Obtaining integrals

Forte2 uses the Libint2 integral engine. It provides a straightforward way of accessing atomic integrals through its API. You can obtain the integrals for a given molecular system by first specifying a molecular system and using the forte2.ints module. Almost all operators supported by Libint2 (see Libint2 documentation) are available. Forte2 also provides an interface with libcint, with a templated API that supports easy addition of new integral types. Currently, only a small subset (mainly two-center integrals) of libcint integrals are imported, but more can be easily added as needed. Here are some examples of how to obtain the most common integrals. First one needs to set up the molecular system:

import forte2

# Set up your molecular system
system = forte2.System(
    xyz="""C 0 0 0
    N 0 0 1.4""",
    basis_set={"C": "cc-pvdz", "N": "cc-pvtz"},
    auxiliary_basis_set="cc-pvdz-jkfit",
    minao_basis_set="ano-r0",
)

The system object will now contain parsed geometry under atoms, the basis set under basis, the auxiliary basis set under auxiliary_basis, and the minimal atomic basis set under minao_basis.

There are two ways of obtaining integrals: using the forte2.ints module (direct C++ API calls to Libint2 and libcint), or using the forte2.integrals module (Python wrappers around the C++ API calls). The two ways are equivalent, but the latter can be more user-friendly.

Getting integrals through forte2.ints can be achieved as follows:

# overlap integrals
overlap = forte2.ints.overlap(system.basis)

# "mixed basis" overlap integrals are available simply as:
mixed_overlap = forte2.ints.overlap(system.minao_basis, system.basis)

# kinetic energy integrals
kinetic = forte2.ints.kinetic(system.basis)

# potential energy integrals
potential = forte2.ints.nuclear(system.basis, system.atoms)

# dipole integrals (ordered x,y,z)
# the zeroth element is the overlap
dipole = forte2.ints.emultipole1(system.basis, system.atoms)[1:]

# 4-center-2-electron integrals
eri = forte2.ints.coulomb_4c(system.basis)

# 3-center-2-electron integrals (for density-fitting)
B = forte2.ints.coulomb_3c(system.auxiliary_basis, system.basis, system.basis)

Equivalently, getting integrals through forte2.integrals can be achieved as follows:

# overlap integrals
overlap = forte2.integrals.overlap(system)

# "mixed basis" overlap integrals are available simply as:
mixed_overlap = forte2.integrals.overlap(system, system.minao_basis, system.basis)

# kinetic energy integrals
kinetic = forte2.integrals.kinetic(system)

# potential energy integrals
potential = forte2.integrals.nuclear(system)

# dipole integrals (ordered x,y,z)
# the zeroth element is the overlap
dipole = forte2.integrals.emultipole1(system)[1:]

# 4-center-2-electron integrals
eri = forte2.integrals.coulomb_4c(system)

# 3-center-2-electron integrals (for density-fitting)
B = forte2.integrals.coulomb_3c(system)

As shown above, the forte2.integrals module automatically supplies sensibly default basis sets and geometry information from the system object, making it more convenient to use in many cases.