1/20

物質科学のための計算数理 II
Numerical Analysis for Material Science II
12th: Density Functional Theory (5)Dec. 25 (Tue)

Lecturer: Mitsuaki Kawamura (河村光晶)

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Schedule (This semester W1, W2)

1. Sep. 28 (Fri) Guidance Y
2. Oct. 5 (Fri) Monte Carlo method O
3. Oct. 12 (Fri) Monte Carlo method O
4. Oct. 19 (Fri) Monte Carlo method O
5. Oct. 26 (Fri) Exact diagonalization Y
6. Nov. 2 (Fri) Exact diagonalization Y
7. Nov. 9 (Fri) Molecular dynamics O (1st report problem will be announced.)
8. Nov. 30 (Fri) Density functional theory K
9. Dec. 7 (Fri) Density functional theory K
10. Dec. 14 (Fri) Density functional theory K
11. Dec. 21 (Fri) Density functional theory K
12. Dec. 25 (Tue) Density functional theory K (遠隔講義室)
13. Jan. 11 (Fri) (2nd)Report problem K※ Lecturers: Y … Yamaji, K … Kawamura, O… Ohgoe

3/20
1.

2.

3.

4.

5.

6.

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. 7 (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)
• Magnetism
Jan. 11 (Fri) Parctice4/20

Today’s Schedule

Spin density functional theory
Collinear magnetism
Non-collinear magnetism
Beyond local density approximation
Generalized gradient approximation
Non-local pseudopotential
Norm conserving pseudopotential
Ultrasoft pseudopotential
Projector augmented wave (PAW)
Pseudopotential library5/20

Density functional theory innon-spin polarized case𝐸 𝜌 = න 𝑑 3 𝑟𝜌 𝒓 𝑣 𝒓 + 𝐸𝑢𝑛𝑖𝑣 𝜌′1𝜌(𝒓)𝜌(𝒓)33 ′𝐸𝑢𝑛𝑖𝑣 𝜌 = 𝑇𝐾𝑆 𝜌 + ඵ 𝑑 𝑟𝑑 𝑟
+ 𝐸𝑋𝐶 𝜌′2𝒓−𝒓

𝛁2−+ 𝑣𝐾𝑆 [𝜌] 𝒓
2𝜑𝑛 𝒓 = 𝜀𝑛 𝜑𝑛 𝒓

𝑁/2

𝜌 𝒓 = 2 ෍ 𝜑𝑛 𝒓
𝑛=12

𝛿𝐸𝑋𝐶 𝜌𝑣𝑋𝐶 𝜌 𝒓 ≡𝛿𝜌 𝒓

6/20

Spin density functional theory𝐸 𝜌↑ , 𝜌↓ = න 𝑑3 𝑟𝜌 𝒓 𝑣 𝒓 + 𝐸𝑢𝑛𝑖𝑣 𝜌↑ , 𝜌↓′1𝜌(𝒓)𝜌(𝒓)33 ′𝐸𝑢𝑛𝑖𝑣 𝜌↑ , 𝜌↓ = 𝑇𝐾𝑆 𝜌 + ඵ 𝑑 𝑟𝑑 𝑟+ 𝐸𝑋𝐶 𝜌↑ , 𝜌↓′2𝒓−𝒓

𝛁2−+ 𝑣𝐾𝑆,𝜎 [𝜌↑ , 𝜌↓ ] 𝒓
2𝜑𝑛𝜎 𝒓 = 𝜀𝑛𝜎 𝜑𝑛𝜎 𝒓

𝑣𝑋𝐶,𝜎 𝜌↑ , 𝜌↓𝛿𝐸𝑋𝐶 𝜌↑ , 𝜌↓𝒓 ≡𝛿𝜌𝜎 𝒓∞

𝜌𝜎 𝒓 = ෍ 𝜑𝑛𝜎 𝒓
𝑛=12𝜃 𝜀𝐹 − 𝜀𝑛𝜎𝜌(𝒓) = ෍ 𝜌𝜎 (𝒓)
𝜎=↑,↓

7/20

Exchange correlation energy

𝐸𝑋𝐶 𝜌↑ , 𝜌↓𝐸𝑋 2𝜌↑ + 𝐸𝑋 2𝜌↓
𝐸𝑋 𝜌↑ , 𝜌↓ =2
The "exchange" effect acts only the same spin.
Local Spin Density Approximation (LSDA)
𝐸𝑋𝐶 𝜌↑ , 𝜌↓ ≈ න 𝑑3 𝑟෍ 𝜌𝜎 𝒓 𝜀𝑋 2𝜌𝜎 𝒓+ 𝜌 𝒓 𝜀𝐶 𝜌 𝒓 , 𝜁 𝒓𝜎=↑,↓𝜁 = 𝜌↑ − 𝜌↓ /𝜌𝜀𝐶 𝜌, 𝜁 = 𝜀𝐶 𝜌, 0 1 − 𝑓 𝜁+ 𝜀𝐶 𝜌, 1 𝑓 𝜁

