Transcript 投影片 1 - 臺灣大學化學系

Chemical Bonding
Part 1：Properties of Atom
☆ Schrödinger’s Equation
☆ Description of Eigenfunctions
-- Angular Function Yl,ml(,) <s, p, d, …>, Angular Nodes
-- Radial Function Rn,l(r) , Radial Nodes
☆ Orbital Energy (Eigenvalues)
-- H-like Atom ; Emission Spectrum of H Atom
-- Aufbau Process ; Pauli Exclusion Principle ; Hund’s Rule
-- Ionization Potential (IP) ; Electron Affinity (EA)
Part 2：Molecular orbitals (MO) from Atomic orbitals (AO)
☆ Linear Combination of Atomic Orbitals (LCAO)
☆ Dinuclear Molecules
-- Homo-nuclear：H2 ; A2
-- Hetero-nuclear：AH ; AB
-- Poly-nuclear：Hn ; AHn ; AXn
Part 3：Chemical Bonding of 3d Transition Metal Complexes
☆ Lewis Structure ; VSEPR ; Hybridization
☆ Crystal Field Theory (CFT)
☆ Ligand Field Theory (LFT)
Properties of Atom
☆ Schrödinger’s Equation：
H  E
H：Hamiltonian (HK.E.P.E.)
：Eigenfunctions
E：Eigenvalues (Total Energy)
H   H 
Hi  Ei i
  d  1
2
i
space
Hi   Hi   Ei i  Ei i 
2




 i d  1
space
i.e. 2
2

 i d  1
space
2  1 or   1
☆ Description of the Eigenfunctions：
z
M (x, y, z)

x
Orthogonal Coordinates → Spherical Coordinates
(x, y, z)
(r, , )
x  r sin cos r  [0, ]
y  r sin sin   [0, ]
z  r cos
  [0, 2]
r
y

Analytical Form
n,l,m(r, , )  Rn,l(r)Yl,ml(, )
The Angular Part：
The Radial Part：

2
0
0



0
Quantum Numbers：
n：Principal
l：Angular Momentum
ml：Magnetic
2
Yl,m
(θ, φ) sin  θ  dθ dφ  1
l
R 2n,l (r )r 2dr  1
Angular Functions, Yl,ml(,)
☆ The s function (l  0, ml  0)：
Symbol：s
Angular Nodes  l  0
Angular → Y0,0(, )  1
Spherical shape
ml  0
☆ The p function (l  1, ml  1, 0, 1)：
z
Symbol：p
y
x
Angular Nodes  l  1
xy plane
pz
pz：Yl,ml(, )  cos
Y1,0(, )  0; when   90
i.e. xy plane
ml  1
z
z
y
Y1,1  sin θ eiφ
y
x
x
px
px：Yl,ml(, )  sin cos
py
py：Yl,ml(, )  sin sin
Yl,ml(, )  0
Yl,ml(, )  0
  0,   90
  0,   0
yz plane
xz plane
Y1,1  sin θ e iφ
i
Y1,1  Y1,1 
2
i
py  Y1,1  Y1, 1 
2
px 
☆ The d function (l  2, ml  2, 1, 0, 1, 2)：
ml  0
z
y
Symbol：d
x
Angular Nodes  l  2
d z2
Yl,ml(, )  3cos2 
1
when
ml  2
y
x
Yl,ml(, )  sin2
cos2
Y =0 when
  0,   45
1
3
ml  1
z
y
d x 2  y2
cos  
x
x
d xy
Yl,ml(, )  sin2
sin2
  0,   0 or 90
z
d xz
Yl,ml(, )  sin cos
sin
  0 or 90,   0
y
d yz
Yl,ml(, )  sin cos
cos
  0 or 90,   90
Radial Nodes  n  l  1
Total Nodes  n  1
 nl 1 i  r
R n,l r     a i r e
 i 0

 
R2 r 2 a01
0.6
0.5
0.4
0.3
0.2
0.1
Radial Probability
Radial Functions, Rn,l(r)
1s
1
4
3
2
r
3s
Distance from the Nucleus
a0
 
