Quantum mechanics calculation - tse.ens.titech.ac.jp

Quantum mechanics calculation for material science

By

Kunio TAKAHASHI

Tokyo Institute of Technology,

Tokyo 152-8552, JAPAN

Phone/Fax +81-3-5734-3915

[email protected]

http://www.mep.titech.ac.jp/KTakahashi/

This is a material for the class of Takahashi. Bonus point will be given to the student who has found some mistype etc… and report it to Takahashi. Thank you.

1. Classical mechanics to quantum mechanics ( keywords : analytical mechanics, matrix mechanics, wave mechanics )

Equation of motion

– Newton

– Hamilton’s canonical equations

Matter wave ( de Broglie wave )

Uncertainty relation

Real number to linear operator as quantity

Eigenvalue as observable and

Wave function which express a state of system

Schrödinger equation

When Hamiltonian is independent of time.

xmF i

i

dp

dH

dt

dq

i

i

dq

dH

dt

dp

H

ti )()( qq H

ph /

2/ pq

zip

yip

xip zyx

,,

dxdydzzyxzyx ),,(1),,(*

1),,(1),,(* dxdydzzyxzyx

dxdydzzyxzyx ),,(),,(* Χ

2. Equation to be solved Hartree Fock approximation: Materials we are considering are ...


Hamiltonian

– Even if assuming only Coulomb interaction

Wave function; Slater determinant

– Anti-symmetric property of Fermion

– Molecular orbitals must be arbitrary.

Do you know exact solution ?

H

M

baba ba

baM

a

N

i ia

aN

jiji ji

N

i

iR

eZZ

r

eZ

R

e

m)(1, ,,i

2

1 1 ,

2

)(1, ,,e

2

01

22

4

1

2

H

Electrons

Ions

1, 2, 3 ,,, i ,,, j ,,, N-1, N

1, 2, 3 ,,, a ,,, b ,,, M-1, M

a+Z e

-e

ra,i

i,a,bR

e,i,jR

)()(

)()(

!

1),,(

1

111

1

NNN

N

NN

rr

rr

rr

3. Simple examples (1) Materials we are considering are …

One electron systems

– One ionic core : H ( neutral atom )

: He+, Li

2+, Be

3+ , B

4+ , C

5+ , N

6+ , O

7+ ,,, ( ions )

– Many ionic cores : H2

+ ( ion )

: charged molecules

Many electron systems

– One ionic core : He, Li, Be, B, C, N, O,,, ( isolated atoms )

– Many ionic cores : general problem we want to solve.

Jellium : approximation for metals

a+Z e

-e

a+Z e

-e

a+Z e-e

b+Z e

-e

-e

)()(

)()(

!

1),,(

1

111

1

NNN

N

NN

rr

rr

rr

M

baba ba

baM

a

N

i ia

aN

jiji ji

N

i

iR

eZZ

r

eZ

R

e

m)(1, ,,i

2

1 1 ,

2

)(1, ,,e

2

01

22

4

1

2

H

E

Positive charge density distribution

0

3. Simple examples (2) Materials we are considering are …

One electron systems

– One ionic core : H ( neutral atom )

: He+, Li

2+, Be

3+ , B

4+ , C

5+ , N

6+ , O

7+ ,,, ( ions )

– Many ionic cores : H2

+ ( ion )

: charged molecules

Many electron systems

– One ionic core : He, Li, Be, B, C, N, O,,, ( isolated atoms )

– Many ionic cores : general problem we want to solve.

Jellium : approximation for metals

M

baba ba

baM

a

N

i ia

aN

jiji ji

N

i

iR

eZZ

r

eZ

R

e

m)(1, ,,i

2

1 1 ,

2

)(1, ,,e

2

01

22

4

1

2

H

)()(

)()(

!

1),,(

1

111

1

NNN

N

NN

rr

rr

rr

4. Isolated H atom (1) as a one electron and one ion system

Born-Oppenheimer approximation r

e

m 0

22

2

42

H EH

)exp(2

1cos

!

!

2

12

exp2

})!{(2

)!1(2,,

0

12

00

3

3

0

,,

imPml

mll

na

rL

na

r

na

r

lnn

ln

nar

m

l

l

ln

l

mln

1,3,2,1 ln

ml ,2,1,0

,2,1,0m

2

2

00

4

ema

e

3

2)1(12

!2

211

!1

1

!1

!1

ln

lnlnlnl

ln

lnlnlnln

ln

ln

lnL

l

ml

ml

l

m

m

l zdz

d

l

zzP 1

!2

1 222

0

12

00

3

3

0

exp2

!2

!12

na

rL

na

r

na

r

lnn

ln

narR l

ln

l

rRrmln ,,,,

cos!

!

2

12 m

lpml

mll

)exp(2

1

im

2

2

0

2

4 1

42

n

meEn

+e

-e

; principal q. n.

; azimuthal q. n.

; magnetic q.n.

quantum numbers

;associated Legendre polynomials

;associated Laguerre polynomials

0 10 20 30 40

Distance r/a0

0

0.05

0.1

0.15Y

-AX

IS

0

12

00

3

3

0

exp2

!2

!12

na

rL

na

r

na

r

lnn

ln

narR l

ln

l

mlnmlnn ,,,,densityelectron *

Ele

ctro

n d

ensi

ty o

f s-

orb

ital

s

n=1

n=2

n=3 n=4

l=m=0

Spectral lines emitted from hydrogen plasma Balmer series (1885, Johann Jakob Balmer) n =2 ( Q: Why this is first ? )

Lymann series (1906, Theodore Lyman) n =1

Paschen series (1908, Ritz Paschen) n =3

Ref:Wikipedia

Spectral lines emitted from hydrogen plasma Balmer series (1885, Johann Jakob Balmer) n =2

Lymann series (1906, Theodore Lyman) n =1

Paschen series (1908, Ritz Paschen) n =3

Rydberg formula for hydrogen

Brackett series (1922) n =4

Pfunt series (1924) n =5

Humphreys series (1953) n =6

2

2

2

1

111

nnR

2

2

0

2

4 1

42

n

meEn

2

2

2

1

2

0

3

4

photonphoton

11

28

2

1

2

11112

nnc

me

EEc

Ec

Ehc

nn

chhE photon

5. Isolated ionic core with one electron (1) as a one electron and one ion system

Born-Oppenheimer approximation r

eZ

m

a

0

22

2

42

H EH

0

1

a 0a

Za

2

2

0

2

4

42

n

ZmeE a

n

0

12

00

3

3

0

exp2

!2

!12

na

rZL

na

rZ

na

rZ

lnn

ln

na

ZrR al

lna

l

aa rRrmln ,,,,

m

lpml

mll

!