1 1 + 𝜁 4/3 + 1 − 𝜁 4/3 − 2𝑓 𝜁 ≈221/3 − 1

8/20Collinear and non-collinear spin𝜌↑ (𝒓), 𝜌↓ (𝒓)𝜌(𝒓), 𝑚𝑧 (𝒓)𝜌 𝒓 , 𝑚𝑥 𝒓 , 𝑚𝑦 𝒓 , 𝑚𝑧 (𝒓)𝛁2−+ 𝑣𝐾𝑆,↑↑ [𝜌, 𝒎] 𝒓
2𝑣𝐾𝑆,↓↑ [𝜌, 𝒎] 𝒓𝑣𝐻 [𝜌] 𝒓𝑣ො𝐻 𝒓 =0𝑣𝐾𝑆,↑↓ [𝜌, 𝒎] 𝒓
𝛁2−+ 𝑣𝐾𝑆,↓↓ [𝜌, 𝒎] 𝒓
20𝑣𝐻 [𝜌] 𝒓𝑣ො 𝒓 : Spin-orbit coupling𝑣ො𝑋𝐶 𝒓 : Non-collinear spin texture𝜑𝑛↑ (𝒓)𝜑𝑛↑ (𝒓)= 𝜀𝑛𝜑𝑛↓ (𝒓)𝜑𝑛↓ (𝒓)9/20

Beyond LDA :Generalized Gradient Approximation (GGA)𝐿𝐷𝐴𝐸𝑋𝐶𝜌 ≡ න 𝑑 3 𝑟𝜌 𝒓 𝜀𝑋𝐶 𝜌 𝒓• Underestimate bandgaps• Overestimate cohesive energy → Underestimate lattice constatnt• Magnetic state of 3d metals (e.g. the ferromagnetic bcc structure of
Fe does not become the most stable state.)
𝐸𝑋𝐶 𝜌 ≈ න 𝑑3 𝑟𝜌 𝒓 𝜀𝑋𝐶 𝜌 𝒓 , 𝛁𝜌 𝒓
Gradient expansion approximation (GEA)

is not improved from LDA.

Gradient is large in practice → The expansion is not so appropriate.
Generalized Gradient approximation• More flexible form
• Preserve known desired conditions.

10/20

GGA functional

𝐺𝐺𝐴𝐸𝑋𝐶𝜌 = න 𝑑 3 𝑟𝜌 𝒓 𝜀𝑋 𝜌 𝒓 𝐹𝑋𝐶 𝜌 𝒓 , 𝑠 𝒓𝛁𝜌𝛁𝑟𝑠𝑠≡=2𝑘𝐹 𝜌 2 2𝜋/3 1/3 𝑟𝑠

Fig. 8.1 of "Electronic Structure"
by R. M. Martin

Fig. 8.2 of "Electronic Structure"
by R. M. Martin

11/20

GGA XC potential
𝐿𝐷𝐴/𝐺𝐺𝐴

𝛿𝐸𝑋𝐶 𝜌𝑣𝑋𝐶 𝜌 𝒓 ≡𝛿𝜌 𝒓𝐸𝑋𝐶𝜌 ≡ න 𝑑 3 𝑟𝜌 𝒓 𝜀𝑋𝐶 𝜌 𝒓 , 𝛁𝜌 𝒓𝑑𝜀𝑋𝐶 𝜌+𝜌 𝒓𝑑𝜌𝐿𝐷𝐴𝑣𝑋𝐶𝒓= 𝜀𝑋𝐶 𝜌 𝒓𝐺𝐺𝐴𝛿𝐸𝑋𝐶𝜌𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌= න 𝑑 𝑟 𝜀𝑋𝐶 𝛿𝜌 𝒓 + 𝜌 𝒓
𝛿𝜌 𝒓 + 𝜌 𝒓
𝛿𝛁𝜌 𝒓
𝜕𝜌(𝑟)
𝜕𝛁𝜌 𝒓
3𝐺𝐺𝐴𝑣ො𝑋𝐶= 𝜀𝑋𝐶

