Quantization function for potentials with $-1/r^4$ tails

For deep potentials with attractive tails proportional to $-1/r^4$, as occur in the interaction between a charged ion and a neutral polarizable atom (or molecule), we present an analytical expression for the tail contribution to the quantization function $F\left(E\right)$ which determines the near-threshold bound-state energies $E_n$ according to the quantization rule $n_{\text{th}}-n=F\left(E_n\right)$. Its usefulness is demonstrated in applications to a model Lennard-Jones potential and to the $\text{H}_2{}^{+}$ molecular ion.

Figure 1

Difference (30) of the high-$\kappa$ expansion (29) and the exact numerically calculated values of the outer reflection phase, multiplied by ${\left(\kappa {\beta }_4\right)}^{7/2}$, as a function of $1/\left(\kappa {\beta }_4\right)$.

Figure 2

Reflection phase ${\varphi }_{\text{out}}\left(E\right)$ for a homogeneous potential tail (15) with $\alpha =4$ (solid line) together with the low-$\kappa$ expansion (31) (long-dashed line) and the high-$\kappa$ expansion (29) truncated after various values of $j$ (short-dashed lines).

Figure 3

Absolute value of the deviation of the exact numerically calculated outer reflection phase for a $-1/r^4$ potential from the fitted expression (25) (solid line) from the low-$\kappa$ expression (31) (long-dashed line) and from the high-$\kappa$ expression (29), truncated after various values of $j$ (short-dashed lines).

Figure 4

Tail contribution to the quantization function for a homogeneous potential tail (15) with $\alpha =4$ [see Eq. (38) (solid line)]. The dot-dashed line shows the LeRoy-Bernstein result (39).

Figure 5

Tail contribution to the quantization function for a homogeneous potential tail (15) with $\alpha =4$. The solid line shows the quantization function (38) and the dot-dashed line shows the LeRoy-Bernstein result (39). The long-dashed line shows the low-$\kappa$ approximation (36) and the short-dashed line shows the high-$\kappa$ approximation (37).

Figure 6

This figure shows a plot of $n+F_{\text{tail}}\left(E_n\right)$ vs $E_n/\mathcal{E}$ for all 24 bound states in the (12,4) Lennard-Jones potential (40) with strength constant $B_{\text{LJ}}=3500$. (The scaled eigenenergies $E_n/\mathcal{E}$ are listed in Table .) The filled black dots represent results obtained with the correct tail contribution (38) of the $-1/r^4$ potential tail to the quantization function, while the empty circles are based on the LeRoy-Bernstein formula (39).

Figure 7

This figure shows a plot of $n+F_{\text{tail}}\left(E_n\right)$ vs $E_n/\mathcal{E}$ for the highest six bound states in the (12,4) Lennard-Jones potential (40) with strength constant $B_{\text{LJ}}=3500$, $n=18,\dots ,23$. The scale on the ordinate is the same in both panels, but there is a gap of 0.205 between the top of the lower panel and the bottom of the upper panel. The filled black dots in the lower panel represent results obtained with the correct tail contribution (38) of the $-1/r^4$ potential tail to the quantization function, while the empty circles in the upper panel are based on the LeRoy-Bernstein formula (39).

Figure 8

$D_{\text{diss}}^{\left(n-1,n,n+1\right)}$ is the value obtained for the dissociation energy $D_{\text{diss}}$ by solving the simultaneous equations (47) for a triplet of spectroscopic energies centered around $D_n$. The filled black dots show the results obtained with the explicit analytical expression (38) for $F_{\alpha =4}$, the empty circles are obtained by using, instead, the LeRoy-Bernstein function (39).