!

2

12

)exp(2

1

im

a+Z e

-e

Quantum numbersn l m

Orbitals Shell

1 0 0 1s K K

2 0 0 2s L1

1 0

±12p L23

L

3 0 0 3s M1

1 0

±13p M23

2 0

±1

±2

3d M45

M

4 0 0 4s N1

1 0

±14p N23

2 0

±1

±2

4d N45

3 0

±1

±2

±3

4f N67

N

5 0 0 5s O1

1 0

±15p O23

2 0

±1

±2

5d O45

… … …

O

6 0 0 6s P 1

… … … …P

… … … … … …

s d p

1 2 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6IA IIA IIIA IVA VA VIA VIIA VIII IB IIB IIIB IVB VB VIB BIIB 0

1 H He

2 Li Be B C N O F Ne

3 Na Mg Al Si P S Cl Ar

4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

6 Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14

6 * La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

7 ** Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

Extrapolation of

“one ion - one electron model”

to isolated other neutral atoms

Quantum number and periodic table

Order of occupation

Energy depends on only principal quantum number n

2

20

2

4

42

n

ZmeE a

n a

a

r

eZ

m

2

0

22

4

1

2

H

Extrapolation of

“one ion - one electron model”

to isolated other neutral atoms

Order of occupation

considering electron-electron interaction

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



a

a

r

eZ

m

2

0

22

4

1

2

H

N

jiji ji

N

i ia

ai

N

jiji ji

N

i ia

aN

i

i

R

e

r

eZ

m

R

e

r

eZ

m

)(1, ,,e

2

01 ,

2

0

22

)(1, ,,e

2

1 ,

2

01

22

4

1

4

1

2

4

1

2

H

1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p

Roothaan Hartree Fock by Clementi and Roetti

Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

2

20

2

4

42

n

ZmeE a

n

Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0

Orbilals 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 7s

L. Z

1 1 H 1

2 He 2

2 3 Li 2 1

4 Be 2 2

5 B 2 2 1

6 C 2 2 2

7 N 2 2 3

8 O 2 2 4

9 F 2 2 5

10 Ne 2 2 6

3 11 Na 2 2 6 1

12 Mg 2 2 6 2

13 Al 2 2 6 2 1

14 Si 2 2 6 2 2

15 P 2 2 6 2 3

16 S 2 2 6 2 4

17 Cl 2 2 6 2 5

18 Ar 2 2 6 2 6

4 19 K 2 2 6 2 6 1

20 Ca 2 2 6 2 6 2

21 Sc 2 2 6 2 6 1 2

22 Ti 2 2 6 2 6 2 2

23 V 2 2 6 2 6 3 2

24 Cr 2 2 6 2 6 5 1

25 Mn 2 2 6 2 6 5 2

26 Fe 2 2 6 2 6 6 2

27 Co 2 2 6 2 6 7 2

28 Ni 2 2 6 2 6 8 2

29 Cu 2 2 6 2 6 10 1

30 Zn 2 2 6 2 6 10 2

31 Ga 2 2 6 2 6 10 2 1

32 Ge 2 2 6 2 6 10 2 2

33 As 2 2 6 2 6 10 2 3

34 Se 2 2 6 2 6 10 2 4

35 Br 2 2 6 2 6 10 2 5

36 Kr 2 2 6 2 6 10 2 6

5 37 Rb 2 2 6 2 6 10 2 6 1

…

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



Electron configuration for neutral isolated atoms

1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0


L. Z

…

36 Kr 2 2 6 2 6 10 2 6

5 37 Rb 2 2 6 2 6 10 2 6 1

38 Sr 2 2 6 2 6 10 2 6 2

39 Y 2 2 6 2 6 10 2 6 1 2

40 Zr 2 2 6 2 6 10 2 6 2 2

41 Nb 2 2 6 2 6 10 2 6 4 1

42 Mo 2 2 6 2 6 10 2 6 5 1

43 Tc 2 2 6 2 6 10 2 6 5 2

44 Ru 2 2 6 2 6 10 2 6 7 1

45 Rh 2 2 6 2 6 10 2 6 8 1

46 Pd 2 2 6 2 6 10 2 6 10

47 Ag 2 2 6 2 6 10 2 6 10 1

48 Cd 2 2 6 2 6 10 2 6 10 2

49 In 2 2 6 2 6 10 2 6 10 2 1

50 Sn 2 2 6 2 6 10 2 6 10 2 2

51 Sb 2 2 6 2 6 10 2 6 10 2 3

52 Te 2 2 6 2 6 10 2 6 10 2 4

53 I 2 2 6 2 6 10 2 6 10 2 5

54 Xe 2 2 6 2 6 10 2 6 10 2 6

6 55 Cs 2 2 6 2 6 10 2 6 10 2 6 1

…

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0


L. Z

…

54 Xe 2 2 6 2 6 10 2 6 10 2 6

6 55 Cs 2 2 6 2 6 10 2 6 10 2 6 1

56 Ba 2 2 6 2 6 10 2 6 10 2 6 2

57 La 2 2 6 2 6 10 2 6 10 2 6 1 2

58 Ce 2 2 6 2 6 10 2 6 10 1 2 6 1 2

59 Pr 2 2 6 2 6 10 2 6 10 3 2 6 2

60 Nd 2 2 6 2 6 10 2 6 10 4 2 6 2

61 Pm 2 2 6 2 6 10 2 6 10 5 2 6 2

62 Sm 2 2 6 2 6 10 2 6 10 6 2 6 2

63 Eu 2 2 6 2 6 10 2 6 10 7 2 6 2

64 Gd 2 2 6 2 6 10 2 6 10 7 2 6 1 2

