Bonding and Antibonding

(Cf. Atkins and Friedman [5], ch. 8.3, for this section). We now apply the Rayleigh-Ritz variational method to the Hydrogen Molecule Ion $H_2^+$, restricting ourselves to just $n=2$ real wave functions (atomic orbitals) $\psi_i$ ( $i=1,2$), i.e.

$$= \Psi = = x_1 \psi_1+ x_2 \psi_2$$ MO (3.11)

$$\psi_1({\bf r}) = \psi_{n=1,l=0,m=0}({\bf r}-{\bf r}_a),\quad \psi_2({\bf r}) = \psi_{n=1,l=0,m=0}({\bf r}-{\bf r}_b)$$

with two hydrogen groundstate $s$-orbitals for nuclei at ${\bf r}_a$ and ${\bf r}_b$, respectively.