3

H2

HΨ = EΨ (7.1.1)

H Ψ E

H K V

H = K + V (7.1.2)

K = −

2

2m∇ a2 +∇b

2( ) −2

2M∇A2 +∇B

2( )

V = −ke21

rAa+1

rAb+1

rBa+1

rBb−1

rab−1

rAB

⎛

⎝ ⎜

⎞

⎠ ⎟

A B a b

m M rXY X

Y

(7.1.1)

1

L

−

2

2m

∂2

∂x 2+∂2

∂y 2+∂2

∂z2⎛

⎝ ⎜

⎞

⎠ ⎟ +V (x,y,z)

⎡

⎣ ⎢

⎤

⎦ ⎥ Ψ = EΨ (7.1.3)

V (x,y,z) = 0 0 ≤ x,y,z ≤ L

V (x,y,z) =∞

Ψ

Ψ =ϕx (x)ϕy (y)ϕz(z) (7.1.4)

(7.1.3)

−

2

2mϕy (y)ϕz(z)

∂2

∂x 2ϕx (x) +ϕx (x)ϕz(z)

∂2

∂y 2ϕy (y) +ϕx (x)ϕy (y)

∂2

∂z2ϕz(z)

⎛

⎝ ⎜

⎞

⎠ ⎟

+V (x,y,z)ϕx (x)ϕy (y)ϕz (z) = Eϕx (x)ϕy (y)ϕz (z) (7.1.5)

(7.1.4) 0

(7.1.4)

−

2

2m

1

ϕx (x)

∂2

∂x 2ϕx (x) +

1

ϕy (y)

∂2

∂y 2ϕy (y) +

1

ϕz (z)

∂2

∂z2ϕz(z)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

+V (x,y,z) = E (7.1.6)

V(x,y,z) (7.1.3) α = x y z

vα (α) = 0 0 ≤ α ≤ L (7.1.7)

vα (α) =∞ α < 0 L < α

(7.1.6)

−2

2m

d2

dα 2 + vα (α)⎛

⎝ ⎜

⎞

⎠ ⎟ ϕα (α)

α

∑ = εαϕα (α)α

∑ (7.1.8)

(5.1.3)

−

2

2m

d2

dα 2 + vα (α)⎛

⎝ ⎜

⎞

⎠ ⎟ ϕα (α) = εαϕα (α) (7.1.9)

E

E = εx + εy + εz (7.1.10)

(7.1.9) (7.1.3)

0

−

2

2m

d2

dα 2 ϕ(α) = εϕ(α) (7.1.11)

0 ≤ α ≤ L 0

ϕ(0) =ϕ(L) = 0 (7.1.12)

(7.1.11) 2

sin(kα) cos(kα)

(7.1.12) ϕ(0) = 0 cos(kα)

ϕ(α) = Asin(kα) (7.1.13)

(7.1.12) ϕ(L) = 0 n

kL = nπ

ϕ(α) = AsinnπLα

⎛

⎝ ⎜

⎞

⎠ ⎟ (7.1.14)

(7.1.14) (7.1.11)

ε =

π2 2

2mL2n2 =

h2

8mL2n2 (7.1.15)

n n (7.1.14) (7.1.12)

n = 0 (7.1.14) 0

ϕ(α) = Asin0 • πL

α⎛

⎝ ⎜

⎞

⎠ ⎟ = 0 (7.1.16)

n = 0

n (7.1.14) n A

A

n

(7.1.1) (7.1.14) n

(7.1.15) n

n

n = 1

ε =h2

8mL2 (7.1.17)

0.1 nm L (7.1.17)

3.6•10-20 kJ ≈ 38 eV

(7.1.17)

0

m

L

Ψ(x,y,z) = C sinnxπL

x⎛

⎝ ⎜

⎞

⎠ ⎟ sin

nyπ

Ly

⎛

⎝ ⎜

⎞

⎠ ⎟ sin

nzπ

Lz

⎛

⎝ ⎜

⎞

⎠ ⎟ (7.1.18)

E =h2

8mL2nx2 + ny

2 + nz2( ) (7.1.19)

C nx ny nz

2

1800

r 2

Ekb = EkA + EkB =3h2

8mr2+3h2

8mr2=3h2

4mr2 (7.2.1)

r × r × 2r

2

Eka = 2h2

8mr2(12 +12) +

h2

8m(2r)2(12)

⎡

⎣ ⎢

⎤

⎦ ⎥ =

9h2

16mr2 (7.2.2)

ΔEk = Eka − Ekb =9

16−3

4

⎛

⎝ ⎜

⎞

⎠ ⎟ h2

mr2= −

3h2

16mr2 (7.2.3)

ke2

r (7.2.4)

(7.2.4)

ΔEp = 2ke2

r (7.2.5)

r (7.2.3) (7.2.5)

7.1

ΔE = ΔEk + ΔEp (7.2.6)

7.1ΔE ΔEk

ΔEp r

2 Å

0.1 nm = 1 Å

2 1927

7.2 b

7.20 a b

c n = 2 d n = 1 b d a c

n = 1 d

1s

0

a c 2 0

2 0

2 2

a c 2

2

nb nab BO

BO =nb − nab2

(7.2.7)

2

2 2

7.3

3

10 5

7.3HOMO

highest occupied molecular orbital HOMO

lowest unoccupied molecular orbital LUMO

HOMO LUMO

HOMO LUMO

R.

1981

7.3 10 5

2

7.3

HOMO 3 xy

2pz

HOMO

2pz

HOMO

HOMO 7.4 xy

2 2pz 2pz

xy 2

7.4 HOMO

7.4

HOMO

3

A

B

3

5

1 5.3

2

7.5

CnH2n+2 + 3n +12

O2 nCO2 + (n + 1)H2O

7.5

p = ∫ rρ(r)dV (7.4.1)

ρ(r)

2 SI C m

D

1 D = 10-18 esu cm ≈ 3.33564 10-30 C m (7.4.2)

0

p–

3

2

p p = |p|

AB

(7.4.1)

pionized = e•rA-B (7.5.1)

e rA-B

(7.5.1) AB

= p

e • rA−B (7.5.2)

HCl

20%

7.6

6

A B xA xB

7.6

D.A. J.D.

1999

L. 3

1974

© K. Saito, 11/30/2017