1. 物質科学のための計算数理 II Numerical Analysis for Material Science II
9th: Density Functional Theory (2)
Dec. 7 (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) Standard DFT code K
Dec (Fri) Density functional theory K
Dec. 14 (Fri) Density functional theory K
Dec. 21 (Fri) Density functional theory K
Dec. 25 (Tue) (2nd)Report problem K (遠隔講義室)
Jan (Fri) Density functional theory 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.)
Brillouin-zone integral (Tetrahedron method)
Visualization (T)
Dec. 21 (Fri) Total Energy
Coulomb potential for periodic point charge (Ewald sum)
Dec. 25 (Tue) (2nd)Report problem K (遠隔講義室)
Jan (Fri) Density functional theory K

4. Today’s Schedule Kohn-Sham eq. in periodic system
Plane-wave representation
Iterative diagonalization
Matrix-vector product
Kinetic energy term
Potential term
Visualize

5. Kohn-Sham method last week 𝐸 𝜌 = 𝑑 3 𝑟𝜌 𝒓 𝑣 𝒓 + 𝐸 𝑢𝑛𝑖𝑣 𝜌
𝐸 𝜌 = 𝑑 3 𝑟𝜌 𝒓 𝑣 𝒓 + 𝐸 𝑢𝑛𝑖𝑣 𝜌
𝑣 𝒓 ≡ 𝐼=1 𝑁 atom 𝑍 𝐼 𝒓− 𝑹 𝐼
𝐸 𝑢𝑛𝑖𝑣 𝜌 = 𝑇 𝐾𝑆 𝜌 𝑑 3 𝑟 𝑑 3 𝑟 ′ 𝜌(𝒓)𝜌( 𝒓 ′ ) 𝒓− 𝒓 ′ + 𝐸 𝑋𝐶 𝜌
Kinetic energy of non-interacting system whose charge density is 𝜌(𝒓)
− 𝛁 𝑣 𝐾𝑆 [𝜌] 𝒓 𝜑 𝑛 𝒓 = 𝜀 𝑛 𝜑 𝑛 𝒓
𝜌 𝒓 =2 𝑛=1 𝑁/2 𝜑 𝑛 𝒓 2
𝑣 𝐾𝑆 [𝜌] 𝒓 =𝑣 𝒓 + 𝑑 3 𝑟 ′ 𝜌 𝒓 ′ 𝒓− 𝒓 ′ + 𝑣 𝑋𝐶 𝜌 𝒓
𝑣 𝑋𝐶 𝜌 𝒓 ≡ 𝛿 𝐸 𝑋𝐶 𝜌 𝛿𝜌 𝒓
Self-consistent field (SCF)

6. Kohn-Sham eq. for periodic system (1)
last week
𝑣 𝐾𝑆 𝒓 is periodic,
i.e., 𝑣 𝐾𝑆 𝒓+ 𝑛 1 𝒂 1 + 𝑛 2 𝒂 2 + 𝑛 3 𝒂 3 = 𝑣 𝐾𝑆 𝒓
− 𝛁 𝑣 𝐾𝑆 𝒓 𝜑 𝒓 =𝜀𝜑 𝒓
Unit lattice vectors (Not unique)
𝜌 𝒓 = 𝑛=1 𝑁× 𝑁 𝑐 𝜑 𝑛 𝒓 2
𝑁 electrons per unit cell
𝑁 𝑐 cells → ∞
Equation to solve in the whole region of bulk crystal
𝒂 1
𝒂 1
𝒂 2
Unit cell

7. Kohn-Sham eq. for periodic system (2)
last week
𝜑 𝒓 = 1 𝑁 𝐶 𝑒 𝑖𝒌⋅𝒓 𝑢 𝑛𝒌 (𝒓)
𝜑(𝒓) can be written as
Bloch’s theorem
𝑈.𝐶. 𝑑 3 𝑟𝜌 𝑟 =𝑁
− 𝛁+𝒌 𝑉 𝐾𝑆 𝒓 𝑢 𝑛𝒌 𝒓 =𝜀 𝑢 𝑛𝒌 𝒓
𝜌 𝒓 =2 1 𝑉 𝐵𝑍 𝐵𝑍 𝑑 3 𝑘 𝑛=1 ∞ 𝑢 𝑛𝒌 𝒓 2 𝜃( 𝜀 F − 𝜀 𝑛𝑘 )
Equation to solve only in the unit cell
𝜀 𝑛𝑘
Band structure
𝒂 1 𝒂 2 𝒂 𝒃 1 , 𝒃 2 , 𝒃 3 =2𝜋 𝐼
Unit reciprocal lattice vectors
𝜀 𝑛𝑘 and 𝑢 𝑛𝒌 𝒓 is
periodic with 𝒃 𝛼 in the k space. 𝒌
𝒃 1

