On the emergence of molecular structure

The structure of $\{a^{\pm },a^{\pm },b^{\mp }\}$-type Coulombic systems is characterized by the effective ground-state density of the $a$-type particles, computed via nonrelativistic quantum mechanics without introduction of the Born-Oppenheimer approximation. A structural transition is observed when varying the relative mass of the $a$- and $b$-type particles, e.g., between atomic ${\mathrm{H}}^{-}$ and molecular ${\mathrm{H}}_2$${}^{+}$. The particle-density profile indicates a molecular-type behavior for the positronium ion, Ps${}^{-}$.

Figure 1

Transition from an atomic- to a molecular-type distribution of the $a$ particles in $\{a^{\pm },a^{\pm },b^{\mp }\}$-type Coulombic systems. In each graph the normalized density plot of ${\rho }_a(\mathit{R})$ is shown for $\mathit{R}=(X,Y,0)$ in terms of $X,Y\in [-2,2]$ bohr. The assembled particles are specified as $\{m_a^{\pm },m_a^{\pm },m_b^{\mp }\}$, where $m_a$ and $m_b$ are the masses and $m_{\mathrm{p}}=1836.15\hspace{0.5em}267\hspace{0.5em}247$ [47] in units of the mass of an electron. The center of mass is the center of each plot.

Figure 2

Emergence of molecular structure in $\{a^{\pm },a^{\pm },b^{\mp }\}$-type Coulombic systems. ${\rho }_a(\mathit{R})$ is shown along the ray $\mathit{R}=(X,0,0)$ and $X\in [-10,10]$ bohr for ratios $m_a/m_b\in [0.4,2.0]$. For $m_a/m_b⩽1$: $m_a=1$ and for $m_a/m_b>1$: $m_b=1$ were fixed, measured in units of the mass of an electron. Note the mass-scale similarity of Coulombic systems.

Figure 3

$a$-particle density profile in $\{a^{\pm },a^{\pm },b^{\mp }\}$-type Coulombic systems. ${\rho }_a(\mathit{R})$ is plotted along the ray $\mathit{R}=(X,0,0)$ within the interval $X\in [-10,10]$ bohr. The origin coincides with the center of mass, and $m_p/m_e=1836.15\hspace{0.5em}267\hspace{0.5em}247$ [47].