level of a cc-pVQZ basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are corrected for size-extensivity errors by means of the Davidson modification (MRCI+Q). These PECs are extrapolated to the complete basis set limit by the total-energy extrapolation scheme. The spectroscopic parameters are determined by fitting the vibrational levels, which are calculated in a direct forward manner from the analytic potential by solving the ro-vibrational Schrödinger equation using Numerov's method. The spectroscopic results have been compared in detail with those reported in the literature. Excellent agreement has been found between the present spectroscopic results and the experimental ones. Using the Breit-Pauli operator, the spin–orbit coupling effect on the spectroscopic parameters is included in the X3Π and D3Π electronic states. The vibrational level, inertial rotation and centrifugal distortion constants are calculated for each vibrational state of each electronic state. And those of the first 20 vibrational states of each electronic state are reported when the rotational quantum number equals zero. Comparison with the measurements shows that the present results are accurate. The spectroscopic parameters of the c1Δ, 15Π, 31Σ+, 33Π, 21Π, 23Σ+, 13Δ, 15Σ+, 43Π and 15Σ− electronic states and the vibrational manifolds of all the electronic states obtained here are expected to be reliable predicted results.

Investigations on spectroscopic parameters and molecular constants of SO+ (b4∑-) cation

The potential energy curve (PEC) of b4Σ- electronic state of the SO+ cation is calculated using the internally contracted multireference configuration interaction approach with the Davidson modification (MRCI+Q) for internuclear separations from 0.103 to 1.083 nm. The basis set used is a correlation- consistent aug-cc-pV5Z basis set. The spin-orbit coupling effect on the spectroscopic parameters is taken into account by the state interaction approach with the full Breit-Pauli operator with all-electron basis set, aug-cc-pCVTZ. To improve the quality of PEC and spin-orbit coupling constant, core-valence correlation and relativistic corrections are included. Core-valence correlation correction is calculated using a cc-pCVTZ basis set. Relativistic corrections are included by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. At the MRCI+Q/aug-cc-pV5Z+CV+DK level, the spin-orbit coupling constant of the SO+ (b4Σ-1/2,3/2) is 1 cm-1 when the aug-cc-pCVTZ basis set is used for the spin-orbit coupling calculations The spectroscopic parameters are determined and compared with those reported in the literature. Excellent agreement is found between the present results and the measurements. The vibrational level G(v) inertial rotation constant Bv and centrifugal distortion constant Dv are predicted for each vibrational state of the b4Σ- electronic state by solving the ro-vibrational Schrödinger equation of nuclear motion using Numerov's method and those of the first 2 vibrational states are reported for the non-rotation SO+ cation. Comparison with the measurements demonstrates that the present vibrational manifolds are both reliable and accurate. They should be good predictions for future experimental or theoretical research.

A modified Numerov method for solving singularly perturbed differential–difference equations arising in science and engineering

In this paper a modified fourth order Numerov method is presented for singularly perturbed differential–difference equation of mixed type, i.e., containing both terms having a negative shift and terms having positive shift. Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. To handle the negative and positive shift terms, we construct a special type of mesh, so that the terms containing shift lie on nodal points after discretization. The proposed finite difference method works nicely when the shift parameters are smaller or bigger to perturbation parameter. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameters on the boundary layer behavior or oscillatory behavior of the solution of the problem.

Transient Stability Constrained Optimal Power Flow Based on Second-Order Differential Equations

This paper directly uses the second-order differential equations to denote the swing equations of generators and adopts numerov method to convert the differential equations to a series of algebraic equations. Considering the truncation error of specific numerical integration algorithm, the transient differential equations are dispersed to inequality constraints instead of equality constraints. The proposed method can reduce the dimension of correction equation by reducing the number of the equality constraints and variables, so as to improve the computational efficiency and reduce the memory usage. Numerical simulations on the four test systems, from 9 to118 buses, show that the proposed method is robust and efficient.

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

This study employs the use of the fourth order Numerov scheme to determine the eigenstates and eigenvalues of particles, electrons in particular, in single and double delta function potentials. For the single delta potential, it is found that the eigenstates could only be attained by using specific potential depths. The depth of the delta potential well has a value that varies depending on the delta strength. These depths are used for each well on the double delta function potential and the eigenvalues are determined. There are two bound states found in the computation, one with a symmetric eigenstate and another one which is antisymmetric. Keywords—Double Delta Potential, Eigenstates, Eigenvalue, Numerov Method, Single Delta Potential

Spectroscopic investigations on NO+(X1Σ+, a3Σ+, A1Π) ion using multi-reference configuration interaction method and correlation-consistent sextuple basis set augmented with diffuse functions

Three low-lying electronic states (X1Σ+, a3Σ+, and A1Π) of NO+ ion are studied using the complete active space self-consistent-field (CASSCF) method followed by highly accurate valence internally contracted multi-reference configuration interaction (MRCI) approach in combination of the correlation-consistent sextuple basis set augmented with diffuse functions, aug-cc-pV6Z. The potential energy curves (PECs) of the NO+(X1Σ+, a3Σ+, A1Π) are calculated. Based on the PECs, the spectroscopic parameters Re, De, ωe, ωeχe, αe, Be, and D0 are reproduced, which are in excellent agreement with the available measurements. By numerically solving the radial Schrödinger equation of nuclear motion using the Numerov method, the first 20 vibrational levels, inertial rotation and centrifugal distortion constants of NO+(X1Σ+, a3Σ+, A1Π) ion are derived when the rotational quantum number J is equal to zero (J = 0) for the first time, which accord well with the available measurements. Finally, the analytical potential energy functions of these states are fitted, which are used to accurately derive the first 20 classical turning points when J = 0. These results are compared in detail with those of previous investigations reported in the literature.

