electronic structure, density- functional theory, plane waves

electronic structure, density-functional theory, plane waves

Stefano BaroniScuola Internazionale Superiore di Studi Avanzati and

DEMOCRITOS National Simulation CenterTrieste - Italy

a quick overview of terms and concepts

the saga of time and length scales

10-15 10-12 10-9 10-6 10-3

time [s]

length [m]

10-9

10-6

10-3

nano scale! = 1

! = 0macro scale


10-15 10-12 10-9 10-6 10-3

time [s]

length [m]

10-9

10-6

10-3

nano scale! = 1

! = 0macro scale

hic sunt leones


10-15 10-12 10-9 10-6 10-3

time [s]

length [m]

10-9

10-6

10-3

nano scale! = 1

! = 0macro scale

hic sunt leones

thermodynamics& finite elements

kinetic Monte Carlo

electronic structure methods

classical moleculardynamics

size vs. accuracy

quantum many-body methods quantum Monte Carlo MP2, CCSD(T), CI GW, BSE

accuracy

size

/du

rati

on

classical empirical methods pair potentials force fields shell models

quantum empirical methods tight-binding embedded atom

quantum self-consistent methods density Functional Theory Hartree-Fock

size vs. accuracy

quantum many-body methods quantum Monte Carlo MP2, CCSD(T), CI GW, BSE

accuracy

size

/du

rati

on

classical empirical methods pair potentials force fields shell models

quantum empirical methods tight-binding embedded atom

quantum self-consistent methods density Functional Theory Hartree-Fock

ab initio simulations

i!∂Φ(r,R; t)∂t

=(− !2

2M

∂2

∂R2− !2

2m

∂2

∂r2+ V (r,R)

)Φ(r,R; t)

MR = −∂E(R)∂R(

− !2

2m

∂2

∂r2+ V (r,R)

)Ψ(r|R) = E(R)Ψ(r|R)

ab initio simulations

The Born-Oppenheimer approximation (M≫m)

i!∂Φ(r,R; t)∂t

=(− !2

2M

∂2

∂R2− !2

2m

∂2

∂r2+ V (r,R)

)Φ(r,R; t)

density-functional theory

V (r,R) =e2

2ZIZJ

|RI −RJ | −ZIe2

|ri −RI | +e2

21

|ri − rj |


V (r,R) =e2

2ZIZJ

|RI −RJ | −ZIe2

|ri −RI | +e2

21

|ri − rj |


DFT

V (r,R) =e2

2ZIZJ

|RI −RJ | −ZIe2

|ri −RI | +e2

21

|ri − rj |

V (r,R)→ e2

2ZIZJ

|RI −RJ | + v[ρ](r)

ρ(r) =∑

v

|ψv(r)|2


DFT

V (r,R) =e2

2ZIZJ

|RI −RJ | −ZIe2

|ri −RI | +e2

21

|ri − rj |

V (r,R)→ e2

2ZIZJ

|RI −RJ | + v[ρ](r)

ρ(r) =∑

v

|ψv(r)|2Kohn-Sham Hamiltonian


DFT

V (r,R) =e2

2ZIZJ

|RI −RJ | −ZIe2

|ri −RI | +e2

21

|ri − rj |

V (r,R)→ e2

2ZIZJ

|RI −RJ | + v[ρ](r)

(− !2

2m

∂2

∂r2+ v[ρ](r)

)ψv(r) = εvψv(r)

functionals

G[f ] : f !→ R

functionals

G[f ] : f !→ Rexamples:G[f ] = f(x0)

G[f ] =∫ b

af2(x)dx

G[f ] =∫ b

a|f ′(x)|2dx

· · ·

functionals

x1

f(x)

x2 ... xi xN

f1f2

... fi

fN

G[f ] ≈ g(f1, f2, · · · , fN )G[f ] ≈ g(c1, c2, · · · , cN )

approximations:


G[f ] =∫ b

af2(x)dx

G[f ] =∫ b

a|f ′(x)|2dx

· · ·

functionals

x1

f(x)

x2 ... xi xN

f1f2

... fi

fN

f(x) ≈∑

n

cnφn(x)