8. How to solve Kohn-Sham eq. : Basis
last week
− 𝛁+𝑖𝒌 𝑣 𝐾𝑆 𝒓 𝑢 𝑛𝒌 𝒓 = 𝜀 𝑛𝑘 𝑢 𝑛𝒌 𝒓
𝑢 𝑛𝒌 𝒓 = 𝐺 𝑢 𝑛𝒌 𝑮 𝑒 𝑖𝑮⋅𝒓 𝑉 𝑢𝑐
𝑮 ′ 𝑮+𝒌 𝛿 𝑮 𝑮 ′ + 𝑣 𝐾𝑆 (𝑮− 𝑮 ′ ) 𝑢 𝑛𝒌 𝑮 ′ = 𝜀 𝑛𝑘 𝑢 𝑛𝒌 𝑮
𝑣 𝐾𝑆 𝑮 = 𝑢𝑐 𝑑 3 𝑟 𝑒 −𝑖𝑮⋅𝒓 𝑉 𝑢𝑐 𝑣 𝐾𝑆 (𝒓)

9. Unit in this lecture Input and output : Length : Angstrom
Energy : Electron volt
Useful to read by VESTA
Inside program code:
Length : Atomic unit (Bohr radius). 1 [Bohr] = [Å]
Energy : Atomic unit (Hartree). 1 [Eh] = [eV]
[Bohr-1]
𝑮 ′ 𝑮+𝒌 𝛿 𝑮 𝑮 ′ + 𝑣 𝐾𝑆 (𝑮− 𝑮 ′ ) 𝑢 𝑛𝒌 𝑮 = 𝜀 𝑛𝑘 𝑢 𝑛𝒌 𝑮
[Eh]
[Eh]

10. Fourier Transformation
𝑣 𝐾𝑆 𝑮 = 𝑢𝑐 𝑑 3 𝑟 𝑒 −𝑖𝑮⋅𝒓 𝑉 𝑢𝑐 𝑣 𝐾𝑆 (𝒓)
Discretize (considering one dimension)
𝑣 2𝜋 𝐿 𝑚 = 𝑛=1 𝑁 𝑃𝑊 1 𝑁 𝑃𝑊 𝑒 −𝑖 2𝜋 𝐿 𝑚×𝐿 𝑛 𝑁 𝑃𝑊 𝑣 𝐿 𝑛 𝑁 𝑃𝑊
𝑁 𝑃𝑊 → ∞ : Exact
𝑚=− 𝑁 𝑃𝑊 2 ,⋯,−1,0,1,⋯, 𝑁 𝑃𝑊 2
𝑁 𝑃𝑊 × 𝑁 𝑃𝑊 𝑉
𝑂 𝑁 𝑃𝑊 2 numerical cost
Fast Fourier Transformation (FFT) : 𝑂 𝑁 𝑃𝑊 ln 𝑁 𝑃𝑊 numerical cost
We do not detail in this lecture. We just use FFT numerical library.

11. Cutoff frequency 𝑢 𝑛𝒌 𝒓 = 𝑮 𝑮 2 2 < 𝐸 𝑐𝑢𝑡 𝑢 𝑛𝒌 𝑮 𝑒 𝑖𝑮⋅𝒓 𝑉 𝑢𝑐
𝑢 𝑛𝒌 𝒓 = 𝑮 𝑮 < 𝐸 𝑐𝑢𝑡 𝑢 𝑛𝒌 𝑮 𝑒 𝑖𝑮⋅𝒓 𝑉 𝑢𝑐
𝑢 2𝜋 𝐿 𝑁 𝑃𝑊 −𝑚 = 𝑛=1 𝑁 𝑃𝑊 1 𝑁 𝑃𝑊 𝑒 −𝑖2𝜋 𝑛 𝑁 𝑃𝑊 −𝑚 𝑁 𝑃𝑊 𝑉 𝐿 𝑛 𝑁 𝑃𝑊 = 𝑉 − 2𝜋 𝐿 𝑚

12. Very simple plane-wave DFT program
$ cd ~
$ git clone git://git.osdn.net/gitroot/educational-pwdft/pwdft.git
$ cd pwdft
$ cp make.inc.mac make.inc
$ make
$ cd sample/Al/
$ ../../src/pwdft.x < direct.in
&CONTROL
calculation = 'direct' /
&SYSTEM
nbnd = 10
nat = 1
ntyp = 1
ecutwfc =
ecutrho =
&ELECTRONS
CELL_PARAMETERS
ATOMIC_SPECIES
Al al.lda.lps
ATOMIC_POSITIONS Al
K_POINTS 1
Aluminum
Face centered cubic (fcc)
Metal
Almost free electron
keep
"ecutrho" ≧ 4*ecutwfc
Only perform first SCF step
𝜌 𝑟 = 𝑁 𝑉 𝑢𝑐
Initial density is
uniform

