The hydrogen molecular ion and bonding

At fixed proton separation R, use the electronic Hamiltonian excluding proton-proton repulsion, consistent with the preceding lesson. It contains electron kinetic energy and attraction to each proton. The electronic eigenvalue is then supplemented by positive nuclear repulsion to obtain the nuclear potential curve. At very small R the nuclear repulsion grows strongly; at large R the appropriate dissociation channel is a hydrogen atom plus a proton. Overall center-of-mass and finite-nuclear-mass corrections are omitted at this level.

Choose normalized 1s-like functions $\varphi_A$ and $\varphi_B$ centered on the two nuclei. They overlap, so their inner product S is generally nonzero. Their sum has enhanced amplitude between the nuclei, while their difference has a node on the median plane for equivalent centers. The sign of an orbital amplitude is not an electric charge: it specifies the relative quantum phase needed when amplitudes are added. Density is obtained only after squaring the complete normalized wavefunction.

For equivalent centers, symmetric and antisymmetric trial states have normalization factors involving $1+S$ and $1-S$, respectively. Evaluating the electronic expectation value gives energies $(H_{AA}\pm H_{AB})/(1\pm S)$ when the matrix elements are real and equal-center symmetry applies. Dividing only by two would be incorrect for overlapping functions. The same calculation is a generalized eigenproblem with an overlap matrix, not an ordinary eigenproblem in an orthonormal atomic basis.

The symmetric state is lower than the diagonal electronic expectation when $H_{AB}-S H_{AA}<0$. The overlap contribution means that the sign of an off-diagonal element by itself is insufficient without its convention and normalization. A variational calculation provides an upper bound to the exact ground electronic energy for the chosen Hamiltonian and geometry. Enlarging the trial space can improve that bound, but a two-function result does not give an exact equilibrium bond length, dissociation energy, or complete spectrum.

Enhanced internuclear electron density can favor bonding, but the complete energy balance also includes electronic kinetic energy, electron-nucleus attraction, and nuclear repulsion. A bound minimum requires the total surface to fall below an appropriate separated-fragment threshold over a range of geometries. Comparing one electronic eigenvalue with a bare atomic number while omitting nuclear repulsion can falsely suggest excessive binding. Even a total surface minimum lies below the lowest vibrational state by its zero-point energy.

The elementary molecular-orbital bond-order convention assigns one half to a single electron in a bonding orbital, but that count is a qualitative model label, not a directly measured bond energy. Spectroscopy tests actual energy differences and nuclear motion. For the example below, all matrix elements and the separated threshold are deliberately hypothetical to make the arithmetic transparent. They illustrate the calculation’s logic rather than quote computed or measured constants of the physical hydrogen molecular ion.