Molecular Hamiltonian generation

This tutorial is for demonstrating how to obtain the molecular orbital (MO) electron integrals ($h_{ij}$ and $h_{ijkl}$) as well as the molecular Hamiltonian ($H$).

Here, we adopt the physicists' convention for the electron integrals, so that they are related to the Hamiltonian by the equation:

$$\begin{equation} H = E_{\text{nuc}} + \sum_{i, j = 1}^{N} h_{ij} c_i^{\dagger} c_j + \frac{1}{2} \sum_{i, j, k, l = 1}^{N} h_{ijkl} c_i^{\dagger} c_j^{\dagger} c_k c_l \end{equation}$$

where

• $E_{\text{nuc}}$ is the nuclear repulsion energy.
• $h_{ij}$ is the 1-electron MO integral (physicist's convention).
• $h_{ijkl}$ is the 2-electron MO integral (physicist's convention).
• $c_i^{\dagger}$, $c_i$ are the fermionic creation and annihilation operators on the i-th spin orbtial.
• $N$ is the number of spin oribtals.

Overview

We give a brief overview on how to easily generate the Hamiltonian for a water molecule with QURI Parts. The parameters we use are:

• charge: 0
• spin: 0
• basis: sto-3g
• fermion-qubit mapping: Jordan-Wigner

We first generate the hamiltonian where all spatial orbitals are active. This configuration is referred to as the "full space" through out this tutorial.

from pyscf import gto, scf
from quri_parts.pyscf.mol import get_spin_mo_integrals_from_mole
from quri_parts.openfermion.mol import get_qubit_mapped_hamiltonian

mole = gto.M(atom="O 0 0 0; H 0.2774 0.8929 0.2544; H 0.6068, -0.2383, -0.7169")
mf = scf.RHF(mole).run(verbose=0)

hamiltonian, jw_mapping = get_qubit_mapped_hamiltonian(
    *get_spin_mo_integrals_from_mole(mole, mf.mo_coeff)
)

We can also compute the active space hamiltonian with active space configuration

• number of active electrons: 6
• number of active orbitals: 4

from quri_parts.chem.mol import cas

cas_hamiltonian, cas_jw_mapping = get_qubit_mapped_hamiltonian(
    *get_spin_mo_integrals_from_mole(mole, mf.mo_coeff, cas(6, 4))
)

Note that a mapping object is returned along with the qubit hamiltonian.

Generate spin MO electron integrals

The molecular hamiltonian is computed based on spin MO electron integrals. They are the $h_{ij}$ and $h_{ijkl}$ coefficients in eq.(1). We will have detailed tutorials explaining how they are computed in QURI Parts and the objects that hold them. For now, we can generate them with the get_spin_mo_integrals_from_mole function, which only requires a pyscf.gto.Mole object and the corresponding mo coefficients.

Here we show how we can generate the full space and active space electron integrals

full_space, mo_eint_set = get_spin_mo_integrals_from_mole(mole, mf.mo_coeff)
active_space, cas_mo_eint_set = get_spin_mo_integrals_from_mole(mole, mf.mo_coeff, cas(6, 4))

get_spin_mo_integrals_from_mole returns the active space along with the electron integral The full_space variable here is an ActiveSpace object with n_active_electron being the total number of electrons in the molecule, and n_active_orbitals being the total number of spatial orbitals in the molecule.

Obtaining the fermionic hamiltonian

As the fermionic hamiltonian is directly related to the electron integrals via eq.(1), we can compute the fermionic hamiltonian with them with the get_fermionic_hamiltonian function.

from quri_parts.openfermion.mol import get_fermionic_hamiltonian

fermionic_hamiltonian = get_fermionic_hamiltonian(mo_eint_set)
cas_fermionic_hamiltonian = get_fermionic_hamiltonian(cas_mo_eint_set)

Mapping the fermionic Hamiltonian to qubit hamiltonian

The fermionic hamiltonian is represented by a openfermion.InteractionOperator, where we can use an operator mapper to map it to QURI Parts Operator. There are 2 ways this can be done:

1. Construct a mapping object by hand and use it to convert the FermionicOperator to Operator.
2. Use the operator_from_of_fermionic_op function.

We first show method 1.

from quri_parts.openfermion.transforms import jordan_wigner

jw_mapping = jordan_wigner(2 * full_space.n_active_orb, full_space.n_active_ele)
qubit_hamiltonian = jw_mapping.of_operator_mapper(fermionic_hamiltonian)

cas_jw_mapping = jordan_wigner(2 * active_space.n_active_orb, active_space.n_active_ele)
cas_qubit_hamiltonian = cas_jw_mapping.of_operator_mapper(cas_fermionic_hamiltonian)

This computes the correct qubit Hamiltonian, but is a bit cumbersome because the mapping object is computed by hand. The operator_from_of_fermionic_op function bypasses this shortcoming by taking in the active space and generates the mapping object automatically.

from quri_parts.openfermion.mol import operator_from_of_fermionic_op

qubit_hamiltonian, jw_mapping = operator_from_of_fermionic_op(
    fermionic_hamiltonian,
    full_space,
    sz = None,  # default to None
    fermion_qubit_mapping=jordan_wigner  # default to jordan_wigner
)
cas_qubit_hamiltonian, cas_jw_mapping = operator_from_of_fermionic_op(
    cas_fermionic_hamiltonian,
    active_space,
    sz = None,  # default to None
    fermion_qubit_mapping=jordan_wigner  # default to jordan_wigner
)

Qubit hamiltonian from MO integral

Finally, we introduce the get_qubit_mapped_hamiltonian function demonstrated in the overview section. It serves as a shortcut that completely bypasses the fermionic hamiltonian

qubit_hamiltonian, jw_mapping = get_qubit_mapped_hamiltonian(
    full_space,
    mo_eint_set,
    sz = None,  # default to None
    fermion_qubit_mapping=jordan_wigner  # default to jordan_wigner
)

cas_qubit_hamiltonian, cas_jw_mapping = get_qubit_mapped_hamiltonian(
    active_space,
    cas_mo_eint_set,
    sz = None,  # default to None
    fermion_qubit_mapping=jordan_wigner  # default to jordan_wigner
)