# Source code for pyscf.pbc.tdscf.krhf_slow_gamma

```
# Author: Artem Pulkin
# flake8: noqa
"""
This and other `_slow` modules implement the time-dependent Hartree-Fock procedure. The primary performance drawback is
that, unlike other 'fast' routines with an implicit construction of the eigenvalue problem, these modules construct
TDHF matrices explicitly via an AO-MO transformation, i.e. with a O(N^5) complexity scaling. As a result, regular
`numpy.linalg.eig` can be used to retrieve TDHF roots in a reliable fashion without any issues related to the Davidson
procedure. Several variants of TDHF are available:
* `pyscf.tdscf.rhf_slow`: the molecular implementation;
* `pyscf.pbc.tdscf.rhf_slow`: PBC (periodic boundary condition) implementation for RHF objects of `pyscf.pbc.scf`
modules;
* `pyscf.pbc.tdscf.krhf_slow_supercell`: PBC implementation for KRHF objects of `pyscf.pbc.scf` modules. Works with
an arbitrary number of k-points but has a overhead due to an effective construction of a supercell;
* (this module) `pyscf.pbc.tdscf.krhf_slow_gamma`: A Gamma-point calculation resembling the original `pyscf.pbc.tdscf.krhf`
module. Despite its name, it accepts KRHF objects with an arbitrary number of k-points but finds only few TDHF roots
corresponding to collective oscillations without momentum transfer;
* `pyscf.pbc.tdscf.krhf_slow`: PBC implementation for KRHF objects of `pyscf.pbc.scf` modules. Works with
an arbitrary number of k-points and employs k-point conservation (diagonalizes matrix blocks separately).
"""
from pyscf.pbc.tdscf import krhf_slow_supercell as td
from pyscf.tdscf import rhf_slow
import numpy
# Convention for these modules:
# * PhysERI, PhysERI4, PhysERI8 are 2-electron integral routines computed directly (for debug purposes), with a 4-fold
# symmetry and with an 8-fold symmetry
# * vector_to_amplitudes reshapes and normalizes the solution
# * TDRHF provides a container
[docs]
class PhysERI(td.PhysERI):
def __init__(self, model, frozen=None):
"""
The TDHF ERI implementation performing a full transformation of integrals to Bloch functions. No symmetries are
employed in this class. Only a subset of transformed ERI is returned, corresponding to oscillations without a
momentum transfer. The k-points of the returned ERIs come in two pairs :math:`(k_1 k_1 | k_2 k_2)`. As a result,
one of the diagonal blocks of the full TDHF matrix can be reconstructed. For other blocks, look into `PhysERI`
class of this momdule.
Args:
model (KRHF): the base model;
frozen (int, Iterable): the number of frozen valence orbitals or the list of frozen orbitals for all
k-points or multiple lists of frozen orbitals for each k-point;
"""
super(PhysERI, self).__init__(model, frozen=frozen)
[docs]
def tdhf_diag(self):
"""
Retrieves the merged diagonal block with equal pairs of k-indexes (k, k) only.
Returns:
The diagonal block.
"""
return super(PhysERI, self).tdhf_diag(pairs=((i, i) for i in range(len(self.model.kpts))))
def __calc_block__(self, item, k):
if k in self.__full_eri_k__:
return super(PhysERI, self).__calc_block__(item, k)
else:
raise RuntimeError("The block k = {:d} + {:d} - {:d} - {:d} does not conserve momentum".format(*k))
[docs]
def eri_mknj(self, item):
"""
Retrieves the merged ERI block using 'mknj' notation with pairs of k-indexes (k1, k1, k2, k2).
Args:
item (str): a 4-character string of 'mknj' letters;
Returns:
The corresponding block of ERI (phys notation).
