Numerov-Cooley Method Research Articles

산화질소 분자 퍼텐셜에 적용한 Numerov-Cooley 방법

Numerov-Cooley 방법을 적용할 때 양쪽에서 전파된 파동함수를 연결시키는 것이 관건인데, 잘 알려진 스프레드시트인 엑셀에서 해찾기 도구를 활용하여 Numerov 방법의 파동함수 연결을 해결하였다. 한쪽방향으로 전파하여 얻은 에너지 값과 반대쪽으로 전파하여 얻은 에너지 값들의 평균을 이용해 상태함수의 고유값을 정하는 방법도 같은 결과를 보임을 확인하였다. 산화질소 분자 퍼텐셜 위에서 이 방법을 적용하였고, 진동에너지에 따른 원자간 평균거리와 터널링 변화를 계산하였다. 진동에너지가 증가하면서 분자 결합 길이는 비례하여 늘어나지만, 터널 효과는 모든 진동 상태가 어느 정도의 확률을 가지며 에너지 증가에 둔감하다는 것을 확인하였다. In applying Numerov-Cooley method, Excel tool ‘Solver' is used to match those two wave functions propagated inward and outward, respectively. It is numerically confirmed that the same eigenvalue is obtained by using the average of two energy values of each inward and outward wave functions. This method is applied to a NO molecule potential, and we calculated the variations of the average bond distance and tunneling for a given vibrational energy. It is found that the average bond lengths increase proportionately to the vibrational energy, while the tunneling is not so sensitive to the energy changes. Rather substantial amount of tunnel effect is found for every vibrational state.

Nonadiabatic and Born–Oppenheimer calculations of the polarizabilites of LiH and LiD

We present fully nonadiabatic as well as Born–Oppenheimer calculations of the dipole moments and static polarizabilites of LiH and LiD. The nonadiabatic calculations are done in a basis of explicitly correlated gaussian functions. The electronic Born–Oppenheimer values are corrected for vibrational contributions using the numerical Numerov–Cooley method including all orders of anharmonicity. A comparison between the two methods does not suggest any substantial disagreement in absolute values or the isotopic differences of the calculated properties.

On the effect of potential-energy cubic-spline interpolation error on Numerov-Cooley diatomic rovibrational levels

A heuristic procedure is presented to estimate cubic-spline interpolation error for diatomic-molecule potential-energy curves and assess its effect on rovibrational energy levels computed by the Numerov-Cooley method. The approach is tested on the experimental potential curve for Li 2 (X 1∑ g +) as defined by its RKR turning points, and allows the identification of a few misprints in what is currently the most up-to-date reference for the latter.

Vibrational frequency shifts of diatomic molecules in interaction with a Na + cation by ab initio calculations. Comparison with experiment on H 2 and N 2 adsorbed in NaA zeolite

Ab initio calculations on H 2 and N 2 and on the same molecules in the presence of a Na + cation are presented. The equilibrium configuration and the vibrational frequency shift due to the interaction are calculated. The potential energy surfaces are obtained by local osculatory interpolations and extrapolations. The vibrational frequencies are calculated by the Numerov—Cooley method. The direction of the frequency shift is found to be related to the orientation of the diatomic molecule with respect to the cation. The results are compared with experimental data on induced infrared bands of H 2 and N 2 adsorbed in NaA zeolite.

Finite element method versus finite difference methods : theoretical investigations of some models of potential of biological interest

In view of evaluating the specific advantages of the finite difference methods and the finite element method (FEM) in the resolution of the Schrödinger nuclear equation, we performed accurate investigations of some one-dimensional multi-well potentials. Two finite difference methods were used : an Adams method and the powerful Numerov-Cooley method. The FEM investigations were carried out with quadratic interpolation functions and the shooting method was used in place of the usual algorithms.

A note on Cooley's energy correction formula

A simple alternative derivation based on standard first-order perturbation theory in quantum mechanics is given for Cooley's energy correction formula which is employed in the popular iterative Numerov-Cooley method for numerically solving the one-dimensional eigenvalue Schrödinger equation.

New numerical approaches to the solution of the N‐well Schrödinger equation

AbstractTwo new numerical methods for solving the Schrödinger equation for an N‐well periodic potential are presented. One is a diagonalization method based on Floquet–Bloch formalism, and the other is a renormalized Numerov–Cooley method with periodic boundary conditions. The numerical superiority of these methods over the standard diagonalization technique is demonstrated. The methods are illustrated by applications to internal rotation.

Numerical experiments in quantum physics: Finite-element method

A new numerical approach (finite-element method) to solving one-dimensional Schrödinger problems is discussed. Its effectiveness is demonstrated in solving the nontrivial finite well and Morse potential problems. Moreover, it is shown by a case study of 16 low-lying vibrational states of Li2 that the finite-element method can achieve the same order of accuracy more easily than the better known Numerov–Cooley method.

