1. 物質科学のための計算数理 II Numerical Analysis for Material Science II
11th: Density Functional Theory (4)
Dec. 21 (Fri)
Lecturer: Mitsuaki Kawamura (河村光晶)

2. Schedule (This semester W1, W2)
Sep. 28 (Fri) Guidance Y
Oct (Fri) Monte Carlo method O
Oct. 12 (Fri) Monte Carlo method O
Oct. 19 (Fri) Monte Carlo method O
Oct. 26 (Fri) Exact diagonalization Y
Nov (Fri) Exact diagonalization Y
Nov (Fri) Molecular dynamics O (1st report problem will be announced.)
Nov. 30 (Fri) Density functional theory K
Dec (Fri) Density functional theory K
Dec. 14 (Fri) Density functional theory K
Dec. 21 (Fri) Density functional theory K
Dec. 25 (Tue) Density functional theory K (遠隔講義室)
Jan (Fri) (2nd)Report problem K
※ Lecturers: Y … Yamaji, K … Kawamura, O… Ohgoe

3. Schedule in this section (DFT)
Nov. 30 (Fri) Standard DFT code
First-principles calculation and Density functional theory (Lecture)
One-body Schrödinger eq. for periodic system and Bloch theorem (L)
Numerical solution of Kohn-Sham (one-body Schrödinger) eq. (L)
Hands-on DFT code (Tutorial)
Version control system : Git (T)
Dec (Fri) Kohn-Sham eq.
Plane-wave basis and Pseudopotentials (L)
Iterative eigenvalue solution method (L & T)
Dec. 14 (Fri) Self-Consistent loop
Hartree potential (Poisson eq.), Atomic potential, XC potential
Update (Broyden's method)
Visualization of grid data (T)
Dec. 21 (Fri) Total Energy
Total energy
Brillouin-zone integral (Tetrahedron method)
Coulomb potential for periodic point charge (Ewald sum)
Dec. 25 (Tue) Advanced subjects for productive calculation (遠隔講義室)
Generalized gradient correction
Non-local pseudopotentials (Norm-conserving, ultrasoft, PAW)
Procedure
Jan (Fri) (2nd) Report problem, Question time

4. Today’s Schedule Total energy Hartree term Exchange correlation term
Brillouin-zone integration
Basics
Smearing method
Tetrahedron method and its improvement
Fermi surface plot

5. Total energy functional for DFT
𝐸 𝑡𝑜𝑡 =𝐸 𝜌 𝑅, 𝑅 ′ 𝑁 𝑐 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 1− 𝛿 𝑅 𝑅 ′ 𝛿 𝝉 𝝉 ′ 𝝉+𝑹− 𝝉 ′ − 𝑹 ′
𝐸 𝜌 = 𝑑 3 𝑟𝜌 𝒓 𝑣 𝒓 + 𝐸 𝑢𝑛𝑖𝑣 𝜌
𝐸 𝑢𝑛𝑖𝑣 𝜌 = 𝑇 𝐾𝑆 𝜌 𝑑 3 𝑟 𝑑 3 𝑟 ′ 𝜌(𝒓)𝜌( 𝒓 ′ ) 𝒓− 𝒓 ′ + 𝐸 𝑋𝐶 𝜌
𝑇 𝐾𝑆 𝜌 = 𝑁 𝑐 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝑛 𝑑 3 𝑟 𝜑 𝑛𝒌 ∗ 𝑟 − 𝛁 𝜑 𝑛𝒌 𝑟
𝜑 𝑛𝒌 𝒓 = 1 𝑁 𝐶 𝑒 𝑖𝒌⋅𝒓 𝑢 𝑛𝒌 (𝒓)
𝑢 𝑛𝒌 𝒓 = 𝐺 𝑢 𝑛𝒌 𝑮 𝑒 𝑖𝑮⋅𝒓 𝑉 𝑢𝑐
= 𝑁 𝑐 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝑛 𝑮 𝒌+𝑮 𝑢 𝑛𝒌 𝑮 2
Local density approximation
𝐸 𝑋𝐶 𝜌 = 𝑑 3 𝑟𝜌 𝑟 𝜀 𝑋𝐶 𝜌 𝑟 = 𝑁 𝑐 𝑢𝑐 𝑑 3 𝑟𝑟𝜌 𝑟 𝜀 𝑋𝐶 𝜌 𝑟