R2 r 2 a01
0.2
2s
0.1
2
Combine angular and radial nodes
r
8
6
4
a0
 
R2 r 2 a01
0.2
2p
0.1
3p
Radial Probability
2
6
4
8
3p
Distance from the Nucleus
r
a0
Nomenclature for the Eigenfunctions, i
l  0, 1, 2, 3, …
↓↓↓↓
s p d f
The Principal
Quantum Number
n
The Angular
Quantum Number
l
The Magnetic
Quantum Number
Ml
n  1, 2, 3, …
0ln
l  ml  l
n1
l0
ml  0
1s
l0
ml  0
2s
l1
ml  1
ml  0
ml  1
2p1
2p0
2p1
l0
ml  0
3s
l1
ml  1
ml  0
ml  1
3p1
3p0
3p1
l2
ml  2
ml  1
ml  0
ml  1
ml  2
3d2
3d1
3d0
3d1
3d2
n2
n3
☆ Orbital Energy (Eigenvalues)：
For H
En = - Ry / n2
Degeneracy = n2;
e.g.: 2s; 2px，2py，2pz
Degenerate states : different orbitals. Have same eigenvalue
Hydrogen-like Atoms
RyZe ff2
Eigenvalues → E n,l  
n2
He (Z  2), Li2 (Z  3), and Be3 (Z  4)…
Slater’s Rules for The Calculation of The Screening Constant
n’  n  1
n’  n  1
n’  n
n’  n
1s


0.30
0
ns, np
1
0.85
0.35
0
nd, nf
1

0.35
0
Emission Spectrum of H Atom
En  
Ry
n2
n  1, 2, 3, ...
E1  Ry
(n  1)
ground state
E2  Ry/4
(n  2)
first excited state
E3  Ry/9
(n  3)
second excited
state
etc.
Ry  Ry 
1 

E   2    2   Ry 1  2   h
n  1 
 n 
E (eV)
0
Paschen

Ry  1
1 
 2 2
n  n'
n 
Balmer
n∞
n5
n4
n3
n2
E  Ry  13.6 eV
13.6
Lyman
n1
The Aufbau Process — Klechkowsky’s Rule
1s
nl
1


2s
2
2p

3
3s

3
3p

4s
4
1
4
2

3d
5
n/l
0
3
1
1s
2
2s
2p
3
3s
3p
3d
4
4s
4p
4d
4f
5
5s
5p
5d
5f
6
6s
6p
6d
7
7s

4p
…
5
…
Pauli Exclusion Principle
☆ 4th Quantum Number：Spin Quantum → ms  
N
S
e
e
S
1
2
Two electrons can be populated on each orbital.
Each electron is identified with unique quantum # n , l , ml , ms
N
Hund’s Rule
For degenerate states, the electron is filled in with the same spin to
set the maximum total spin quantum S.
S   ms
orbital electron
Ionization Potential (IP)
A → A  e
H  IP = -orbital energy
0
10.7 eV
N
O
O
2p
1s
2s
Successive Ionization Energies in Kilojoules per Mole for the Elements in Period 3
General Increase
General Decrease
13.6 eV
N
Elemen
t
I1
I2
Na
495
4560
Mg
735
1445
7730
Al
580
1815
2740
11,600
Si
780
1575
3220
4350
16,100
P
1060
1890
2905
4950
6270
21,200
S
1005
2260
3375
4565
6950
8490
27,000
Cl
1255
2295
3850
5160
6560
9360
11,000
I3
I4
I5
I6
I7
Core electrons*
Ar
1527
2665
3945
5770
7230
8780
12,000
*Note the large jump in ionization energy in going from removal of valence electrons
to removal of core electrons. (kJ/mol)
Electron Affinity (EA)
X(g)  e → X(g)
F
H  B.E.
F
Electron Affinities of the Halogens
Atom
Electron Affinity
(kJ/mol)
F
 327.8
Cl
 348.7
Br
 324.5
I
 295.2