Quantum effects of nuclear motion in three-particle diatomic ions

A high-accuracy, nonrelativistic wave function is used to study nuclear motion in the ground state of three-particle $\left\{a_1^{+}{a}_2^{+}{a}_3^{-}\right\}$ electronic and muonic molecular systems without assuming the Born-Oppenheimer approximation. Intracule densities and center-of-mass particle densities show that as the mass ratio $m_{a_i}/m_{a_3}, i=1,2$, becomes smaller, the localization of the like-charged particles (nuclei) $a_1$ and $a_2$ decreases. A coordinate system is presented to calculate center-of-mass particle densities for systems where $a_1\ne a_2$. It is shown that the nuclear motion is strongly correlated and depends on the relative masses of the nuclei $a_1$ and $a_2$ rather than just their absolute mass. The heavier particle is always more localized and the lighter the partner mass, the greater the localization. It is shown, for systems with $m_{a_1}<m_{a_2}$, that the ratio of (i) the density maximum and (ii) the FWHM of the radial distribution of each nucleus from the center of mass is directly proportional to the mass ratio of the nuclei: $m_{a_1}/m_{a_2}$ for the former and $m_{a_2}/m_{a_1}$ for the latter, thus quantifying a quantum effect of nuclear correlation.

Figure 1

Coordinate systems used in this work for $\left\{a_1^{+}{a}_2^{+}{a}_3^{-}\right\}$ systems: (a) interparticle coordinates $r_1, r_2$, and $r_3$, and (b) center-of-mass coordinates $r_1, r_2$, and $s_i, i=1$ or 2, where $c$ is the center of mass.

Figure 2

Intracule densities $h\left(r\right)$ for (a) electronic systems (in atomic units) and (b) muonic systems (in muon-atomic units, where 1 m.a.u. = $\frac{1}{206.7682826}$ a.u.). The inset corresponds to $\left\{{\mu }^{+}{\mu }^{+}{\mu }^{-}\right\}$.

Figure 3

Density plots of ${\rho }_{c,a_i}\left(s\right)$ for $s=(s_x,s_y,0)$. The density scale is given on the right-hand side color bar. The center of each plot corresponds to the center of mass. (Note scale, 1 m.a.u. = $\frac{1}{206.7682826}$ a.u.)

Figure 4

For homonuclear diatomic ions, center-of-mass particle densities ${\rho }_{c,a_i}\left(s\right)$ for (a) electronic systems and (b) muonic systems, and radial center-of-mass particle density distributions $4\pi s^2{\rho }_{c,a_i}\left(s\right)$ for (c) electronic systems and (d) muonic systems. For (c) and (d) the area under each peak is equal to 1. The inset corresponds to $\left\{{\mu }^{+}{\mu }^{+}{\mu }^{-}\right\}$. The center of mass coincides with the origin and the left peak corresponds to ${\rho }_{c,a_1}\left(s\right)$ and the right peak ${\rho }_{c,a_2}\left(s\right),$ where in each case $m_{a_1}\le m_{a_2}$. (Note scale, 1 m.a.u. = $\frac{1}{206.7682826}$ a.u.)

Figure 5

Center-of-mass particle densities for heteronuclear diatomic ions for (a) electronic systems and (b) muonic systems, and radial center-of-mass particle density distributions for (c) electronic systems and (d) muonic systems. For (c) and (d) the area under each peak is equal to 1. The center of mass coincides with the origin and the left peak corresponds to ${\rho }_{c,a_1}\left(s\right)$ and the right peak ${\rho }_{c,a_2}\left(s\right)$ where in each case $m_{a_1}\le m_{a_2}$. (Note scale, 1 m.a.u. = $\frac{1}{206.7682826}$ a.u.)