"""
return super(PhysERI, self).eri_mknj(
item,
pairs_row=((i, i) for i in range(len(self.model.kpts))),
pairs_column=((i, i) for i in range(len(self.model.kpts))),
)
[docs]
class PhysERI4(PhysERI):
def __init__(self, model, frozen=None):
"""
The TDHF ERI implementation performing partial transformations of integrals to Bloch functions. A 4-fold
symmetry of complex-valued functions is employed in this class. Only a subset of transformed ERI is returned,
corresponding to oscillations without a momentum transfer. The k-points of the returned ERIs come in two pairs
:math:`(k_1 k_1 | k_2 k_2)`. As a result, one of the diagonal blocks of the full TDHF matrix can be
reconstructed. For other blocks, look into `PhysERI` class of this momdule.
Args:
model (KRHF): the base model;
frozen (int, Iterable): the number of frozen valence orbitals or the list of frozen orbitals for all
k-points or multiple lists of frozen orbitals for each k-point;
"""
super(PhysERI4, self).__init__(model, frozen=frozen)
symmetries = [
((0, 1, 2, 3), False),
((1, 0, 3, 2), False),
((2, 3, 0, 1), True),
((3, 2, 1, 0), True),
]
def __calc_block__(self, item, k):
if k in self.__full_eri_k__:
return td.PhysERI4.__calc_block__.im_func(self, item, k)
else:
raise RuntimeError("The block k = {:d} + {:d} - {:d} - {:d} does not conserve momentum".format(*k))
[docs]
class PhysERI8(PhysERI4):
def __init__(self, model, frozen=None):
"""
The TDHF ERI implementation performing partial transformations of integrals to Bloch functions. An 8-fold
symmetry of real-valued functions is employed in this class. Only a subset of transformed ERI is returned,
corresponding to oscillations without a momentum transfer. The k-points of the returned ERIs come in two pairs
:math:`(k_1 k_1 | k_2 k_2)`. As a result, one of the diagonal blocks of the full TDHF matrix can be
reconstructed. For other blocks, look into `PhysERI` class of this momdule.
Args:
model (KRHF): the base model;
frozen (int, Iterable): the number of frozen valence orbitals or the list of frozen orbitals for all
k-points or multiple lists of frozen orbitals for each k-point;
"""
super(PhysERI8, self).__init__(model, frozen=frozen)
symmetries = [
((0, 1, 2, 3), False),
((1, 0, 3, 2), False),
((2, 3, 0, 1), False),
((3, 2, 1, 0), False),
((2, 1, 0, 3), False),
((3, 0, 1, 2), False),
((0, 3, 2, 1), False),
((1, 2, 3, 0), False),
]
[docs]
def vector_to_amplitudes(vectors, nocc, nmo):
"""
Transforms (reshapes) and normalizes vectors into amplitudes.
Args:
vectors (numpy.ndarray): raw eigenvectors to transform;
nocc (tuple): numbers of occupied orbitals;
nmo (int): the total number of orbitals per k-point;
Returns:
Amplitudes with the following shape: (# of roots, 2 (x or y), # of kpts, # of kpts, # of occupied orbitals,
# of virtual orbitals).
"""
if not all(i == nocc[0] for i in nocc):
raise NotImplementedError("Varying occupation numbers are not implemented yet")
nk = len(nocc)
nocc = nocc[0]
if not all(i == nmo[0] for i in nmo):
raise NotImplementedError("Varying AO spaces are not implemented yet")
nmo = nmo[0]
vectors = numpy.asanyarray(vectors)
# Compared to krhf_slow_supercell, only one k-point index is present here. The second index corresponds to
# momentum transfer and is integrated out
vectors = vectors.reshape(2, nk, nocc, nmo-nocc, vectors.shape[1])
norm = (abs(vectors) ** 2).sum(axis=(1, 2, 3))
norm = 2 * (norm[0] - norm[1])
vectors /= norm ** .5
return vectors.transpose(4, 0, 1, 2, 3)
[docs]
class TDRHF(rhf_slow.TDRHF):
eri4 = PhysERI4
eri8 = PhysERI8
v2a = staticmethod(vector_to_amplitudes)
```