Concept 2 of 7 · What we’re looking for

Hamiltonians and ground states

A Hamiltonian is the equation that tells you the energy of every possible state of a system; the ground state is whichever state has the lowest energy, and finding it is what chemistry is really about.

The Hamiltonian defines this whole energy landscape. The ground state is the valley bottom; VQE’s job is to slide down into it.

What it actually means

The Hamiltonian (written Ĥ) is the mathematical object that encodes every interaction in a system. For a molecule, it sums up:

The kinetic energy of each electron
The attraction between each electron and each nucleus
The repulsion between each pair of electrons
The repulsion between each pair of nuclei

Once you know where the atoms are sitting, the Hamiltonian is fully determined. It’s not something you guess — it falls out of the geometry.

The Hamiltonian assigns an energy to every possible state of the system. Some states have high energy (electrons piled on top of each other, unfavourable arrangements), some have low energy (electrons settled into the optimal pattern for that geometry).

The ground state is whichever state has the lowest possible energy. It’s the state the molecule actually occupies at zero temperature — and it’s what determines all the chemistry: bond strengths, geometries, reactivity.

Why brute force fails

In matrix language: the Hamiltonian is a (huge) matrix, and the ground state is its smallest eigenvalue’s eigenvector. For 50 electrons that matrix has on the order of 10²⁰ rows. You cannot store it, let alone diagonalize it. So we need a smarter approach than brute force — that’s what VQE is.

Why it matters for our problem

When we ask “how strongly does tungsten bind hydrogen?”, what we’re really asking is:

What’s the lowest-energy state of (W + H together), and how does that compare to the lowest-energy states of (W alone) plus (H alone)?

The difference is the binding energy. So everything in this repo — classical pyscf calculations, VQE on simulator, VQE on QPU — is ultimately a ground-state-finding exercise on different Hamiltonians.

In 03_wh_binding.ipynb we build the Hamiltonian for WH⁻ at one geometry, then we scan through different W–H bond distances and rebuild it each time. The curve we plot (results/wh_minus_pec.png) is just the ground-state energy of each Hamiltonian as a function of distance — the potential energy curve. Its minimum tells us the equilibrium bond length; its depth tells us how much energy you’d need to pull the atoms apart.