Atomic orbital & Hydrogen

Report
Atomic orbital
&
Hydrogen-atom wave function
原子軌道と水素原子波動関数
Derivation of hydrogen-atom wave function
@ Schrödinger eq. of H atom(水素原子のシュレディンガー方程式)
Derivation of hydrogen-atom wave function
@ Schrödinger eq. of H atom(水素原子のシュレディンガー方程式)
convert to the atomic unit(原子単位へ変換する)
convert from the xyz to the rθφ(polar) coordinate(xyz座標から極座標へ変換する)
separate the variables of ( r, θ, φ) (変数分離する)
@ Solution(解)
1
ψn,l,m (r,θ,φ) = Rn,l (r) x Yl,m (θ,φ) , E = - —
2
2n
Hydrogen-atom wave function
Spherical Harmonics
(水素原子波動関数)
(球面調和関数)
Radial wave function
(動径波動関数)
where
Energy
(エネルギー)
n = 1, 2, 3, . . .
l = 0, 1, 2, . . . , n-1
m = 0, ±1, ±2, . . . , ±l
eliminate a imaginary by linear combination(足し算引き算して虚数を消去する)
@ realized Hydrogen-atom wave function (実数化した水素原子波動関数)
real
±
ψn,l,m (r,θ,φ) = Rn,l (r) x Yl,m (θ,φ) where
Realized Spherical Harmonics
(実数化した球面調和関数)
n = 1, 2, 3, . . .
l = 0, 1, 2, . . . , n-1
m = 0, 1, 2, . . . , l
Load to the Atomic Orbitals
real
ψn,l,m
(r,θ,φ) = Rn,l (r) x
±
Yl,m (θ,φ)
l=0
n=1
R1,0
r
where
θ
π
Y0,0
l=1
n = 1, 2, 3, . . .
l = 0, 1, 2, . . . , n-1
m = 0, 1, 2, . . . , l
2π
φ
n=2
R2,0
R2,1
Y-1,1
Y1,0
Y+1,1
Y-2,1
Y2,0
Y+2,1
l=2
n=3
R3,0 R3,1
R3,2
Y-2,2
Y+2,2
Load to the Atomic Orbitals
real
ψn,l,m
(r,θ,φ) = Rn,l (r) x
±
Yl,m (θ,φ)
where
θ
n=1
R1,0
n = 1, 2, 3, . . .
l = 0, 1, 2, . . . , n-1
m = 0, 1, 2, . . . , l
π
Y0,0
3D representation
r
2π
φ
n=2
R2,0
R2,1
Y-1,1
Y1,0
Y+1,1
Y-2,1
Y2,0
Y+2,1
n=3
R3,0 R3,1
R3,2
Y-2,2
Y+2,2
This transformation is similar to that of the world map.
(この変形は世界地図の変形に似ている)
l=2 : d orbital
dz2
z2
conversion
Y2,0 (r,x,y,z)
orbital
Load to the Atomic Orbitals
real
ψn,l,m
(r,θ,φ) = Rn,l (r) x
±
Yl,m (θ,φ)
n=1
R1,0
+
Y0,0
r
n=2
R2,0
where
n = 1, 2, 3, . . .
l = 0, 1, 2, . . . , n-1
m = 0, 1, 2, . . . , l
+ Y-1,1
R2,1
n=3
R3,0 R3,1
R3,2
+ - + -
Y-2,2
+
+
-
Y-2,1
+
+
Y2,0
+
Y+1,1
Y1,0
+
+
-
+
+ - +
Y+2,1
+
- + -
Y+2,2
+
Load to the Atomic Orbitals
real
ψn,l,m
(r,θ,φ) = Rn,l (r) x
±
Yl,m (θ,φ)
±
Yl,m
(r,x,y,z)
n=1
R1,0
s
r
n=2
R2,0
R2,1
py
pz
px
dyz
dz2
dzx
n=3
R3,0 R3,1
R3,2
dxy
dx2-y2
s orbital
Rn,l (r)
R1,0
x
±
Yl,m
(r,x,y,z)
=
real
ψn,l,m
(r,x,y,z)
n=1
1s
R2,0
n=2
x
s
=
2s
n=3
R3,0
3s
p orbital
Rn,l (r)
x
±
Yl,m
(r,x,y,z)
real
ψn,l,m
=
(r,x,y,z)
l=1
n=2
pz
R2,1
n=3
R3,1
x
py
=
2pz
3pz
2py
3py
px
2px
3px
d orbital
Rn,l (r)
x
±
Yl,m
(r,x,y,z)
=
real
ψn,l,m
(r,x,y,z)
3dz2
l=2
dz2
dyz
n=3
R3,2
3dyz
x
dzx
dxy
dx2-y2
=
3dzx
3dxy
3dx2-y2
Atomic Orbitals and Energies of the Hydrogen Atom
(水素原子の原子軌道とエネルギー)
Atomic Orbitals and Energies of the Hydrogen Atom
1
E=-—
2n2
l=0
l=1
l=2
l=3
( N shell ) 0
n=4
n=3
( M shell )
m=0
n=2
m=±1
m=±2
m=0
m=±1
m=±2
m=±3
( L shell )
m=0
m=±1
ψn,l,m (r,θ,φ)
principal quantum number
n = 1, 2, 3, . . .
(主量子数)
orbital angular momentum quantum number
l = 0, 1, 2, . . . , n-1
(軌道角運動量量子数)
magnetic quantum number m = 0, ±1, ±2, . . . , ±l
n=1-0.5
( K shell )
(磁気量子数)
m=0
Atomic Orbitals and Energies of the Hydrogen Atom
1
E=-—
2n2
n=4
n=3
4dx2-y2
4fz(5z2-3r2)
4dxy
4s
4pz 4px 4py 4dz2 4dzx 4dyz
3s
3pz 3px 3py 3dz2 3dzx 3dyz 3dxy 3dx2-y2
2s
2pz 2px 2py
n=2
4fx(5z2-r2)
4fy(5z2-r2)
ψn,l,m (r,θ,φ)
real
ψn,l,m (r,x,y,z)
1s
n=1
4fz(x2-y2)
4fxyz
4fx(x2-3y2)
4fy(3x2-y2)
Atomic Orbitals and Energies of the Hydrogen Atom
E=
real
1 ψn,l,m (r,x,y,z)
-—
2n2
4s
4pz 4px 4py 4dz2 4dzx 4dyz
4dx2-y2
4fz(5z2-3r2)
4dxy
n=4
n=3
4fx(5z2-r2)
4fy(5z2-r2)
n=2
3dxy
2py
n=1
4fz(x2-y2)
4fxyz
1s
2pz
4fx(x2-3y2)
4fy(3x2-y2)
3dyz
3dz2
3dzx
3py
3pz
3px
2px
2s
3s
3dx2-y2
4s
4fy(3x2-y2)
4py
4pz
4px
4dxy
4dyz
4dz2
4dzx
4dx2-y2
4fxyz
4fx(5z2-r2)
4fz(5z2-3r2)
4fy(5z2-r2)
4fz(x2-y2)
4fx(x2-3y2)
4s
2py
2pz
2px
3py
3pz
3px
4py
4pz
4px
3dyz
3dz2
3dzx
3s
2s
3dxy
3dx2-y2
1s
4fy(3x2-y2)
4dxy
4dyz
4dz2
4dzx
4dx2-y2
4fxyz
4fx(5z2-r2)
4fz(5z2-3r2)
4fy(5z2-r2)
4fz(x2-y2)
4fx(x2-3y2)

similar documents