Symmetrizing displacements
A key task is to symmetrize the basis such that the vectors transform like irreducible representations.
Categories:
Here we illustrate how to symmtrize displacements at a given q-point according to irreducible representations of the little group using the script pm-disp-qrep. Take the example of \(\bm{q}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})\) in cubic SrTiO\(_3\). We will use pm-prototype-xtal to generate the crystal structure file.
$ pm-prototype-xtal --perovskite > xtal.yml
Printing out the irreducible representations, we have
$ pm-disp-qrep xtal.yml --point-group Oh --qpoint 1/2,1/2,1/2 --print-irreps
OrbitKey(name='Sr', index=0) [0]
T2g
OrbitKey(name='Ti', index=0) [0]
T1u
OrbitKey(name='O', index=0) [0, 1, 2]
A1g+Eg+T1g+T2g
We can also print out the irreducible representation vectors.
$ pm-disp-qrep xtal.yml --point-group Oh --qpoint 1/2,1/2,1/2 --print-vectors
T2g[Sr]
0.0000 0.0000 1.0000
1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
T1u[Ti]
1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
0.0000 0.0000 1.0000
A1g[O]
0.5774 0.0000 0.0000 0.0000 0.5774 0.0000 0.0000 0.0000 0.5774
Eg[O]
0.7071 0.0000 0.0000 0.0000 -0.7071 0.0000 0.0000 0.0000 0.0000
-0.4082 0.0000 0.0000 0.0000 -0.4082 0.0000 0.0000 0.0000 0.8165
T1g[O]
0.0000 0.0000 0.0000 0.0000 0.0000 0.7071 0.0000 -0.7071 0.0000
0.0000 0.0000 -0.7071 0.0000 0.0000 0.0000 0.7071 0.0000 0.0000
0.0000 0.7071 0.0000 -0.7071 0.0000 0.0000 0.0000 0.0000 0.0000
T2g[O]
0.0000 0.7071 0.0000 0.7071 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.7071 0.0000 0.7071 0.0000
0.0000 0.0000 0.7071 0.0000 0.0000 0.0000 0.7071 0.0000 0.0000
This script will block diagonalize a dynamical matrix which has been provided. Consider the dyanmical matrix at the gamma point for STO:
$ cat dmat.yaml
dynamical_matrix_real:
3.6228 0 0 -2.990 0 0 0.2751 0 0 0.2751 0 0 -1.1822 0 0
0 3.6228 0 0 -2.990 0 0 -1.1822 0 0 0.2751 0 0 0.2751 0
0 0 3.6228 0 0 -2.990 0 0 0.2751 0 0 -1.1822 0 0 0.2751
-2.990 0 0 2.060 0 0 0.2442 0 0 0.2442 0 0 0.4416 0 0
0 -2.990 0 0 2.060 0 0 0.4416 0 0 0.2442 0 0 0.2442 0
0 0 -2.990 0 0 2.060 0 0 0.2442 0 0 0.4416 0 0 0.2442
0.2751 0 0 0.2442 0 0 4.0322 0 0 0.90 0 0 -5.460 0 0
0 -1.1822 0 0 0.4416 0 0 11.6616 0 0 -5.460 0 0 -5.460 0
0 0 0.2751 0 0 0.2442 0 0 4.0322 0 0 -5.460 0 0 0.90
0.2751 0 0 0.2442 0 0 0.90 0 0 4.0322 0 0 -5.460 0 0
0 0.2751 0 0 0.2442 0 0 -5.460 0 0 4.0322 0 0 0.90 0
0 0 -1.1822 0 0 0.4416 0 0 -5.460 0 0 11.6616 0 0 -5.4605
-1.1822 0 0 0.4416 0 0 -5.460 0 0 -5.460 0 0 11.6616 0 0
0 0.2751 0 0 0.2442 0 0 -5.460 0 0 0.90 0 0 4.0322 0
0 0 0.2751 0 0 0.2442 0 0 0.90 0 0 -5.460 0 0 4.0322
We can then block diagonalize as follows:
pm-prototype-xtal --perovskite2 | pm-disp-qrep --point-group Oh --qpoint 0,0,0 --dynamical-matrix ~/dmat.yaml
T1u-block : [('T1u', 0), ('T1u', 1), ('T1u', 2), ('T1u', 3)]
Dynamical matrix sub-block - first row
[[ 0. 0. 0. 0. ]
[ 0. 2.57 3.2 -0.18]
[ 0. 3.2 3.01 -1.33]
[ 0. -0.18 -1.33 16.7 ]]
Mass matrix sub-block - first row
[[ 36.7 5.59 -27.74 0. ]
[ 5.59 45.08 13.87 0. ]
[-27.74 13.87 69.71 0. ]
[ 0. 0. 0. 16. ]]
Mass renormalized Eigvecs
[[-0.324 0.931 -0.169 -0.017]
[ 0.692 0.111 -0.713 0. ]
[-0.644 -0.348 -0.68 -0.046]
[-0.035 0. -0.034 0.999]]
Unique Frequencies (meV)
[-8.19 -0.58 19.56 66.13]
T2u-block : [('T2u', 0)]
Dynamical matrix sub-block - first row
[[3.13]]
Mass matrix sub-block - first row
[[16.]]
Mass renormalized Eigvecs
[[1.]]
