VQE for HeH+ Molecule Ground State
Overview
Helium hydride cation (HeH⁺) is a simple 2-electron molecular system with asymmetric nuclear charges. It's historically significant as the first molecule to form in the early universe after the Big Bang.
The Molecule
HeH⁺ consists of:
- Helium atom (nuclear charge +2)
- Hydrogen atom (nuclear charge +1)
- 2 electrons (ionic bond)
The asymmetry makes the electronic structure different from H₂.
The Ansatz
UCCSD-inspired ansatz for 2 qubits:
CODE┌──────────┐ ┌──────────┐ q_0: ┤ RY(θ₀) ├────■─────┤ ├────■──── ├──────────┤ ┌─┴─┐ │ RY(θ₂) │ ┌─┴─┐ q_1: ┤ RY(θ₁) ├──┤ X ├───┤ ├──┤ X ├── └──────────┘ └───┘ └──────────┘ └───┘
Running the Circuit
PYTHONfrom circuit import run_circuit result = run_circuit() print(f"Energy: {result['energy']:.4f} Ha") print(f"Exact: {result['exact_ground_state']} Ha")
Expected Output
| Metric | Value |
|---|---|
| Exact ground state | -2.862 Ha |
| VQE estimate | ~-2.85 to -2.87 Ha |
| Bond length | 0.775 Å |
Comparison with H2
| Property | H₂ | HeH⁺ |
|---|---|---|
| Electrons | 2 | 2 |
| Symmetry | Homonuclear | Heteronuclear |
| Ground state | -1.137 Ha | -2.862 Ha |
| Bond type | Covalent | Ionic |
Cosmological Significance
HeH⁺ was the first molecule to form in the universe:
- Formed ~100,000 years after Big Bang
- He + H⁺ → HeH⁺
- First detected in space in 2019 (planetary nebula NGC 7027)