SlideShare a Scribd company logo
計算材料学 
Computational Materials Science 
東京工業大学大学院総合理工学研究科2014 年度冬学期 
Tokyo Tech Winter Semester, AY2014 
October 17, 2014 
合田義弘 
Y. Gohda 
はじめに/Introduction 
計算機を用いたシミュレーションは実験と相補的な立場にある材料学研究に欠かせないツールで 
ある。本講義の内、合田担当分では第一原理計算を中心に電子状態の数値解析について扱う。第 
一原理計算とは、実験的に決定されたパラメーターを用いず、素電荷e、Planck 定数! 等の基礎 
的な物理定数と計算精度をコントロールする数値解析的パラメーターのみによる非経験的電子状 
態計算の事である。 
この講義ノートは 
http://www.cms.materia.titech.ac.jp/lecture_cms.pdf 
においてダウンロード可能である(要パスワード)。質問は歓迎する。講義中にどんどんしてほし 
い。参考書としては 
押山淳ほか:岩波講座計算科学3 「計算と物質」 
R.M. Martin: Electronic Structure 日本語訳あり 
J.M. ティッセン:「計算物理学」 
J.J. Sakurai: Modern Quantum Mechanics 日本語訳あり 
を勧める。 
第一原理計算を実際に試す事も可能である。Mac あるいはLinux にコードを自分でインストー 
ルしても良いが、“MateriApps LIVE!” のUSB を入手あるいは作成すれば、手持ちのコンピュー 
ターで計算環境が整備済のLinux をUSB ブートする事が出来る。“MateriApps LIVE!” に収録さ 
れているOpenMX コードを勧める。 
東京工業大学の国際化推進に向けた方針に鑑み、以下の講義ノートは原則英語で記述する。 
1
Contents 
1 Born-Oppenheimer approximation and many-electron system 3 
1.1 Summary of basic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 3 
1.2 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 
1.3 Molecular dynamics and structure optimization. . . . . . . . . . . . . . . . . . . . 6 
1.4 Quantum many-body theory of electrons . . . . . . . . . . . . . . . . . . . . . . . 9 
2 Hartree-Fock approximation and wave-function theory 12 
2.1 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 
2.2 Post-Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 
2.3 Diffusion Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 
3 Density functional theory 21 
3.1 Hohenberg–Kohn theorem and the Kohn-Sham equation . . . . . . . . . . . . . . 21 
3.2 Exchange-correlation functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 
3.3 Beyond standard DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 
4 Basis sets 28 
4.1 Without basis sets: real-space discretization . . . . . . . . . . . . . . . . . . . . . 28 
4.2 Plane-wave basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 
4.3 Pseudo-atomic orbitals as a basis set . . . . . . . . . . . . . . . . . . . . . . . . . 29 
5 Examples 33 
5.1 Diatomic molecules: N2 and O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 
5.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 
5.3 Microstructure interfaces of magnetic materials . . . . . . . . . . . . . . . . . . . 41 
2
5.2 Graphene 
Brillouin zone of the hexagonal lattice 
lattice vectors: 
a = a 
! 
1 
2 
ex − 
√3 
2 
ey 
" 
, b = a 
! 
1 
2 
ex + 
√3 
2 
ey 
" 
, c = cez . 
reciprocal-lattice vectors: 
Ga = 2π 
b × c 
a · (b × c) 
= 
4π 
√3a2c 
b × c = 
4π 
√3a 
!√3 
2 
ex − 
1 
2 
ey 
" 
, 
Gb = 2π 
c × a 
a · (b × c) 
= 
4π 
√3a2c 
c × a = 
4π 
√3a 
!√3 
2 
ex + 
1 
2 
ey 
" 
, 
Gc = 2π 
a × b 
a · (b × c) 
= 
4π 
√3a2c 
a × b = 
2π 
c 
ez . 
Γ: Center of the Brillouin zone 
K: Middle of an edge joining two rectangular faces→ (Ga + Gb)/3 
M: Center of a rectangular face → Ga/2 
A: Center of a hexagonal face → Gc/2 
H: Corner point → (Ga + Gb)/3 + Gc/2 
L: Middle of an edge joining a hexagonal and a rectangular face → Ga/2 + Gc/2 
10 
5 
0 
−5 
Energy relative to EF [eV] 
−10 
Γ KM Γ A H L Γ 
Figure 6: Band structure of the normal state 
of a superconductor, MgB2. 
34
Unit cell and atomic structure of graphene 
The computational cell for graphene is hexagonal one. 
(a) (b) 
Figure 7: The unit cell and the atomic struc-ture 
of graphene used for first-principles calcu-lations: 
(a) the side view and (b) the top view. 
Input file of OpenMX to calculate electronic states of graphene 
# 
# OpenMX Inputfile 
# 
# File Name 
# 
System.Name graphene 
DATA.PATH /home/gohda/openmx/DFT_DATA13 
# Definition of Atomic Species 
# 
Species.Number 2 
<Definition.of.Atomic.Species 
C C6.0-s2p2d1 C_PBE13 
E Rn13.0-s2p2d2f1 E 
Definition.of.Atomic.Species> 
# Atoms 
# 
Atoms.Number 3 
Atoms.SpeciesAndCoordinates.Unit FRAC # Ang|AU 
<Atoms.SpeciesAndCoordinates 
1 C 0.33333333333333 0.66666666666667 0.50000000000000 2.0 2.0 
2 C 0.66666666666667 0.33333333333333 0.50000000000000 2.0 2.0 
3 E 0.00000000000000 0.00000000000000 0.50000000000000 0.0 0.0 
Atoms.SpeciesAndCoordinates> 
35
Atoms.UnitVectors.Unit Ang # Ang|AU 
<Atoms.UnitVectors 
1.23000000000000 -2.13042249330972 0.00000000000000 
1.23000000000000 2.13042249330972 0.00000000000000 
0.00000000000000 0.00000000000000 10.00000000000000 
Atoms.UnitVectors> 
# SCF or Electronic System 
# 
scf.XcType GGA-PBE # LDA|LSDA-CA|LSDA-PW|GGA-PBE 
scf.SpinPolarization off # On|Off|NC 
scf.energycutoff 300.0 # default=150 (Ry) 
scf.maxIter 100 # default=40 
