basic band structure conceptsdegironc/es/lectures/band...14 bravais lattices: 7 crystal systems!!...
TRANSCRIPT
![Page 1: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/1.jpg)
Basic Band Structure Concepts
![Page 2: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/2.jpg)
Crystals: periodic boundary conditions, Bravais lattices, Reciprocal Lattice, Brillouin Zone, Bloch Theorem.
Energy in an infinite solid: PlaneWave expansion,Ewald sums, BZ sampling.
Band structure of materials: metal, insulators, partial occupations, ( fictitious ) temperature
All-electrons vs Pseudopotential methods
Cohen-Bergstressen empirical pseudopotential,Norm-Conserving PP, Ultra-Soft PP, PAW
![Page 3: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/3.jpg)
Infinite Solid with Periodic Boundary Conditions
Real crystals are finite in size and can present many defects (vacancies, impurities, ...). However many properties of a solid, the bulk properties, can be studied in the ideal model of an infinite crystal where an elementary unit (one or more atoms) is repeated throughout space. The resulting periodicity greatly simplifies the problem.
![Page 4: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/4.jpg)
Infinite Solid with Periodic Boundary Conditions
In three dimensions the infinite periodic solid is defined by three fundamental lattice vectors a1, a2, a3, and by the positions of the atoms in the unit cell (nat = # of atoms in the unit cell).
The general atom position Rion is given by
NB: atoms at the same position Ts in different unit cells are of the same atomic kind.
![Page 5: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/5.jpg)
Born-von Kármán Periodic Boundary Conditions
It is convenient to view the infinite solid as a three dimensional thorus containing N = N1 x N2 x N3 cells with periodic boundary conditions. Crystal sums are finite, wfc are normalized.The limit of large Ni is understood.
![Page 6: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/6.jpg)
Crystal latticeA regular array of atoms periodically repeated in space
The crystal structure is defined by
1) Bravais lattice : regular periodic arrangement of pointsin space defined by the primitive translation vectors (cell shape)
![Page 7: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/7.jpg)
Crystal latticeA regular array of atoms periodically repeated in space
The vectors R=n1 a1 + n2 a2 + n3 a3 are the direct lattice vectors; they define the crystal lattice. Translation by any lattice vector leaves the crystal unchanged. This is by far the most important symmetry of the crystal => Bloch Theorem.Notice that this symmetry only applies to infinite crystals (or crystals with BvK periodic boundary conditions) which is why they have been introduced !
![Page 8: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/8.jpg)
Primitive unit cellAny shape that repeated by lattice vectors covers the space.
- can be defined by three fundamentale lattice vectors a1, a2, a3
three lenghts (a,b,c) and three angles ( )- contains exactly one lattice point- !! the choice of primitive vectors is not unique !! - volume of the primitive cell is independent on that choice
The Wigner-Seitz cell (the region closer to the origin than to any other lattice point) is the most symmetric choice
![Page 9: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/9.jpg)
BCC
FCC
Wigner-Seitz cell
![Page 10: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/10.jpg)
Crystal latticeA regular array of atoms periodically repeated in space
The crystal structure is defined by
1) Bravais lattice : regular periodic arrangement of pointsin space defined by the primitive translation vectors (cell shape)
2) Basis : position of the atoms in the primitive cell
![Page 11: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/11.jpg)
Crystal = Bravais Lattice + basis
![Page 12: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/12.jpg)
Crystal Symmetry
!! symmetry of the crystal goes beyond translational invariance => point group operations !!
![Page 13: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/13.jpg)
Crystal Symmetry
32 point groups
230 space groups: 73 symmorphic, 157 non-symmorphic
14 Bravais lattices: 7 crystal systems
!! symmetry of the crystal goes beyond translational invariance => point group operations !!
The complete group of coordinate transformation of a crystal (its space group) is smaller than the space group of its lattice The presence of more than one atom in the unit cell can only reduce the number of symmetry operations of the lattice.