65 Tb 2 2 6 2 6 10 2 6 10 9 2 6 2

66 Dy 2 2 6 2 6 10 2 6 10 10 2 6 2

67 Ho 2 2 6 2 6 10 2 6 10 11 2 6 2

68 Er 2 2 6 2 6 10 2 6 10 12 2 6 2

69 Tm 2 2 6 2 6 10 2 6 10 13 2 6 2

70 Yb 2 2 6 2 6 10 2 6 10 14 2 6 2

71 Lu 2 2 6 2 6 10 2 6 10 14 2 6 1 2

72 Hf 2 2 6 2 6 10 2 6 10 14 2 6 2 2

73 Ta 2 2 6 2 6 10 2 6 10 14 2 6 3 2

74 W 2 2 6 2 6 10 2 6 10 14 2 6 4 2

75 Re 2 2 6 2 6 10 2 6 10 14 2 6 5 2

76 Os 2 2 6 2 6 10 2 6 10 14 2 6 6 2

77 Ir 2 2 6 2 6 10 2 6 10 14 2 6 7 2

78 Pt 2 2 6 2 6 10 2 6 10 14 2 6 9 1

79 Au 2 2 6 2 6 10 2 6 10 14 2 6 10 1

80 Hg 2 2 6 2 6 10 2 6 10 14 2 6 10 2

81 Tl 2 2 6 2 6 10 2 6 10 14 2 6 10 2 1

82 Pb 2 2 6 2 6 10 2 6 10 14 2 6 10 2 2

83 Bi 2 2 6 2 6 10 2 6 10 14 2 6 10 2 3

84 Po 2 2 6 2 6 10 2 6 10 14 2 6 10 2 4

85 At 2 2 6 2 6 10 2 6 10 14 2 6 10 2 5

86 Rn 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6

7 87 Fr 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 1

…

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0


L. Z

…

54 Xe 2 2 6 2 6 10 2 6 10 2 6

6 55 Cs 2 2 6 2 6 10 2 6 10 2 6 1

56 Ba 2 2 6 2 6 10 2 6 10 2 6 2

57 La 2 2 6 2 6 10 2 6 10 2 6 1 2

58 Ce 2 2 6 2 6 10 2 6 10 1 2 6 1 2

59 Pr 2 2 6 2 6 10 2 6 10 3 2 6 2

60 Nd 2 2 6 2 6 10 2 6 10 4 2 6 2

61 Pm 2 2 6 2 6 10 2 6 10 5 2 6 2

62 Sm 2 2 6 2 6 10 2 6 10 6 2 6 2

63 Eu 2 2 6 2 6 10 2 6 10 7 2 6 2

64 Gd 2 2 6 2 6 10 2 6 10 7 2 6 1 2

65 Tb 2 2 6 2 6 10 2 6 10 9 2 6 2

66 Dy 2 2 6 2 6 10 2 6 10 10 2 6 2

67 Ho 2 2 6 2 6 10 2 6 10 11 2 6 2

68 Er 2 2 6 2 6 10 2 6 10 12 2 6 2

69 Tm 2 2 6 2 6 10 2 6 10 13 2 6 2

70 Yb 2 2 6 2 6 10 2 6 10 14 2 6 2

71 Lu 2 2 6 2 6 10 2 6 10 14 2 6 1 2

72 Hf 2 2 6 2 6 10 2 6 10 14 2 6 2 2

73 Ta 2 2 6 2 6 10 2 6 10 14 2 6 3 2

74 W 2 2 6 2 6 10 2 6 10 14 2 6 4 2

75 Re 2 2 6 2 6 10 2 6 10 14 2 6 5 2

76 Os 2 2 6 2 6 10 2 6 10 14 2 6 6 2

77 Ir 2 2 6 2 6 10 2 6 10 14 2 6 7 2

78 Pt 2 2 6 2 6 10 2 6 10 14 2 6 9 1

79 Au 2 2 6 2 6 10 2 6 10 14 2 6 10 1

80 Hg 2 2 6 2 6 10 2 6 10 14 2 6 10 2

81 Tl 2 2 6 2 6 10 2 6 10 14 2 6 10 2 1

82 Pb 2 2 6 2 6 10 2 6 10 14 2 6 10 2 2

83 Bi 2 2 6 2 6 10 2 6 10 14 2 6 10 2 3

84 Po 2 2 6 2 6 10 2 6 10 14 2 6 10 2 4

85 At 2 2 6 2 6 10 2 6 10 14 2 6 10 2 5

86 Rn 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6

7 87 Fr 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 1

…

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0


L. Z

…

86 Rn 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6

7 87 Fr 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 1

88 Ra 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 2

89 Ac 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 1 2

90 Th 2 2 6 2 6 10 2 6 10 14 2 6 10 2 6 2 2

91 Pa 2 2 6 2 6 10 2 6 10 14 2 6 10 2 2 6 1 2

92 U 2 2 6 2 6 10 2 6 10 14 2 6 10 3 2 6 1 2

93 Np 2 2 6 2 6 10 2 6 10 14 2 6 10 4 2 6 1 2

94 Pu 2 2 6 2 6 10 2 6 10 14 2 6 10 6 2 6 2

95 Am 2 2 6 2 6 10 2 6 10 14 2 6 10 7 2 6 2

96 Cm 2 2 6 2 6 10 2 6 10 14 2 6 10 7 2 6 1 2

97 Bk 2 2 6 2 6 10 2 6 10 14 2 6 10 9 2 6 2

98 Cf 2 2 6 2 6 10 2 6 10 14 2 6 10 10 2 6 2

99 Es 2 2 6 2 6 10 2 6 10 14 2 6 10 11 2 6 2

100 Fm 2 2 6 2 6 10 2 6 10 14 2 6 10 12 2 6 2

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Quantum numbersn l m

Orbitals Shell

1 0 0 1s K K

2 0 0 2s L1

1 0

±12p L23

L

3 0 0 3s M1

1 0

±13p M23

2 0

±1

±2

3d M45

M

4 0 0 4s N1

1 0

±14p N23

2 0

±1

±2

4d N45

3 0

±1

±2

±3

4f N67

N

5 0 0 5s O1

1 0

±15p O23

2 0

±1

±2

5d O45

… … …

O

6 0 0 6s P 1

… … … …P

… … … … … …

Spectroscopic naming for orbitals

Principle of analyses

Electron Probe Micro Analyzer (EPMA)