scf.EigenvalueSolver band # DC|GDC|Cluster|Band 
scf.Kgrid 11 11 1 # means n1 x n2 x n3 
scf.Mixing.Type rmm-diisk # Simple|Rmm-Diis|Gr-Pulay|Kerker|Rmm-Diisk 
scf.Init.Mixing.Weight 0.30 # default=0.30 
scf.Min.Mixing.Weight 0.001 # default=0.001 
scf.Max.Mixing.Weight 0.700 # default=0.40 
scf.Mixing.History 10 # default=5 
scf.Mixing.StartPulay 5 # default=6 
scf.Mixing.EveryPulay 1 # default=5 
scf.criterion 1.0e-6 # default=1.0e-6 (Hartree) 
scf.lapack.dste dstevx # dstegr|dstedc|dstevx, default=dstevx 
# DOS and PDOS 
# 
Dos.fileout on # on|off, default=off 
Dos.Erange -20.0 10.0 # default = -20 20 
Dos.Kgrid 48 48 1 # default = Kgrid1 Kgrid2 Kgrid3 
# Band dispersion 
# 
Band.dispersion on # on|off, default=off 
Band.Nkpath 4 
<Band.kpath 
200 0.0 0.0 0.0 0.33333333333333 0.33333333333333 0.0 G K 
100 0.33333333333333 0.33333333333333 0.0 0.5 0.0 0.0 K M 
173 0.5 0.0 0.0 0.0 0.0 0.0 M G 
37 0.0 0.0 0.0 0.0 0.0 0.5 G A 
Band.kpath> 
# Others 
# 
scf.restart on 
scf.fixed.grid 0.00000000000000 0.00000000000000 0.00000000000000 
36
Band strucuture and density of states 
You can excecute OpenMX by typing on a UNIX (MacOSX, Linux, etc.) terminal as follows: 
$ openmx INCAR >log 2>&1 & 
5 
(a) (b) 
0 
−10 
Γ K M Γ A 
−5 
Energy relative to EF [eV] 
−15 
−20 
0 
−20 −15 −10 −5 0 5 10 
Energy relative to EF [eV] 
Density of states [arb. unit] 
w EA 
w/o EA 
w EA 
w/o EA 
(c) 
Figure 8: (a) The energy band and (b) the de-sity 
of states for graphene obtained by first-principles 
calculations with enpty-atom basis 
functions (red) and without them (black). (c) 
Nearly-free-electron states that cannot be de-scribed 
without the empty atom. 
Comparison with the tight-binding model with the H¨uckel approximation 
In the Huckel ¨approximation, only the |φpz ⟩ state is considered for the frontier orbitals of benzene 
rings. The tight-binding Hamiltonian using the notation of |I⟩ = |φI 
pz ⟩ is 
H = 
# 
I 
εI |I⟩⟨I|− 
# 
<IJ> 
t|I⟩⟨J| = − 
# 
<IJ> 
t|I⟩⟨J| , (23) 
where t is the transfer integral and we define εI = 0. Using the Bloch sum 
φB 
i,k(r) = 
1 
√Nk 
# 
T 
eik·T φpz (r − Ri − T ) (24) 
with the approximation of ⟨I|J⟩ = δIJ (Thus Sij = δij), the matrix elements are 
⟨φB 
i,k|H|φB 
i,k⟩ = 0 , and (25) 
⟨φB 
i,k|H|φB 
j,k⟩ = ⟨φB 
i,k⟩∗ = −t(eik·T0 + eik·T1 + eik·T2) 
j,k|H|φB 
= −t(1 + eik·a + e−ik·b) ≡ −tf . (26) 
37
Thus the transformed Hamiltonian becomes 
H = −t(f|φB1 
⟩⟨φB2 
| + f∗|φB2 
⟩⟨φB1 
|) . (27) 
Diagonalizing H through ε2 − t2|f|2 = 0, we obtain 
ε = ±t|f| 
= ±t 
$ 
(1 + cos k · a + cos k · b)2 + (sink · a − sin k · b)2 
= ±t 
$ 
3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) . (28) 
At the M point (k = Ga/2), 
ε = ±t 
$ 
3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) 
= ±t 
$ 
5 + 4 cosGa/2 · a 
= ±t√5 + 4 cos π 
= ±t . (29) 
As for the Γ–K path, i.e. k = κ(Ga + Gb), 
ε = ±t 
$ 
3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) 
= ±t 
$ 
3 + 2 cos κGa · a + 2 cos κGb · b + 2 cos κ(Ga + Gb) · (a + b) 
= ±t 
$ 
3 + 4 cos (2πκ) + 2 cos(4πκ) (30) 
Γ K 
10 
5 
0 
Energy relative to EF [eV] 
−5 
w EA 
w/o EA 
TB 
Figure 9: Comparison of energy bands ob-tained 
by first-principles calculations and the 
tight-binding model with the H¨uckel approxi-mation. 
The hopping integral was chosen as 
t = 2.7 eV. 
38
Γ K M 
0 0 
0 0 
0 
0 
0 
0 0 
0 
0 
π 0 
π 
π 
0 0 
π π 
0 
0 
π 
π 
π 
π 
π 
π 
B1 
Figure 10: The phase θ = k · T appear in 
the Bloch sum for the A sublattice (i = 1), 
% 
φk(r) = √1 ,Nk 
T eik·T φpz (r − R1 − T ). 
Let us take a closer look of the wave functions: 
At the Γ point: 
H = −3t(|φB1 
⟩⟨φB2 
| + |φB2 
⟩⟨φB1 
|) , ε= ±3t , (31) 
|π⟩ = 
1 
√2 
(|φB1 
⟩ + |φB2 
⟩) , |π∗⟩ = 
1 
√2 
(|φB1 
⟩ − |φB2 
⟩) . (32) 
In general: 
H = −t(f|φB1 
⟩⟨φB2 
| + f∗|φB2 
⟩⟨φB1 
|) , ε= ±t|f| , (33) 
|π⟩ = 
1 
√2 
& 
|φB1 
⟩ + 
f∗ 
|f||φB2 
⟩ 
' 
, |π∗⟩ = 
1 
√2 
& 
|φB1 
⟩ − 
f∗ 
|f||φB2 
⟩ 
' 
. (34) 
At the K point: 
H = 0 , ε= 0 , (35) 
|π⟩ = cπ,1|φB1 
⟩ + cπ,2|φB2 
⟩ , |π∗⟩ = cπ∗,1|φB1 ⟩ + cπ∗,2|φB2 
⟩ . (36) 
Exercise 
For k vectors around the K point, k = 13 
(Ga + Gb) + δkxex + δkyey, derive the effective 
Hamiltonian using the Talor expansion starting from the form 
H = 
⎛ 
⎝ 0 −tf 
−tf ∗ 0 
⎞ 
⎠ . 
Use the Pauli matrices σ: 
σx = 
⎛ 
⎝ 0 1 
1 0 
⎞ 
⎠ , σy 
= 
⎛ 
⎝ 0 −i 
i 0 
⎞ 
⎠ , and σz = 
⎛ 
⎝ 1 0 
0 1 
⎞ 
⎠ . 
39
Answer: 
Exercise 
Calculate i) the eigenenergy and ii) the sign of the linear combination for π orbitals in a butadiene 
molecule CH2CHCHCH2, a virtual fragment of graphene, using the H¨uckel approximation. 
Hint: 2 < √5 < 3 
Answer: 
40