G[f ] ≈ g(f1, f2, · · · , fN )G[f ] ≈ g(c1, c2, · · · , cN )

approximations:


G[f ] =∫ b

af2(x)dx

G[f ] =∫ b

a|f ′(x)|2dx

· · ·

functional derivatives

G[f0 + εf1] = G[f0] + ε

∫f1(x)

δG

δf(x)

∣∣∣∣f=f0

dx +O(ε2

)


G[f0 + εf1] = G[f0] + ε

∫f1(x)

δG

δf(x)

∣∣∣∣f=f0

dx +O(ε2

)

δG

δf(x)

∣∣∣∣f=f0

≈ 1h

∂g

∂fix1

f(x)

x2 ... xi xN

f1f2

... fi

fN


δG

δf(x)“ = ” lim

ε→0

G[f(•)− εδ(•− x)]−G[f(•)]ε

G[f0 + εf1] = G[f0] + ε

∫f1(x)

δG

δf(x)

∣∣∣∣f=f0

dx +O(ε2

)

δG

δf(x)

∣∣∣∣f=f0

≈ 1h

∂g

∂fix1

f(x)

x2 ... xi xN

f1f2

... fi

fN

the Hellmann-Feynman theorem

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


H(λ)Ψλ = E(λ)Ψλ

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]


E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]


E(R) = minΨ

(E[Ψ, R]

)

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]∂G

∂x

∣∣∣∣x=x(λ)

= 0


E(R) = minΨ

(E[Ψ, R]

)

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]∂G

∂x

∣∣∣∣x=x(λ)

= 0

g(λ) = G[x(λ), λ]


E(R) = minΨ

(E[Ψ, R]

)

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]∂G

∂x

∣∣∣∣x=x(λ)

= 0

g(λ) = G[x(λ), λ] g′(λ) = x′(λ)∂G

∂x

∣∣∣∣x=x(λ)

+∂G

∂λ


E(R) = minΨ

(E[Ψ, R]

)

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]∂G

∂x

∣∣∣∣x=x(λ)

= 0

g(λ) = G[x(λ), λ] g′(λ) = x′(λ)∂G

∂x

∣∣∣∣x=x(λ)

+∂G

∂λ


E(R) = minΨ

(E[Ψ, R]

)

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1


g(λ) = minx

G[x, λ]∂G

∂x

∣∣∣∣x=x(λ)

= 0

g(λ) = G[x(λ), λ] g′(λ) = x′(λ)∂G

∂x

∣∣∣∣x=x(λ)

+∂G

∂λ

E′(λ) = 〈Ψλ|H ′(λ)|Ψλ〉


E(R) = minΨ

(E[Ψ, R]

)

E(λ) = minΨ

〈Ψ|H(λ)|Ψ〉

〈Ψ|Ψ〉 = 1

conjugate variables & Legendre transforms

E = E(V, x)

V


E = E(V, x)

V

P = −∂E

∂V

V


E = E(V, x)

V

P = −∂E

∂V

V

Legendre transform: H(P,X) = E + PV

properties: • E convex⇒ V ! P

• H(P,X) = maxV

(E(V,X) + PV

)

• Hellmann-Feynman:∂H

∂x=

∂E

∂x• H concave

• E(V,X) = minP

(H(P,X)− PV

)


E = E(V, x)

V

P = −∂E

∂V

V



• H(P,X) = maxV

(E(V,X) + PV

)


∂x=

∂E

∂x• H concave

• E(V,X) = minP

(H(P,X)− PV

)


E = E(V, x)

V

P = −∂E

∂V

V



• H(P,X) = maxV

(E(V,X) + PV

)


∂x=

∂E

∂x• H concave

• E(V,X) = minP

(H(P,X)− PV

)


E = E(V, x)

V

P = −∂E

∂V

V



• H(P,X) = maxV

(E(V,X) + PV

)


∂x=

∂E

∂x• H concave

• E(V,X) = minP

(H(P,X)− PV

)


E = E(V, x)

V

P = −∂E

∂V

V



• H(P,X) = maxV

(E(V,X) + PV

)


∂x=

∂E

∂x• H concave

• E(V,X) = minP

(H(P,X)− PV

)


E = E(V, x)