𝜌 𝒓 , 𝛁𝜌 𝒓

𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌+𝜌 𝒓+𝜌 𝒓⋅𝛁𝜕𝜌(𝑟)𝜕𝛁𝜌 𝒓

𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌+𝜌 𝒓−𝛁 𝜌 𝒓
𝜕𝜌(𝑟)
𝜕𝛁𝜌 𝒓

𝐺𝐺𝐴𝑣𝑋𝐶𝒓= 𝜀𝑋𝐶 𝜌 𝒓 , 𝛁𝜌 𝒓

𝐺𝐺𝐴𝑣𝑋𝐶𝒓𝑖

𝜕𝜀𝑋𝐶
𝜕𝜀𝑋𝐶 𝜌, 𝛁𝜌 𝛁𝜌 𝒓
= 𝜀𝑋𝐶 + 𝜌+ ෍𝜌 𝒓𝐶𝑗𝑖𝜕𝜌𝜕𝛁𝜌 𝒓𝛁𝜌 𝒓𝑗𝛁𝜌 𝒓𝑖 = ෍ 𝜌 𝒓𝑖 𝐶𝑖𝑗𝑗

12/20Beyond LDA : Hybrid functionalCoupling constant integral (adiabatic connection) form𝑁

11Ψ𝜆
෍2 ′𝐸𝑋𝐶 𝜌 = න 𝑑𝜆0𝑛,𝑛 =11
Ψ𝜆 − 𝐸𝐻 𝜌𝒓𝑛 − 𝒓𝑛 ′𝑁/2𝜆=1

∗∗′′′
𝜑𝒓
𝜑𝒓
𝜑𝒓
𝜑𝒓
1′𝑛𝑛𝑛𝑛
𝐸𝑋𝐹𝑜𝑐𝑘 = − ෍ ඵ 𝑑3 𝑟𝑑3 𝑟 ′2 ′𝑟 − 𝑟′𝑛,𝑛𝜆=0

𝐺𝐺𝐴𝐸𝑋𝐶𝜌

𝐸𝑋𝐹𝑜𝑐𝑘 + 𝐸𝑋𝐶
𝐸𝑋𝐶 =2𝑁/2∗′′
𝜑𝒓
𝜑𝒓′𝑛𝑣ො𝑋𝐹𝑜𝑐𝑘 𝜑𝑛 𝒓 = − ෍ 𝜑𝑛′ 𝒓 න 𝑑3 𝑟 ′ 𝑛′𝑟−𝑟′𝑛

For PW basis, it can be done with FFT. 𝑂 𝑁𝑘2 𝑁𝑏2 𝑁𝑃𝑊 log 𝑁𝑃𝑊

13/20Beyond LDA : Other functional• van der Waals functional
𝐸𝑋𝐶 𝜌 = ඵ 𝑑3 𝑟𝑑 3 𝑟 ′ 𝜌 𝒓 𝑔 𝒓 − 𝒓′ 𝜌 𝒓′
Organic solid
Graphite (layer distance)
Mirror potential for surface• DFT+U𝐺𝐺𝐴𝐸𝑋𝐶 𝜌 = 𝐸𝑋𝐶+ ෍ 𝜌𝜏𝑎 𝜌𝜏𝑎 𝑈𝜏𝑎 − 𝐸𝐷𝐶𝜏𝑎• meta-GGA

𝐸𝑋𝐶 𝜌 = න 𝑑3 𝑟𝜌 𝒓 𝜀𝑋𝐶 𝜌 𝒓 , 𝛁𝜌 𝒓 , 𝑡 𝑟
Kinetic energy density (Relates to 𝛁 𝟐 𝜌 𝒓 )14/20Modern pseudopotential

Concept of modern pseudopotential : Orthogonalized Plane Wave (OPW)𝛻2−+ 𝑣𝐾𝑆
2𝜙𝑐 𝜙𝑐 ′ = 𝛿𝑐𝑐 ′𝜑𝑛 = 𝜀𝑛 𝜑𝑛𝜙𝑐 𝜑𝑛 = 0𝜑𝑛 ≡ 𝜑෤𝑛 − ෍ 𝜙𝑐 𝜙𝑐 𝜑෤𝑛𝑐𝛻2−+ 𝑣𝐾𝑆2𝜑෤𝑛 − ෍ 𝜙𝑐 𝜙𝑐 𝜑෤𝑛𝛻2−+ 𝑣𝐾𝑆2= 𝜀𝑛𝜑෤𝑛 − ෍ 𝜙𝑐 𝜙𝑐 𝜑෤𝑛𝑐𝛻2−+ 𝑣𝐾𝑆 − ෍ 𝜙𝑐 𝜙𝑐2𝑐𝑐𝜑෤𝑛 = 𝜀𝑛 1 − ෍ 𝜙𝑐 𝜙𝑐𝑐

𝜑෤𝑛 1 − ෍ 𝜙𝑐 𝜙𝑐 𝜑෤ 𝑛′ = 𝛿𝑛𝑛′𝑐𝜙𝑐 = 𝜀𝑐 𝜙𝑐

𝜑෤𝑛

15/20Norm conserving pseudopotential

What is the good pseudopotential (PS)• Only requires small cutoff for plane wave. ↔ Shallow• Reproduce the result of the all-electron (AE) calculation atvarious cases (atoms, molecule, solid, ...) → Transferability