More Related Content

What's hot

C++による数値解析の並列化手法
C++による数値解析の並列化手法C++による数値解析の並列化手法
C++による数値解析の並列化手法
dc1394
 
深層ニューラルネットワークの積分表現(Deepを定式化する数学)
深層ニューラルネットワークの積分表現(Deepを定式化する数学)深層ニューラルネットワークの積分表現(Deepを定式化する数学)
深層ニューラルネットワークの積分表現(Deepを定式化する数学)
Katsuya Ito
 
ゲート方式量子コンピュータの概要
ゲート方式量子コンピュータの概要ゲート方式量子コンピュータの概要
ゲート方式量子コンピュータの概要
Yahoo!デベロッパーネットワーク
 
Python 機械学習プログラミング データ分析演習編
Python 機械学習プログラミング データ分析演習編Python 機械学習プログラミング データ分析演習編
Python 機械学習プログラミング データ分析演習編
Etsuji Nakai
 
パターン認識と機械学習 13章 系列データ
パターン認識と機械学習 13章 系列データパターン認識と機械学習 13章 系列データ
パターン認識と機械学習 13章 系列データ
emonosuke
 
A summary on “On choosing and bounding probability metrics”
A summary on “On choosing and bounding probability metrics”A summary on “On choosing and bounding probability metrics”
A summary on “On choosing and bounding probability metrics”
Kota Matsui
 
PRML 8.2 条件付き独立性
PRML 8.2 条件付き独立性PRML 8.2 条件付き独立性
PRML 8.2 条件付き独立性
sleepy_yoshi
 
「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料
「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料
「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料
Ken'ichi Matsui
 
ディープボルツマンマシン入門
ディープボルツマンマシン入門ディープボルツマンマシン入門
ディープボルツマンマシン入門
Saya Katafuchi
 
第一原理計算と密度汎関数理論
第一原理計算と密度汎関数理論第一原理計算と密度汎関数理論
第一原理計算と密度汎関数理論
dc1394
 
パターン認識と機械学習6章(カーネル法)
パターン認識と機械学習6章(カーネル法)パターン認識と機械学習6章(カーネル法)
パターン認識と機械学習6章(カーネル法)
Yukara Ikemiya
 
SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...
SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...
SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...
harmonylab
 
PRML輪読#2
PRML輪読#2PRML輪読#2
PRML輪読#2
matsuolab
 
「統計的学習理論」第1章
「統計的学習理論」第1章「統計的学習理論」第1章
「統計的学習理論」第1章
Kota Matsui
 
条件付き確率場の推論と学習
条件付き確率場の推論と学習条件付き確率場の推論と学習
条件付き確率場の推論と学習
Masaki Saito
 
はじめてのパターン認識8章サポートベクトルマシン
はじめてのパターン認識8章サポートベクトルマシンはじめてのパターン認識8章サポートベクトルマシン
はじめてのパターン認識8章サポートベクトルマシン
NobuyukiTakayasu
 
クラシックな機械学習の入門  8. クラスタリング
クラシックな機械学習の入門  8. クラスタリングクラシックな機械学習の入門  8. クラスタリング
クラシックな機械学習の入門  8. クラスタリング
Hiroshi Nakagawa
 
機械学習と機械発見:自然科学研究におけるデータ利活用の再考
機械学習と機械発見:自然科学研究におけるデータ利活用の再考機械学習と機械発見:自然科学研究におけるデータ利活用の再考
機械学習と機械発見:自然科学研究におけるデータ利活用の再考
Ichigaku Takigawa
 
密度汎関数法 Density Functional Theory (DFT)の基礎第6回
密度汎関数法 Density Functional Theory (DFT)の基礎第6回密度汎関数法 Density Functional Theory (DFT)の基礎第6回
密度汎関数法 Density Functional Theory (DFT)の基礎第6回
SATOH daisuke, Ph.D.
 

What's hot (20)

C++による数値解析の並列化手法
C++による数値解析の並列化手法C++による数値解析の並列化手法
C++による数値解析の並列化手法
 
深層ニューラルネットワークの積分表現(Deepを定式化する数学)
深層ニューラルネットワークの積分表現(Deepを定式化する数学)深層ニューラルネットワークの積分表現(Deepを定式化する数学)
深層ニューラルネットワークの積分表現(Deepを定式化する数学)
 
ゲート方式量子コンピュータの概要
ゲート方式量子コンピュータの概要ゲート方式量子コンピュータの概要
ゲート方式量子コンピュータの概要
 
Python 機械学習プログラミング データ分析演習編
Python 機械学習プログラミング データ分析演習編Python 機械学習プログラミング データ分析演習編
Python 機械学習プログラミング データ分析演習編
 
パターン認識と機械学習 13章 系列データ
パターン認識と機械学習 13章 系列データパターン認識と機械学習 13章 系列データ
パターン認識と機械学習 13章 系列データ
 
A summary on “On choosing and bounding probability metrics”
A summary on “On choosing and bounding probability metrics”A summary on “On choosing and bounding probability metrics”
A summary on “On choosing and bounding probability metrics”
 
PRML 8.2 条件付き独立性
PRML 8.2 条件付き独立性PRML 8.2 条件付き独立性
PRML 8.2 条件付き独立性
 
「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料
「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料
「ベータ分布の謎に迫る」第6回 プログラマのための数学勉強会 LT資料
 
ディープボルツマンマシン入門
ディープボルツマンマシン入門ディープボルツマンマシン入門
ディープボルツマンマシン入門
 
第一原理計算と密度汎関数理論
第一原理計算と密度汎関数理論第一原理計算と密度汎関数理論
第一原理計算と密度汎関数理論
 
パターン認識と機械学習6章(カーネル法)
パターン認識と機械学習6章(カーネル法)パターン認識と機械学習6章(カーネル法)
パターン認識と機械学習6章(カーネル法)
 
SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...
SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...
SegFormer: Simple and Efficient Design for Semantic Segmentation with Transfo...
 