![Page 14: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/14.jpg)
14 Bravais Lattices grouped in 7 crystal systems
![Page 15: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/15.jpg)
![Page 16: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/16.jpg)
Primitive vs conventional cellConventional cells are often used instead of the primitive cell
1 atom/cell 4 atoms/cell
![Page 17: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/17.jpg)
Reciprocal LatticeBesides the Bravais lattice it is useful to introduce its dual the reciprocal lattice
Defined by
if
then
with
![Page 18: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/18.jpg)
Reciprocal Lattice
- spacing in the RL is inversely proportional to the spacing in the BL
- the primitive cell contains exacly one G point
- the choice of primitive cell is not unique but defined by the choice made for the direct lattice- the volume of the primitive cell is
- points defined as
![Page 19: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/19.jpg)
b1
b2
Brillouin zone
Reciprocal Lattice cell
![Page 20: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/20.jpg)
![Page 21: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/21.jpg)
![Page 22: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/22.jpg)
Kohn-Sham equations in a periodic solid
Periodic solid: atoms in periodically repeated positions
is periodic, hence so is
![Page 23: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/23.jpg)
Periodic effective potential
The effective potential represents the effect of the ions on the electrons and has the same periodicity of the crystal
can be expanded in plane waves COMPATIBLE with the periodicity of the crystal
![Page 24: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/24.jpg)
Bloch Theorem
is a good quantum number to label the eigenfunctions of a periodic crystal
is a symmetry op.
BvK boundary conds:
There are as many k points in the BZ as cells in the BL
![Page 25: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/25.jpg)
Bloch Theorem
is a good quantum number to label the eigenfunctions of a periodic crystal
Wavefunctions are PWs modulated by a periodic function
with manifestly periodic
Crystal periodicity imposes that the crystal wavefunctions are linear combinations of PWs with wave-vector k+G
![Page 26: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/26.jpg)
![Page 27: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/27.jpg)
![Page 28: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/28.jpg)
Dual Space Formalism
![Page 29: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/29.jpg)
Bloch Theorem
![Page 30: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/30.jpg)
- each completely filled band contains 2 electrons/cell -
![Page 31: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/31.jpg)
![Page 32: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/32.jpg)
Charge Density of infinite periodic solid
where
is normalized in the whole crystal volume
where
is normalized in the unit cell volume
![Page 33: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/33.jpg)
BZ integrals
Assuming f(k) has the full symmetry of the crystal if we can find a point where f(k)=<f> we only need to compute the function f in that single point !
![Page 34: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/34.jpg)
![Page 35: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/35.jpg)
BZ integrals
Actually it is zero up to !
![Page 36: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/36.jpg)
![Page 37: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/37.jpg)
BZ integrals
A regular grid of k-points with integrates
exactly up to (in a cubic system)
![Page 38: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/38.jpg)
Monkorst-Pack meshes
●Regular equispaced meshes in the Brillouin Zone
![Page 39: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/39.jpg)
Monkorst-Pack meshes
●Symmetry is exploited in order to reduce the number of inequivalent points to be considered
![Page 40: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/40.jpg)
Brillouin Zone integration●In insulators the quantity to be integrated is smooth across the BZ (exponentially localized Wannier functions)
insulator
●The integral is computed exactly if the WF are localized inside the SuperCell corresponding to the k-point grid
![Page 41: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/41.jpg)
Brillouin Zone integration
●Sampling at the Gamma point (isolated molecules, clusters)●Sampling on a regular grid (Baldereschi,Chadi-Cohen, Monkhorst-Pack)
●In metallic systems, the function to be intergrated is sharply varying across the Fermi surface. Integration can be improved by introducing a finite (ficticious) temperature.
metal insulator
![Page 42: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/42.jpg)
Brillouin Zone sampling
●In metals the quantity to be integrated is discontinuous due to the presence of the Fermi surface
![Page 43: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/43.jpg)
Brillouin Zone sampling●Integration is improved by introducing a finite (ficticious) temperature. [Fermi-Dirac, Gaussian, Methfessel-Paxton, Marzari-Vanderbilt]
![Page 44: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/44.jpg)
The larger is the smoother is the function and therefore smaller is the number of k-points in the grid needed to integrate accurately
The step function makes the integral badly convergent
![Page 45: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/45.jpg)
Unfortunately, the larger is the more the function differs from the original one and the integral deviates from the limit that corresponds to the physical situation.