V

P = −∂E

∂V

V


Hohenberg-Kohn DFT

H = − !2

2m

∑

i

∂2

∂r2i

+12

∑

i !=j

e2

|ri − rj | +∑

i

V (ri)

Hohenberg-Kohn DFT

H = − !2

2m

∑

i

∂2

∂r2i

+12

∑

i !=j

e2

|ri − rj | +∑

i

V (ri)

E[V ] = minΨ

〈Ψ|K + W + V |Ψ〉

= minΨ

[〈Ψ|K + W |Ψ〉 +

∫ρ(r)V (r)dr

]

Hohenberg-Kohn DFT

H = − !2

2m

∑

i

∂2

∂r2i

+12

∑

i !=j

e2

|ri − rj | +∑

i

V (ri)

properties: • E[V ] is convex (requires some work to demonstrate)

• ρ(r) =δE

δV (r)(from Hellmann-Feynman)

E[V ] = minΨ

〈Ψ|K + W + V |Ψ〉

= minΨ

[〈Ψ|K + W |Ψ〉 +

∫ρ(r)V (r)dr

]

Hohenberg-Kohn DFT

H = − !2

2m

∑

i

∂2

∂r2i

+12

∑

i !=j

e2

|ri − rj | +∑

i

V (ri)


• ρ(r) =δE


E[V ] = minΨ

〈Ψ|K + W + V |Ψ〉

= minΨ

[〈Ψ|K + W |Ψ〉 +

∫ρ(r)V (r)dr

]

consequences: • V (r) ! ρ(r) (1st HK theorem)

• F [ρ] = E −∫

V (r)ρ(r)dr is the Legendre transform of E

• E[V ] = minρ

[F [ρ] +

∫V (r)ρ(r)dr

](2nd HK theorem)

Hohenberg-Kohn DFT

H = − !2

2m

∑

i

∂2

∂r2i

+12

∑

i !=j

e2

|ri − rj | +∑

i

V (ri)


• ρ(r) =δE


E[V ] = minΨ

〈Ψ|K + W + V |Ψ〉

= minΨ

[〈Ψ|K + W |Ψ〉 +

∫ρ(r)V (r)dr

]

E[V ] = minρ

[F [ρ] +

∫V (r)ρ(r)dr

]

consequences: • V (r) ! ρ(r) (1st HK theorem)

• F [ρ] = E −∫

V (r)ρ(r)dr is the Legendre transform of E

• E[V ] = minρ

[F [ρ] +

∫V (r)ρ(r)dr

](2nd HK theorem)

Kohn-Sham DFT

F [ρ] = T0[ρ] +e2

2

∫ρ(r)ρ(r′)|r− r′| drdr′ + Exc[ρ]

Kohn-Sham DFT

F [ρ] = T0[ρ] +e2

2


δT0

δρ(r)+ e2

∫ρ(r′)

|r− r′|dr′ +

δExc

δρ(r)+ V (r) = µ

Kohn-Sham DFT

F [ρ] = T0[ρ] +e2

2


δT0

δρ(r)+

vKS [ρ](r)︷︸︸︷

e2

∫ρ(r′)

|r− r′|dr′ +

δExc

δρ(r)+ V (r) = µ

Kohn-Sham DFT

F [ρ] = T0[ρ] +e2

2


(− !2

2m∇2 + vKS [ρ](r)

)ψv(r) = εvψv(r)

δT0

δρ(r)+

vKS [ρ](r)︷︸︸︷

e2

∫ρ(r′)

|r− r′|dr′ +

δExc

δρ(r)+ V (r) = µ

ρ(r) =∑

v

|ψv(r)|2θ(εv − µ)

KS equations from functional minimization

E[ψ,R] = − !2

2m

∑

v

∫ψ∗

v(r)∂2ψv(r)

∂r2dr +

∫V (r,R)ρ(r)dr+

e2

2



E[ψ,R] = − !2

2m

∑

v

∫ψ∗

v(r)∂2ψv(r)

∂r2dr +

∫V (r,R)ρ(r)dr+

e2

2


E(R) = minψ

(E[ψ,R]

)