6. Tutorial for total energy
$ cd ~/pwdft/
$ git checkout master
$ git pull
$ make clean; make
$ cd sample/Al/
$ ../../src/pwdft.x < scf.in
Energy per u.c. [eV]
Kinetic energy :
Hartree energy :
rho*V :
Ewald eta [Bohr^-1] :
Ewald grid (G) :
Ewald grid (R) :
Eward energy :
XC energy :
Total energy :
$ bash vol.sh

7. Hartree and atomic potential energy.F90
𝑑 3 𝑟 𝑑 3 𝑟 ′ 𝜌(𝒓)𝜌( 𝒓 ′ ) 𝒓− 𝒓 ′ = 1 2 𝑮, 𝑮 ′ 𝜌 𝑮 𝜌 𝑮 ′ 𝑑 3 𝑟 𝑑 3 𝑟 ′ 𝑒 𝑖𝑮⋅𝒓 𝑒 𝑖 𝑮 ′ ⋅ 𝒓 ′ 𝒓− 𝒓 ′
= 1 2 𝑮, 𝑮 ′ 𝜌 𝑮 𝜌 𝑮 ′ 𝑑 3 𝑟 ′ 𝑒 𝑖 𝑮+ 𝑮 ′ ⋅ 𝒓 ′ 𝑑 3 𝑟 𝑒 𝑖𝑮⋅ 𝒓− 𝒓 ′ 𝒓− 𝒓 ′
= 𝑁 𝑐 𝑉 𝑢𝑐 2 𝑮 𝜌 𝑮 𝜌 −𝑮 4𝜋 𝑮 𝟐 = 𝑁 𝑐 𝑉 𝑢𝑐 2 𝑮 𝜌 𝑮 𝜋 𝑮 𝟐
lim 𝑮→0 𝑁 𝑐 𝑉 𝑢𝑐 2 4𝜋 𝑮 𝟐 𝑁 𝑉 𝑢𝑐 2
Atomic potential term
𝑑 3 𝑟𝜌 𝒓 𝑣 𝒓 = 𝑮, 𝑮 ′ 𝜌 𝑮 𝑣 𝑮 ′ 𝑑 3 𝑟 𝑒 𝑖𝑮⋅𝒓 𝑒 𝑖 𝑮 ′ ⋅ 𝒓 ′ = 𝑁 𝑐 𝑉 𝑢𝑐 𝑮, 𝑮 ′ 𝜌 𝑮 𝑣 −𝑮
− lim 𝑮→0 𝑁 𝑐 𝑉 𝑢𝑐 4𝜋 𝑮 𝟐 𝑁 𝑉 𝑢𝑐 2
Compensating
divergent term

8. Ion-ion Coulomb repulsion
1 2 𝑅, 𝑅 ′ 𝑁 𝑐 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 1− 𝛿 𝑅 𝑅 ′ 𝛿 𝝉 𝝉 ′ 𝝉+𝑹− 𝝉 ′ − 𝑹 ′ = 𝑁 𝑐 2 𝑅 𝑁 𝑐 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 1− 𝛿 𝑅𝟎 𝛿 𝝉 𝝉 ′ 𝝉− 𝝉 ′ +𝑹
= 𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 𝑅 𝑁 𝑐 1− 𝛿 𝑅𝟎 𝛿 𝝉 𝝉 ′ 𝝉− 𝝉 ′ +𝑹
𝑑 3 𝑟 𝑒 −𝑖𝑮⋅𝒓 𝒓
𝑅 𝑁 𝑐 1 𝒓+𝑹 = 𝐺 𝑒 𝑖𝑮⋅𝒓 𝑢𝑐 𝑑 3 𝑟 ′ 𝑉 𝑢𝑐 𝑒 −𝑖𝑮⋅ 𝒓 ′ 𝑅 𝑁 𝑐 𝒓 ′ +𝑹 = 𝐺 𝑒 𝑖𝑮⋅𝒓 4𝜋 𝑉 𝑢𝑐 𝑮 2
= 𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 𝐺 𝑒 𝑖𝑮⋅ 𝝉− 𝝉 ′ 4𝜋 𝑉 𝑢𝑐 𝑮 2 − 𝑅 𝑁 𝑐 𝛿 𝑅𝟎 𝛿 𝝉 𝝉 ′ 𝝉− 𝝉 ′ +𝑹
𝑅 1 𝑹
𝐺 𝑒 𝑖𝑮⋅𝒓 𝑮 2
lim 𝑮→0 𝑁 𝑐 𝑉 𝑢𝑐 2 4𝜋 𝑮 𝟐 𝑁 𝑉 𝑢𝑐 2
Slow decay in sum