Spectroscopic investigations on BH+ (X2 Σ+ ) ion using MRCI method and correlation-consistent sextuple basis set augmented with diffuse functions

The highly accurate valence internally contracted multireference configuration interaction (MRCI) approach in combination with a series of correlation-consistent basis sets, cc-pVnZ and aug-cc-pVnZ (n = 2, 3, 4, 5, 6), in the valence range is employed to investigate the equilibrium internuclear separations, harmonic frequencies and potential energy curves (PECs) of the BH+(X2Σ+) ion. The PECs are all fitted to the Murrell–Sorbie function, which are used to reproduce the spectroscopic parameters, such as De, ωeχe, αe, Be, and D0. By comparison with the available experiments, The PEC, which is determined at the sextuple basis set augmented with diffuse functions, aug-cc-pV6Z, is found to most suitably determine the spectroscopic properties of the ground-state BH+ ion. The D0, De, Re, ωe, ωeχe, αe, and Be at this basis set are of 1.9823 eV, 2.1364 eV, 0.12041 nm, 2520.81 cm−1, 61.80 cm−1, 0.4824 cm−1, and 12.5932 cm−1, respectively, which are in excellent agreement with the available measurements. With the interaction potential determined at the MRCI/aug-cc-pV6Z level of theory, by numerically solving the radial Schrodinger equation of nuclear motion using the Numerov method, the total of 16 vibrational states is predicted when the rotational quantum number J is equal to zero (J = 0). The complete vibrational levels, classical turning points, inertial rotation and centrifugal distortion constants are determined when J = 0 for the first time, which agree well with the available experiments. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Elastic Scattering between Ultracold 23Na and 85Rb Atoms in the Triplet State

The elastic scattering properties between ultracold 23Na and 85Rb atoms for the triplet state (a3 Σ+u) are researched. The s-wave scattering lengths of 23Na and 85Rb are calculated by the Numerov and semiclassical method with two kinds of interatomic potentials, which are the interpolation potential and Lennard—Jones potential (LJ12,6) by the same phase Φ. Shape resonances appear clearly in the l = 5 partial waves for the a3 Σ+u state. Moreover, the s-wave scattering cross section, total cross section and energy positions of shape resonances are also discussed.

Accurate potential energy function and spectroscopic study of the X2Σ+, A2Π and B2Σ+ states of the CP radical**Project supported by the National Natural Science Foundation of China (Grant No. 10874064) and the Program for Science & Technology Innovation Talents in Universities of Henan Province in China (Grant No. 2008HASTIT008).

This paper calculates the equilibrium internuclear separations, the harmonic frequencies and the potential energy curves of the X2Σ+, A2Π and B2Σ+ states of the CP radical by the highly accurate valence internally contracted multireference configuration interaction method with correlation-consistent basis sets (aug-cc-pV6Z for C atom and aug-cc-pVQZ for P atom). The potential energy curves are all fitted with the analytic potential energy function by the least-square fitting. Employing the analytic potential energy function, we determine the spectroscopic constants (Be, αe and ωeχe) of these states. For the X2Σ+ state, the obtained values of De, Be, αe, ωeχe, Re and ωe are 5.4831 eV, 0.792119 cm−1, 0.005521 cm−1, 6.89653 cm−1, 0.15683 nm, 12535.11 cm−1, respectively. For the A2Π state, the present values of De, Be, αe, ωeχe, Re and ωe are 4.586 eV, 0.703333 cm−1, 0.005458 cm−1, 6.03398 cm−1, 0.16613 nm, 1057.89 cm−1, respectively. For the B2Σ+ state, the present values of De, Be, αe, ωeχe, Re and ωe are 3.506 eV, 0.677561 cm−1, 0.00603298 cm−1, 5.68809 cm−1, 0.1696 nm, 822.554 cm−1, respectively. For these states, the vibrational states with the rotational quantum number J equals zero (J = 0) are studied by solving the radial nuclear Schrödinger equation using the Numerov method. For each vibrational state, the vibrational level, the classical turning points, the rotational inertial and the centrifugal distortion constants are calculated. Comparison is made with recent theoretical and experimental results.

Spectroscopic investigations on AlH(X1Σ+) radical using multireference configuration interaction theory in combination with correlation‐consistent quintuple basis set augmented with diffuse functions

AbstractThe equilibrium internuclear separations, harmonic frequencies, and potential energy curves (PECs) of the AlH(X1Σ+) radical have been investigated using the highly accurate valence internally contracted multireference configuration interaction (MRCI) approach in combination with a series of correlation‐consistent basis sets in the valence range. The PECs are all fitted to the Murrell‐Sorbie function, which are used to reproduce the spectroscopic parameters (De, D0, ωeχe, αe, and Be). By comparison with the available experimental results, the PEC obtained at the basis set, aug‐cc‐pV5Z, is selected to investigate the vibrational manifolds. The present D0, De, Re, ωe, ωeχe, αe, and Be are of 3.0731 eV, 3.1775 eV, 0.16510 nm, 1683.37 cm−1, 29.3786 cm−1, 0.18876 cm−1, and 6.3663 cm−1, respectively, which almost perfectly conform to the available measurements. With the potential determined at the MRCI/aug‐cc‐pV5Z level of theory, the total of 26 vibrational states is predicted when the rotational quantum number J is set to equal zero (J = 0) by numerically solving the radial Schrödinger equation of nuclear motion with the Numerov method. The complete vibrational levels, classical turning points, inertial rotation, and centrifugal distortion constants are determined when J = 0 for the first time, which are in excellent agreement with the available experimental data. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011