This tutorial can be downloaded link.
Advanced tutorial 1: Symmetry analysis in Quantum Defect Embedding Theory (QDET)¶
We explain how to identify the symmetry of excited many-body states calculated with the Quantum Defect Embedding Theory (QDET) using the point symmetry of the defect. In this tutorial, we focus on the \(\mathrm{NV^-}\) center in diamond.
Step 1: Mean-field starting point¶
As for a standard QDET calculation, we need to determine the mean-field starting point first. In order to perform the symmetry analysis, we will additionally write the single-particle wavefunctions to cube files, in order to determine their symmetry character.
Download the following files in your working directory:
[1]:
%%bash
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/pw.in
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/pp.in
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/C_ONCV_PBE-1.0.upf
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/N_ONCV_PBE-1.0.upf
We can now inspect the pw.in file, the input for the pw.x code:
[2]:
%%bash
cat pw.in
&CONTROL
calculation = 'scf'
wf_collect = .true.
pseudo_dir = './'
/
&SYSTEM
input_dft = 'PBE'
ibrav = 0
ecutwfc = 50
nosym = .true.
tot_charge = -1
nspin = 1
nbnd = 176
occupations = 'from_input'
nat = 63
ntyp = 2
/
&ELECTRONS
conv_thr = 1D-08
/
K_POINTS gamma
CELL_PARAMETERS angstrom
7.136012 0.000000 0.000000
0.000000 7.136012 0.000000
0.000000 0.000000 7.136012
ATOMIC_SPECIES
C 12.01099968 C_ONCV_PBE-1.0.upf
N 14.00699997 N_ONCV_PBE-1.0.upf
ATOMIC_POSITIONS crystal
C 0.99996000 0.99996000 0.99996000
C 0.12495000 0.12495000 0.12495000
C 0.99905000 0.25039000 0.25039000
C 0.12350000 0.37499000 0.37499000
C 0.25039000 0.99905000 0.25039000
C 0.37499000 0.12350000 0.37499000
C 0.25039000 0.25039000 0.99905000
C 0.37499000 0.37499000 0.12350000
C 0.00146000 0.00146000 0.50100000
C 0.12510000 0.12510000 0.62503000
C 0.00102000 0.24944000 0.74960000
C 0.12614000 0.37542000 0.87402000
C 0.24944000 0.00102000 0.74960000
C 0.37542000 0.12614000 0.87402000
C 0.24839000 0.24839000 0.49966000
C 0.37509000 0.37509000 0.61906000
C 0.00146000 0.50100000 0.00146000
C 0.12510000 0.62503000 0.12510000
C 0.00102000 0.74960000 0.24944000
C 0.12614000 0.87402000 0.37542000
C 0.24839000 0.49966000 0.24839000
C 0.37509000 0.61906000 0.37509000
C 0.24944000 0.74960000 0.00102000
C 0.37542000 0.87402000 0.12614000
C 0.99883000 0.50076000 0.50076000
C 0.12502000 0.62512000 0.62512000
C 0.99961000 0.74983000 0.74983000
C 0.12491000 0.87493000 0.87493000
C 0.25216000 0.50142000 0.74767000
C 0.37740000 0.62659000 0.87314000
C 0.25216000 0.74767000 0.50142000
C 0.37740000 0.87314000 0.62659000
C 0.50100000 0.00146000 0.00146000
C 0.62503000 0.12510000 0.12510000
C 0.49966000 0.24839000 0.24839000
C 0.61906000 0.37509000 0.37509000
C 0.74960000 0.00102000 0.24944000
C 0.87402000 0.12614000 0.37542000
C 0.74960000 0.24944000 0.00102000
C 0.87402000 0.37542000 0.12614000
C 0.50076000 0.99883000 0.50076000
C 0.62512000 0.12502000 0.62512000
C 0.50142000 0.25216000 0.74767000
C 0.62659000 0.37740000 0.87314000
C 0.74983000 0.99961000 0.74983000
C 0.87493000 0.12491000 0.87493000
C 0.74767000 0.25216000 0.50142000
C 0.87314000 0.37740000 0.62659000
C 0.50076000 0.50076000 0.99883000
C 0.62512000 0.62512000 0.12502000
C 0.50142000 0.74767000 0.25216000
C 0.62659000 0.87314000 0.37740000
C 0.74767000 0.50142000 0.25216000
C 0.87314000 0.62659000 0.37740000
C 0.74983000 0.74983000 0.99961000
C 0.87493000 0.87493000 0.12491000
N 0.48731000 0.48731000 0.48731000
C 0.49191000 0.76093000 0.76093000
C 0.62368000 0.87476000 0.87476000
C 0.76093000 0.49191000 0.76093000
C 0.87476000 0.62368000 0.87476000
C 0.76093000 0.76093000 0.49191000
C 0.87476000 0.87476000 0.62368000
OCCUPATIONS
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00
2.00 2.00 2.00 2.00 2.00 2.00 1.00 1.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
We can now run pw.x on 2 cores.