∫ψ∗

u(r)ψv(r)dr = δuv


E[ψ,R] = − !2

2m

∑

v

∫ψ∗

v(r)∂2ψv(r)

∂r2dr +

∫V (r,R)ρ(r)dr+

e2

2


E(R) = minψ

(E[ψ,R]

)

∫ψ∗

u(r)ψv(r)dr = δuv

δEKS

δψ∗v(r)

=∑

uv

Λvuψu(r)


E[ψ,R] = − !2

2m

∑

v

∫ψ∗

v(r)∂2ψv(r)

∂r2dr +

∫V (r,R)ρ(r)dr+

e2

2


E(R) = minψ

(E[ψ,R]

)

∫ψ∗

u(r)ψv(r)dr = δuv

(− !2

2m∇2 + vKS [ρ](r)

)ψv(r) = εvψv(r)

δEKS

δψ∗v(r)

=∑

uv

Λvuψu(r)

solving the Kohn-Sham equations

ψv(r) ! c(v, j)

ψv(r) =∑

j

c(j, v)ϕj(r)


ψv(r) ! c(v, j)

ψv(r) =∑

j

c(j, v)ϕj(r)

δEKS

δψ∗v(r)

=∑

uv

Λvuψu(r)


ψv(r) ! c(v, j)

∑

j

hKS [c](i, j)c(j, v) = εvc(i, v)

ψv(r) =∑

j

c(j, v)ϕj(r)

δEKS

δψ∗v(r)

=∑

uv

Λvuψu(r)


ψv(r) ! c(v, j)

∑

j

hKS [c](i, j)c(j, v) = εvc(i, v)

c(i, v) = −∑

j

hKS [c](i, j)c(j, v)+

∑

u

Λvuc(i, v)

ψv(r) =∑

j

c(j, v)ϕj(r)

δEKS

δψ∗v(r)

=∑

uv

Λvuψu(r)

requirements

requirements

‣ (effective) completeness easily checked and systematically improved

requirements


‣ matrix elements easy to calculate and/or Hψ products easily calculated on the fly

requirements



‣ Hartree and XC potentials easy to represent and calculate

requirements



‣ Hartree and XC potentials easy to represent and calculate

‣ orthogonality is a plus

plane waves

periodic boundary conditions

ϕj(r) =1√Ω

eiqj ·r !2

2mq2

j ≤ Ecut

ϕ(x + ") = ϕ(x)→ qn =2π

"n

plane waves


ϕj(r) =1√Ω

eiqj ·r !2

2mq2

j ≤ Ecut

ϕ(x + ") = ϕ(x)→ qn =2π

"n

q = G

finite systems (! = a)

plane waves


ϕj(r) =1√Ω

eiqj ·r !2

2mq2

j ≤ Ecut

ϕ(x + ") = ϕ(x)→ qn =2π

"n

q = G

finite systems (! = a)

ψvk(r) = eik·rukv(r)k ∈ BZ u(x + a) = u(x)

infinite crystals (! = L)

using plane waves

−∇2ψ(r) #−→ |k + G|2cnk(G)

V (r)ψ(r) #−→∫

e−iG·rV (r)unk(r)dr

ρ(r) =∑

vk

|uvk(r)|2

Vxc(r) = µxc

(ρ(r)

)

VH(r) = e2

∫ρ(r′)

|r− r′|dr′

= e2∑

G"=0

eiG·r 4π

G2ρ(G)

PWs: pros & cons

PWs: pros & cons

approach to completeness easily and systematically checked (|k+G|2<Ecut)

PWs: pros & cons


basis set independent of nuclear positions (no Pulay forces)

PWs: pros & cons



matrix elements and Hψ poducts easily calculated

PWs: pros & cons




density, Hartree, and XC potentials easily calculated

PWs: pros & cons





orthonormality

PWs: pros & cons





orthonormality

basis set depends on volume shape/size (Pulay stress)

PWs: pros & cons





orthonormality

basis set depends on volume shape/size (Pulay stress)

uniform spatial resolution (no core states!)

treating core states

1

2

3

4

5

6

7

Perio

d

* The systematic names and symbols for elements greater than 110 will be used until the approval of trivial names by IUPAC.