The step function makes the integral badly convergent
![Page 46: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/46.jpg)
![Page 47: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/47.jpg)
![Page 48: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/48.jpg)
![Page 49: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/49.jpg)
![Page 50: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/50.jpg)
![Page 51: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/51.jpg)
![Page 52: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/52.jpg)
Brillouin Zone sampling●Integration is improved by introducing a finite (ficticious) temperature. [Fermi-Dirac, Gaussian, Methfessel-Paxton, Marzari-Vanderbilt]
![Page 53: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/53.jpg)
Brillouin Zone sampling
![Page 54: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/54.jpg)
Step 1 : defining V_ext
![Page 55: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/55.jpg)
The external potential
Electrons experience a Coulomb potential due to the nuclei.
This has a known simple form.
For a single atom it is
![Page 56: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/56.jpg)
Periodic potential
![Page 57: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/57.jpg)
Periodic potential
atomic form factor crystal structure factor
![Page 58: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/58.jpg)
nuclear potential
The Coulomb potential due to any single atom is
The direct use of this potential in a Plane Wave codeleads to computational difficulties!
![Page 59: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/59.jpg)
Problems for a Plane-Wave based code
Core wavefunctions:Sharply peaked close to nuclei due to deepCoulomb potential.
Valence wavefunctions:Lots of wiggles near nuclei due to orthogonality to core wavefunctions
High Fourier components are presenti.e. large kinetic energy cutoff needed
![Page 60: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/60.jpg)
![Page 61: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/61.jpg)
![Page 62: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/62.jpg)
![Page 63: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/63.jpg)
Solutions for a Plane-Wave based code
Core wavefunctions:Sharply peaked close to nuclei due to deepCoulomb potential.
Valence wavefunctions:Lots of wiggles near nuclei due to orthogonality to core wavefunctions
Don't solve forcore wavefunction
Remove wiggles fromvalence wavefunctions
Replace hard Coulomb potential by smooth PseudoPotentials
This can be done on an empirical basis by fitting experimental band structure data ..
![Page 64: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/64.jpg)
Empirical PseudoPotentials
Cohen & Bergstresser, PRB 141, 789 (1966)
transferability to other systems is problematic
![Page 65: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/65.jpg)
ab initio Norm Conserving PseudoPotentials
Let's consider an atomic problem ...
… in the frozen core approximation:
if and do not overlap significantly:
![Page 66: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/66.jpg)
ab initio Norm Conserving PseudoPotentials
... hence
or in case of overlap we have (non-linear core correction)
with
with
with a Coulomb tail corresponding to
![Page 67: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/67.jpg)
ab initio Norm Conserving PseudoPotentials
is further modified in the core region so that the reference valence wavefunctions are nodeless and smooth and properly normalized (norm conservation) so that the valence charge density (outside the core) is simply:
The norm-conservation condition ensures correct electrostatics outside the core region and that atomic scattering properties are reproduced correctly
this determines transferability
![Page 68: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/68.jpg)
An example: Mo
l-dependent potential
Hamann, schlueter & Chiang, PRL 43, 1494 (1979)
![Page 69: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/69.jpg)
An example: Mo
Hamann, schlueter & Chiang, PRL 43, 1494 (1979)
![Page 70: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/70.jpg)
ab initio Norm Conserving PseudoPotentials
semilocal form
is local with a Coulomb tail
is local in the radial coordinate, short ranged and l-dependent
where projects over L = l(l+1)
is a full matrix ! NO use of dual-space approach
2
![Page 71: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/71.jpg)
ab initio Norm Conserving PseudoPotentials
… to Kleinman-Bylander fully non-local form
is local with a Coulomb tail
are localized radial functions such that the transformed pseudo acts in the same way as the original form on the reference config.
from semilocal form ...
