BID and HS-BID¶
We use the chainrule analysis (Derivative_Analysis class) to find the minimum amount of displacements that can probe all the irreducible derivatives. Additionally, hierarchical supercell approach isolates the irreducible derivatives to the smallest supercells that accomondate them respectively. This does not improves the efficiency, but also allows the calculations in the smaller systems to be utilized when computing for the larger systems, reducing redundant calculations. It also
incentivize starting calculations from a smaller system and gradually increase the system size.
BID¶
Using the chainrule analysis provided by derivative_analysis module, we can, for a given system size and order, find the smallest number of measurements required to compute all the irreducible derivatives of that system. We currently use the simple bundled basis (SBB) descrived in the paper Phys. Rrev. B 100, 014303 (2019). One is also free to provide any measurement basis to probe the system.
HS-BID¶
The core idea of the HS-BID method is similar to BID, with the added requirement to compute each irreducible derivatives in their repective smallest supercells.
To begin with, we can group irreducble derivatives with their corresponding Q. Given we have the algorithm to determine the smallest cell that each Q can fit in, our job now becomes finding the smallest amount of supercells that there is one supercell in the list that for at least one member of each Q-start, this supercell is the smallest cell this Q can fit in. Given an exhaustive search for the optimum scale badly ( As a set cover problem, it is a NP-complete problem), we use a greedy algorithm to find the solution. Finally, we have a list of sub-cells (They are still supercells of the primitive cell), rank by cell size (cells with the same size will still be ranked by which is found first, though the order here does not matter that much).
We then generate the analytical dynamic tensors for the irredcuible Q for the system we try to compute, which would ensure consistent phase and symbol conventions across the sub-cells we will perform first-principle calculations in. We then pass the Q’s that are need in each of the sub-cells into each cell to construct the
derivative_analysisobject, and for each cell we subtract all lower ranked cells from it so that the irreducible derivatives in the lower ranked cells are treated as known variables in the higher ranked ones.With the above setup, BID analysis can be performed on each cell to create calculations. When computing irreducible derivatives, the ID’s from lower ranked cells will be used to solve for the ID’s that are supposed to be extracted from the given cell. The mechanism is similar to the equation:
\[a_1 x_1 + a_2 x_2 = b\]where \(a_1\), \(a_2\) are predetermined coefficients and \(b\) is obtained with first-principle calculations. When \(x_1\) is known from a lower ranked system, we can solve for \(x_2\) with
\[x_2 = (b - a_1 x_1) / a_2\].
These modules only obtains the dynamic tensors for the given system. Fourier interpolation is handled in fourier_interp_Dqn module.
[1]:
import numpy as np
from finite_difference import Finite_Difference
from hierarchical import Hierarchical
from coordinate_tools import coord_to_string as c2s, table_text
np.set_printoptions(linewidth=180, suppress=True)
[2]:
poscar = """\
graphene
1.00000000000000
2.1217113285639000 1.2249706066897641 0.0000000000000000
2.1217113285639000 -1.2249706066897641 0.0000000000000000
0.0000000000000000 0.0000000000000000 -15.0000000000000000
2
Direct
0.00000000 0.00000000 0.0000000000000000 C:p
0.33333333 0.33333333 0.0000000000000000 C:p
""".strip()
Examples:
- BID
[3]:
fd = Finite_Difference(
pos=poscar,
supa="2 -1 0 -1 2 0 0 0 1",
order=3,
pgn="D6h",
pg_loc="1/3 1/3 0",
)
[4]:
fd.symmetry_analysis(
nocache=True, # Do not cache analytic tensor not in the irreducible Q
use_symadt=True, # Use full-group approach
verbose=True,
)
Symadt mode
((0, 0, 0), (0, 0, 0), (0, 0, 0))
((2/3, 1/3, 0), (2/3, 1/3, 0), (2/3, 1/3, 0))
((0, 0, 0), (2/3, 1/3, 0), (1/3, 2/3, 0))
Number of irreducible terms: 12
Rank of a single pull set: 12
Try 0 picks 1 rank 12
