1. Computational Solid State Physics 計算物性学特論 第６回
6. Pseudopotential

2. Potential energy in crystals
:periodic potential
a,b,c: primitive vectors of the crystal
n,l,m: integers
Fourier transform of the periodic potential energy
G: reciprocal lattice vectors

3. Summation over ionic potentials
Zj: atomic number
:position of j-th atom in (n,l,m) unit cell

4. Bragg reflection
: position of the j-th atom in a unit cell
Assume all the atoms in a unit cell are the same kind.
:structure factor The Bragg reflection disappears when SG vanishes.

5. Valence states
We are interested in behavior of valence electrons, since it determines main electronic properties of solids.
Valence states must be orthogonal to core states.
Core states are localized near atoms in crystals and they are described well by the tight-binding approximation.
Which kinds of base set is appropriate to describe the valence state?

6. Orthogonalized Plane Wave (OPW)
: core Bloch function

7. Core Bloch function
・Tight-binding approximation

8. Inner product of OPW

9. Expansion of valence state by OPW
:Extra term due to OPW base set
orthogonalization of valence Bloch functions to core functions

10. Pseudo-potential: OPW method
Fc(r’)
generalized pseudo-potential

11. Generalized pseudopotential
:pseudo wave function
:real wave function

12. Empty core model
Core region
completeness

13. Empty core pseudopotential
(r<rc)
(r>rc)
Ω: volume of a unit cell

14. Screening effect by free electrons
dielectric susceptibility for metals
n: free electron concentration εF: Fermi energy

15. Screening effect by free electrons
・screening length in metals
・Debye screening length
　in semiconductors

16. Empty core pseudopotential and screened empty core pseudopotential

17. Brillouin zone for fcc lattice

18. Pseudopotential for Al

19. Energy band structure of metals

20. Merits of pseudopotential
The valence states become orthogonal to the core states.
The singularity of the Coulomb potential disappears for pseudopotential.
Pseudopotential changes smoothly and the Fourier transform approaches to zero more rapidly at large wave vectors.

21. The first-principles norm-conserving pseudopotential (1)
: Norm conservation
First order energy dependence of the scattering
logarithmic derivative

22. The first-principle norm- conserving pseudopotential (2)
: spherical harmonics

23. The first-principle norm conserving pseudo-potential(3)

24. The first-principles norm-conserving pseudopotential (4)
Pseudo wave function has no nodes, while the true wave function has nodes within core region.
Pseudo wave function coincides with the true wave function beyond core region.
Pseudo wave function has the same energy eigenvalue and the same first energy derivative of the logarithmic derivative as the true wave function.

25. Flow chart describing the construction of an ionic pseudopotential

26. First-principles pseudopotential and pseudo wave function
Pseudopotential of Au

27. Pseudopotential of Si

28. Pseudo wave function of Si(1)

29. Pseudo wave function of Si(2)

30. Siの各種定数 計算値 実験値 計算値と実験値のずれ 格子定数 5.4515[Å] 5.429 [Å] +0.42% 凝集エネルギー
5.3495[eV/atom]
4.63[eV/atom] %
体積弾性率
0.925[Mbar]
0.99 [Mbar]
-7.1%
エネルギーギャップ
0.665[eV]
1.12[eV] %
凝集エネルギー=Total Energy－2×EXC(非線形内殻補正による分)－2×ATOM Energy－(ゼロ点振動エネルギー)
Total Energy = E+01 [HR]
EXC = E+00 [HR]
ATOM TOTAL = [HT]
Siのゼロ点振動エネルギー = [eV]

31. Lattice constant vs. total energy of Si

32. Energy band of Si

33. Problems 6
Calculate Fourier transform of Coulomb potential and obtain inverse Fourier transform of the screened Coulomb potential.
Calculate both the Bloch functions and the energies of the first and second bands of Al crystal at X point in the Brillouin zone, considering the Bragg reflection for free electrons.
Calculate the structure factor SG for silicon and show which Bragg reflections disappear.