9. Ewald's method (1) 1 𝒓+𝑹 = 2 𝜋 0 ∞ 𝑑𝑥 𝑒 − 𝒓+𝑹 2 𝑥 2
1 𝒓+𝑹 = 2 𝜋 0 ∞ 𝑑𝑥 𝑒 − 𝒓+𝑹 2 𝑥 2
𝑹 𝑁 𝑐 𝑒 − 𝒓+𝑹 2 𝑥 2 = 𝑮 𝑒 𝑖𝑮⋅𝒓 𝑢𝑐 𝑑 3 𝑟 ′ 𝑉 𝑢𝑐 𝑒 −𝑖𝑮⋅ 𝒓 ′ 𝑹 𝑁 𝑐 𝑒 − 𝒓 ′ +𝑹 2 𝑥 2 = 𝑮 𝑒 𝑖𝑮⋅𝒓 𝜋 𝑒 − 𝑮 𝑥 𝑉 𝑢𝑐 𝑥 3
𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 𝑅 𝑁 𝑐 1− 𝛿 𝑅𝟎 𝛿 𝝉 𝝉 ′ 𝝉− 𝝉 ′ +𝑹
= 𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 𝑅 𝑁 𝑐 𝜋 𝜂 ∞ 𝑑𝑥 𝑒 − 𝝉− 𝝉 ′ +𝑹 2 𝑥 2 − 𝛿 𝑅𝟎 𝛿 𝝉 𝝉 ′ 0 ∞ 𝑑𝑥
+ 𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 𝑮 𝑒 𝑖𝑮⋅ 𝝉− 𝝉 ′ 2𝜋 𝑉 𝑢𝑐 0 𝜂 𝑑𝑥 𝑥 3 𝑒 − 𝑮 𝑥 2
2𝑒 − 𝑮 𝜂 𝑮 2
cos 𝑮⋅ 𝝉− 𝝉 ′

10. Ewald's method (2) Complementary error function
= 𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑠 𝑍 𝝉 𝑍 𝝉 ′ 𝑅 𝑁 𝑐 1− 𝛿 𝑅𝟎 𝛿 𝝉 𝝉 ′ 𝜋 𝜂 ∞ 𝑑𝑥 𝑒 − 𝝉− 𝝉 ′ +𝑹 2 𝑥 2
− 𝑁 𝑐 2 𝝉 𝑁 𝑍 𝝉 𝜋 0 𝜂 𝑑𝑥
− 𝑁 𝑐 𝝉 𝑁 𝜂 𝑍 𝝉 2 𝜋
erfc 𝝉− 𝝉 ′ +𝑹 𝜂 𝝉− 𝝉 ′ +𝑹
+ 𝑁 𝑐 2 𝝉, 𝝉 ′ 𝑁 𝑍 𝝉 𝑍 𝝉 ′ 𝑮≠𝟎 cos 𝑮⋅ 𝝉− 𝝉 ′ 𝜋𝑒 − 𝑮 𝜂 𝑉 𝑢𝑐 𝑮 − 𝑁 𝑐 2 𝜋 𝑁 2 𝑉 𝑢𝑐 𝜂 lim 𝑮→0 𝑁 𝑐 𝑉 𝑢𝑐 2 4𝜋 𝑮 𝟐 𝑁 𝑉 𝑢𝑐 2

