Multicomponent density-functional theory for electrons and nuclei

We present a general multicomponent density-functional theory in which electrons and nuclei are treated completely quantum mechanically, without the use of a Born-Oppenheimer approximation. The two fundamental quantities in terms of which our theory is formulated are the nuclear $N$-body density and the electron density expressed in coordinates referring to the nuclear framework. For these two densities, coupled Kohn-Sham equations are derived, and the electron-nuclear correlation functional is analyzed in detail. The formalism is tested on the hydrogen molecule ${\mathrm{H}}_2$ and its positive ion $\mathrm{H}_2{}^{+}$, using several approximations for the electron-nuclear correlation functional.

Figure 1

Effective nuclear potential $V_{S,n}\left(R\right)$ for the ${\mathrm{H}}_2$ molecule obtained from self-consistent solutions of the MCKS scheme employing various approximations. For comparison, results from a BO calculations is added. To facilitate comparison with the OAO and SAO approaches the BO and Hartree curves are shifted with $-F^e\left[\rho \right]$. Measurements are in atomic units.

Figure 2

Effective nuclear potential $V_{S,n}\left(R\right)$ for the $\mathrm{H}_2{}^{+}$ molecule obtained from self-consistent solutions of the MCKS scheme employing various approximations. The methods denoted OAO (optimized atomic orbital) and SAO (scaled atomic orbital) are explained further in the text. For comparison, results from BO calculations are added. To facilitate comparison with the OAO and SAO approaches the BO and Hartree curves are shifted by $-F^e\left[\rho \right]$. Measurements are in atomic units.

Figure 3

Radial nuclear density $4\pi R^2\Gamma \left(R\right)$ for the $\mathrm{H}_2{}^{+}$ molecule obtained from self-consistent solutions of the MCKS scheme employing various approximations. For comparison, results from BO calculations are added. Measurements are in atomic units.

Figure 4

Radial nuclear density $4\pi R^2\Gamma \left(R\right)$ for the ${\mathrm{H}}_2$ molecule obtained from self-consistent solutions of the MCKS scheme employing various approximations. For comparison, results from BO calculations are added. Measurements are in atomic units.

Figure 5

Typical behavior of the electronic density $\rho \left(z\right)$ and the conditional density $\gamma (z\mid R)\approx 6\mathrm{a.u.}$. for a diatomic molecule, plotted along the internuclear $\left(z\right)$ axis. Measurements are in atomic units.

Figure 6

Atomic orbital obtained for the $\mathrm{H}_2{}^{+}$ molecule from a self-consistent solution of the OAO scheme explained in the text and compared to a hydrogenic $\left(1s\right)$ orbital ${\varphi }_H\left(z\right)$. Both curves are plotted along the electronic $z$ axis. Measurements are in atomic units.

Figure 7

Contributions to the nuclear conditional potential (124) as obtained from a self-consistent solution of the MCKS-OAO scheme for the $\mathrm{H}_2{}^{+}$ molecule. They are compared to the corresponding BO curves and to results provided by a simple LCAO employing hydrogenic $\left(1s\right)$ orbitals. In addition, the mean (equilibrium) internuclear distance $⟨R⟩$ is marked. Measurements are in atomic units.

Figure 8

Scaling function $\lambda \left(R\right)$, Eq. (143), obtained from a self-consistent solution of the MCKS scheme for the ${\mathrm{H}}_2$ molecule. Measurements are in atomic units.

Figure 9

Comparison of different contributions to the nuclear potential, $V_{\mathrm{cond},W}^{\mathrm{SAO}}$ and $V_{\mathrm{cond},T}^{\mathrm{SAO}}$, as obtained for the $\mathrm{H}_2{}^{+}$ molecule from the BO, the SAO, and the OAO methods. Measurements are in atomic units.