21.7. Mixing PBC and molecular modules

The post-HF methods, as a standalone numerical solver, do not require the knowledge of the boundary condition. The calculations of finite-size systems and extend systems are distinguished by the boundary condition of integrals (and basis). The same post-HF solver can be used for both the finite-size problem and the periodic boundary problem if they have the similar Hamiltonian structure.

In PySCF, many molecular post-HF solvers has two implementations: incore and outcore versions. They are differed by the treatments on the 2-electron integrals. The incore solver takes the _eri (or with_df, see df — Density fitting) from the underlying mean-field object as the two-electron interaction part of the Hamiltonian while the outcore solver generates the 2-electron integrals (with free boundary condition) on the fly. To use the molecular post-HF solvers in PBC code, we need ensure the incore version solver being called.

Generating _eri in mean-filed object is the straightforward way to trigger the incore post-HF solver. If the allowed memory is big enough to hold the entire 2-electron integral array, the gamma point HF solver always generates and holds this array. A second choice is to set incore_anyway in cell which forces the program generating and holding _eri in mean-field object.

Note

If the problem is big, incore_anyway may overflow the available physical memory.

Holding the full integral array _eri in memory limits the problem size one can treat. Using the density fitting object with_df to hold the integrals can overcome this problem. This architecture has been bound to PBC and molecular mean-field modules. But the relevant post-HF density fitting solvers are still in development thus this feature is not available in PySCF 1.2 or older.

Aside from the 2-electron integrals, there are some attributes and methods required by the post-HF solver. They are get_hcore(), and get_ovlp() for 1-electron integrals, _numint, grids for the numerical integration of DFT exchange-correlation functionals. They are all overloaded in PBC mean-field object to produce the PBC integrals.

21.7.1. Examples

#!/usr/bin/env python

'''
Gamma point post-HF calculation needs only real integrals.
Methods implemented in finite-size system can be directly used here without
any modification.
'''

import numpy
from pyscf.pbc import gto, scf

cell = gto.M(
    a = numpy.eye(3)*3.5668,
    atom = '''C     0.      0.      0.    
              C     0.8917  0.8917  0.8917
              C     1.7834  1.7834  0.    
              C     2.6751  2.6751  0.8917
              C     1.7834  0.      1.7834
              C     2.6751  0.8917  2.6751
              C     0.      1.7834  1.7834
              C     0.8917  2.6751  2.6751''',
    basis = '6-31g',
    verbose = 4,
)

mf = scf.RHF(cell).density_fit()
mf.with_df.mesh = [10]*3
mf.kernel()

#
# Import CC, TDDFT moduel from the molecular implementations
#
from pyscf import cc, tddft
mycc = cc.CCSD(mf)
mycc.kernel()

mytd = tddft.TDHF(mf)
mytd.nstates = 5
mytd.kernel()