A team at Lawrence Berkeley National Laboratories reported the discovery of elements 116 and 118 in June 1999.The same team retracted the discovery in July 2001. The discovery of elements 113, 114, and 115 has been reported but not confirmed.

HydrogenSemiconductors(also known as metalloids)

MetalsAlkali metalsAlkaline-earth metalsTransition metalsOther metals

NonmetalsHalogensNoble gasesOther nonmetals

Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass

Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9

Group 1 Group 2 Group 14 Group 15 Group 16 Group 17

Group 18

Group 10 Group 11 Group 12

Group 13

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium

Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton

Aluminum Silicon Phosphorus Sulfur Chlorine Argon

Carbon Nitrogen Oxygen Fluorine Neon

Helium

Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon

Platinum

Darmstadtium Unununium Ununbium Ununtrium UnunpentiumUnunquadium

Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon

Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf

Ni Cu Zn Ga Ge As Se Br Kr

Al Si P S Cl Ar

C N O F Ne

He

Pd Ag Cd In Sn Sb Te I Xe

Pt

Ds Uuu* Uub* Uut* Uup*Uuq*

Au Hg Tl Pb Bi Po At Rn

Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

BLi

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5

The atomic masses listed in this table reflect the precision of current measurements. (Values listed in parentheses are the mass numbers of those radioactive elements’ most stable or most common isotopes.)

1

2

3

4

5

6

7

Perio

d






Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass



Group 18


Group 13

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium




Helium


Platinum



Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf


Al Si P S Cl Ar

C N O F Ne

He


Pt



Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

BLi

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5


ε1s ∼ Z2 a1s ∼1Z

Ecut ∼ Z2


1

2

3

4

5

6

7

Perio

d






Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass



Group 18


Group 13

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium




Helium


Platinum



Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf


Al Si P S Cl Ar

C N O F Ne

He


Pt



Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

BLi

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5


1

2

3

4

5

6

7

Perio

d






Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass



Group 18


Group 13

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium




Helium


Platinum



Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf


Al Si P S Cl Ar

C N O F Ne

He


Pt



Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

BLi

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5


ε1s ∼ Z2 a1s ∼1Z

Ecut ∼ Z2

NPW =4π

3k3

cutΩ

(2π)3

∼ Z3


1

2

3

4

5

6

7

Perio

d






Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass



Group 18


Group 13

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium




Helium


Platinum



Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf


Al Si P S Cl Ar

C N O F Ne

He


Pt



Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

BLi

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5


1

2

3

4

5

6

7

Perio

d






Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass



Group 18


Group 13

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium




Helium


Platinum



Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf


Al Si P S Cl Ar

C N O F Ne

He


Pt



Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

BLi

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5


ε1s ∼ Z2 a1s ∼1Z

Ecut ∼ Z2

NPW =4π

3k3

cutΩ

(2π)3

∼ Z3


1

2

3

4

5

6

7

Perio

d


Key:6

CCarbon12.0107

Atomic number

Symbol

Name

Average atomic mass


Group 1 Group 2

6.941

22.989 770

39.0983

85.4678

132.905 43

(223)

140.116

232.0381

140.907 65

231.035 88

144.24

238.028 91

(145)

(237)

150.36

(244)

9.012 182

24.3050

40.078

87.62

137.327

(226)

44.955 910

88.905 85

138.9055

(227)

47.867

91.224

178.49

(261)

50.9415

92.906 38

180.9479

(262)

51.9961

95.94

183.84

(266)

54.938 049

(98)

186.207

(264)

55.845

101.07

190.23

(277)

58.933 200

102.905 50

192.217

(268)