Prml 2.3
Prml 2.3Prml 2.3
Prml 2.3
 
PRML輪読#2
PRML輪読#2PRML輪読#2
PRML輪読#2
 
「統計的学習理論」第1章
「統計的学習理論」第1章「統計的学習理論」第1章
「統計的学習理論」第1章
 
条件付き確率場の推論と学習
条件付き確率場の推論と学習条件付き確率場の推論と学習
条件付き確率場の推論と学習
 
はじめてのパターン認識8章サポートベクトルマシン
はじめてのパターン認識8章サポートベクトルマシンはじめてのパターン認識8章サポートベクトルマシン
はじめてのパターン認識8章サポートベクトルマシン
 
クラシックな機械学習の入門  8. クラスタリング
クラシックな機械学習の入門  8. クラスタリングクラシックな機械学習の入門  8. クラスタリング
クラシックな機械学習の入門  8. クラスタリング
 
機械学習と機械発見:自然科学研究におけるデータ利活用の再考
機械学習と機械発見:自然科学研究におけるデータ利活用の再考機械学習と機械発見:自然科学研究におけるデータ利活用の再考
機械学習と機械発見:自然科学研究におけるデータ利活用の再考
 
密度汎関数法 Density Functional Theory (DFT)の基礎第6回
密度汎関数法 Density Functional Theory (DFT)の基礎第6回密度汎関数法 Density Functional Theory (DFT)の基礎第6回
密度汎関数法 Density Functional Theory (DFT)の基礎第6回
 

Similar to 計算材料学

Subquad multi ff
Subquad multi ffSubquad multi ff
Subquad multi ff
Fabian Velazquez
 
Computing the masses of hyperons and charmed baryons from Lattice QCD
Computing the masses of hyperons and charmed baryons from Lattice QCDComputing the masses of hyperons and charmed baryons from Lattice QCD
Computing the masses of hyperons and charmed baryons from Lattice QCD
Christos Kallidonis
 
Optimal L-shaped matrix reordering, aka graph's core-periphery
Optimal L-shaped matrix reordering, aka graph's core-peripheryOptimal L-shaped matrix reordering, aka graph's core-periphery
Optimal L-shaped matrix reordering, aka graph's core-periphery
Francesco Tudisco
 
ACME2016-extendedAbstract
ACME2016-extendedAbstractACME2016-extendedAbstract
ACME2016-extendedAbstract
Zhaowei Liu
 
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...
Shizuoka Inst. Science and Tech.
 
Solutions modern particlephysics
Solutions modern particlephysicsSolutions modern particlephysics
Solutions modern particlephysics
itachikaleido
 
14 34-46
14 34-4614 34-46
14 34-46
idescitation
 
Generalized Nonlinear Models in R
Generalized Nonlinear Models in RGeneralized Nonlinear Models in R
Generalized Nonlinear Models in R
htstatistics
 
A Parallel Branch And Bound Algorithm For The Quadratic Assignment Problem
A Parallel Branch And Bound Algorithm For The Quadratic Assignment ProblemA Parallel Branch And Bound Algorithm For The Quadratic Assignment Problem
A Parallel Branch And Bound Algorithm For The Quadratic Assignment Problem
Mary Calkins
 
Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...
Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...
Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...
Ola Carmen
 
Hyperon and charmed baryon masses and axial charges from Lattice QCD
Hyperon and charmed baryon masses and axial charges from Lattice QCDHyperon and charmed baryon masses and axial charges from Lattice QCD
Hyperon and charmed baryon masses and axial charges from Lattice QCD
Christos Kallidonis
 
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionData sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansion
Alexander Litvinenko
 
Data sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve ExpansionData sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve Expansion
Alexander Litvinenko
 
Slides
SlidesSlides
Hyperon and charm baryons masses from twisted mass Lattice QCD
Hyperon and charm baryons masses from twisted mass Lattice QCDHyperon and charm baryons masses from twisted mass Lattice QCD
Hyperon and charm baryons masses from twisted mass Lattice QCD
Christos Kallidonis
 
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...
Alexander Litvinenko
 
EM_Theory.pdf
EM_Theory.pdfEM_Theory.pdf
EM_Theory.pdf
ssuser9ae06b
 
PhD work on Graphene Transistor
PhD work on Graphene TransistorPhD work on Graphene Transistor
PhD work on Graphene Transistor
Southern University and A&M College - Baton Rouge
 
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Alexander Litvinenko
 
Bayesian Inference and Uncertainty Quantification for Inverse Problems
Bayesian Inference and Uncertainty Quantification for Inverse ProblemsBayesian Inference and Uncertainty Quantification for Inverse Problems
Bayesian Inference and Uncertainty Quantification for Inverse Problems
Matt Moores
 

Similar to 計算材料学 (20)

Subquad multi ff
Subquad multi ffSubquad multi ff
Subquad multi ff
 
Computing the masses of hyperons and charmed baryons from Lattice QCD
Computing the masses of hyperons and charmed baryons from Lattice QCDComputing the masses of hyperons and charmed baryons from Lattice QCD
Computing the masses of hyperons and charmed baryons from Lattice QCD
 
Optimal L-shaped matrix reordering, aka graph's core-periphery
Optimal L-shaped matrix reordering, aka graph's core-peripheryOptimal L-shaped matrix reordering, aka graph's core-periphery
Optimal L-shaped matrix reordering, aka graph's core-periphery
 
ACME2016-extendedAbstract
ACME2016-extendedAbstractACME2016-extendedAbstract
ACME2016-extendedAbstract
 
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...
Talk at SciCADE2013 about "Accelerated Multiple Precision ODE solver base on ...
 
