The solution of coupled Schrödinger equations using an extrapolation method

Abstract We apply extrapolation to the limit in a finite-difference method to solve a system of coupled Schrodinger equations. This combination results in a method that only requires knowledge of the potential energy functions for the system. This numerical procedure has several distinct advantages over the more conventional methods such as Numerov's method or the method of finite differences without extrapolation. These advantages are: (i) initial guesses for the term values are not needed; (ii) no assumptions need be made about the behaviour of the wavefunctions such as the slope or magnitude in the non-classical region; (iii) the algorithm is easy to implement, has a firm mathematical foundation and provides error estimates; (iv) the method is less sensitive to round-off error than other methods since a small number of mesh points is used; (v) it can be implemented on small computers. We solve the coupled Schrodinger equation for the X2Π state of OH. Our algorithm results in term values that agree with experimentally-derived values within 6 parts in 104. The calculated wavefunctions are compared indirectly through experimentally-derived rotational constants and are found to be accurate to better than 3 parts in 103. A comparison of our method with another numerical method shows results agreeing within 1 part in 104.

[1]  W. J. Stevens,et al.  Study of the ground state potential curve and dipole moment of OH by the method of optimized valence configurations , 1974 .

[2]  Ralph W. Nicholls,et al.  Reinvestigation of Rotational-Line Intensity Factors in Diatomic Spectra , 1974 .

[3]  D. Albritton,et al.  Equivalent methods of calculating diatomic centrifugal distortion constants , 1975 .

[4]  B. R. Johnson.  The renormalized Numerov method applied to calculating bound states of the coupled‐channel Schroedinger equation , 1978 .

[5]  R. Zare.  Calculation of Intensity Distribution in the Vibrational Structure of Electronic Transitions: The B3Π0+u—X1Σ0+g Resonance Series of Molecular Iodine , 1964 .

[6]  J. A. Coxon,et al.  Rotational analysis of hydroxyl vibration–rotation emission bands: Molecular constants for OH X2Π, 6 ≤ ν ≤ 10 , 1982 .

[7]  F. Mies.  Calculated vibrational transition probabilities of OH(X2Π) , 1974 .

[8]  Ronald F. Boisvert,et al.  Guide to Available Mathematical Software. , 1984 .

[9]  A. Goldman,et al.  Spectral line parameters for the X2II-X2II (1,0) bands of OH and ClO for atmospheric applications , 1981 .

[10]  Roy G. Gordon,et al.  New Method for Constructing Wavefunctions for Bound States and Scattering , 1969 .

[11]  W. Richards,et al.  A Reanalysis of the A2?+ - X2?i System of OH , 1971 .

[12]  I. Kovacs,et al.  Rotational structure of the spectra of diatomic molecules , 1972 .

[13]  K. Lawley.  An efficient algorithm for solving the virbronic coupling problem , 1987 .

[14]  David J. Nesbitt,et al.  The dipole moment function and vibrational transition intensities of OH , 1989 .

[15]  R. F. Wallis,et al.  Influence of Vibration‐Rotation Interaction on Line Intensities in Vibration‐Rotation Bands of Diatomic Molecules , 1955 .