Unique Frequencies (meV)
[28.61]
It may be desirable to perform a unitary transformation within some irreducible representation subspace. Consider a yam file called unitary.yaml with the following content:
# unitary transformation to nearly diagonize the T1u subspace
- block: ['Optical',0]
irr_rep: 'T1u'
instances: [0,1]
U:
[[ -0.73059222, 0.68281403 ],
[ 0.68281403, 0.73059222 ]]
The conventional output is:
$ pm-prototype-xtal --perovskite2 | pm-disp-qrep --point-group Oh --qpoint 0,0,0 --print-vectors
('Acoustic', 0) [0, 1, 2, 3, 4]
T1u
('T1u', 0)
[[0.447 0. 0. 0.447 0. 0. 0.447 0. 0. 0.447 0. 0. 0.447 0. 0. ]
[0. 0.447 0. 0. 0.447 0. 0. 0.447 0. 0. 0.447 0. 0. 0.447 0. ]
[0. 0. 0.447 0. 0. 0.447 0. 0. 0.447 0. 0. 0.447 0. 0. 0.447]]
('Optical', 0) [0, 1, 2, 3, 4]
2T1u
('T1u', 0)
[[-0.224 0. 0. 0.894 0. 0. -0.224 0. 0. -0.224 0. 0. -0.224 0. 0. ]
[ 0. -0.224 0. 0. 0.894 0. 0. -0.224 0. 0. -0.224 0. 0. -0.224 0. ]
[ 0. 0. -0.224 0. 0. 0.894 0. 0. -0.224 0. 0. -0.224 0. 0. -0.224]]
('T1u', 1)
[[-0.866 0. 0. 0. 0. 0. 0.289 0. 0. 0.289 0. 0. 0.289 0. 0. ]
[ 0. -0.866 0. 0. 0. 0. 0. 0.289 0. 0. 0.289 0. 0. 0.289 0. ]
[ 0. 0. -0.866 0. 0. 0. 0. 0. 0.289 0. 0. 0.289 0. 0. 0.289]]
('O', 0) [2 3 4]
T1u+T2u
('T1u', 0)
[[ 0.408 0. 0. 0.408 0. 0. -0.816 0. 0. ]
[ 0. -0.816 0. 0. 0.408 0. 0. 0.408 0. ]
[ 0. 0. 0.408 0. 0. -0.816 0. 0. 0.408]]
('T2u', 0)
[[ 0. 0. 0.707 0. 0. 0. 0. 0. -0.707]
[-0.707 0. 0. 0.707 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. -0.707 0. 0. 0.707 0. ]]
Now we can print the relevant output after the unitary transformation:
$ pm-prototype-xtal --perovskite2 | pm-disp-qrep --point-group Oh --qpoint 0,0,0 --unitary-transformation ~/unitary.yaml --print-vectors
('Optical', 0) [0, 1, 2, 3, 4]
2T1u
('T1u', 0)
[[-0.428 0. 0. -0.653 0. 0. 0.36 0. 0. 0.36 0. 0. 0.36 0. 0. ]
[ 0. -0.428 0. 0. -0.653 0. 0. 0.36 0. 0. 0.36 0. 0. 0.36 0. ]
[ 0. 0. -0.428 0. 0. -0.653 0. 0. 0.36 0. 0. 0.36 0. 0. 0.36 ]]
('T1u', 1)
[[-0.785 0. 0. 0.611 0. 0. 0.058 0. 0. 0.058 0. 0. 0.058 0. 0. ]
[ 0. -0.785 0. 0. 0.611 0. 0. 0.058 0. 0. 0.058 0. 0. 0.058 0. ]
[ 0. 0. -0.785 0. 0. 0.611 0. 0. 0.058 0. 0. 0.058 0. 0. 0.058]]
Given some distorted crystal structure, one can project the displacement onto symmetrized displacements. Consider the primitive unit cell of STO where the ferroelectric mode is allowed to condense by treating the nuclear motion as classical:
$ cat xtal_ferro.yaml
# Crystal
vec:
[[ 3.89736730, 0.00092044, 0.00092044],
[ 0.00092044, 3.89736730, 0.00092044],
[ 0.00092044, 0.00092044, 3.89736730]]
atoms:
- Sr:
[[ 0.02422827, 0.02422827, 0.02422827]]
- Ti:
[[ 0.52563170, 0.52563170, 0.52563170]]
- O:
[[ 0.51647453, 0.01719097, 0.51647453],
[ 0.51647453, 0.51647453, 0.01719097],
[ 0.01719097, 0.51647453, 0.51647453]]
Projecting onto irreducible representations gives:
pm-prototype-xtal --perovskite2 | pm-disp-qrep --point-group Oh --qpoint 0,0,0 --unitary-transformation unitary.yaml --project-xtal xtal_ferro.yaml
Projection onto irreducible representations
Block : Acoustic
----------------
T1u 0.1744 0.1744 0.1744
Block : Optical
---------------
T1u -0.0353 -0.0353 -0.0353
1.T1u -0.0018 -0.0018 -0.0018
Block : O
---------
T1u -0.0023 -0.0023 -0.0023
T2u 0 0 0
Feedback
Was this page helpful?
Glad to hear it! Please tell us how we can improve.
Sorry to hear that. Please tell us how we can improve.