Solutions modern particlephysics
Solutions modern particlephysicsSolutions modern particlephysics
Solutions modern particlephysics
 
14 34-46
14 34-4614 34-46
14 34-46
 
Generalized Nonlinear Models in R
Generalized Nonlinear Models in RGeneralized Nonlinear Models in R
Generalized Nonlinear Models in R
 
A Parallel Branch And Bound Algorithm For The Quadratic Assignment Problem
A Parallel Branch And Bound Algorithm For The Quadratic Assignment ProblemA Parallel Branch And Bound Algorithm For The Quadratic Assignment Problem
A Parallel Branch And Bound Algorithm For The Quadratic Assignment Problem
 
Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...
Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...
Coherence incoherence patterns in a ring of non-locally coupled phase oscilla...
 
Hyperon and charmed baryon masses and axial charges from Lattice QCD
Hyperon and charmed baryon masses and axial charges from Lattice QCDHyperon and charmed baryon masses and axial charges from Lattice QCD
Hyperon and charmed baryon masses and axial charges from Lattice QCD
 
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionData sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansion
 
Data sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve ExpansionData sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve Expansion
 
Slides
SlidesSlides
Slides
 
Hyperon and charm baryons masses from twisted mass Lattice QCD
Hyperon and charm baryons masses from twisted mass Lattice QCDHyperon and charm baryons masses from twisted mass Lattice QCD
Hyperon and charm baryons masses from twisted mass Lattice QCD
 
Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...Computation of electromagnetic fields scattered from dielectric objects of un...
Computation of electromagnetic fields scattered from dielectric objects of un...
 
EM_Theory.pdf
EM_Theory.pdfEM_Theory.pdf
EM_Theory.pdf
 
PhD work on Graphene Transistor
PhD work on Graphene TransistorPhD work on Graphene Transistor
PhD work on Graphene Transistor
 
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...
 
Bayesian Inference and Uncertainty Quantification for Inverse Problems
Bayesian Inference and Uncertainty Quantification for Inverse ProblemsBayesian Inference and Uncertainty Quantification for Inverse Problems
Bayesian Inference and Uncertainty Quantification for Inverse Problems
 

More from Computational Materials Science Initiative

MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
MateriApps LIVE!の設定
MateriApps LIVE!の設定MateriApps LIVE!の設定
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
Computational Materials Science Initiative
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
Computational Materials Science Initiative
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
MateriApps LIVE!の設定
MateriApps LIVE!の設定MateriApps LIVE!の設定
ALPSチュートリアル
ALPSチュートリアルALPSチュートリアル
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
Computational Materials Science Initiative
 
MateriApps LIVE!の設定
MateriApps LIVE!の設定MateriApps LIVE!の設定
MateriApps: OpenMXを利用した第一原理計算の簡単な実習
MateriApps: OpenMXを利用した第一原理計算の簡単な実習MateriApps: OpenMXを利用した第一原理計算の簡単な実習
MateriApps: OpenMXを利用した第一原理計算の簡単な実習
Computational Materials Science Initiative
 
CMSI計算科学技術特論C (2015) ALPS と量子多体問題②
CMSI計算科学技術特論C (2015) ALPS と量子多体問題②CMSI計算科学技術特論C (2015) ALPS と量子多体問題②
CMSI計算科学技術特論C (2015) ALPS と量子多体問題②
Computational Materials Science Initiative
 
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
Computational Materials Science Initiative
 

More from Computational Materials Science Initiative (20)

MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
MateriApps LIVE!の設定
MateriApps LIVE!の設定MateriApps LIVE!の設定
MateriApps LIVE!の設定
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
MateriApps LIVE! の設定
MateriApps LIVE! の設定MateriApps LIVE! の設定
MateriApps LIVE! の設定
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
MateriApps LIVE!の設定
MateriApps LIVE!の設定MateriApps LIVE!の設定
MateriApps LIVE!の設定
 
ALPSチュートリアル
ALPSチュートリアルALPSチュートリアル
ALPSチュートリアル
 
How to setup MateriApps LIVE!
How to setup MateriApps LIVE!How to setup MateriApps LIVE!
How to setup MateriApps LIVE!
 
MateriApps LIVE!の設定
MateriApps LIVE!の設定MateriApps LIVE!の設定
MateriApps LIVE!の設定
 
MateriApps: OpenMXを利用した第一原理計算の簡単な実習
MateriApps: OpenMXを利用した第一原理計算の簡単な実習MateriApps: OpenMXを利用した第一原理計算の簡単な実習
MateriApps: OpenMXを利用した第一原理計算の簡単な実習
 
CMSI計算科学技術特論C (2015) ALPS と量子多体問題②
CMSI計算科学技術特論C (2015) ALPS と量子多体問題②CMSI計算科学技術特論C (2015) ALPS と量子多体問題②
CMSI計算科学技術特論C (2015) ALPS と量子多体問題②
 
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
 

Recently uploaded

Lake classification and Morphometry.pptx
Lake classification and Morphometry.pptxLake classification and Morphometry.pptx
Lake classification and Morphometry.pptx
boobalanbfsc
 
MARIGREEN PROJECT - overview, Oana Cristina Pârvulescu
MARIGREEN PROJECT - overview, Oana Cristina PârvulescuMARIGREEN PROJECT - overview, Oana Cristina Pârvulescu
MARIGREEN PROJECT - overview, Oana Cristina Pârvulescu
Faculty of Applied Chemistry and Materials Science
 
atom, elements, molecule and compounds in telugu.pptx
atom, elements, molecule and compounds in telugu.pptxatom, elements, molecule and compounds in telugu.pptx
atom, elements, molecule and compounds in telugu.pptx
ManjulaVani3
 
Fish in the Loop: Exploring RAS - Julie Hansen Bergstedt
Fish in the Loop: Exploring RAS - Julie Hansen BergstedtFish in the Loop: Exploring RAS - Julie Hansen Bergstedt
Fish in the Loop: Exploring RAS - Julie Hansen Bergstedt
Faculty of Applied Chemistry and Materials Science
 
Pancreas_functional anatomy_enzymes.pptx
Pancreas_functional anatomy_enzymes.pptxPancreas_functional anatomy_enzymes.pptx
Pancreas_functional anatomy_enzymes.pptx
muralinath2
 
PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...
PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...
PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...
Thane Heins
 
SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...
SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...
SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...
Sérgio Sacani
 
Potential of Marine Renewable and Non renewable energy.pptx
Potential of Marine Renewable and Non renewable energy.pptxPotential of Marine Renewable and Non renewable energy.pptx
Potential of Marine Renewable and Non renewable energy.pptx
J. Bovas Joel BFSc
 