One has
![Page 72: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/72.jpg)
ab initio Norm Conserving PseudoPotentials
Kleinman-Bylander fully non-local form
is local with a Coulomb tail
are localized radial functions such that the transformed pseudo acts in the same way as the original form on the reference config.
The pseudopotential reduces to a sum of dot products
One has
![Page 73: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/73.jpg)
ab initio Norm Conserving PseudoPotentials
Kleinman-Bylander fully non-local form
When this happens the pseudopotential has GHOST states and should not be used.
The KB form is more efficiently computed than the original semi-local form.
By construction it behaves as the original form on the reference configuration … but … there is no guarantee that the reference configuration is the GS of the modified potential.
![Page 74: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/74.jpg)
ab initio Norm Conserving PseudoPotentials
Desired properties of a pseudopotential are
- Transferability (norm-conservation, small core radii, non-linear core correction, multi projectors)
- Softness (various optimization/smoothing strategies, large core radii)
For some elements it's easy to obtain “soft” Norm-ConservingPseudoPotentials.
For some elements it's instead very difficult!
Expecially for first row elements (very localized 2p orbitals) and 1st row transition metals (very localized 3d orbitals)
![Page 75: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/75.jpg)
Norm-Conserving PseudoPotentials basic literature
<1970 empirical PP. es: Cohen & Bergstresser, PRB 141, 789 (1966) 1979 Hamann, Schlueter & Chang, PRL 43, 1494 (1979), ab initio NCPP1982 Bachelet, Hamann, Schlueter, PRB 26, 4199 (1982), BHS PP table 1982 Louie, Froyen & Cohen, PRB 26, 1738 (1982), non-linear core corr.1982 Kleinman & Bylander, PRL 48, 1425 (1982), KB fully non local PP
1985 Vanderbilt, PRB 32, 8412 (1985), optimally smooth PP1990 Rappe,Rabe,Kaxiras,Joannopoulos, PRB 41, 1227 (1990), optm. PP1990 Bloechl, PRB 41, 5414 (1990), generalized separable PP1991 Troullier & Martins, PRB 43, 1993 (1991), efficient PP….
1990 Gonze, Kackell, Scheffler, PRB 41, 12264 (1990), Ghost states
1991 King-Smith, Payne, Lin, PRB 44, 13063 (1991), PP in real space
![Page 76: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/76.jpg)
Ultra Soft PseudoPotentials
In spite of the devoted effort NCPP’s are still “hard”and require a large plane-wave basis sets (Ecut > 70Ry) forfirst-row elements (in particular N, O, F) and for transitionmetals, in particular the 3d row: Cr, Mn, Fe, Co, Ni, …
Copper 3d orbital nodeless
RRKJ, PRB 41,1227 (1990)
![Page 77: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/77.jpg)
Ultra Soft PseudoPotentials
Even if just one atom is “hard”, a high cutoff is required.
UltraSoft (Vanderbilt) PseudoPotentials (USPP) are devised to overcome such a problem.
Oxygen 2p orbital nodeless
Vanderbilt, PRB 41, 7892 (1991)
![Page 78: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/78.jpg)
Ultra Soft PseudoPotentials
where the “augmentation charges” are
are projectors
are atomic states (not necessarily bound)
are pseudo-waves (coinciding with beyond some core radius)
![Page 79: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/79.jpg)
Ultra Soft PseudoPotentials
where
leading to a generalized eigenvalue problem
Orthogonality with USPP:
![Page 80: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/80.jpg)
Ultra Soft PseudoPotentials
where
There are additional terms in the density, in the energy, in the hamiltonianin the forces, ...
![Page 81: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/81.jpg)
Ultra Soft PseudoPotentials
Electronic states are orthonormal with a (configuration dependent) overlap matrix
There are additional terms in the density, in the energy, in the hamiltonianin the forces, ...
The “augmentation charges” typically require a larger cutoff for the charge density:
QE Input parameter: ecutrho (SYSTEM namelist)
Default value is ecutrho = 4 × ecutwfc (OK for NC PP)
For USPP a larger value ecutrho is often needed.
