# 1‧以電腦為工具探討化學反應。

```實驗九. 預測化學反應途徑與反應速率

49712048

49712025

49712016

1‧以電腦為工具探討化學反應。
2‧解分子的 electronic Schrödinger equation

3‧熟悉

4‧計算反應速率常數，並得出產量。

B ORN -O PPENHEIMER A PPROXIMATION
Molecular Hamiltonian：
First
term: The kinetic energy of the nuclei
Second term: The kinetic energy of the electrons
Third
term: The potential energy of the repulsions between
the nuclei
Fourth term: The potential energy of the attractions between
the electrons and nuclei
Fifth
term: The potential energy of the repulsions between
the electrons

A S AN EXAMPLE , CONSIDER
H2
ˆ 
2
 -
2
2
2m p
 -
2
2
2m p
2me
 2
1
2
2me

2
2
e e e e e e

- - - 
r r1 r1 r2 r2 r12
2
2
2
2
2
2
α、 β為兩氫原子之原子核，下標1、2

:電子
:原子核
Purely electronic Hamiltonian:

：
The nuclear repulsion
Purely electronic energy:
ˆ  E 

el el
el el

Re：位能達最低點時之平衡距離
De：達平衡時之解離能
D0：分子的振動能階之最低能量(零點能量)
ν=0：zero point energy=1/2hν
Purely nuclear Hamiltonian:
ˆ   E

N N
N

  qi ,q   el  qi ;q  N  q 

H ARTREE - FOCK
APPROXIMATION
◎ Consider a simpler N-el system:
(neglect the el-el repulsion)
N
H   h(i )
i 1
Recall:
→ where h(i) is the oper-
ator describing the K.E
and P.E of electron i.
Now we can write:
◎ Hartree product:
because H is the sum of one-el Hamiltonians ,
a wave function which is a simple product of
the wave functions for each electron .
ψHP(x1,x2,…,xN) = χi(x1) χj(x2)˙˙˙χk(xN)
An eigenfunction of H :
Such a many-electron wave function is termed
a “Hartree product”.
NOTE:
The Hartree product does not satisfy
the “antisymmetry principle” .
◎If we put electron-one in χi and electron-two
in χj ,we have:
ψHP12(x1,x2)= χi(x1) χj(x2)---------------(1)
ψHP21(x1,x2)= χi(x2) χj(x1)---------------(2)
We can obtain a wave function to satisfy the
antisymmetry principle by taking the appropriate linear combination of these two HP.
It can be rewritten as a determinant and is
called
a“slater determinant”.
i( x1) j ( x1)
i( x2) j ( x2)
1/ 2
( x1, x2)  2
-1/2
ψ(x1, x2,...,xN) = (N!)
i ( x1)
j ( x1)
 k ( x1)
i ( x 2)
j ( x 2 )
 k ( x 2 )
i ( xN )
j ( xN )
 k ( xN )
Clearly,
ψ(x1,x2) = -ψ(x2,x1)
(fermion)
◎Hartree-Fock equation
where f(i) is an effec-
tive one-el operator
called the Fock operator
1 2 M ZA
f (i)   i    HF (i)
2
A1 riA
Where νHF is the average potential experienced by the i-th electron due to the presence
of the other electrons.
The potential energy of interaction between
Q1Q 2
Point charges Q1 and Q2 is V 12 
4 0 r12
Q1  2
V 12 
dv 2

4 0 r 12
 2  e s 2
Q1  e
2
V 12  e'2 
s2
2
r 12
2
e
e' 2 
4 0
dv2
Adding in the interactions w/ the other el’
we have:
n
V 12  V 13  ...  V 1n   e'
j 2
2
sj
r
2
1j
dvj
 The
Born-Oppenheimer approximation is inherently
assumed.
 Relativistic
effects are completely neglected.
 The
variational solution is assumed to be a
linear combination of a finite number of basis
functions.
 Each
energy eigenfunction is assumed to be
describable by a single Slater determinant.
 The
mean field approximation is implied.
6-31G
◎basis
BASIS SET
set: a mathematical description of orb-
itals of a system, which is used for approximate theoretical calculation or modeling. It is
a set of basic functional building blocks can
be stacked or added to have the features we
need.
  a11  a2 2  ... ann
   curgu
u
χ稱為收斂高斯函數(contracted Gaussian)
g為初始高斯函數( primitive Gaussian)
→ 例如STO- 3G：

◎6-31G
→內殼層（inner shell）：每個原子軌域(AO)以一個

→價殼層（valence shell）：由兩個基底函數組合而

“＊” 號代表極化函數(polarized functions)，第一個*號表

Eyring equation

k P
A  B  C 
d [ P]
V
 k  [C  ]
dt

C ：activated com plex or transition state

1. C is in pre  equilibrium with A、B
if gases,

P C  / P
P C  P
RT[C  ] P
[C  ]
P



 
( PA / P )(PB / P ) PAPB RT[ A]RT[ B] RT [ A][B ]
nA
(ideal gas: P V  nRT  PA  RT  [A]RT )
V
RT
 [C  ]     [ A][B ]
P

d[P ]
RT
 k  [C ]  φ k  Κ  [A][B] k2[A][B]
dt
P
RT
 k2  φ k  Κ   t o find k2: get (1) k  (2) Κ 
P
2. V 
(1) t o get k ：

k  κ
κ : t ransmission coefficient
(2) t o get Κ  ：　(equilibrium const ant　for A  B  C )

φ
q J, m νJ 
 ΔrE0 /RT
recall, K  (
) e
J
NA


NA q φc  ΔrE /RT
0
Κ  φ φ e
qA qB

where ΔrE0  E 0 (C ) - E 0 (A) - E 0 (B)
φ
q J  st andardmolarpart it ionfunct ion
vibrat ional mode
q
1
1  e  h / kT

h
1
kT

x2
h
e  1 x 
  , nowthat x  
2!
kT
1
1
kT
q


h
h
1  (1 
 ) 1  (1 
) h
kT
kT
x





qC   qC  ,T qC  , qC  ,R qC  , E
kT 
( qC  ,T qC  , R qC  , E qC  , )
h
kT 


qC 
h

 
kT 

qC 
N
q
kT A C   rE 0 / RT kT 
 rE 0 / RT
h
e

e


 
 
h q A q B
h
q A qB
NA
RT   RT
kT 
k






h
P
P
kT RT 



h P
kT 

C

h

```

– Cards

– Cards