Phytoremediation: Harnessing Nature's Power with Phytoremediation
Phytoremediation: Harnessing Nature's Power with PhytoremediationPhytoremediation: Harnessing Nature's Power with Phytoremediation
Phytoremediation: Harnessing Nature's Power with Phytoremediation
Gurjant Singh
 
Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...
Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...
Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...
Sérgio Sacani
 
NuGOweek 2024 Ghent programme__flyer.pdf
NuGOweek 2024 Ghent programme__flyer.pdfNuGOweek 2024 Ghent programme__flyer.pdf
NuGOweek 2024 Ghent programme__flyer.pdf
pablovgd
 
17. 20240529_Ingrid Olesen_MariGreen summer school.pdf
17. 20240529_Ingrid Olesen_MariGreen summer school.pdf17. 20240529_Ingrid Olesen_MariGreen summer school.pdf
17. 20240529_Ingrid Olesen_MariGreen summer school.pdf
marigreenproject
 
Composting blue materials - Joshua Cabell
Composting blue materials - Joshua CabellComposting blue materials - Joshua Cabell
Composting blue materials - Joshua Cabell
Faculty of Applied Chemistry and Materials Science
 
Classification and role of plant nutrients - Roxana Madjar
Classification and role of plant nutrients - Roxana MadjarClassification and role of plant nutrients - Roxana Madjar
Classification and role of plant nutrients - Roxana Madjar
Faculty of Applied Chemistry and Materials Science
 
Surface properties of the seas of Titan as revealed by Cassini mission bistat...
Surface properties of the seas of Titan as revealed by Cassini mission bistat...Surface properties of the seas of Titan as revealed by Cassini mission bistat...
Surface properties of the seas of Titan as revealed by Cassini mission bistat...
Sérgio Sacani
 
Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...
Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...
Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...
US Environmental Protection Agency (EPA), Center for Computational Toxicology and Exposure
 
Bioconversion of sago waste and oil cakes into biobutanol using Environmental...
Bioconversion of sago waste and oil cakes into biobutanol using Environmental...Bioconversion of sago waste and oil cakes into biobutanol using Environmental...
Bioconversion of sago waste and oil cakes into biobutanol using Environmental...
Dr NEETHU ASOKAN
 
Data Visualization Workshop for Summer Interns
Data Visualization Workshop for Summer InternsData Visualization Workshop for Summer Interns
Data Visualization Workshop for Summer Interns
Zachary Labe
 
Rice Genome Project a complete saga .(1).pptx
Rice Genome  Project a complete saga .(1).pptxRice Genome  Project a complete saga .(1).pptx
Rice Genome Project a complete saga .(1).pptx
SoumyaDixit11
 
bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...
bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...
bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...
muralinath2
 

Recently uploaded (20)

Lake classification and Morphometry.pptx
Lake classification and Morphometry.pptxLake classification and Morphometry.pptx
Lake classification and Morphometry.pptx
 
MARIGREEN PROJECT - overview, Oana Cristina Pârvulescu
MARIGREEN PROJECT - overview, Oana Cristina PârvulescuMARIGREEN PROJECT - overview, Oana Cristina Pârvulescu
MARIGREEN PROJECT - overview, Oana Cristina Pârvulescu
 
atom, elements, molecule and compounds in telugu.pptx
atom, elements, molecule and compounds in telugu.pptxatom, elements, molecule and compounds in telugu.pptx
atom, elements, molecule and compounds in telugu.pptx
 
Fish in the Loop: Exploring RAS - Julie Hansen Bergstedt
Fish in the Loop: Exploring RAS - Julie Hansen BergstedtFish in the Loop: Exploring RAS - Julie Hansen Bergstedt
Fish in the Loop: Exploring RAS - Julie Hansen Bergstedt
 
Pancreas_functional anatomy_enzymes.pptx
Pancreas_functional anatomy_enzymes.pptxPancreas_functional anatomy_enzymes.pptx
Pancreas_functional anatomy_enzymes.pptx
 
PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...
PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...
PART 1 & PART 2 The New Natural Principles of Newtonian Mechanics, Electromec...
 
SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...
SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...
SOFIA/HAWC+ FAR-INFRARED POLARIMETRIC LARGE-AREA CMZ EXPLORATION (FIREPLACE) ...
 
Potential of Marine Renewable and Non renewable energy.pptx
Potential of Marine Renewable and Non renewable energy.pptxPotential of Marine Renewable and Non renewable energy.pptx
Potential of Marine Renewable and Non renewable energy.pptx
 
Phytoremediation: Harnessing Nature's Power with Phytoremediation
Phytoremediation: Harnessing Nature's Power with PhytoremediationPhytoremediation: Harnessing Nature's Power with Phytoremediation
Phytoremediation: Harnessing Nature's Power with Phytoremediation
 
Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...
Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...
Detection of the elusive dangling OH ice features at ~2.7 μm in Chamaeleon I ...
 
NuGOweek 2024 Ghent programme__flyer.pdf
NuGOweek 2024 Ghent programme__flyer.pdfNuGOweek 2024 Ghent programme__flyer.pdf
NuGOweek 2024 Ghent programme__flyer.pdf
 
17. 20240529_Ingrid Olesen_MariGreen summer school.pdf
17. 20240529_Ingrid Olesen_MariGreen summer school.pdf17. 20240529_Ingrid Olesen_MariGreen summer school.pdf
17. 20240529_Ingrid Olesen_MariGreen summer school.pdf
 
Composting blue materials - Joshua Cabell
Composting blue materials - Joshua CabellComposting blue materials - Joshua Cabell
Composting blue materials - Joshua Cabell
 
Classification and role of plant nutrients - Roxana Madjar
Classification and role of plant nutrients - Roxana MadjarClassification and role of plant nutrients - Roxana Madjar
Classification and role of plant nutrients - Roxana Madjar
 
Surface properties of the seas of Titan as revealed by Cassini mission bistat...
Surface properties of the seas of Titan as revealed by Cassini mission bistat...Surface properties of the seas of Titan as revealed by Cassini mission bistat...
Surface properties of the seas of Titan as revealed by Cassini mission bistat...
 
Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...
Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...
Accessing Data to Support Pesticide Residue and Emerging Contaminant Analysis...
 
Bioconversion of sago waste and oil cakes into biobutanol using Environmental...
Bioconversion of sago waste and oil cakes into biobutanol using Environmental...Bioconversion of sago waste and oil cakes into biobutanol using Environmental...
Bioconversion of sago waste and oil cakes into biobutanol using Environmental...
 