Number of distinct pull sets: 1
[5]:
print "chainrule.real"
print table_text(fd.chainrulemat_unknown.real, 4)
print "chainrule.imag"
print table_text(fd.chainrulemat_unknown.imag, 4)
chainrule.real
0.0055 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0106 0.0480 0.0316
0.0271 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0602 -0.0263 0.0212
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0365 0.0000 0.0000 0.0000 0.0000
-0.0055 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0106 -0.0480 -0.0316
-0.0271 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0602 0.0263 -0.0212
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0365 0.0000 0.0000 0.0000 0.0000
0.0000 -0.0006 -0.0154 -0.0110 -0.0092 -0.0056 0.0203 -0.0083 -0.0199 0.0733 0.0330 0.0000
0.0000 0.0112 0.0199 -0.0290 0.0002 -0.0074 0.0138 0.0210 -0.0173 0.0219 -0.0039 0.0000
0.0000 0.0000 0.0000 0.0000 -0.0164 0.0000 -0.0209 0.0065 0.0000 0.0000 0.0000 0.0152
0.0000 0.0006 -0.0372 -0.0008 0.0092 0.0102 -0.0045 0.0031 -0.0110 0.0377 0.0105 0.0000
0.0000 0.0112 0.0103 -0.0363 0.0002 -0.0025 0.0088 0.0069 -0.0466 -0.0800 -0.0031 0.0000
0.0000 0.0000 0.0000 0.0000 0.0106 0.0000 0.0249 -0.0088 0.0000 0.0000 0.0000 0.0016
0.0000 -0.0006 -0.0154 -0.0110 -0.0092 0.0056 -0.0203 -0.0083 0.0199 0.0733 0.0330 0.0000
0.0000 0.0112 0.0199 -0.0290 0.0002 0.0074 -0.0138 0.0210 0.0173 0.0219 -0.0039 0.0000
0.0000 0.0000 0.0000 0.0000 -0.0164 0.0000 0.0209 0.0065 0.0000 0.0000 0.0000 0.0152
0.0000 0.0006 -0.0372 -0.0008 0.0092 -0.0102 0.0045 0.0031 0.0110 0.0377 0.0105 0.0000
0.0000 0.0112 0.0103 -0.0363 0.0002 0.0025 -0.0088 0.0069 0.0466 -0.0800 -0.0031 0.0000
0.0000 0.0000 0.0000 0.0000 0.0106 0.0000 -0.0249 -0.0088 0.0000 0.0000 0.0000 0.0016
chainrule.imag
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0406 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0158 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0406 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0158 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0112 0.0199 -0.0077 0.0002 0.0074 -0.0138 -0.0210 -0.0116 0.0179 0.0039 0.0000
0.0000 0.0006 0.0154 0.0001 0.0092 -0.0056 0.0203 -0.0083 -0.0445 0.0668 0.0330 0.0000
0.0000 0.0000 0.0000 0.0000 -0.0455 0.0000 -0.0201 0.0005 0.0000 0.0000 0.0000 -0.0519
0.0000 -0.0112 -0.0103 0.0004 -0.0002 -0.0025 0.0088 0.0069 -0.0177 -0.0403 -0.0031 0.0000
0.0000 0.0006 -0.0372 0.0119 0.0092 -0.0102 0.0045 -0.0031 -0.0136 -0.0312 -0.0105 0.0000
0.0000 0.0000 0.0000 0.0000 0.0143 0.0000 -0.0046 -0.0200 0.0000 0.0000 0.0000 0.0279
0.0000 -0.0112 -0.0199 0.0077 -0.0002 0.0074 -0.0138 0.0210 -0.0116 -0.0179 -0.0039 0.0000
0.0000 -0.0006 -0.0154 -0.0001 -0.0092 -0.0056 0.0203 0.0083 -0.0445 -0.0668 -0.0330 0.0000
0.0000 0.0000 0.0000 0.0000 0.0455 0.0000 -0.0201 -0.0005 0.0000 0.0000 0.0000 0.0519
0.0000 0.0112 0.0103 -0.0004 0.0002 -0.0025 0.0088 -0.0069 -0.0177 0.0403 0.0031 0.0000
0.0000 -0.0006 0.0372 -0.0119 -0.0092 -0.0102 0.0045 0.0031 -0.0136 0.0312 0.0105 0.0000
0.0000 0.0000 0.0000 0.0000 -0.0143 0.0000 -0.0046 0.0200 0.0000 0.0000 0.0000 -0.0279
In order to compute irreducible derivatives from first-principle calculations, one simply use the following methods:
fd.gather_forces(dft_force_io = vasp.get_forces) # Collect forces from the first-principle calculations
fd.compute_derivatives() # Compute irreducible derivatives and dynamic tensors
- BID
[6]:
h = Hierarchical(
pos=poscar,
supa="2 -1 0 -1 2 0 0 0 1",
order=3,
pgn="D6h",
pg_loc="1/3 1/3 0",
nocache=True,
use_symadt=True,
)
Symadt mode
((0, 0, 0), (0, 0, 0), (0, 0, 0))
((2/3, 1/3, 0), (2/3, 1/3, 0), (2/3, 1/3, 0))
((0, 0, 0), (2/3, 1/3, 0), (1/3, 2/3, 0))
[7]:
h.construct_hierarchical()
h.symmetry_analysis()
Construct hierarchical cells
Isolation of terms
Symmetry analysis for cell 1 0 0 0 1 0 0 0 1
Number of irreducible terms: 1
Rank of a single pull set: 1
Number of distinct pull sets: 1
Symmetry analysis for cell 1 1 0 0 3 0 0 0 1
Number of irreducible terms: 11
Rank of a single pull set: 11
Number of distinct pull sets: 1
To compute irreducible derivatives from first-principle calculations:
h.compute_irrterms(dft_force_io=vasp.get_forces)