Energy Dispersive X-ray Spectroscopy (EDS)

X-ray Photo-electron Spectroscopy (XPS)

Auger Electron Spectroscopy (AES)

1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Electron Probe Micro Analyzer (EPMA)

Energy Dispersive X-ray Spectroscopy (EDS)

Secondary Electron Microscope (SEM)

X-ray Photo-electron Spectroscopy (XPS)

= Electron Spectroscopy for Chemical Analysis (ESCA）

Hemispherical Electrostatic Analyzer (HEA)

Ultra High Vacuum (UHV) and Introduction chamber

Ion sputtering

Auger Electron Spectroscopy (AES)

Secondary Electron Microscope (SEM)

Cylindrical Mirror Analyzer (CMA) or HEA

Ultra High Vacuum (UHV) and Introduction chamber

Ion sputtering

Chemical properties

Atomic radius

Ionic radius

Covalent radius

Metallic radius

Van der Waars radius

Ionization energy

( Work function )

Electro negativity

Pauling

Mulliken

Etc…

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Ref:Wikipedia

Do you have systematic feature of elements ? Calculation by yourself is essential. Please try it.

Material which shows …

the highest melting point ( or cohesive energy ),

lager electro-negativity ( or smaller ionization energy ),

close-pack structure ( fcc,hcp,bcc,sc,dia.,,, ),

chemical activity ( or non-activity )

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

6. H2

+

LCAO approximation

+e

-e

+e

ra rb

R

bbaa CC

min

d

dE

H0

ba C

E

C

E

12 d

R

e

r

e

r

e

m ba

222

0

22

4

1

2

H EH

0,0,1

d

dE

H

dCC

dCCCC

bbaa

bbaabbaa

2

H

22

22

2

2

bbabaa

bbbaabaaaa

CdCCC

dCdCCdC

HHH

22

22

2

2

bbaa

bbbabbaaaa

CSCCC

HCHCCHC

dH aaaa H

d

R

e

r

e

r

e

ma

ba

a

0

2

0

2

0

22

2

4442

d

r

e

ma

a

a

0

22

2

42

d

r

ea

b

a

0

2

4

d

R

eaa

0

2

4

ISHE

d

r

ea

b

a

0

2

4 R

e

0

2

4

R

eJE ISH

0

2

4

d

r

eJ a

b

a

0

2

4

0222222

bbbabbaaaabbaa HCHCCHCECSCCC

R

ed

r

eEH b

a

bISHbb

0

2

0

2

44

R

eJE ISH

0

2

4

dH baab H

d

R

e

r

e

r

e

mb

ba

a

0

2

0

2

0

22

2

4442

d

r

e

mb

b

a

0

22

2

42

dr

eb

a

a

0

2

4

dR

eba

0

2

4

dE baISH

d

r

eb

a

a

0

2

4

dR

eba

0

2

4

SR

eKSE ISH

0

2

4

d

r

eK b

a

a

0

2

4

dS ba

d

r

eK b

a

a

0

2

4 dS ba

d

r

eJ a

b

a

0

2

4

0

3

0

exp1

a

r

a

aa

R

rr ba R

rr ba dddR

d 22

3

2

2

000 3

11exp

a

R

a

R

a

R

000

2 2exp11

4 a

R

a

R

R

e

0000

2

1exp4 a

R

a

R

a

e

0222222

bbbabbaaaabbaa HCHCCHCECSCCC

0 babaaa CHSECHE

0 aabbbb CHSECHE0

b

a

bbab

abaa

C

C

HEHSE

SEHE

12222

bbaa CSCCCdt 02

abbbaa

bbab

abaaHSEHEHE

HEHSE

HSEHE

0 abaaabaa HSEHEHSEHE

abaa

abaa

HSEH

HSEHE

12222

SCC aa

ba CS

C

22

112222

SCC aa

ba CS

C

22

1

S

ba

22

S

ba

22

bbaa HH

00

2

000

2

exp13

21exp

14 a

R

a

R

a

R

a

R

SR

eE ISH

00

2

000

2

exp13

21exp

14 a

R

a

R

a

R

a

R

SR

eE ISH

E

2

000 3

11exp

a

R

a

R

a

RS

0

ba C

E

C

E

2 4 6 8

Atomic distance R/a0 (a.u.)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Tota

l en

ergy

of

H2+ s

yst

em E

H2+/E

H (

a.u.)

Bonding orbital

Anti-bonding orbital

00

2

H4 a

eE

00

2

000

2

exp13

21exp

14 a

R

a

R

a

R

a

R

SR

eE ISH

00

2

000

2

exp13

21exp

14 a

R

a

R

a

R

a

R

SR

eE ISH

E

kcal/mol 52H2H 2

kcal/mol 42.7

eV 84.1

a.u. 8064.0

kcal/mol 4.8527.42

: : Bonding state

: : Anti-bonding state

S

ba

22

S

ba

22

2

2

00

4

ema

e

: Bonding state

: Anti-bonding state

S

ba

22

S

ba

22

7. Two ionic core with one electron

LCAO approximation a+Z e b+Z e

-e

ra rb

R

Large binding energy is due to ...

large difference between energy levels

or

large overlap of wave functions

R

eZZ

r

eZ

r

eZ

m

ba

b

b

a

a

222

0

22

4

1

2

H

EH

bbaa CC

min

d

dE

H0

ba C

E

C

E

12 d

222

411212

2

bbaa

bbababaaabab

ab

bbaa

ab

ababbbaa

HH

HSHHSH

S

HH

S

HSHHE

dH aaaa

dH baab

dS baab

dH bbbb

a+Z e b+Z e

-e

ra rb

R

When : covalent bond

When : ionic bond



or


R

eZZ

r

eZ

r

eZ

m

ba

b

b

a

a

222

0

22

4

1

2

H EH

222

411212

2

bbaa

bbababaaabab

ab

bbaa

ab

ababbbaa

HH

HSHHSH

S

HH

S

HSHHE

dH aaaa

dH baab

dS baab

dH bbbb

bbaa HH

EEab

1abS

abbbaa

ab HHH

EEE

2

bbaaab HHH 1abS

bbaa

abbb

bbaa

abaa

ab

HH

HH

HH

HH

EEE2

2

Ionic bond, Covalent bond, and Metallic bond consideration with “Tow ionic core with one electron model”

Models for each limitation

How understood with Q.M. ?