Data Visualization Workshop for Summer Interns
Data Visualization Workshop for Summer InternsData Visualization Workshop for Summer Interns
Data Visualization Workshop for Summer Interns
 
Rice Genome Project a complete saga .(1).pptx
Rice Genome  Project a complete saga .(1).pptxRice Genome  Project a complete saga .(1).pptx
Rice Genome Project a complete saga .(1).pptx
 
bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...
bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...
bloodclotfactorsprocoagulantsexstrinsicintrinsicfactors-240607054610-6895d6e5...
 

計算材料学

  • 1. 計算材料学 Computational Materials Science 東京工業大学大学院総合理工学研究科2014 年度冬学期 Tokyo Tech Winter Semester, AY2014 October 17, 2014 合田義弘 Y. Gohda はじめに/Introduction 計算機を用いたシミュレーションは実験と相補的な立場にある材料学研究に欠かせないツールで ある。本講義の内、合田担当分では第一原理計算を中心に電子状態の数値解析について扱う。第 一原理計算とは、実験的に決定されたパラメーターを用いず、素電荷e、Planck 定数! 等の基礎 的な物理定数と計算精度をコントロールする数値解析的パラメーターのみによる非経験的電子状 態計算の事である。 この講義ノートは http://www.cms.materia.titech.ac.jp/lecture_cms.pdf においてダウンロード可能である(要パスワード)。質問は歓迎する。講義中にどんどんしてほし い。参考書としては 押山淳ほか:岩波講座計算科学3 「計算と物質」 R.M. Martin: Electronic Structure 日本語訳あり J.M. ティッセン:「計算物理学」 J.J. Sakurai: Modern Quantum Mechanics 日本語訳あり を勧める。 第一原理計算を実際に試す事も可能である。Mac あるいはLinux にコードを自分でインストー ルしても良いが、“MateriApps LIVE!” のUSB を入手あるいは作成すれば、手持ちのコンピュー ターで計算環境が整備済のLinux をUSB ブートする事が出来る。“MateriApps LIVE!” に収録さ れているOpenMX コードを勧める。 東京工業大学の国際化推進に向けた方針に鑑み、以下の講義ノートは原則英語で記述する。 1
  • 2. Contents 1 Born-Oppenheimer approximation and many-electron system 3 1.1 Summary of basic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Molecular dynamics and structure optimization. . . . . . . . . . . . . . . . . . . . 6 1.4 Quantum many-body theory of electrons . . . . . . . . . . . . . . . . . . . . . . . 9 2 Hartree-Fock approximation and wave-function theory 12 2.1 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Post-Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Diffusion Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Density functional theory 21 3.1 Hohenberg–Kohn theorem and the Kohn-Sham equation . . . . . . . . . . . . . . 21 3.2 Exchange-correlation functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Beyond standard DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Basis sets 28 4.1 Without basis sets: real-space discretization . . . . . . . . . . . . . . . . . . . . . 28 4.2 Plane-wave basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Pseudo-atomic orbitals as a basis set . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Examples 33 5.1 Diatomic molecules: N2 and O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 Microstructure interfaces of magnetic materials . . . . . . . . . . . . . . . . . . . 41 2
  • 3. 5.2 Graphene Brillouin zone of the hexagonal lattice lattice vectors: a = a ! 1 2 ex − √3 2 ey " , b = a ! 1 2 ex + √3 2 ey " , c = cez . reciprocal-lattice vectors: Ga = 2π b × c a · (b × c) = 4π √3a2c b × c = 4π √3a !√3 2 ex − 1 2 ey " , Gb = 2π c × a a · (b × c) = 4π √3a2c c × a = 4π √3a !√3 2 ex + 1 2 ey " , Gc = 2π a × b a · (b × c) = 4π √3a2c a × b = 2π c ez . Γ: Center of the Brillouin zone K: Middle of an edge joining two rectangular faces→ (Ga + Gb)/3 M: Center of a rectangular face → Ga/2 A: Center of a hexagonal face → Gc/2 H: Corner point → (Ga + Gb)/3 + Gc/2 L: Middle of an edge joining a hexagonal and a rectangular face → Ga/2 + Gc/2 10 5 0 −5 Energy relative to EF [eV] −10 Γ KM Γ A H L Γ Figure 6: Band structure of the normal state of a superconductor, MgB2. 34
  • 4. Unit cell and atomic structure of graphene The computational cell for graphene is hexagonal one. (a) (b) Figure 7: The unit cell and the atomic struc-ture of graphene used for first-principles calcu-lations: (a) the side view and (b) the top view. Input file of OpenMX to calculate electronic states of graphene # # OpenMX Inputfile # # File Name # System.Name graphene DATA.PATH /home/gohda/openmx/DFT_DATA13 # Definition of Atomic Species # Species.Number 2 <Definition.of.Atomic.Species C C6.0-s2p2d1 C_PBE13 E Rn13.0-s2p2d2f1 E Definition.of.Atomic.Species> # Atoms # Atoms.Number 3 Atoms.SpeciesAndCoordinates.Unit FRAC # Ang|AU <Atoms.SpeciesAndCoordinates 1 C 0.33333333333333 0.66666666666667 0.50000000000000 2.0 2.0 2 C 0.66666666666667 0.33333333333333 0.50000000000000 2.0 2.0 3 E 0.00000000000000 0.00000000000000 0.50000000000000 0.0 0.0 Atoms.SpeciesAndCoordinates> 35
