DqN analysis

With the capability of forming dynamic tensor analytically at any given Q, we can use the analytic tensors of Q’s from a given Brillouin Zone (BZ), to perform more advanced analysis, since we will be able to express quantities in terms of irreducible derivatives.

The core feature of this analysis is situated in the abstract class abstract_dqn_analysis, with which different analysis can be devised, by deriving from the abstract and adding required features.

The Core feature contains:

• Forming analytic tensors at irreducible Q’s with either little group or full-group method.
• Rotating the tensors from irreducible space the the full BZ, thus obtaining the DqN of the BZ in terms of the irreducible derivatives.

Currently we have the following analysis available:

• Analytic PhiN:

  The Fourier Transformation is added the abtain PhiN of the real space in terms of the irreducible derivatives.

• Derivative Analysis:

  The chain-rule feature is added to compute the transformation matrix to the derivatives of an arbitrary basis from the symmetrized basis (i.e. in terms of irreducible derivatives).

• Analytic Gruneisen parameter - using Analytic PhiN:

  With the existing formula to compute the analytical formula of gruneisen parameters in terms of the irreducible derivatives.

• Analytic formula of the Born-Oppenheimer potential:

  This is basically a formatter for the DqN of the BZ. In writing the formula out, it assists back-of-envelope type simple analysis, and is also helps pedagogy. Or just a show off.

Here examples for Abstract_Dqn_Analysis, Analytic_PhiN and Derivative_Analysis will be shown.

from abstract_dqn_analysis import Abstract_Dqn_Analysis
from analytic_phin import Analytic_PhiN
from derivative_analysis import Derivative_Analysis

First of all, the constructors of all above classes are the same.

class Abstract_Dqn_Analysis(pos=None, supa=None, pgn=None, pg_loc=None, order=2, D=None, tensor_vec_indices=None, known_tensor_vec_indices=None, nocache=False, parallel=False, tol=1.0E-8, use_star=False, symadt=False, quiet=False)

This class simply constructs analytic dynamic tensors of the chosen approach for all irreducible Q’s at the given order and given system.

• pos: A structure object or a string that can construct the object.
• supa: A coordinate_parse compatible input for super cell. Usually a 3x3 matrix or length 9 array, but if using diagonal super cell, one can simply give a length 3 array for the diagonal components.
• pgn: The name of the point group of the structure.
• pg_loc: The site of the symmetry, (default: None).
• order: The order of the interaction, (default: 2).
• nocache: Do not cache the vectorized tensor and store those in IBZ only.
• parallel: Use parallelization in generating tensor vectors.
• quiet: Reduce verbosity.
• tol: Tolerance in numerical comparison.
• D: An existing existing vectorized tensor to instantiate the object with along with irreducible term lables, all three need to be present at the same time.
• tensor_vec_indices: An existing list irreducible terms corresponding to the vectorized tensor above.
• known_tensor_vec_indices: All the known irreducieble terms from the list above.
• use_star: It will default use little group analysis, with this being True, it will use star analysis.
• symadt: With this True, it will use symmetric ADT which exploits full symmetry.

Let’s use the following made-up test structure

poscar="""\
test
1.0
0 0.5 0.5
0.5 0 0.5
0.5 0.5 0
2
D
0 0 0 A:p
0.5 0.5 0.5 B:p"""

---

Analytic PhiN

Let’s construct the analytic PhiN in terms of the irreducible derivatives at 2nd order.

aph = Analytic_PhiN(
    pos=poscar,
    supa='1 0 0  0 1 0  0 0 1',
    pgn='Oh',
    order=2,
    symadt=True)
aph.fourier_transform()

Symadt mode
((0, 0, 0), (0, 0, 0))

aph.tensor_vec_indices

[((((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)),
    ('T1u', (('T1u', 0), 0))),
   ((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)),
    ('T1u', (('T1u', 0), 0)))),
  ('A1g', 0))]

print "shape of the analytic PhiN", aph.phin_WS[((0, 0, 0),)].shape
print "Analytic PhiN at (0, 0, 0) in terms of the only one irreducible derivative:"
print aph.phin_WS[((0, 0, 0),)][:, :, 0]

shape of the analytic PhiN (6, 6, 1)
Analytic PhiN at (0, 0, 0) in terms of the only one irreducible derivative:
[[0.5+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j]
 [0. +0.j 0.5+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j]
 [0. +0.j 0. +0.j 0.5+0.j 0. +0.j 0. +0.j 0. +0.j]
 [0. +0.j 0. +0.j 0. +0.j 0.5+0.j 0. +0.j 0. +0.j]
 [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.5+0.j 0. +0.j]
 [0. +0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.5+0.j]]

---

Derivative Analysis

da = Derivative_Analysis(
    pos=poscar,
    supa='1 0 0  0 1 0  0 0 1',
    pgn='Oh',
    order=2,
    symadt=True)

Symadt mode
((0, 0, 0), (0, 0, 0))

Let’s test for a simply displacement

\[d_{example} = \left[1, 0, 0, 0, 0, 0\right]\]

displcement = [np.zeros_like(da.pos.atc.flatten())]
displcement[0][0] = 1
displcement