Difference between energy levels.

Large difference seems charge transfer.

Distribution of frontier electrons

Spherical or localized to directions

Ionic bond

Covalent bond Metallic bond

FrF

NaCl

ZnFe Cu

SnPb

SiO2

AlN

Si

1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



Ionic bond

Covalent bond Metallic bond

CsF

NaCl

ZnFe Cu

SnPb

SiO2

AlN

Si

There is neither

- purely-ionic-bond material,

- purely-covalent-bond material,

nor

- purely-metallic-bond material.

Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0


L. Z

1 1 H 1

2 He 2

2 3 Li 2 1

4 Be 2 2

5 B 2 2 1

6 C 2 2 2

7 N 2 2 3

8 O 2 2 4

9 F 2 2 5

10 Ne 2 2 6

3 11 Na 2 2 6 1

12 Mg 2 2 6 2

13 Al 2 2 6 2 1

14 Si 2 2 6 2 2

15 P 2 2 6 2 3

16 S 2 2 6 2 4

17 Cl 2 2 6 2 5

18 Ar 2 2 6 2 6

4 19 K 2 2 6 2 6 1

20 Ca 2 2 6 2 6 2

21 Sc 2 2 6 2 6 1 2

22 Ti 2 2 6 2 6 2 2

23 V 2 2 6 2 6 3 2

24 Cr 2 2 6 2 6 5 1

25 Mn 2 2 6 2 6 5 2

26 Fe 2 2 6 2 6 6 2

27 Co 2 2 6 2 6 7 2

28 Ni 2 2 6 2 6 8 2

29 Cu 2 2 6 2 6 10 1

30 Zn 2 2 6 2 6 10 2

31 Ga 2 2 6 2 6 10 2 1

32 Ge 2 2 6 2 6 10 2 2

33 As 2 2 6 2 6 10 2 3

34 Se 2 2 6 2 6 10 2 4

35 Br 2 2 6 2 6 10 2 5

36 Kr 2 2 6 2 6 10 2 6

5 37 Rb 2 2 6 2 6 10 2 6 1

…

Difference between energy levels Large difference seems charge transfer.

Distribution of frontier electrons Spherical or localized to directions

Cohesion energy and Periodic table s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



Shell K L M N O P Q

Principal Q.N. 1 2 3 4 5 6 7

Azimuthal Q.N. 0 0 1 0 1 2 0 1 2 3 0 1 2 3 0 1 2 0


L. Z

1 1 H 1

2 He 2

2 3 Li 2 1

4 Be 2 2

5 B 2 2 1

6 C 2 2 2

7 N 2 2 3

8 O 2 2 4

9 F 2 2 5

10 Ne 2 2 6

3 11 Na 2 2 6 1

12 Mg 2 2 6 2

13 Al 2 2 6 2 1

14 Si 2 2 6 2 2

15 P 2 2 6 2 3

16 S 2 2 6 2 4

17 Cl 2 2 6 2 5

18 Ar 2 2 6 2 6

4 19 K 2 2 6 2 6 1

20 Ca 2 2 6 2 6 2

21 Sc 2 2 6 2 6 1 2

22 Ti 2 2 6 2 6 2 2

23 V 2 2 6 2 6 3 2

24 Cr 2 2 6 2 6 5 1

25 Mn 2 2 6 2 6 5 2

26 Fe 2 2 6 2 6 6 2

27 Co 2 2 6 2 6 7 2

28 Ni 2 2 6 2 6 8 2

29 Cu 2 2 6 2 6 10 1

30 Zn 2 2 6 2 6 10 2

31 Ga 2 2 6 2 6 10 2 1

32 Ge 2 2 6 2 6 10 2 2

33 As 2 2 6 2 6 10 2 3

34 Se 2 2 6 2 6 10 2 4

35 Br 2 2 6 2 6 10 2 5

36 Kr 2 2 6 2 6 10 2 6

5 37 Rb 2 2 6 2 6 10 2 6 1

…

Difference between energy levels Large difference seems charge transfer.

Distribution of frontier electrons Spherical or localized to directions

Cohesion energy and Periodic table

1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

You have obtain a systematic feature of elements ? Calculation by yourself is essential. Please try it.

Material which shows …

the highest melting point ( or cohesive energy ),

lager electro-negativity ( or smaller ionization energy ),

close-pack structure ( fcc,hcp,bcc,sc,dia.,,, ),

chemical activity ( or non-activity )

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



8. Neutral He atom (1) as an example for

isolated neutral atoms

Spin-orbital

Two electrons in 1s orbital.

Integration with spin coordinate

Ortho-normality

– orbital term

– spin term

dxdydzdsdVdsd ***

s11 s12

1)()( iii dsrr 1)()( iii dsrr 0)()( iii dsrr

dxdydzdsdVdsd

lkiilik d ,)()( rr

nmiinim dV ,)()( rr

)()(

)()(

!

1),,(

1

111

1

NNN

N

NN

rr

rr

rr

8. Neutral He atom (2) as an example for isolated neutral atoms


Remember “one ion - one electron model”

