H₂ molecular qubit Hamiltonian (STO-3G, 2-qubit tapered)

In a minimal STO-3G basis, H₂ has two spatial molecular orbitals (bonding $\sigma_g$ and antibonding $\sigma_u^*$), i.e. four spin-orbitals. Jordan-Wigner encoding these four spin-orbitals gives a 4-qubit fermionic Hamiltonian; applying a parity mapping and exploiting the two $\mathbb{Z}_2$ symmetries of a fixed 2-electron singlet state (particle-number and spin-parity conservation) tapers this down to the standard 2-qubit operator

$H = g_0\, I\!I + g_1\, Z_0 I + g_2\, I Z_1 + g_3\, Z_0 Z_1 + g_4\, Y_0 Y_1 + g_5\, X_0 X_1,$

with the following coefficients (in Hartree) at the H₂ equilibrium bond length $R=0.7414\,\text{Å}$, curated from O'Malley et al. (2016), Table I:

$g_0=-0.4804,\quad g_1=0.3435,\quad g_2=-0.4347,\quad g_3=0.5716,\quad g_4=g_5=0.0910.$

Structural facts checked here. Every Pauli string appearing ($II,ZI,IZ,ZZ,YY,XX$) is Hermitian, and all six coefficients are real, so $H$ is Hermitian by construction — a property that must hold for any physical Hamiltonian and is trivial to confirm directly from the term list. The absence of any $XY$, $YX$, single-$X$, or single-$Y$ terms is a direct consequence of the residual symmetry after tapering (the untapered operator has more terms; the surviving six are exactly the ones commuting with both retained $\mathbb{Z}_2$ symmetry generators), and is the standard structural signature of this specific reduction scheme rather than a general property of arbitrary two-qubit molecular Hamiltonians.

Provenance of the numbers. The form of the operator (six terms, this Pauli-string set) is a structural consequence of the mapping and is checked directly; the specific numeric values $g_0,\dots,g_5$ come from a classical Hartree-Fock plus configuration-interaction calculation in the literature and are curated here as literature values, not independently recomputed — this catalog verifies the mathematical object is well-formed, not the underlying quantum-chemistry calculation that produced its coefficients.

Why it is the canonical VQE target. Because the exact ground-state energy of this specific 2-qubit matrix is obtainable by classical diagonalization (trivial at 2 qubits), any VQE run on this Hamiltonian has a known correct answer to compare against — which is exactly why H₂ (and the closely related minimal-basis diatomics) became the field's de facto small-molecule benchmark for early superconducting- and photonic-qubit VQE demonstrations.

Bond-length dependence. The six coefficients above are only valid at $R=0.7414\,\text{Å}$; scanning the bond length produces a different $g_0,\dots,g_5$ at each point, tracing out the H₂ potential energy surface used to validate dissociation-curve accuracy — a standard downstream use of this exact operator family.