![Page 82: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/82.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
all-electron wave function
pseudo wave function
all-electron atomic partial waves
pseudo atomic partial waves
localized projectors on the atomic partial waves such that
It is always possible to express the AE wfc via augmentationof a smooth (pseudo) wfc using atomic reference states
an all-electron method !
where...
![Page 83: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/83.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
It is always possible to express the AE wfc via augmentationof a smooth (pseudo) wfc using atomic reference states
an all-electron method !
pictorially
's coincide outside the core region and we can truncate them
's and
The 's projectors are localized in the core region...
is a localized operator !
![Page 84: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/84.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
AE matrix elements of any operator can then be computed as
an all-electron method !
for local operators (kinetic energy, potential,...) one can show
if the expansion is complete
and normalization of wfc is computed with
![Page 85: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/85.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
an all-electron method !
AE results can be computed from the PS matrix elements augmented by KB-like contributions that can be computed from atomic AE and PS reference calculations.
![Page 86: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/86.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
an all-electron method !
The charge density is therefore
![Page 87: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/87.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
an all-electron method !
The charge density is therefore
but it is convenient to add/subtract a compensating charge so that the AE and PS atomic references have the same Multipole expansion
![Page 88: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/88.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
an all-electron method !
The charge density is therefore
![Page 89: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/89.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
an all-electron method !
…
The different energy contributions so become
![Page 90: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/90.jpg)
Projector Augmented Waves Bloechl, PRB 50, 17953 (1994)
an all-electron method !
Finally the KS eigenvalue problem is as for USPP
with
where
![Page 91: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/91.jpg)
Step 2 : initial guess for rho_in
![Page 92: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/92.jpg)
Initial choice of rho_in
Various possible choices, e.g.,:
● Superpositions of atomic densities.● Converged n(r) from a closely related calculation (e.g., one where ionic positions slightly different).● Approximate n(r) , e.g., from solving problem in a smaller/different basis.● Random numbers.
![Page 93: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/93.jpg)
Initial choice of rho_in
Various possible choices, e.g.,:
● Superpositions of atomic densities.● Converged n(r) from a closely related calculation (e.g., one where ionic positions slightly different).● Approximate n(r) , e.g., from solving problem in a smaller/different basis.● Random numbers.
Initial guess of wfc
QE input parameter startingwfc
'atomic' | 'atomic+random' | 'random' | 'file'
![Page 94: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/94.jpg)
Pseudopotentials in Quantum ESPRESSO
Go to http://www.quantum-espresso.org/
![Page 95: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/95.jpg)
Pseudopotentials for Quantum ESPRESSO Click on the element for which the PP is desired
![Page 96: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/96.jpg)
Pseudopotentials for Quantum ESPRESSO
Pseudopotential's name givesInformation about
-exchange correlation functional
-type of pseudopotential
![Page 97: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/97.jpg)
Atomic and V_ion info for QE
ATOMIC_SPECIESBa 137.327 Ba.pbe-nsp-van.UPFTi 47.867 Ti.pbe-sp-van_ak.UPFO 15.999 O.pbe-van_ak.UPF
QE input card ATOMIC_SPECIES example:
NOTEshould use the same XC functional for all pseudopentials.ecutwfc, ecutrho depend on type of pseudopotentials used (should test for system & property of interest).
![Page 98: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/98.jpg)
DFT Total Energy in an infinite periodic solid
![Page 99: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/99.jpg)
DFT Total Energy in an infinite periodic solid
![Page 100: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/100.jpg)
DFT Total Energy in an infinite periodic solid
![Page 101: Basic Band Structure Conceptsdegironc/ES/lectures/band...14 Bravais lattices: 7 crystal systems!! symmetry of the crystal goes beyond translational invariance => point group operations](https://reader030.vdocument.in/reader030/viewer/2022040617/5f2151da1178b179fa0b1ff6/html5/thumbnails/101.jpg)
THE END