e21

21

2

02

2

0

22

2

1

2

0

21

2

21

2

02

2

1

2

0

22

21

2

4

12

4

1

2

2

4

1

2

4

122

4

1

2

HHH

rrrr

rrrrH

ee

m

e

m

eee

m

)2()1(

)2()1(

!2

1

)()(

)()(

!2

1),(

22

11

2212

211121

rr

rrrr

H )()()( 11 iii s

)()()( 12 iii s

+2e

-e

r1r2

1 2

a+Z e

-e sssssss CCC 3322111

,3221 , , ,

2

spss

aZ

G3G3G2G2G1G11 CCCs

ddddE e21 **** HHHH

8. Neutral He atom (3) as an example for isolated neutral atoms

Total energy

d

d

d

d

d

dd

ssss

ssss

ssss

ssss

ssssssss

)2()2()1()1(*)1()1()2()2(2

1

)1()1()2()2(*)2()2()1()1(2

1

)1()1()2()2(*)1()1()2()2(2

1

)2()2()1()1(*)2()2()1()1(2

1

)1()1()2()2()2()2()1()1(*)1()1()2()2()2()2()1()1(2

1

)1()2()2()1(*)1()2()2()1(2

1*

11111

11111

11111

11111

111111111

2121121211

H

H

H

H

H

HH

)2()1(

)2()1(

!2

1

)()(

)()(

!2

1),(

22

11

2212

211121

rr

rrrr

s11

s12

1111

111112211

1111211

11111

)1(*)1(2

1

)1(*)1()1()1()2()2()2()2(2

1

)1()1(*)1()1()2()2()2()2(2

1

)2()2()1()1(*)2()2()1()1(2

1

dV

dVdsdsdV

dd

d

ss

ssss

ssss

ssss

H

H

H

H

1111

111112211

1111211

11111

)1(*)1(2

1

)1(*)1()1()1()2()2()2()2(2

1

)1()1(*)1()1()2()2()2()2(2

1

)1()1()2()2(*)1()1()2()2(2

1

dV

dVdsdsdV

dd

d

ss

ssss

ssss

ssss

H

H

H

H

ddddE e21 **** HHHH

11111 )1(*)1(* dd ss HH

21212 )2(*)2(* dd ss HH

d

d

d

d

d

ddH

ssss

ssss

ssss

ssss

ssssssss

)2()2()1()1(*)1()1()2()2(2

1

)1()1()2()2(*)2()2()1()1(2

1

)1()1()2()2(*)1()1()2()2(2

1

)2()2()1()1(*)2()2()1()1(2

1

)1()1()2()2()2()2()1()1(*)1()1()2()2()2()2()1()1(2

1

)1()2()2()1(*)1()2()2()1(2

1*

11111

11111

11111

11111

111111111

2121121211

H

H

H

H

H

H

0

)1(*)1()1()1()2()2()2()2(2

1

)1()1(*)1()1()2()2()2()2(2

1

)1()1()2()2(*)2()2()1()1(2

1

111112211

1111211

11111

dVdsdsdV

dd

d

ssss

ssss

ssss

H

H

H

0

)1(*)1()1()1()2()2()2()2(2

1

)1()1(*)1()1()2()2()2()2(2

1

)2()2()1()1(*)1()1()2()2(2

1

111112211

1111211

11111

dVdsdsdV

dd

d

ssss

ssss

ssss

H

H

H

1111 )1(*)1(2

1dVss H

)2()1(

)2()1(

!2

1

)()(

)()(

!2

1),(

22

11

2212

211121

rr

rrrr

2111e11

2111e11

11e11

11e11

2111e1111e11

2111e1111e11

1111e1111

2121e2121e

)1()2(*)1()2(2

1

)2()1(*)2()1(2

1

0)2()2()1()1(*)1()1()2()2(2

1

0)1()1()2()2(*)2()2()1()1(2

1

)1()2(*)1()2(2

1)1()1()2()2(*)1()1()2()2(

2

1

)2()1(*)2()1(2

1)2()2()1()1(*)2()2()1()1(

2

1

)1()1()2()2()2()2()1()1(*)1()1()2()2()2()2()1()1(2

1

)1()2()2()1(*)1()2()2()1(2

1*

dVdV

dVdV

d

d

dVdVd

dVdVd

d

dd

ssss

ssss

ssss

ssss

ssssssss

ssssssss

ssssssss

H

H

H

H

HH

HH

H

HH

ddddE e21 **** HHHH

2111e11

2111e11

2111e11e

)2()1(*)2()1(

)1()2(*)1()2(2

1

)2()1(*)2()1(2

1*

dVdV

dVdV

dVdVd

ssss

ssss

ssss

H

H

HH

2111e111111

e21

)2()1(*)2()1()1(*)1(2

****

dVdVdV

ddddE

ssssss

HH

HHHH

2111e11

2111e11

2111e11e

)2()1(*)2()1(

)1()2(*)1()2(2

1

)2()1(*)2()1(2

1*

dVdV

dVdV

dVdVd

ssss

ssss

ssss

H

H

HH

11111 )1(*)1(* dd ss HH 21212 )2(*)2(* dd ss HH

e21

21

2

02

2

0

22

2

1

2

0

21

2

21

2

02

2

1

2

0

22

21

2

4

12

4

1

2

2

4

1

2

4

122

4

1

2

HHH

rrrr

rrrrH

ee

m

e

m

eee

m

)()()()( 22,111,1

1

,11 icicici ss

L

j

jjss

9. Neutral isolated atoms Roothaan Hartree Fock approximation

E.Clementi and C.Roetti, Atomic Data and Nuclear Data Tables, 14, 177-478 (1974)

1s

2s

3s

2p

3p 3d

4s 4p

4d

5s 5p


Atomic number Z

En

erg

y lev

el o

f o

rbit

als

fo

r is

ola

ted

ato

ms

0 5 10 15 20 25 30 35 40 45 50 55

-0.1

-1

-10

-100

-1000

Energy level of neutral atoms

Effect of e--e- interaction

One electron systems Multi-electron systems G.C.Pimentel, R.D.Spratley, “Chemical Bonding clarified through quantum mechanics”, Holden-day Inc.

2

2

0

2

4

42

n

ZmeE a

n

10. Diatomic molecules

LCAO

One-electron approximation

Multi-electron effect

We can imagine based on

Overlap of atomic wave functions

Sign of atomic wave functions

Do you know “-bond and -bond” ?

a+Z e b+Z e

-e

ra rb

R

a+Z e b+Z e

-e

ra rb

R

“-bond and -bond” At first, remember

When : covalent bond

When : ionic bond



or


R

eZZ

r

eZ

r

eZ

m

ba

b

b

a

a

222

0

22

4

1

2

H EH

222

411212

2

bbaa

bbababaaabab

ab

bbaa

ab

ababbbaa

HH

HSHHSH

S

HH

S

HSHHE

dH aaaa

dH baab

dS baab

dH bbbb

bbaa HH

EEab

1abS

abbbaa

ab HHH

EEE

2

bbaaab HHH 1abS

bbaa

abbb

bbaa

abaa

ab

HH

HH

HH

HH

EEE

2

22

2

-bond”

“-bond”

s d p


1 H He






7 Fr Ra **

f

1 2 3 4 5 6 7 8 9 10 11 12 13 14



0 10 20 30 40

Distance r/a0

0

0.05

0.1

0.15

Y-A

XIS

11. Organic molecules

Any Liner combination of atomic orbital is a solution of the wave equation.