Required conditions for PS atomic orbital• 𝜀𝑙𝑃𝑆 = 𝜀𝑙𝐴𝐸 at the atomic case.
• 𝜑𝑙𝑃𝑆 𝑟𝑐 = 𝜑𝑙𝐴𝐸 𝑟𝑐• 𝐷𝑙𝑃𝑆 𝑟𝑐 , 𝜀 = 𝐷𝑙𝐴𝐸 𝑟𝑐 , 𝜀𝑑𝐷𝑙 𝑟, 𝜀𝑙 ≡ 𝑟 ln 𝜑𝑙 𝑟
𝑑𝑟𝑑 𝑃𝑆𝑑•
𝐷𝑙 𝑟𝑐 , 𝜀 = 𝐷𝑙𝐴𝐸 𝑟𝑐 , 𝜀𝑑𝜀𝑟𝑐

𝑑𝜀𝜑𝑙𝐴𝐸 𝑟
𝜑𝑙𝑃𝑆 𝑟𝑟𝑟𝑐

න 𝑑𝑟𝑟 2 𝜑𝑙𝑃𝑆 𝑟 = න 𝑑𝑟𝑟 2 𝜑𝑙𝐴𝐸 𝑟00𝑣𝑙𝑃𝑆 𝑟 = 𝑣𝑙𝐴𝐸 𝑟 − 𝑣𝐻𝑋𝐶 [𝜌𝑃𝑆 ](𝑟)𝑣𝑙𝑃𝑆 𝑟𝑣𝑙𝐴𝐸 𝑟

16/20Ultrasoft pseudopotential andpartial core correction

Problem : Norm-conserving condition for 1s (H), 2p (B,C,..), 3d(Ti,V,..), 4f, ...
𝜑𝑙𝑁𝐶𝑃𝑆 𝑟

𝜑෤𝑛 1 + ෍ Δ𝑄𝑠𝑠′ 𝛽𝑠 𝛽𝑠′ 𝜑෤ 𝑛′ = 𝛿𝑛𝑛′𝑠,𝑠′𝜑𝑙𝐴𝐸 𝑟𝑟𝑐

Δ𝑄𝑠𝑠′ = න 𝑑𝑟 Δ𝑄𝑠𝑠′ 𝑟

𝜑𝑙𝑈𝑆𝑃𝑆 𝑟0𝑃𝑆∗
Δ𝑄𝑠𝑠′ 𝑟 = 𝜑𝑠𝐴𝐸∗ (𝑟)𝜑𝑠𝐴𝐸(𝑟)𝜑𝑠𝑃𝑆′ (𝑟) − 𝜑𝑠′ (𝑟)𝑟

Partial core correction (nonlinear core correction)𝑣𝑋𝐶 𝜌𝑐 + 𝜌𝑣 ≠ 𝑣𝑋𝐶 𝜌𝑐 + 𝑣𝑋𝐶 𝜌𝑣
𝑣𝑙𝑃𝑆 𝑟 = 𝑣𝑙𝐴𝐸 𝑟 − 𝑣𝐻𝑋𝐶 𝜌𝑃𝑆 𝑟 − 𝑣𝑋𝐶 [𝜌𝑝𝑐 ]𝑣𝑋𝐶 = 𝑣𝑋𝐶 𝜌𝑃𝑆 + 𝜌𝑝𝑐

17/20

Projector augmented wave

Orthogonalized Plane Wave (OPW) again
𝜑𝑛 ≡ 𝜑෤𝑛 − ෍ 𝜙𝑐 𝜙𝑐 𝜑෤𝑛 = 𝑇෠ 𝜑෤ 𝑛𝑐

The final equation form is the same as US pseudopotential,but ..
• We can obtain true KS orbital rather than the pseudo
orbitals• (Very ?) slightly different result for the magnetic system.

18/20Pseudopotential library for QE

Although we can generate PS by ourselves ...

→ Huge effort !
Good pseudopotential• Requires small cutoff energy• Well verified (compared with the AE results)• Partial core correction• Semicore (becomes harder)
Pseudopotential libraries• SG15 library• http://www.quantum-simulation.org/potentials/sg15_oncv/• Norm conserving pseudopotential• PS library• https://www.quantum-espresso.org/pseudopotentials• QE official• Standard Solid State library (SSSP)• https://www.materialscloud.org/discover/sssp/table/efficienc• Strictly verified

19/20Standard Solid State Pseudopotentials

https://www.materialscloud.org/discover/sssp/table/efficiency

20/20Today's summary Magnetics system
 Spin density functional theory
 Local Spin Density Approximation (LSDA)
 Collinear and non-collinear magnetism
 Beyond LDA
 Generalized Gradient Approximation
 Hybrid functional
 VdW, DFT+U