Molecular Potential Energy

Within BO approximation, the energies $E_{\pm}(R)$ enter the nuclear Hamiltonian (cf. Eq. () with $\varepsilon_\alpha=E_{\pm}$) for the wave functions $\chi$

$$\left[ \sum_{i=a,b}\frac{{\bf P}_i^2}{2M} + E_{\pm}(R)\right]\chi_{\pm}({\bf x}_a,{\bf x}_b) = E \chi_{\pm}({\bf x}_a,{\bf x}_b)$$ (3.25)

of the nuclear system with $R =\vert{\bf x}_a-{\bf x}_b\vert$, cf. Eq. (V.3.1). Clearly, a separation in center-of mass and relative motion is easily done here. The potential energy for the nuclei is given by the function $E_{\pm}(R)$, cf. Eq. (V.3.22),

$$E_{\pm}(R)= E_{1s}+\frac{e^2}{4\pi\varepsilon_0 a_0}\left[\frac{1}{R}\mp \frac{j(R)\pm k(R)}{1\pm S(R)}\right],$$ (3.26)

with the explicit expression for $j(R)$, $k(R)$, $S(R)$, in Eq. (V.3.22). The parametric eigenenergies of the electronic system become the potential for the nuclei, which is the characteristic feature of the BO approximation. The corresponding potential curves are shown in Fig.(V.3.3.4).

Figure: $E_{\pm}(R)$, Eq. (V.3.26), for the $H_2^+$-ion in Born-Oppenheimer approximation and using the MO-LCAO Rayleigh-Ritz method, from Weissbluth [4].

• The potential energy $E_{+}(R)$ of the bonding molecular orbital has a minimum at $R=R_0$. This determines the equilibrium position of the two nuclei. Occupation of the bonding MO helps to bond the nuclei together and thereby form the molecule.
• The potential energy $E_{-}(R)$ of the antibonding molecular orbital has no local minimum. Therefore, the antibonding state is an excited state in which the molecule dissociates.