This tutorial can be downloaded link.

Advanced Tutorial 1: Symmetry Analysis in QDET Calculations¶

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¶

Similarly to the case of a standard QDET calculation, the workflow starts with the calculation of the mean-field starting point. In order to perform the symmetry analysis, we additionally write the single-particle wavefunctions to cube files.

Download the following files in your working directory:

[1]:

%%bash
wget -N -q https://west-code.org/doc/training/nv_diamond_63/pw.in
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/C_ONCV_PBE-1.2.upf
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/N_ONCV_PBE-1.2.upf

We can now inspect the pw.in file, the input for the pw.x code:

[2]:

%%bash
cat pw.in

&CONTROL
calculation = 'scf'
pseudo_dir = './'
/
&SYSTEM
input_dft = 'PBE'
ibrav = 0
ecutwfc = 50
tot_charge = -1
nspin = 1
nbnd = 176
occupations = 'from_input'
nat = 63
ntyp = 2
/
&ELECTRONS
diago_full_acc = .true.
/
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-1.2.upf
N  14.00699997  N_ONCV_PBE-1.2.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 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 1 1 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0

We run pw.x on 2 cores.

[ ]:

%%bash
mpirun -n 2 pw.x -i pw.in > pw.out

We use westpp.x to write the wavefunctions to cube files. Create the input file westpp.in:

[3]:

import yaml

d = {}
d["westpp_control"] = {}
d["westpp_control"]["westpp_calculation"] = "W"
d["westpp_control"]["westpp_range"] = [1, 176]
d["westpp_control"]["westpp_format"] = "C"
d["westpp_control"]["westpp_sign"] = True

with open("westpp.in", "w") as f:
    yaml.dump(d, f, sort_keys=False)

[4]:

%%bash
cat westpp.in

westpp_control:
  westpp_calculation: W
  westpp_range:
  - 1
  - 176
  westpp_format: C
  westpp_sign: true

We run westpp.x on 2 cores.

[ ]:

%%bash
mpirun -n 2 westpp.x -i westpp.in > westpp.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

[5]:

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 FFT 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     :  6.0.0
Today            :  2024-06-19 10:30:15.939195

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 output file of the QDET calculation. To learn how to perform QDET calculations, see Basic Tutorial 5.

[6]:

%%bash
mkdir -p west.wfreq.save
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfreq.json -O west.wfreq.save/wfreq.json

To add point group analysis to QDET, we 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.

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfcK000001B000087.cube
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfcK000001B000122.cube
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfcK000001B000123.cube
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfcK000001B000126.cube
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfcK000001B000127.cube
wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfcK000001B000128.cube

[7]:

# the diagonalization is the same as without the symmetry analysis
from westpy.qdet import QDETResult

# list of cube files for the wavefunctions
wfct_filenames = ["wfcK000001B000087.cube", "wfcK000001B000122.cube", "wfcK000001B000123.cube",
                  "wfcK000001B000126.cube", "wfcK000001B000127.cube", "wfcK000001B000128.cube"]

# 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(1.00)', 'E(1.00)']
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	0.004	0.091
2	0.436	1E	-0.001	-0.009	-0.018	-0.067	0.092	0.002
3	1.251	1A1	-0.002	-0.019	-0.023	-0.067	0.054	0.057
4	1.939	3E	-0.003	-0.010	-0.127	-0.860	1.000	0.000
5	1.940	3E	-0.003	-0.010	-0.127	-0.860	0.000	1.000
6	2.935	1E	-0.000	-0.032	-0.043	-0.855	0.929	0.002
7	2.936	1E	-0.000	-0.032	-0.043	-0.855	0.002	0.929
8	4.661	1A1	-0.006	-0.054	-0.188	-1.672	0.960	0.960
9	5.080	3E	-0.014	-0.698	-0.213	-0.075	1.000	0.000

-----------------------------------------------------