11. Optimize lattice constant
($ bash vol.sh)
$ grep "Total energy" scf*.out | \
sed -e 's/scf//g' -e 's/.out://g' > ene.dat
$ gnuplot
gnuplot> set xlabel "Lattice constant [Ang]"
gnuplot> set ylabel "Total energy [eV]"
gnuplot> plot "ene.dat" u 1:5 w lp

12. Brillouin-zone integration
𝜌 𝑟 = 𝒌 𝑛 𝜑 𝑛𝒌 𝑟 2 𝜃 𝜀 𝐹 − 𝜀 𝑛𝑘
Born-von Karman's boundary condition :
𝜑 𝑛𝒌 𝑟+𝐿 = 𝜑 𝑛𝒌 𝑟 :𝐿→∞
2𝜋 𝐿 3 = 2𝜋 3 𝑁 𝑐 𝑉 𝑢𝑐
The size of each state in the reciprocal space :
𝒌 ⋯ = 𝐵𝑍 𝑑 3 𝑘 2𝜋 3 /( 𝑁 𝑐 𝑉 𝑢𝑐 ) ⋯ = 𝑁 𝑐 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 ⋯
2𝜋 3 𝑉 𝑢𝑐 ≡ 𝑉 𝐵𝑍 = 𝒃 1 × 𝒃 2 ⋅ 𝒃 3
𝑉 𝑢𝑐 = 𝒂 1 × 𝒂 2 ⋅ 𝒂 3
𝜌 𝑟 = 𝑁 𝑐 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝑛 𝜑 𝑛𝒌 𝑟 2 𝜃 𝜀 𝐹 − 𝜀 𝑛𝑘 = 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝑛 𝑢 𝑛𝒌 𝑟 2 𝜃 𝜀 𝐹 − 𝜀 𝑛𝑘

13. 𝒌-integration with 𝜃 𝜀 𝑘 or 𝛿 𝜀 𝑘
Charge density
Density of states
𝜌 𝑟 = 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝑛 𝑢 𝑛𝒌 𝑟 2 𝜃 𝜀 𝐹 − 𝜀 𝑛𝑘
𝐷 𝜀 = 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝑛 𝛿 𝜀− 𝜀 𝑛𝑘
𝑁 𝒌
Using 𝜃( 𝜀 𝐹 − 𝜀 𝑛𝑘 ) as is
Too many k points are required.
Broadening
M. Methfessel, A. T. Paxton, Phys. Rev. B 40, 3616 (1989).
Replace 𝜃( 𝜀 𝐹 − 𝜀 𝑛𝑘 ) with a smeared function such as the error function.
Tetrahedron
O. Jepsen and O. K. Andersen, Solid State Commun. 9, 1763 (1971).

14. Why we need to reduce 𝑘-points?
E.g. 1 DFT calculation of crystals with plane wave and pseudopotencial.
𝑣 𝐾𝑆 𝒓 𝑢 𝑛𝒌 𝒓 (for all orbitals) O 𝑁 𝒌 × 𝑁 𝑏 × 𝑁 𝑃𝑊 log 𝑁 𝑃𝑊
Subspace diagonalization O 𝑁 𝒌 × 𝑁 𝑏 3
Update 𝑢 𝑛𝒌 𝑮 O 𝑁 𝒌 × 𝑁 𝑏 2 × 𝑁 𝑃𝑊
E.g. 2 Hartree-Fock, hybrid DFT, TC, GW
𝑣 𝐹𝑜𝑐𝑘 𝑢 𝑛𝒌 𝒓 (for all orbitals) O 𝑁 𝒌 2 × 𝑁 𝑏 2 × 𝑁 𝑃𝑊 log 𝑁 𝑃𝑊
※ 𝑁 𝑏 ～ The number of electrons
The reduction of k-points without loss of accuracy is the best way for efficient calculation.