13.
lwork = -1
allocate(work(1))
call zheev('V', 'U', npw, ham, npw, eval_full, work, lwork, rwork, info)
lwork = nint(dble(work(1)))
deallocate(work)
allocate(work(lwork))
Workspace (memory size) query : Do not need to care.
Diagonalize Hamiltonian matrix "ham"
Computational cost : 𝑂 𝑁 𝑃𝑊 3
Memory size : 𝑂( 𝑁 𝑃𝑊 3 )
keep ecutrho ≧ 4*ecutwfc
Changing "ecutrho" and "ecutwfc"
and see "Kohn-Sham Time" in Standard output
Large numerical cost for large 𝑁 𝑃𝑊
We only need lower energy (occupied) band.

14. Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method
One of the iterative eigen solution methods
A. V. Knyazev, SIAM J. Sci. Compute. 23, 517 (2001).
山田進 他, 日本計算工学会論文集, (2006).
For computing 𝑁 𝑏 eigenvector, at each step
Construct subspace Hamiltonian (3 𝑁 𝑏 dim.)
Approx. eigenvector 𝒙 1 , 𝒙 2 ,⋯ 𝒙 𝑁 𝑏
𝒙 𝑖 † 𝒘 𝑖 † 𝒑 𝑖 † 𝐻 ( 𝒙 𝑖 , 𝒘 𝑖 ,{ 𝒑 𝑖 })
Residual vector 𝒘 𝑖 = 𝐻 𝒙 𝑖 − 𝜀 𝑖 𝒙 𝑖
CG vector 𝒑 1 , 𝒑 2 ,⋯ 𝒑 𝑁 𝑏
Diagonalize with direct method
Take lowest 𝑁 𝑏 vectors as approximate eigenvectors at next step
Loop until all 𝑟 𝑖 become smaller than the threshold
Computational cost : 𝑂 𝛼 𝑁 𝑏 3 +𝛽 𝑁 𝑏 𝑁 𝑃𝑊 ln 𝑁 𝑃𝑊 +𝛾 𝑁 𝑏 2 𝑁 𝑃𝑊
Memory : 𝑂 𝑁 𝑏 𝑁 𝑃𝑊

15. Algorithm of LOBPCG method lobpcg_main@lobpcg.F90
Initial guess 𝒙 𝑖 (Random or atomic)
𝒑 𝑖 =𝟎
𝑷 𝑖 =𝟎
𝑿 𝑖 = 𝐻 𝒙 𝑖
𝜀 𝑖 = 𝑿 𝑖 † 𝒙 𝑖
do iteration
𝒘 𝑖 = 𝑿 𝑖 − 𝜀 𝑖 𝒙 𝑖
All 𝒘 𝑖 are small enough ? → Exit
𝒘 𝑖 = 𝑃 𝒘 𝑖
𝒘 𝑖 = 𝒘 𝑖 / 𝒘 𝑖
𝑾 𝑖 = 𝐻 𝒘 𝑖
𝐻 𝑠𝑢𝑏 = 𝑾 𝑖 , 𝑿 𝑖 , 𝑷 𝑖 † 𝒘 𝑖 , 𝒙 𝑖 , 𝒑 𝑖
𝑂 𝑠𝑢𝑏 = 𝒘 𝑖 , 𝒙 𝑖 , 𝒑 𝑖 † 𝒘 𝑖 , 𝒙 𝑖 , 𝒑 𝑖
Solve 𝐻 𝑠𝑢𝑏 𝑥 𝑖,𝑠𝑢𝑏 = 𝜀 𝑖,𝑠𝑢𝑏 𝑥 𝑖,𝑠𝑢𝑏
Take lowest 𝑁 𝑏 : 𝑥 𝑖,𝑠𝑢𝑏 𝑙𝑜𝑤
𝒙 𝑖 = 𝒘 𝑖 , 𝒙 𝑖 , 𝒑 𝑖 𝑥 𝑖,𝑠𝑢𝑏 𝑙𝑜𝑤
𝒑 𝑖 = 𝒘 𝑖 , 𝟎 , 𝒑 𝑖 𝑥 𝑖,𝑠𝑢𝑏 𝑙𝑜𝑤
𝑿 𝑖 = 𝑾 𝑖 , 𝑿 𝑖 , 𝑷 𝑖 𝑥 𝑖,𝑠𝑢𝑏 𝑙𝑜𝑤
𝑷 𝑖 = 𝑾 𝑖 , 𝟎 , 𝑷 𝑖 𝑥 𝑖,𝑠𝑢𝑏 𝑙𝑜𝑤
𝒙 𝑖 = 𝒙 𝑖 / 𝒙 𝑖
𝑿 𝑖 = 𝑿 𝑖 / 𝑿 𝑖
𝒑 𝑖 = 𝒑 𝑖 / 𝒑 𝑖
𝑷 𝑖 = 𝑷 𝑖 / 𝑷 𝑖
end do iteration