Derivative Numerov–Cooley theory. A method for finding vibrational state properties of diatomic molecules

The difference equation method for finding one-dimensional vibrational state wave functions and energies which is usually referred to as the Numerov–Cooley method has been extended to molecular properties which are defined as energy derivatives. By straightforward differentiation of the Schrödinger equation with respect to embedded parameters, for instance, electric field components, a difference equation procedure has been developed that finds derivative vibrational wave functions and derivatives of the electronic–vibrational state energies of diatomic molecules. Representative test calculations are reported.

Solution of the radial Schrödinger equation by a modified “shooting” method

Based on a comparative analysis of various modifications of the “shooting” method, used for solving the radial Schrodinger equation, a new algorithm is proposed, which allows to increase considerably its solution accuracy. This is achieved, first, by rising the order of approximation of the Schrodinger equation fromh4 (in the traditional Numerov-Cooley method) up toh6 and, second, by using the variable integration steph. The computer program has been written in Fortran IV to execute this algorithm for calculating vibrational wave functions and related quantities for diatomic molecules.

ChemInform Abstract: THEORETICAL CALCULATIONS OF THE RING‐PUCKERING VIBRATION FOR TRIMETHYLENE OXIDE (TMO) AND ITS DEUTERATED DERIVATIVES. THE SEMIRIGID MODELS FOR THE LARGE‐AMPLITUDE RING‐PUCKERING VIBRATION

Chemischer InformationsdienstVolume 14, Issue 19 Physical Organic Chemistry ChemInform Abstract: THEORETICAL CALCULATIONS OF THE RING-PUCKERING VIBRATION FOR TRIMETHYLENE OXIDE (TMO) AND ITS DEUTERATED DERIVATIVES. THE SEMIRIGID MODELS FOR THE LARGE-AMPLITUDE RING-PUCKERING VIBRATION I. FOLTYNOWICZ, I. FOLTYNOWICZSearch for more papers by this author I. FOLTYNOWICZ, I. FOLTYNOWICZSearch for more papers by this author First published: May 10, 1983 https://doi.org/10.1002/chin.198319074AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume14, Issue19May 10, 1983 RelatedInformation

Theoretical calculations of the ring-puckering vibration for trimethylene oxide (TMO) and its deuterated derivatives: The rigid model for the large-amplitude ring-puckering vibration

The Hougen-Bunker-Johns model (J. Mol. Spectrosc.34, 136 (1970)) has been extended to a four-membered ring molecule allowing for large-amplitude out-of-plane ring vibration, usually described as a ring-puckering vibration. This model has been used for the trimethylene oxide molecule (TMO) and its symmetrically deuterated derivatives, namely for TMO-β-d2, TMO-α,α′-d4, and TMO-d6. The large-amplitude ring-puckinger vibration energy levels have been calculated using a zeroth-order Hamiltonian in which the small-amplitude vibrations are omitted. The calculations have been carried out using the Numerov-Cooley method for two different forms of the double-minimum potential function: (i) a two-parameter polynomial quadratic-quartic function and (ii) a three-parameter function being a sum of the quadratic and Lorentz terms. It has been shown that: (i) satisfactory agreement between the experimentally observed ring-puckering bands and theoretically calculated ones has been obtained using only two ring-puckering transitions for the fitting of the potential function parameters, (ii) the values of the potential function parameters are different for four TMO molecules, (iii) the model satisfactorily reproduces the intensities of the ring-puckering transitions.

On quantal bound state solutions and potential energy surface fitting. A comparison of the finite element, Numerov-Cooley and finite difference methods

This paper investigates several factors affecting the accuracy and efficiency of numerical determination of the bound state energy eigenvalues of the one dimensional Schrödinger equation. The efficiencies of the finite element method (FEM), the Numerov-Cooley method, and the finite difference method are compared. From this comparison, it is concluded that for potentials containing a single energy minimum, the Numerov-Cooley method is the most efficient, while for the most complex potentials the finite element method is superior due to its better numerical stability in the classically forbidden regions. The effects of various polynomial interpolation schemes on the calculated eigenvalues of potentials known only on a small number of points is examined. It is found that while higher order fits are superior to lower ones when the potential points are known accurately, they can introduce spurious information into the potential for inaccurately known points, and thus produce poor eigenvalues. Likewise, for accurately known potentials, a spline or Hermitian interpolation is better than a Lagrangian fit, but the Lagrangian functions are less susceptible to noise in a less well known case.

Errors in eigenvalues calculated by the Numerov-Cooley method

Formulas are established for the errors in the eigenvalues obtained by using standard Numerov integration methods. These error estimates can be used to derive more accurate eigenvalues without the necessity of increasing the number of iteration steps.

Vibration-inversion-rotation spectra of ammonia. The “inverse” eigenvalue problem for the one-dimensional Schrödinger equation

An iteration procedure based on the Numerov-Cooley method of numerical integration is proposed for the computation of the potential function parameters in a one-dimensional Schrödinger equation from the experimental energies. The procedure has been tested on a calculation of the parameters of a Morse potential function and has been applied to a determination of the parameters of a double-minimum potential for the inversion motion of NH 3 and ND 3.