Also atomic orbital can be approximated to be a LCAO, as is used in Roothaan Hartree Fock approximation.

A set of hybrid orbitals is another set of atomic orbitals.

Hybrid orbitals

sp3

sp2

sp

Hybrid orbitals

sp3

C : (2s)2(2p)2 -> (sp3)4

ex.

Methane (CH4), Ethane (H3C- CH3), Methyl group (H3C-), Si, C (diamond), etc…

sp2

sp

Hybrid orbitals

sp3

sp2

C : (2s)2(2p)2 -> (sp2)3(2pz)1

ex.

Ethylene (Methene, H2C=CH2), Methylene group (H2C=) , etc…

sp

Hybrid orbitals

sp3

sp2

sp

C : (2s)2(2p)2 -> (sp)2(2px)1(2py)

1

ex.

Acetylene (Ethyne, HCCH), Mtehine group (HC), etc…

Hybrid orbitals

sp3

Methane (CH4), Ethane (H3C- CH3), Methyl group (H3C-),

Si, C (diamond), etc…

sp2

Ethylene (Methene, H2C=CH2), Methylene group (H2C=) , etc…

sp

Acetylene (Ethyne, HCCH), Mtehine group (HC), etc…

Others

Benzene (C6H6), Cyclo-hexane (C6H12)

Methyl alcohol (CH3OH)

1-Buten (-Butylene, H2C=CHCH2CH3)

12. Multi-atomic molecules

13. Jellium


Electron distribution -> negative charge density distribution

Total charge density distribution

Electric field distribution

Potential energy that electrons see.

E


0

Surface energy

Surface tension

One electron approximation

Effective potential energy

Potential shift determined from charge neutrality

Analytic wave functions !!

Analytic formula for surface tension !!

Fermi level

Potential energy v

Distance xFermi level

x

F

V

F

V

effveff

H

eff2

2

2v

mH

Distance x/FR

elat

ive

elec

tro

n d

ensi

ty

Positive

back ground

SCF jellium

Shifted

Step Potential

rs = 5

(r = 2.35

= V/F)

-1.0 -0.5 0.0 0.5 1.0

0.0

0.5

1.0

by K.Takahashi, and T.Onzawa,

Physical Review B, 48, 5689 (1993)

%31035.35

)(5

)0( 24

F24

F2

s

m

krK

m

kT r

Theoretical approach for surface tension at 0 K

SCF-Jellium

Stabilized jellium

Al

ZnMg

Pb

Ca

Li

Sr

Ba

Na

K

Rb Cs

Shifted step potential

Effective electron density parameter rm (Bohr)

Su

rfac

e en

ergy

at z

ero

tem

pet

ratu

re

s (

mJ/

m2)

Be

GaCd

In

Hg

2 3 4 5 6

500

1000

1500

2000

2500

3000

100

Shifted step potential K.Takahashi, and T.Onzawa,


Stabilized jellium J.P.Predew, H.Q.Tran and E.D.Smith,

Phys. Rev. B, 42, 11627 (1990).

SCF-jellium N.D.Lang and W.Kohn,

Phys. Rev. B, 1, 4555 (1970).

Surface energy (surface tension at 0 K) by K.F.Wojciechovski, Surface Science, 437, 285-288 (1999)




Phys. Rev. B, 42, 11627 (1990).


Phys. Rev. B, 1, 4555 (1970). SCF-Jellium

Stabilized jellium

Al Al

ZnZn

MgMg

PbPb

Li

CaCa

Li

Sr

BaSr

Ba

Na

K

Rb

Cs

K

RbCs


Density parameter rs( ), rm( ) (Bohr)

Su

rfac

e en

ergy

s (

mJ/

m2)

2 3 4 5 6

200

400

600

800

1000

1200140016001800

SCF-Jellium

Stabilized jellium

Al Al

ZnZn

MgMg

PbPb

Li

CaCa

Li

Sr

BaSr

Ba

Na

K

Rb

Cs

K

RbCs


Density parameter rs( ), rm( ) (Bohr)

Su

rfac

e en

ergy

s (

mJ/

m2)

2 3 4 5 6

200

400

600

800

1000

1200140016001800

Limitation of density functional theory (DFT) by K.F.Wojciechovski, Surface Science, 437, 285-288 (1999)

Extrapolation for liquid as experimental values

Electron density from

Kittel, Introduction to Solid State Physics, Wiley, NewYork, 1986

to

Perrot,F., and Rasolt,M., J.Phys.: Cond.Matter, 6, 1473-1482, 1994

Density parameter

-

Electron density

Electron density correction for jellium approximation Density parameter from Kittel to Perrot

Kittel, Introduction to Solid State Physics, wiley, NewYork, 1986

Perrot,F., and Rasolt,M., J.Phys.: Cond.Matter, 6, 1473-1482, 1994

Valence

no.

Metal Electron

density

x1022

Density

parameter

Fermi

wave

vector

x108

Fermi

velocity

x108

Fermi

energy

Fermi

temp.BFF / kT

x104

(cm-3) (cm-1) (cm/s) (eV) (K)