1.007 94

Lithium

Sodium

Potassium

Rubidium

Cesium

Francium

Cerium

Thorium

Praseodymium

Protactinium

Neodymium

Uranium

Promethium

Neptunium

Samarium

Plutonium

Beryllium

Magnesium

Calcium

Strontium

Barium

Radium

Scandium

Yttrium

Lanthanum

Actinium

Titanium

Zirconium

Hafnium

Rutherfordium

Vanadium

Niobium

Tantalum

Dubnium

Chromium

Molybdenum

Tungsten

Seaborgium

Manganese

Technetium

Rhenium

Bohrium

Iron

Ruthenium

Osmium

Hassium

Cobalt

Rhodium

Iridium

Meitnerium

Hydrogen

Li

V

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

Hf

Rf

Nb

Ta

Db

Cr

Mo

W

Sg

Mn

Tc

Re

Bh

IrOs

Ce

Th

Pr

Pa

Nd

U

Pm

Np

Sm

Pu

Fe

Ru

Hs

Co

Rh

Mt

H

3

11

19

37

55

87

4

12

20

38

56

88

21

39

57

89

22

40

72

104

23

41

73

105

24

42

74

106

25

43

75

107

26

44

76 77

108

27

45

109

1

58

90

59

91

60

92

61

93

62

94


Semiconductors(also known as metalloids)



Group 14 Group 15 Group 16 Group 17

Group 18


Group 13

151.964

(243)

157.25

(247)

158.925 34

(247)

162.500

(251)

164.930 32

(252)

167.259

(257)

168.934 21

(258)

173.04

(259)

174.967

(262)

58.6934 63.546 65.409 69.723 72.64 74.921 60 78.96 79.904 83.798

26.981 538 28.0855 30.973 761 32.065 35.453 39.948

12.0107 14.0067 15.9994 18.998 4032 20.1797

4.002 602

106.42 107.8682 112.411 114.818 118.710 121.760 127.60 126.904 47 131.293

195.078

(281) (272) (285) (284) (288)(289)

196.966 55 200.59 204.3833 207.2 208.980 38 (209) (210) (222)

10.811

Europium

Americium

Gadolinium

Curium

Terbium

Berkelium

Dysprosium

Californium

Holmium

Einsteinium

Erbium

Fermium

Thulium

Mendelevium

Ytterbium

Nobelium

Lutetium

Lawrencium




Helium


Platinum



Boron

Eu

Am

Gd

Cm

Tb

Bk

Dy

Cf


Al Si P S Cl Ar

C N O F Ne

He


Pt



Ho

Es

Er

Fm

Tm

Md

Yb

No

Lu

Lr

B

63

95

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

28 29 30 31 32 33 34 35 36

13 14 15 16 17 18

6 7 8 9 10

2

46 47 48 49 50 51 52 53 54

78

110 111 112 113 115114

79 80 81 82 83 84 85 86

5 ε1s ∼ Z2 a1s ∼1Z

Ecut ∼ Z2

0

10

20

10 20 30 40 50 60 70 80

IP (e

V)

Z

IP ∼ 1 a ∼ 1

first principle of computational physics

first principle of computational physics

when two systems have similar physical and chemical properties, their simulation should

require a similar amount of computer resources

power to the imagination


Chemically similar atoms have potentials with very different, possibly nasty, mathematical properties?


Chemically similar atoms have potentials with very different, possibly nasty, mathematical properties?

Imagine pseudo-atoms whose chemical properties are very similar to those of real atoms, but whose pseudo-potentials are as gentle as possible.

Well ...

properties of pseudopotentials


Vps does not have core states: valence states of any given angular symmetry are the lowest-lying states of that symmetry:

φpsval is nodeless and smooth


Vps does not have core states: valence states of any given angular symmetry are the lowest-lying states of that symmetry:

φpsval is nodeless and smooth

φpsval(r) = φae

val(r) for r > rc

The chemical properties of the pseudo-atom are the same as those of the true atom:

εpsval = εae

val

0 1 2 3 4 5 6 7 8r (a.u.)

Si

3s

0 1 2 3 4 5 6 7 8r (a.u.)

0 1 2 3 4 5 6 7 8r (a.u.)

Si

3s

0 1 2 3 4 5 6 7 8r (a.u.)

0 1 2 3 4 5 6 7 8r (a.u.)

Si0 1 2 3 4 5 6 7 8

r (a.u.)

3p

0 1 2 3 4 5 6 7 8r (a.u.)

0 1 2 3 4 5 6 7 8r (a.u.)

Si0 1 2 3 4 5 6 7 8

r (a.u.)

3p

0 1 2 3 4 5 6 7 8r (a.u.)

0 1 2 3 4 5 6 7 8r (a.u.)

0 1 2 3 4 5 6 7 8r (a.u.)