15. Broadening method We can calculate only on discrete k-points. 𝜀 𝒌
Replace 𝛿(𝑥) with smeared function such as (Hermite) Gaussian.
𝛿 𝑥 = 𝑛=0 ∞ −1 𝑛 𝑛!4 𝑛 𝜋 𝐻 2𝑛 𝑥 𝜎 𝑒 − 𝑥 2 / 𝜎 2
M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989).
This method becomes worse when smeared function not cover all energy range. 𝜎 𝒌
True band structure

16. Interpolation method libtetrabz_occ_main@libtetrabz/libtetrabz_occ_mod
Interpolation method
𝜀 1
O. Jepsen and O. K. Andersen, Solid State Commun. 9, 1763 (1971)
𝜀 4
𝑉 𝑇
𝑉 𝑇 = 𝑉 𝐵𝑍 6 𝑁 𝑘
𝜀 2
𝜀 3
When we use linear interpolation, the next is satisfied.
If 𝜀 1 ≤𝜀≤ 𝜀 2 ≤ 𝜀 3 ≤ 𝜀 4 ,
𝑇 𝜃 𝜀− 𝜀 𝑘 = 𝑉 𝜀 𝑘 ≤𝜀 ≈ 𝑉 𝑇 𝜀− 𝜀 1 𝜀 2 − 𝜀 1 𝜀− 𝜀 1 𝜀 3 − 𝜀 1 𝜀− 𝜀 1 𝜀 4 − 𝜀 1
𝑇 𝛿 𝜀− 𝜀 𝑘 = 𝜕 𝜕𝜀 𝑇 𝜃 𝜀− 𝜀 𝑘
These expressions are not applicable for higher order (2nd , 3rd ・・・) interpolation !
If 𝜀 1 ≤ 𝜀 2 ≤𝜀≤ 𝜀 3 ≤ 𝜀 4 , ⋯
If 𝜀 1 ≤ 𝜀 2 ≤ 𝜀 3 ≤𝜀≤ 𝜀 4 , ⋯
If 𝜀 1 ≤ 𝜀 2 ≤ 𝜀 3 ≤ 𝜀 4 ≤𝜀, ⋯

17. Systematic error in tetrahedron method
(Traditional) tetrahedron method linearly interpolate the function of k (such as wave function, KS energy, etc. ). T T
If the sign of curvature of these function is always negative, we overestimete.
Not Interpolating
But Fitting

18. Optimized tetrahedron method
M. Kawamura et al., Phys. Rev. B 89, (2014).
Optimized tetrahedron method
Conventional tetrahedron method T T 𝒌 𝒌
Higher order interpolation of 𝜀 𝑘 , etc.
Fit this function into a linear function in the tetrahedron
Tetrahedron method with fitted linear function
Linear interpolation of 𝜀 𝑘 , etc.
Tetrahedron method with interpolated linear function

19. Library of tetrahedron method
LibTetraBZ
MPI + OpenMP (Thread safe)
Fortran/C/C++
𝑛, 𝑛 ′ 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝛿 𝜀 𝐹 − 𝜀 𝑛𝒌 𝛿 𝜀 𝐹 − 𝜀 𝑛 ′ 𝒌 ′ 𝑋 𝑛 𝑛 ′ 𝒌
𝑛 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝛿 𝜀− 𝜀 𝑛𝒌 𝑋 𝑛𝒌
𝑛, 𝑛 ′ 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀 𝑛 ′ 𝒌 ′ − 𝜀 𝐹 𝜃 𝜀 𝐹 − 𝜀 𝑛𝒌 𝑋 𝑛 𝑛 ′ 𝒌
𝑛 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀− 𝜀 𝑛𝒌 𝑋 𝑛𝒌
𝑛, 𝑛 ′ 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀 𝑛 ′ 𝒌 ′ − 𝜀 𝐹 𝜃 𝜀 𝐹 − 𝜀 𝑛𝒌 𝜀 𝑛 ′ 𝒌 ′ − 𝜀 𝑛𝒌 𝑋 𝑛 𝑛 ′ 𝒌
𝑛, 𝑛 ′ 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀 𝐹 − 𝜀 𝑛 ′ 𝒌 ′ 𝜃 𝜀 𝐹 − 𝜀 𝑛𝒌 𝛿 𝜀 𝑛 ′ 𝒌 ′ − 𝜀 𝑛𝒌 +𝜔 𝑋 𝑛 𝑛 ′ 𝒌 (𝜔)
𝑛, 𝑛 ′ 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀 𝐹 − 𝜀 𝑛 ′ 𝒌 ′ 𝜃 𝜀 𝐹 − 𝜀 𝑛𝒌 𝜀 𝑛 ′ 𝒌 ′ − 𝜀 𝑛𝒌 +𝑖𝜔 𝑋 𝑛 𝑛 ′ 𝒌 (𝜔)

