Periodic systems¶

Extended solid-state simulations at any level of dimensionality reduction can be performed by taking advantage of PySCF’s periodic boundary conditions (PBC) capabilities. The inquanto-pyscf extension has two types of drivers for periodic calculations: Gamma point drivers, for \(k=0\) calculations, and momentum space drivers, where a \(k\)-point grid is specified.

Below, we show a Gamma point example with the ChemistryDriverPySCFGammaROHF driver for a Pd (111) surface with a 2x2x1 slab and 20 Å of vacuum:

from pyscf.pbc import gto as pgto
from inquanto.extensions.pyscf import ChemistryDriverPySCFGammaRHF

cell_pd221=[
    [5.50129075763134, 0.0, 0.0],
    [2.75064537881567, 4.764257549713281, 0.0],
    [0.0, 0.0, 20.0]
]

geometry_pd221=[
    ['Pd', [ 0.,  0., 10.]],
    ['Pd', [ 2.75064538,  0.        , 10.        ]],
    ['Pd', [ 1.37532269,  2.38212877, 10.        ]],
    ['Pd', [ 4.12596807,  2.38212877, 10.        ]]
]

driver_pd221 = ChemistryDriverPySCFGammaRHF(
    basis="lanl2dz",
    ecp="lanl2dz",
    geometry=geometry_pd221,
    charge=0,
    cell=cell_pd221,
    dimension=2, # SLAB 2D PBC system
    df="GDF",
    output=None,
    verbose=1,
    exp_to_discard = 0.1
)

where we have also used Gaussian density fitting (GDF) to reduce the computational cost of operations with the two-body electronic integrals (see here). From this point, one may use the get_system() method as usual to build the fermionic hamiltonian and construct the quantum problem.

Calculations with a non-trivial \(k\)-point grid proceed similarly. Below, we consider an H2 chain (1D system) with PBC and a [4,1,1] \(k\)-point grid:

from inquanto.extensions.pyscf import ChemistryDriverPySCFMomentumRHF

cell_h2 = pgto.Cell()
cell_h2.atom=[['H', [0., 5., 5.]], ['H', [0.75, 5.  , 5.  ]]]
cell_h2.a=[(1.875,0.,0.), (0.,10.0,0.), (0.,0.,10.0)]

driver_h2 = ChemistryDriverPySCFMomentumRHF(
    basis="sto-3g",
    geometry=cell_h2.atom,
    charge=0,
    cell=cell_h2.a,
    nks=[4, 1, 1],
    dimension=1, # 1D linear chain PBC system
    precision=1e-15,
    df="GDF",
    output=None,
    verbose=1,
    exp_to_discard = 0.1
)
print("Hartree-Fock energy: ", driver_h2.run_hf())

Hartree-Fock energy:  -1.0760400910891406

In this case, get_system() generates a special Fock space class: FermionSpaceBrillouin, which manages the \(k\)-point quantum number in addition to orbital and spin indices. Otherwise, construction of the quantum problem proceeds as normal.