[array([1., 0., 0., 0., 0., 0.])]

da.valid_unknowns

[((((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)),
    ('T1u', (('T1u', 0), 0))),
   ((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)),
    ('T1u', (('T1u', 0), 0)))),
  ('A1g', 0))]

da.chainrule_disp(displcement)

OrderedDict([(((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (0, 'p_x')),
              array([0.5+0.j])),
             (((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (0, 'p_y')),
              array([0.+0.j])),
             (((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (0, 'p_z')),
              array([0.+0.j])),
             (((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (1, 'p_x')),
              array([-0.5+0.j])),
             (((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (1, 'p_y')),
              array([0.+0.j])),
             (((Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (1, 'p_z')),
              array([0.+0.j]))])

This means with the a displacement along the first atom along the x-axis, we will have the following relation:

\[\frac{\partial^2 V}{\partial d_{(\Gamma, atom A, x)} d_{example}} = \frac{1}{2} \textit{ID}_0\]

\[\frac{\partial^2 V}{\partial d_{(\Gamma, atom B, x)} d_{example}} = -\frac{1}{2} \textit{ID}_0\]

where \(\textit{ID}_0\) is the first and only irreducible derivative.

This kind of chainrule analysis allows us to find the minimum amount of displacements that allow us to compute the irreducible derivatives.

Hierarchical supercells

The subtraction operator of the Abstract_Dqn_Analysis has been defined to accomondate hierarchy of supercells. Meaning that irreducible derivatives of a subcell can be treated as known irreducible derivatives in the bigger cells. This knowledge, in the case of derivative analysis, can be used to seperate the chainrule matrix into a known part and an unknown part, allowing the irreducible derivatives being calculated in a hierarchical manner.

The application of this feature will be shown in the HS-BID work code.

da_2 = Derivative_Analysis(
    pos=poscar,
    supa='2 0 0  0 2 0  0 0 2',
    pgn='Oh',
    order=2,
    symadt=True)

Symadt mode
((0, 0, 0), (0, 0, 0))
((1/2, 0, 0), (1/2, 0, 0))
((1/2, 1/2, 0), (1/2, 1/2, 0))

da_21 = da_2 - da

Here we subtract the 1x1x1 mesh from the 2x2x2 mesh, meaning the irreducible derivatives in the 1x1x1 mesh becomes known irreducible derivatives in the 2x2x2 mesh.

map(c2s, da_21.valid_knowns)

['((((0, 0, 0), (T1u, ((T1u, 0), 0))), ((0, 0, 0), (T1u, ((T1u, 0), 0)))), (A1g, 0))']

map(c2s, da_21.valid_unknowns)

['((((1/2, 0, 0), (A1g, ((A1g, 0), 0))), ((1/2, 0, 0), (A1g, ((A1g, 0), 0)))), (A1g, 0))',
 '((((1/2, 0, 0), (Eg, ((Eg, 0), 0))), ((1/2, 0, 0), (Eg, ((Eg, 0), 0)))), (A1g, 0))',
 '((((1/2, 0, 0), (A2u, ((A2u, 0), 0))), ((1/2, 0, 0), (A2u, ((A2u, 0), 0)))), (A1g, 0))',
 '((((1/2, 0, 0), (Eu, ((Eu, 0), 0))), ((1/2, 0, 0), (Eu, ((Eu, 0), 0)))), (A1g, 0))',
 '((((1/2, 1/2, 0), (T1u, ((A2u, 0), 2))), ((1/2, 1/2, 0), (T1u, ((A2u, 0), 2)))), (A1g, 0))',
 '((((1/2, 1/2, 0), (T1u, ((A2u, 0), 2))), ((1/2, 1/2, 0), (T1u, ((A2u, 1), 3)))), (A1g, 0))',
 '((((1/2, 1/2, 0), (T1u, ((A2u, 1), 3))), ((1/2, 1/2, 0), (T1u, ((A2u, 1), 3)))), (A1g, 0))',
 '((((1/2, 1/2, 0), (T1u, ((Eu, 0), 0))), ((1/2, 1/2, 0), (T1u, ((Eu, 0), 0)))), (A1g, 0))',
 '((((1/2, 1/2, 0), (T1u, ((Eu, 0), 0))), ((1/2, 1/2, 0), (T1u, ((Eu, 1), 1)))), (A1g, 0))',
 '((((1/2, 1/2, 0), (T1u, ((Eu, 1), 1))), ((1/2, 1/2, 0), (T1u, ((Eu, 1), 1)))), (A1g, 0))']

---

Substituting in the irreducible derivatives for the actual dynamic tensors

The method substitude_unknownvec allows one to substitute in values of the irreducible derivatives to obtain the actual dynamic tensors from the vectorized analytic tensors.

da.substitude_unknownvec(
    unknowns_vec=unknowns_vec, # An array of numbers that is the unknown irreducible derivatives
    knowns_vec=knowns_vec, # An array of numbers that is the known irreducible derivatives
)