Si0 1 2 3 4 5 6 7 8

r (a.u.)0 1 2 3 4 5 6 7 8

r (a.u.)0 1 2 3 4 5 6 7 8

r (a.u.)

3d

0 1 2 3 4 5 6 7 8r (a.u.)

0 1 2 3 4 5 6 7 8r (a.u.)

Si0 1 2 3 4 5 6 7 8

r (a.u.)0 1 2 3 4 5 6 7 8

r (a.u.)0 1 2 3 4 5 6 7 8

r (a.u.)

3d

0 1 2 3 4 5 6 7 8r (a.u.)

US pseudopotentials

US pseudopotentials

HUSφn = εnSφn 〈φn|S|φm〉 = δnm

using fast Fourier transforms

f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs

ei2π(pk+ql+sm)f(pqs)

f(pqs) =∑

klm

e−i2π(pk+ql+sm)f(klm)

FFT

FFT−1

f(G) f(pG1 + qG2 + sG3) f

(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1


f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1

f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1

f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1


f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1

f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1

f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1

f(G) =1

Ω

∫

f(r)e−iG·rdr ≈

1

N3

∑

klm

f

rklm︷︸︸︷(

k

N1a1 +

l

N2a2 +

m

N3a3

)

e−iG·rklm

f(r) =∑

pqs

f (pG1 + qG2 + sG3)︸︷︷︸

Gpqs

e−iGpqs·r

Gpqs · rklm =2π

N(pk + ql + sm)

f(klm) =∑

pqs


f(pqs) =∑

klm


FFT

FFT−1


(k

N1a1 +

l

N2a2 +

m

N3a3

)scatter

gather

FFT

FFT−1

1

Fourier analysis

Φ(x) =∑

n

ϕ(x− n")

Φ(x + !) = Φ(x)!

Φ

Fourier analysis

Φ(x) =∑

n

ϕ(x− n")

Φ(x + !) = Φ(x)!

Φ

Φ(x) =∑

q

Φ(q)eiqx qk = k2π

"

Fourier analysis

Φ(x) =∑

n

ϕ(x− n")

Φ(x + !) = Φ(x)!

Φ

Φ(x) =∑

q

Φ(q)eiqx qk = k2π

" Φ

2π/"

Φ(q) =1!

∫ !

0Φ(x)e−iqxdx

=1!

∫ ∞

−∞ϕ(x)e−iqxdx

=1!ϕ(q)

sampling theoremϕ

a

ϕ(x) = 0 for |x| >a

2

ϕ

∆q

sampling theoremϕ

a

ϕ(x) = 0 for |x| >a

2

∆q <2π

a

ϕ

∆q

Φ

2π/∆q

sampling theoremϕ

a

ϕ(x) = 0 for |x| >a

2

∆q <2π

a

ϕ

∆q

Φ

2π/∆q

sampling theoremϕ

a

ϕ(x) = 0 for |x| >a

2

ϕ

∆q

∆q <2π

a

∆q >2π

a

ϕ

∆q

Φ

2π/∆q

sampling theoremϕ

a

ϕ(x) = 0 for |x| >a

2

ϕ

∆q

∆q <2π

a

∆q >2π

a

Φ

2π/∆q

using periodic boundary conditions

finite systems


ψ(x + ") = ψ(x)" = a

finite systems

ψ(x) =∑

G

c(G)eiGx

G1

G2


ψ(x + ") = ψ(x)" = a

finite systems

G1

G2

using periodic boundary conditionsinfinite crystals

ψ(x + ") = ψ(x)" = L

ψ(x + a) = eikaψ(x)

ψk(x) = eikxuk(x)uk(x + a) = uk(x)

G1

G2


ψ(x + ") = ψ(x)" = L



uk(x) =∑

G

ck(G)eiGx

kn =2π

Ln

G1

G2


ψ(x + ") = ψ(x)" = L



uk(x) =∑

G

ck(G)eiGx

kn =2π

Ln

G1

G2


ψ(x + ") = ψ(x)" = L



uk(x) =∑

G

ck(G)eiGx

kn =2π

Ln

these slides athttp://stefano.baroni.me/presentations

Documents

electronic structure, density- functional theory, plane waves