16. Hamiltonian vector product h_psi@hamiltonian.F90
𝑾 𝑖 = 𝐻 𝒘 𝑖
H 𝑢 𝑮 = − 𝑮+𝒌 𝑣 𝐾𝑆 𝑢 𝑮
𝑣 𝐾𝑆 𝑢 𝑮 = 𝑢𝑐 𝑑 3 𝑟 𝑒 −𝑖𝑮⋅𝒓 𝑉 𝑢𝑐 𝑣 𝐾𝑆 𝒓 𝑢 𝒓
≈ 𝒓 𝑉 𝑢𝑐 𝑁 𝒓 𝑒 −𝑖𝑮⋅𝒓 𝑣 𝐾𝑆 𝒓 𝑢 𝒓
𝑢 𝑛𝒌 𝒓 = 𝑮 𝑢 𝑛𝒌 𝑮 𝑒 𝑖𝑮⋅𝒓 𝑉 𝑢𝑐
Done in subroutine hpsi in hamiltonian.F90

17. Running iterative method
$ cd ~/pwdft/sample/Al/
$ ../../src/pwdft.x < iterative.in
&CONTROL
calculation = 'iterative' /
&SYSTEM
nbnd = 10
nat = 1
ntyp = 1
ecutwfc =
ecutrho =
&ELECTRONS
CELL_PARAMETERS
ATOMIC_SPECIES
Al al.lda.lps
ATOMIC_POSITIONS Al
K_POINTS 1
See "Kohn-Sham Time" in
Standard output
keep ecutrho ≧ 4*ecutwfc

18. Report problem 1
(1) Compare the computational time of direct method and LOBPCG method by changing ecutrho and ecutwfc.
keep ecutrho ≧ 4*ecutwfc
Time [sec]
ecutwfc [Ry]
(2) Plot the cutoff-energy dependence of Kohn-Sham energy 𝜀 𝑛𝑘 for
𝑘=(0,0,0) and 𝑛=1, 2 (they are outputted to a file band.dat).

19. What is the command "git" Backup code Port program in any system
$ git clone
Backup code
Port program in any system
Merge modifications
Compare diff
Log each modification with comment
Etc.

20. Considered case 1 ・ Where is the latest source code ?
System C
PC cluster
Laptop PC
Labo PC
System B
Home PC
・ Where is the latest source code ?
・ We need to update source code easily.

21. Considered case 2 ・ We need to trace when was it broken.
Program
System A
System B
Speed up with symmetry
Program
System A
OpenMP
Program
System A
OpenMP+MPI
System A
System B
Program
・ We need to trace when was it broken.
・ We need to be available to return the stable (no error) code.

22. Version control $ git pull update modification $ git commit $ git pull
Logging :
Who
When
What
modified
Program (commit 1)
Program (commit 2)
Repository
server
Program (commit 3) ⋮
Program (commit 10)
Latest
$ git pull
update modification
Program (commit 11)
$ git commit
$ git push
$ git pull
Update latest version
local repository
local repository
Modify

23. When modified simultaneously
Program (commit 1)
Program (commit 2)
Repository
server
Program(commit 3)
$ git push ⋮
Program (commit 10)
Program (commit 11)
Latest
Program (commit 12)
$ git pull
commit 10+
commit 11+
local repo
local repo
$ git push
commit

24. Tutorial $ cd ~/pwdft/ $ git branch -a
$ git config --global user.name "user-name"
$ git config --global user.
$ git checkout -b user-name
$ echo "Hello, I am user-name." > user-name.txt
$ git add user-name.txt
$ git commit (vi is used)
$ git push --set-upstream origin user-name

25. Example of branches V0.2 V1.0 V0.1 Master Hotfix (Bug fix) Release
Develop
Feature (new features)
"A successful branch in git"

26. Today's summary Plane-wave representation of Kohn-Sham eq.
Direct method and LOBPCG method
Hamiltonian-vector product with FFT
Git