Adaptive analytical mapping procedure for efficiently solving the radial Schrödinger equation

This paper shows that replacing the usual integration variable $r∊[0,\infty )$ by a reduced radial variable $y\equiv y(r;\vec{\alpha })$ defined analytically on a finite domain $y∊[a,b]$ transforms the conventional radial Schrödinger equation into an equivalent form in which treatment of levels lying extremely close to dissociation becomes just as straightforward and routine as treating levels in the lower part of the potential well. Explicit integral expressions for the eigenvalue error due to the use of a finite step size in finite-difference methods of numerical integration are presented and are used to improve calculated eigenvalues as well as to determine optimal values of the mapping parameters $\vec{\alpha }$. This adaptive mapping procedure is shown to be versatile and efficient for both finite-difference and pseudospectral methods.

Figure 1

(Color online) Radial eigenfunctions ${\psi }_v\left(r\right)$ of the model 15-level LJ (8,4) potential of Table .

Figure 2

(Color online) Radial eigenfunctions ${\varphi }_v(y;\alpha =1)$ for the same five levels of the model LJ (8,4) potential considered in Fig. 1.

Figure 3

(Color online) Unit normalized mesh-point (in $y$) density distribution functions $\rho \left(r\right)$ for the mapping function $y(r;\alpha ,\overline{r}=r_e=1\left[\mathrm{\AA }\right])$ of Eq. (29) for different values of $\alpha$. Numbers in brackets denote the percent of the grid points on the interval $[0.3,\infty )$ which are located in the subinterval $r∕\overline{r}∊[0.3,3]$ [i.e., the number $100\times {\int }_{0.3}^3\rho \left(r\right)dr∕{\int }_{0.3}^{\infty }\rho \left(r\right)dr$].

Figure 4

(Color online) Optimal mapping parameters $\alpha$ for all vibrational levels of our model 15-levels LJ $(2n,n)$ potentials for $n∊\{3,4,5,6\}$ associated with use of the Numerov propagation (Num) method of integration.

Figure 5

(Color online) Differential equation curvature factor $\tilde{Q}(E;r)$ for energies very near the dissociation limit of the model LJ (12,6) potential of Table for different choices of the mapping variable parameter $\alpha$. Solid curves are for the very highest $(v=14)$ bound level of the model potential of Table ; dotted curves are for a hypothetical binding energy of ${10}^{-7}{\mathrm{cm}}^{-1}$.

Figure 6

(Color online) Convergence tests for polynomial (PCM) and Fourier (FCM) collocation method calculations of the binding energy of the very highest level of the 301-level LJ (6,3) potential of Table .

Figure 7

(Color online) Relative errors of eigenvalues obtained by the polynomial collocation method (PCM) and by the Numerov propagation (Numerov) method extrapolated to zero step size $\left({\delta }_v^{\mathrm{h}\to 0}\right)$, for vibrational levels $0⩽v⩽299$ of the 301-level model LJ (6,3) system of Table . $N$ is the number of grid points.