[ ]:
%%bash
mpirun -n 2 pw.x -i pw.in > pw.out
Once the ground state calculation has finished successfully, we can write the single-particle wavefunctions using the pp.x executable. The input file is given as
[3]:
%%bash
cat pp.in
&INPUTPP
filplot = 'wfct'
plot_num = 7
kpoint(1) = 1
kband(1) = 87
kband(2) = 128
lsign = true
/
&PLOT
iflag = 3
output_format = 6
fileout = '.cube'
/
And the calculation is performed with the following command
[ ]:
%%bash
mpirun -n 2 pp.x -i pp.in > pp.out
Step 2: Definition of the point group¶
We now define the point group. A point group is characterized by two quantities: * a set of symmetry operations (identity, reflection, and rotation operations) * a corresponding character table
[4]:
import numpy as np
from westpy.qdet.symm import PointGroup, PointGroupOperation, PointGroupRotation, PointGroupReflection
sq3 = np.sqrt(3)
# the origin is set to the Nitrogen site.
# The prefactor corresponds to the real-space grid along each direction.
origin = 64 * np.array([0.48731, 0.48731, 0.48731])
point_group = PointGroup(
name="C3v",
operations={
"E": PointGroupOperation(T=np.eye(4)),
"C3_1": PointGroupRotation(rotvec=2 * np.pi / 3 * np.array([1/sq3, 1/sq3, 1/sq3]),
origin=origin),
"C3_2": PointGroupRotation(rotvec=4 * np.pi / 3 * np.array([1/sq3, 1/sq3, 1/sq3]),
origin=origin),
"Cv_1": PointGroupReflection(normal=(1, -1, 0), origin=origin),
"Cv_2": PointGroupReflection(normal=(0, -1, 1), origin=origin),
"Cv_3": PointGroupReflection(normal=(-1, 0, 1), origin=origin)
},
ctable={
"A1": [1, 1, 1, 1, 1, 1],
"A2": [1, 1, 1, -1, -1, -1],
"E": [2, -1, -1, 0, 0, 0]})
_ _ _____ _____ _____
| | | | ___/ ___|_ _|
| | | | |__ \ `--. | |_ __ _ _
| |/\| | __| `--. \ | | '_ \| | | |
\ /\ / |___/\__/ / | | |_) | |_| |
\/ \/\____/\____/ \_/ .__/ \__, |
| | __/ |
|_| |___/
WEST version : 5.2.0
Today : 2022-12-13 15:23:23.758988
Step 3: Diagonalization of QDET Hamiltonian¶
In this tutorial, we focus on the symmetry analysis with QDET. To avoid costly calculations of the matrix elements of the effective Hamiltonian, we simply download the input and output files of the QDET calculation. To learn how to perform QDET calculations, see Basic Tutorial 5.
[5]:
%%bash
mkdir -p west.wfreq.save
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfreq.in
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfreq.json -O west.wfreq.save/wfreq.json
To add point-group analysis to QDET, we simply pass the point-group object defined above to the Hamiltonian object. Additionally, we provide the list of filenames for the wavefunctions.
The code will apply the symmetry operations to the Kohn-Sham wavefunctions stored in the cube files and determine the character of these wavefunctions. After the diagonalization of the effective Hamiltonian, the character of the many-body states can then be inferred from the character of the wavefunctions.
[6]:
# Here, we provide the list of file paths for the cube files for each basis function of the active space.
wfct_filenames = ['wfct_K001_B087.cube','wfct_K001_B122.cube', 'wfct_K001_B123.cube',
'wfct_K001_B126.cube', 'wfct_K001_B127.cube', 'wfct_K001_B128.cube']
[7]:
%%bash
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfct_K001_B087.cube
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfct_K001_B122.cube
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfct_K001_B123.cube
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfct_K001_B126.cube
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfct_K001_B127.cube
wget -N -q http://www.west-code.org/doc/training/nv_diamond_63/wfct_K001_B128.cube
[8]:
# the diagonalization is exactly the same as without the symmetry analysis.
from westpy.qdet import QDETResult
# construct object for effective Hamiltonian
effective_hamiltonian = QDETResult(filename='west.wfreq.save/wfreq.json', point_group=point_group, wfct_filenames=wfct_filenames)
# diagonalize Hamiltonian
solution = effective_hamiltonian.solve()
PointGroupRep: rep matrices are orthogonal
Irrep of orbitals: ['A1(1.00)', 'A1(1.00)', 'A1(1.00)', 'A1(1.00)', 'E(0.99)', 'E(0.99)']
Solutions are projected onto irreps of C3v group
-----------------------------------------------------
===============================================================
Building effective Hamiltonian...
nspin: 1
occupations: [[2. 2. 2. 2. 1. 1.]]
===============================================================
| diag[1RDM - 1RDM(GS)] | ||||||||
|---|---|---|---|---|---|---|---|---|
| E [eV] | char | 87 | 122 | 123 | 126 | 127 | 128 | |
| 0 | 0.000 | 3A2 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 1 | 0.436 | 1E | -0.001 | -0.009 | -0.018 | -0.067 | 0.037 | 0.058 |
| 2 | 0.436 | 1E | -0.001 | -0.009 | -0.018 | -0.067 | 0.058 | 0.037 |
| 3 | 1.250 | 1A1 | -0.002 | -0.019 | -0.023 | -0.067 | 0.056 | 0.055 |
| 4 | 1.941 | 3E | -0.003 | -0.010 | -0.127 | -0.860 | 0.000 | 1.000 |
| 5 | 1.941 | 3E | -0.003 | -0.010 | -0.127 | -0.860 | 1.000 | 0.000 |
| 6 | 2.937 | 1E | -0.000 | -0.032 | -0.043 | -0.855 | 0.110 | 0.821 |
| 7 | 2.937 | 1E | -0.000 | -0.033 | -0.043 | -0.855 | 0.821 | 0.109 |
| 8 | 4.662 | 1A1 | -0.005 | -0.055 | -0.188 | -1.672 | 0.960 | 0.960 |
| 9 | 5.073 | 3E | -0.014 | -0.702 | -0.209 | -0.075 | 0.000 | 1.000 |
-----------------------------------------------------
[ ]: