# 水素原子 - 組織・特性計算グループ(Microstructure y z r r y yz ry yz rr z r r z r r zr z rz z rrrr 233 233 22 2 3 2 3 112 r r 1 (4.2) また、 1 tan cos cos sin sin sin cos cos sin sin cos cos cos sin cos cos sin cos φ φ φφ φ θφ θφ φ φ θ φ φφ φ φ θφ φ θ φ φφ = ∂ ∂ =− ∴ ∂ ∂ =− =− =− ∂

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• by T.Koyama

• +LNM

OQP =

22

2mV r E( ) ( ) ( ) r r (1)

x ry rz r

===

sin cossin sincos

(2)

r x y z zr

yx

2 2 2 2= + + = =, cos , tan (3)

r x y z

r rx

x rx

xr

rr

r ry

y ry

yr

rr

r rz

z rz

zr

rr

2 2 2 2

2 2

2 2

2 2

= + +

=

= = =

=

= = =

=

= = =

sin cos sin cos

sin sin sin sin

cos cos

(4.1)

cos

sinsin

sin cos cossin

cos cos

sinsin

sin sin cossin

cos sin

sinsin sin

cossin

sin

=

=

=

= = =

=

=

= = =

=

=

= =

=

zr

xzr

rx

xzr x

xzr r

yzr

ry

yzr y

yzr r

zrr

zr

rz r

zr z

zr r r r

2 3 3

2 3 3

2 2

2

3

2

3

21 1

r

r1

(4.2)

1

• tan

coscos sin sin

sin coscos sin

sin

coscos cos

sin coscossin

cos

=

=

= = =

=

= = =

=

=

yx

xyx x

yx

rr r

yxx x x r r

z z

1

1 1

1 0 0

2 2 22

2 2 22

2 22

2

2

(4.3)

x, y, z

=

+

+

=

+

=

+

+

=

+

+

=

+

+

=

xrx r x x r r r

yry r y y r r r

zrz r z z r r

sin cos cos cos sinsin

sin sin cos sin cossin

cos sin

(5)

x, y, z(5)

FHG

IKJ =

+

FHG

IKJ

+

FHG

IKJ

=

+

FHG

IKJ

+

+

FHG

IKJ

+

x x r r r r r r

r r r r

r r r r

r r

sin cos cos cos sinsin

sin cos cos cos sinsin

sin cos sin cos cos cos sinsin

cos cos sin cos cos cos sinsin

sinsin

sin cos cos cos

r r

r r r r r r r

rr r r r

r r

rr

FHG

IKJ

=

FHG

IKJ +

FHG

IKJ

+

+

+

F

H

GGGG

I

K

JJJJ

+

sinsin

sin cos sin cos cos sin cos

cos coscos cos sin cos sin cos cos cos

sin cossin

sinsin

sinsin

sin sin

2 22

2

2 2 2

2 2

2

2

2

1 1

sin cos cos sin cos cos

cossin

sinsin

+

+

+

+

F

H

GGGG

I

K

JJJJ

2 2

2

2

r r r

r r

2

• =

FHG

IKJ +

FHG

IKJ

+

+

FHG

IKJ +

FHG

IKJ

+

FHG

IKJ

+

+

FHG

IKJ

sin cos sin cos cos sin cos

cos cos sin cos cos

cos sin cossin

cossin

sinsin

sin cos

2 22

2

2 2

2 2 2

2

2 2

2

2

2

2

2

2

1 1

1 1

1

r r r r r r r

r r r r r r

r

r r r r

2

FHG

IKJ

+

FHG

IKJ

2

2

2

2

1

r r

r

sin

cos sinsin

sin cos

(6)

FHG

IKJ =

+

+

FHG

IKJ

+

+

FHG

IKJ

=

+

+

FHG

IKJ

+

+

+

FHG

IKJ

+

+

y y r r r r r r

r r r r

r r r r

r r

sin sin cos sin cossin

sin sin cos sin cossin

sin sin sin sin cos sin cossin

cos sin sin sin cos sin cossin

cossin

sin sin cos sin

r r

r r r r r r r

rr r r r

r r

r

+

FHG

IKJ

=

+

+

FHG

IKJ

+

+

+

+

F

H

GGGG

I

K

JJJJ

+

cossin

sin sin sin sin cos sin cos sin cossin

cossin

cos sincos sin sin sin sin sin cos sin

cos cossin

cossin

cossin

sin cos

2

2 2

2

2

2

2 2

2

2

2

r r r r

r r

+

+

+

+

F

H

GGGG

I

K

JJJJ

sin sin cos cos cos sin

sinsin

cossin

2 2

2

2

=

+

+

FHG

IKJ +

FHG

IKJ

+

+

FHG

IKJ +

FHG

IKJ

FHG

IKJ

+

+

FHG

IKJ +

sin sin sin cos sin sin cos

cos sin sin cos sin

cos sin cossin

cossin

cossin

sin cos

2 22

2

2 2 2

2 2 2

2

2 2

2

2

2

2

2

2

1 1

1 1

1

r r r r r r r

r r r r r r

r

r r r r r r

r

FHG

IKJ

+

+

FHG

IKJ

2

2

2

2

1

sin

cos cossin

cos sin

(7)

