models

`eigenkit.chemistry.models`

Built-in fermionic models with known properties — for validation and demos without any external chemistry dependency.

`fermi_hubbard_dimer(t, u)`

The two-site Fermi-Hubbard model with spin 1/2.

It has four spin-orbital modes, indexed by mode = 2 * site + spin, with spin 0 for up and 1 for down. Site 0 has modes 0 and 1, and site 1 has modes 2 and 3. The Hamiltonian is:

H = -t Σ_σ (a†_{0σ} a_{1σ} + a†_{1σ} a_{0σ})
    + U (n_{0↑} n_{0↓} + n_{1↑} n_{1↓})

The ground-state energy at half filling has the closed form fermi_hubbard_dimer_ground_energy(). This makes the model an exact check of the fermion-to-qubit mapping and of VQE, with no external dependency.

Parameters:

Name	Type	Description	Default
`t`	`float`	The hopping amplitude between the two sites.	required
`u`	`float`	The on-site Coulomb repulsion.	required

Returns:

Type	Description
`FermionOperator`	The Hamiltonian, as a `FermionOperator`.

`fermi_hubbard_dimer_ground_energy(t, u)`

The ground-state energy of the Fermi-Hubbard dimer at half filling.

The energy of fermi_hubbard_dimer() with two electrons in a singlet state is (U - sqrt(U² + 16t²)) / 2. This value is the ground state over all particle numbers only when U < 3t. For a larger U, double occupancy costs so much energy that the state with a single electron, of energy -t, is lower, and the ground state over all particle numbers is min(-t, (U - sqrt(U² + 16t²)) / 2). An ansatz that conserves the number of particles, such as UCCSD, targets the half-filling sector directly. The hardware-efficient ansatz explores all sectors, so use it only where the half-filling state is the overall ground state.

Parameters:

Name	Type	Description	Default
`t`	`float`	The hopping amplitude, as in the model.	required
`u`	`float`	The on-site repulsion, as in the model.	required

Returns:

Type	Description
`float`	The ground-state energy at half filling, from the closed form.