Computing Dissociation Curves

A single energy number is rarely the goal. Chemists want the potential energy surface: total energy as a function of nuclear geometry. The simplest slice is the dissociation curve of a diatomic molecule — energy versus bond length — from which the equilibrium bond length, the dissociation energy, and vibrational frequencies all follow.

Building the curve

To compute a dissociation curve you repeat the whole chemistry pipeline at each bond length $R$:

1. Recompute the molecular integrals $h_{pq}(R)$, $h_{pqrs}(R)$ for the new geometry.
2. Re-map to a qubit Hamiltonian $\hat H(R)$ (the Pauli coefficients change with $R$).
3. Run VQE to obtain the ground-state energy $E_0(R)$.
4. Add the constant nuclear–nuclear repulsion $Z_A Z_B / R$.

Plotting $E_{\text{total}}(R)$ against $R$ traces the curve: it falls to a minimum at the equilibrium bond length $R_e$ (about $0.74\,\text{Å}$ for $\text{H}_2$), then rises and flattens to the dissociation limit as the atoms separate. The depth of the well below the dissociation limit is the bond dissociation energy $D_e$.

The exact ground state of two-qubit H₂

For the two-qubit $\text{H}_2$ Hamiltonian the ground state always lies in the single-excitation subspace $\{|01\rangle, |10\rangle\}$, where $\hat H$ restricts to a $2\times2$ matrix

with diagonal energies $a = \langle 01|\hat H|01\rangle$, $b = \langle 10|\hat H|10\rangle$ (from the $I$, $Z$, $ZZ$ terms) and off-diagonal coupling $c$ from the $X_0X_1 + Y_0Y_1$ terms. Diagonalizing a real symmetric $2\times2$ matrix is exact:

The ground-state energy is the lower root $E_-$. Because VQE's one-parameter ansatz can reach the exact eigenvector of this block, VQE recovers $E_-$ exactly at every bond length — the computed curve has no approximation error beyond the basis set itself.

Why the off-diagonal term matters most at dissociation

Near equilibrium the diagonal gap $|a - b|$ is large, so the ground state is close to the Hartree–Fock determinant $|01\rangle$ and the correlation energy $E_- - \min(a,b)$ is small. As the bond stretches, the bonding and antibonding orbitals become degenerate ($a \to b$), and the ground state becomes an equal mix of $|01\rangle$ and $|10\rangle$. There Hartree–Fock fails badly, and the $\sqrt{(\cdot)^2 + c^2}$ term — pure correlation — dominates.

Try it

For one geometry the $2\times2$ subspace block has diagonal energies $a = -1.0$, $b = -0.5$ and off-diagonal coupling $c = 0.3$ (Hartree). Compute the ground-state energy — the smaller eigenvalue of $\begin{pmatrix} a & c \\ c & b\end{pmatrix}$ — and return it.

Repeating this calculation across bond lengths and connecting the points is the dissociation curve.

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