20. LibTetraBZ 𝜀 𝑘 =− cos 𝑘 𝑥 − cos 𝑘 𝑦 − cos 𝑘 𝑧 $ cd ~/
$ git clone git://git.osdn.net/gitroot/libtetrabz/libtetrabz.git
$ cd libtetrabz
$ touch Makefile.in aclocal.m4 configure
$ ./configure; make
$ cd example
$ ./dos.x
Which tetrahedron method ?(1 = Linear, 2 = Optimized): 2⏎
k-point mesh ?: 10⏎
$ gnuplot
gnuplot> set xlabel "Energy"
gnuplot> set ylabel "DOS"
gnuplot> plot "dos.dat" w l
𝜀 𝑘 =− cos 𝑘 𝑥 − cos 𝑘 𝑦 − cos 𝑘 𝑧

21. Fermi energy with bisection method libtetrabz_fermieng@libtetrabz/src/libtetrabz_occ_mod.F90
𝑁= 𝑛 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀 𝐹 − 𝜀 𝑛𝒌
𝜀 𝑢𝑝 = max { 𝜀 𝑛𝑘 }
𝜀 𝑙𝑜𝑤 = min { 𝜀 𝑛𝑘 }
do iteration
𝜀 𝐹 = 𝜀 𝑢𝑝 + 𝜀 𝑙𝑜𝑤 /2
𝑁 = 𝑛 𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝜃 𝜀 𝐹 − 𝜀 𝑛𝒌
if 𝑁 <𝑁 is small → Finish
else if 𝑁 <𝑁
𝜀 𝑙𝑜𝑤 = 𝜀 𝐹
else if 𝑁 >𝑁
𝜀 𝑢𝑝 = 𝜀 𝐹
end if
end do iteration

22. Iso surface with tetrahedron method
FermiSurfer
Input file format

23. Report problem 3
(1) Modify dos.F90 or dos_c.c to compute DOS with Gaussian broadening method.
𝐵𝑍 𝑑 3 𝑘 𝑉 𝐵𝑍 𝛿 𝜀− 𝜀 𝒌 ≈ 1 𝑁 𝑘 𝑘 𝑁 𝑘 1 𝜎 𝜋 𝑒 − 𝜀− 𝜀 𝒌 𝜎 2
Plot DOS at each 𝑁 𝑘 and 𝜎, then compare with tetrahedron methods (Linear and optimized)
Note:
Replace
call libtetrabz_fermieng(ltetra,bvec,nb,nge,eig,ngw,wght,ef,nelec) to
ef = 0.0d0
(2) Output 𝜀 𝒌 and 𝛁 𝒌 𝜀 𝒌 in the FermiSurfer format (vf.frmsf).
Modified code should be submitted as a diff file generated by
$ cd ~/libtetrabz/
$ git add example
$ git commit
$ git show > diff

24. Today's summary Total energy
Hartree, atomic, XC, kinetic, Ewald energy
Coulomb repulsion of periodically aligned ions is computed efficiently with Ewald's method.
Brillouin zone integration
Tetrahedron method
Optimized tetrahedron method
Library
Fermi surface