3

• FHG

IKJ =

FHG

IKJ

FHG

IKJ

=

FHG

IKJ

FHG

IKJ

=

+

FHG

IKJ

+

FHG

IKJ

=

z z r r r r

r r r r r r

r r r r

r r r r r

cos sin cos sin

cos cos sin sin cos sin

cos cos sin sin

sin sin cos cos sin

cos

2

2 2

2

2

2

2

2

2

2

2

2 2

2

2

1

1 1

+

FHG

IKJ

+

+

FHG

IKJ +

FHG

IKJ

r r r r

r r r r r r

sin cos

sin sin cos

(8)

x, y, z

4

• +

+

=

FHG

IKJ +

FHG

IKJ

+

+

FHG

IKJ +

FHG

IKJ

+

FHG

IKJ +

+

2

2

2

2

2

2

2 22

2

2 2 2

2 2 2

2

2 2

2

2 2

2

1 1

1 1

1

x y z

r r r r r r r

r r r r r r

r r r r

sin cos sin cos cos sin cos

cos cos sin cos cos

cos sin cossin

cossin

sinsin

2

2

2

2

2

2

2 22

2

2 2 2

2 2 2

2

2 2

1

1 1

1 1

FHG

IKJ

FHG

IKJ

+

FHG

IKJ

+

+

+

FHG

IKJ +

FHG

IKJ

+

+

FHG

IKJ +

FH

sin cossin

cos sinsin

sin cos

sin sin sin cos sin sin cos

cos sin sin cos sin

r r r

r

r r r r r r r

r r r r r rGIKJ

FHG

IKJ

+

+

FHG

IKJ +

FHG

IKJ

+

+

FHG

IKJ

+

+

FHG

IKJ

+

+

FHG

IKJ +

cos sin cossin

cossin

cossin

sin cossin

cos cossin

cos sin

cos sin cos

sin sin cos

r

r r r r r r

r

r r r r

r r r r

2

2

2

2

2

2

2

2

2

2

22

2

2

2 2

2

1 1

1

1 1 2

r r

FHG

IKJ

5

• =

+

+

+

+

FHG

IKJ +

+

FHG

IKJ

+

FHG

IKJ +

FHG

IKJ

+

FHG

IKJ +

sin cos sin sin cos

sin cos cos sin cos sin

sin cos cos sin cos sin

sin cos sin cos

2 22

22 2

2

22

2

2

2 2 2 2

2 2 2 2

2

1 1

1 1

1 1

r r r

r r r r r r

r r r r r r

r r r r r

FHG

IKJ

+

FHG

IKJ +

FHG

IKJ

+

+

FHG

IKJ +

+

FHG

IKJ +

+

FHG

IKJ

+

FHG

IKJ +

2

2 2

2 2 2

2

2 2 2

2

2 2

2

2

2

2

1 1

1 1

r

r r r r r r

r r r r r r r r r

r r

sin cos sin cos

cos cos cos sin sin

cos sin cossin

cossin

cos sin cossin

cos

1

sin

sinsin

cossin

sin cossin

sin cossin

cos sinsin

sin cos cos cossin

cos sin

+

FHG

IKJ

+

+

FHG

IKJ +

+

FHG

IKJ

+

+

FHG

IKJ +

FHG

IKJ

+

FHG

IKJ +

+

FHG

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

1 1

1 1

r r r r r r

r r r r r r

r rIKJ

=

+

+

FHG

IKJ +

+

FHG

IKJ +

=

+

+

+

+

=

FHG

IKJ +

FHG

IKJ +

LNM

OQP

2

2

2

2 2

2

2 2

2

2 2

2

2 2 2 2

2

2

2

2 2 2

2

2

1 1 1 1

2 1 1

1 1 1 1

r r r r r r r r

r r r r r r

r rr

r

sincos

sin

cossin sin

sinsin

sin

(9)

1 1 1 1

1 2

2

2

2

2

2

2

2

2

2

2

r rr f

r rr f

r r rr f

r rf r f

r

rfr

fr

r fr

fr r

fr r r r

f

2

FHG

IKJ =

FHG

IKJ =

FHG

IKJ

RSTUVW=

+

FHG

IKJ