  • 5. Atoms.UnitVectors.Unit Ang # Ang|AU <Atoms.UnitVectors 1.23000000000000 -2.13042249330972 0.00000000000000 1.23000000000000 2.13042249330972 0.00000000000000 0.00000000000000 0.00000000000000 10.00000000000000 Atoms.UnitVectors> # SCF or Electronic System # scf.XcType GGA-PBE # LDA|LSDA-CA|LSDA-PW|GGA-PBE scf.SpinPolarization off # On|Off|NC scf.energycutoff 300.0 # default=150 (Ry) scf.maxIter 100 # default=40 scf.EigenvalueSolver band # DC|GDC|Cluster|Band scf.Kgrid 11 11 1 # means n1 x n2 x n3 scf.Mixing.Type rmm-diisk # Simple|Rmm-Diis|Gr-Pulay|Kerker|Rmm-Diisk scf.Init.Mixing.Weight 0.30 # default=0.30 scf.Min.Mixing.Weight 0.001 # default=0.001 scf.Max.Mixing.Weight 0.700 # default=0.40 scf.Mixing.History 10 # default=5 scf.Mixing.StartPulay 5 # default=6 scf.Mixing.EveryPulay 1 # default=5 scf.criterion 1.0e-6 # default=1.0e-6 (Hartree) scf.lapack.dste dstevx # dstegr|dstedc|dstevx, default=dstevx # DOS and PDOS # Dos.fileout on # on|off, default=off Dos.Erange -20.0 10.0 # default = -20 20 Dos.Kgrid 48 48 1 # default = Kgrid1 Kgrid2 Kgrid3 # Band dispersion # Band.dispersion on # on|off, default=off Band.Nkpath 4 <Band.kpath 200 0.0 0.0 0.0 0.33333333333333 0.33333333333333 0.0 G K 100 0.33333333333333 0.33333333333333 0.0 0.5 0.0 0.0 K M 173 0.5 0.0 0.0 0.0 0.0 0.0 M G 37 0.0 0.0 0.0 0.0 0.0 0.5 G A Band.kpath> # Others # scf.restart on scf.fixed.grid 0.00000000000000 0.00000000000000 0.00000000000000 36
  • 6. Band strucuture and density of states You can excecute OpenMX by typing on a UNIX (MacOSX, Linux, etc.) terminal as follows: $ openmx INCAR >log 2>&1 & 5 (a) (b) 0 −10 Γ K M Γ A −5 Energy relative to EF [eV] −15 −20 0 −20 −15 −10 −5 0 5 10 Energy relative to EF [eV] Density of states [arb. unit] w EA w/o EA w EA w/o EA (c) Figure 8: (a) The energy band and (b) the de-sity of states for graphene obtained by first-principles calculations with enpty-atom basis functions (red) and without them (black). (c) Nearly-free-electron states that cannot be de-scribed without the empty atom. Comparison with the tight-binding model with the H¨uckel approximation In the Huckel ¨approximation, only the |φpz ⟩ state is considered for the frontier orbitals of benzene rings. The tight-binding Hamiltonian using the notation of |I⟩ = |φI pz ⟩ is H = # I εI |I⟩⟨I|− # <IJ> t|I⟩⟨J| = − # <IJ> t|I⟩⟨J| , (23) where t is the transfer integral and we define εI = 0. Using the Bloch sum φB i,k(r) = 1 √Nk # T eik·T φpz (r − Ri − T ) (24) with the approximation of ⟨I|J⟩ = δIJ (Thus Sij = δij), the matrix elements are ⟨φB i,k|H|φB i,k⟩ = 0 , and (25) ⟨φB i,k|H|φB j,k⟩ = ⟨φB i,k⟩∗ = −t(eik·T0 + eik·T1 + eik·T2) j,k|H|φB = −t(1 + eik·a + e−ik·b) ≡ −tf . (26) 37
  • 7. Thus the transformed Hamiltonian becomes H = −t(f|φB1 ⟩⟨φB2 | + f∗|φB2 ⟩⟨φB1 |) . (27) Diagonalizing H through ε2 − t2|f|2 = 0, we obtain ε = ±t|f| = ±t $ (1 + cos k · a + cos k · b)2 + (sink · a − sin k · b)2 = ±t $ 3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) . (28) At the M point (k = Ga/2), ε = ±t $ 3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) = ±t $ 5 + 4 cosGa/2 · a = ±t√5 + 4 cos π = ±t . (29) As for the Γ–K path, i.e. k = κ(Ga + Gb), ε = ±t $ 3 + 2 cos k · a + 2 cos k · b + 2 cos k · (a + b) = ±t $ 3 + 2 cos κGa · a + 2 cos κGb · b + 2 cos κ(Ga + Gb) · (a + b) = ±t $ 3 + 4 cos (2πκ) + 2 cos(4πκ) (30) Γ K 10 5 0 Energy relative to EF [eV] −5 w EA w/o EA TB Figure 9: Comparison of energy bands ob-tained by first-principles calculations and the tight-binding model with the H¨uckel approxi-mation. The hopping integral was chosen as t = 2.7 eV. 38
  • 8. Γ K M 0 0 0 0 0 0 0 0 0 0 0 π 0 π π 0 0 π π 0 0 π π π π π π B1 Figure 10: The phase θ = k · T appear in the Bloch sum for the A sublattice (i = 1), % φk(r) = √1 ,Nk T eik·T φpz (r − R1 − T ). Let us take a closer look of the wave functions: At the Γ point: H = −3t(|φB1 ⟩⟨φB2 | + |φB2 ⟩⟨φB1 |) , ε= ±3t , (31) |π⟩ = 1 √2 (|φB1 ⟩ + |φB2 ⟩) , |π∗⟩ = 1 √2 (|φB1 ⟩ − |φB2 ⟩) . (32) In general: H = −t(f|φB1 ⟩⟨φB2 | + f∗|φB2 ⟩⟨φB1 |) , ε= ±t|f| , (33) |π⟩ = 1 √2 & |φB1 ⟩ + f∗ |f||φB2 ⟩ ' , |π∗⟩ = 1 √2 & |φB1 ⟩ − f∗ |f||φB2 ⟩ ' . (34) At the K point: H = 0 , ε= 0 , (35) |π⟩ = cπ,1|φB1 ⟩ + cπ,2|φB2 ⟩ , |π∗⟩ = cπ∗,1|φB1 ⟩ + cπ∗,2|φB2 ⟩ . (36) Exercise For k vectors around the K point, k = 13 (Ga + Gb) + δkxex + δkyey, derive the effective Hamiltonian using the Talor expansion starting from the form H = ⎛ ⎝ 0 −tf −tf ∗ 0 ⎞ ⎠ . Use the Pauli matrices σ: σx = ⎛ ⎝ 0 1 1 0 ⎞ ⎠ , σy = ⎛ ⎝ 0 −i i 0 ⎞ ⎠ , and σz = ⎛ ⎝ 1 0 0 1 ⎞ ⎠ . 39
  • 9. Answer: Exercise Calculate i) the eigenenergy and ii) the sign of the linear combination for π orbitals in a butadiene molecule CH2CHCHCH2, a virtual fragment of graphene, using the H¨uckel approximation. Hint: 2 < √5 < 3 Answer: 40