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Calculation of Eigenvalues, Transitions and Frank-Condon Factors (LEVEL 6.0)

A program is used to solve the Schrödinger equation for bound or quasibound levels of any potential surface. LEVEL6.0 calculates the eigenvalues of the bound and quasibound levels of a smooth one dimensional potential and then calculates centrifugal distortion constants for that potential. LEVEL6.0 locates and calculates expectation values for all vibration-rotation levels and also the width of the quasibound shape resonances. The program optionally calculates the Einstein A coefficients, incorporating the Hönl-London rotational intensity factors.

The core of the calculation is to determine the eigenvalues E$_{\rm v,J}$ of the one dimensional Schrödinger equation (see Equation 3.21) and the eigenfunctions of the potential V$_{\rm J}$(R). This routine is based on the Cooley-Cashion-Zare routine [81], but has the added advantage of being able to determine shape resonances. Numerical integration of Equation 3.21 is performed between two user defined points using the Numerov algorithm. To achieve this, it is necessary to specify initial values of the wavefunction at two points at the end of the range. The final value is normally set to unity, whilst the adjacent mesh point is given by the first order semiclassical or WKB wavefunction.


\begin{displaymath}
{\rm\frac{-\hbar^{2}}{2\mu}~\frac{d^{2}\Psi_{v,J}(R)}{dR^{2}} + V_{J}(R)\Psi_{v,J}(R) = E_{v,J}\Psi_{v,J}(R)}
\end{displaymath} (3.21)

The Cooley procedure is used to find the eigenvalues, using the following method. For any trial energy, the numerical integration proceeds inwards from the outer turning point and outwards from the inner turning point until the two solutions meet. The discontinuity in their slopes at this point is used to estimate a correction in the energy required to make the solutions converge close to the trial energy. This process is repeated until a user set convergence criterion is met. Quasibound levels are found using Airy function boundary conditions [82] to determine their energies and a semiclassical approximation to calculate the width.

Finally `synthetic spectra' can readily be generated using the program in two potential mode, where the Schrödinger equation is solved for both potential surfaces and calculations made for transitions between the two states and the rotational line strength according to selection rules provided in the input.


next up previous contents
Next: Design and construction of Up: Potential energy surface creation Previous: Calculation of Potential Surfaces   Contents
Tim Gibbon
1999-09-06