1 Li 4.70 3.25 1.11 1.29 4.72 5.48

Na 2.65 3.93 0.92 1.07 3.23 3.75

K 1.40 4.86 0.75 0.86 2.12 2.46

Rb 1.15 5.20 0.70 0.81 1.85 2.15

Cs 0.91 5.63 0.64 0.75 1.58 1.83

Cu 8.45 2.67 1.36 1.57 7.00 8.12

Ag 5.85 3.02 1.20 1.39 5.48 6.36

Au 5.90 3.01 1.20 1.39 5.51 6.39

2 Be 24.2 1.88 1.93 2.23 14.14 16.41

Mg 8.60 2.65 1.37 1.58 7.13 8.27

Ca 4.60 3.27 1.11 1.28 4.68 5.43

Sr 3.56 3.56 1.02 1.18 3.95 4.58

Ba 3.20 3.69 0.98 1.13 3.65 4.24

Zn 13.10 2.31 1.57 1.82 9.39 10.9

Cd 9.28 2.59 1.40 1.62 7.46 8.66

3 Al 18.06 2.07 1.75 2.02 11.63 13.49

Ga 15.30 2.19 1.65 1.91 10.35 12.01

In 11.49 2.41 1.50 1.74 8.60 9.98

4 Pb 13.20 2.30 1.57 1.82 9.37 10.87

Sn(w) 14.48 2.23 1.62 1.88 10.03 11.64

Li

3.13

3.07

Be

1.86

1.85

Metal

rm

rs*

Na

3.68

3.67

Mg

2.78

2.58

Al

2.42

2.12

K

4.21

4.37

Ca

3.07

2.96

Sc

2.58

2.33

Ti

2.23

2.00

V

2.12

1.80

Cr

2.05

1.71

Mn

2.04

1.70

Fe

2.06

1.71

Co

2.08

1.74

Ni

2.14

1.83

Cu

2.25

1.95

Zn

2.49

2.11

Ga

2.51

2.28

Rb

4.46

4.68

Sr

3.34

3.16

Y

2.69

2.44

Zr

2.44

2.08

Nb

2.38

1.91

Mo

2.23

1.77

Tc

2.21

1.77

Ru

2.23

1.79

Rh

2.29

1.86

Pd

2.39

2.02

Ag

2.61

2.23

Cd

2.91

2.44

In

2.85

2.57

Sn

2.98

-

Sb

3.33

-

Cs

5.01

-

Ba

3.24

-

La

2.89

-

Hf

2.47

-

Ta

2.36

-

W

2.27

-

Re

2.21

-

Os

2.21

-

Ir

2.26

-

Pt

2.37

-

Au

2.54

-

Hg

2.90

-

Tl

3.04

-

Pb

3.11

-

Bi

3.48

-

rm : determined so as to describe grand state property of arbitrary defects by Perrot

rs* : determined from interstitial region by Moruzzi

Limitation of density functional theory (DFT) by K.F.Wojciechovski, Surface Science, 437, 285-288 (1999)

SCF-Jellium

Stabilized jellium

Al

ZnMg

Pb

Ca

Li

Sr

Ba

Na

K

Rb Cs



Su

rfac

e en

ergy

at z

ero

tem

pet

ratu

re

s (

mJ/

m2)

Be

GaCd

In

Hg

2 3 4 5 6

500

1000

1500

2000

2500

3000

100




Phys. Rev. B, 42, 11627 (1990).


Phys. Rev. B, 1, 4555 (1970).

Limitation of DFT

Surface tension: Free energy required to create unit area of surface

Interfacial tension : Free energy required to create unit area of interface

+2S +

1 1

1

s1

+2S +

2 2

2

s2

1

2 1

2 1 2

+ + +2Si

1

2 1

2

+ +2Si

1

2

+ +2Si

surface

interface

+2S +

1 1

1

s1

+2S +

2 2

2

s2

1

2 1

2 1 2

+ + +2Si

1

2 1

2

+ +2Si

1

2

+ +2Si

1

2 1

2

+

1

2 1

2

+

1

2

+

1

2

1

+

2

+2S(s1 s2 i

interfacesurface2surface1

+S +

1

2

1

2

+)

-------------------------------------

+2S +

1 1

1

s1

+2S +

2 2

2

s2

1

2 1

2 1 2

+ + +2Si

1

2 1

2

+ +2Si

1

2

+ +2Si

1

2 1

2

+

1

2 1

2

+

1

2

+

1

2

1

+

2

+2S(s1 s2 i

interfacesurface2surface1

+S +

1

2

1

2

+)

-------------------------------------

Work of adhesion “Material constant”

from atomic interaction

Work of adhesion

Free energy required to separate unit area of adhered interface

related

Surface tension , Interfacial tension

Free energy required to create unit area of surface (interface)

+S +

1

2

1

2

s i

Temperature dependence

of interfacial/surface tension

Internal energy and entropy

1st law and 2nd law of thermodynamics

For bulk,

therefore

Variables for unit area of surface

For arbitrary area of surface,

Therefore,

p, TA

Vb

dApdVTdSdU btottot

sbtot UUU sbtot SSS

bbb pdVTdSdU

dATdSdU ss

AUU ss ASS

ss

0ssss TdSdUAdATSU

ss TSU ss TdSdU

pTS

s

T

p

dTT

U

TTU

TpTSTpUTp

0

ss

ss

1

),(),(),(

T

p

dTT

U

TTU

TpTSTpUTp

0

ss

ss

1

),(),(),(

surface (interfacial) free energy = surface (interfacial) tension

surface (interfacial) internal energy = surface (interfacial) energy = surface (interfacial) tension at 0 K.

T

p

dTT

U

TTU

TpTSTpUTp

0

ss

ss

1

),(),(),(

Theoretical approach for surface tension at 0 K

SCF-Jellium

Stabilized jellium

Al

ZnMg

Pb

Ca

Li

Sr

Ba

Na

K

Rb Cs



Su

rfac

e en

ergy

at z

ero

tem

pet

ratu

re

s (

mJ/

m2)

Be

GaCd

In

Hg

2 3 4 5 6

500

1000

1500

2000

2500

3000

100




Phys. Rev. B, 42, 11627 (1990).


Phys. Rev. B, 1, 4555 (1970).

T

p

dTT

U

TTUTp

0

ss

1),(

trans.phase s,phonon s,electron s,s UUUU

Temperature dependence

of surface tension

Other contributions

( phonon, phase transformation )

=0, at 0 (zero) K.

T0 Tm

Phase transf.

Phonon +Electron

T0 Tm

Electron

Su

rfac

e en

ergy

U

Su

rfac

e te

nsi

on

s

Solid liquid

0,ext.

0

),(5

)(4

F2

electrons,

rK

m

kTU r

T0 Tm

Phase transf.

Phonon +Electron

T0 Tm

Electron

Su

rfac

e en

ergy

U

Su

rfac

e te

nsi

on

s

Solid liquid

0,ext.

0

Temperature T (K)

Al

Li

Na

K

Cs

Su

rfac

e te

nsi

on

s

(J

/m2)

Rb

Bi

Hg

Pb

Sn

Zn

0 500 1000 15000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

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