# Eyring Equation

```EXPERIMENT 9

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Purpose
• Learn the logical of solving Schrödinger equation :
• Born–Oppenheimer approximation
• Hartree-Fock method
• Predict the optimized structure of transition state and
calculate the rate constant:
• Transition State Theory
• Eyring Equation
• Use computer program to investigate some chemical
phenomena:
• Gaussian 03, Gauss view, ChemDraw
Schrödinger Equation
Eigenfunction
Eigenvector
Eigenvalue
The Born-Oppenheimer Approximation
• Molecular Hamiltonian (time independent form)
Kinetic energy Repulsion of nuclei
of nuclei
• Electronic Hamiltonian
Attraction
between
nuclei and
electrons
Kinetic energy Repulsion of
electrons
of electrons
The Born-Oppenheimer Approximation
too heavy to move
Electron
Nucleus
moves around
very fast
Nucleus and electron have the same momentum(p=mv). While nucleus
is massive(Ma>>Me), relate to electron, it just like nucleus at rest.
The Born-Oppenheimer Approximation
By the Born-Oppenheimer approximation,
 Nuclei wavefunction and electrons wavefunction are independent
from each other.
thus, the Schrödinger Equation can be extended as
Assume we had solved the electronic wavefunction:
The Born-Oppenheimer Approximation
to find out nuclear Hamiltonian, we know
1
1
Kinetic
Energy
Potential
Energy Surface
Nuclear Hamiltonian
The Born-Oppenheimer Approximation
•
The Born-Oppenheimer Approximation
• Limitation
The error comes form the following condition :
1.) The movement of nuclei is too violent.
 So nuclei can’t be viewed as “stationary”.
2.) 1st exciting electronic energy level is too low.
Any condition of a little change of nuclear coordinate leading
severe alternation of electronic wavefunction makes intolerant
errors.
Computational Chemistry
• Scheme for solving many-electron system
A molecule
HF
Computational Chemistry
 ab initio
A method simulate molecule behaviors only by some basic
physical constants and principles instead of by any simplicity
coming from experimental experiences.
 Semi-Empirical
 Molecular Mechanics
 Density Functional Theory (DFT)
Hartree–fock Method (HF)
•
spin orbital
indices
electron
indices
Hartree–fock Method (HF)
• It contains all possible permutations, all of them are
“indistinguishable” because it’s impossible to distinguish
two electron with the difference.
• Interchanging of two rows flips the sign.
asymmetry : electron is fermion
(Pauli principle)
• If with two identical columns, the determination is always
zero.
all electrons with different quantum states
(Pauli exclusion principle)
Hartree–fock Method (HF)
• Purely many-electron Hamiltonian
Kinetic energy Columbic Electron-electron
Attraction Repulsion
• HF Mean-field Hamiltonian
Hartree–fock Method (HF)
•
(Linear combination of primitive functrion)
Hartree–fock Method (HF)
Split-valence Basis Sets – The Pople Basis Sets
• General expression
with diffuse functions
X – YZ + G*
with polarization functions
Gaussian-type
# basic sets for
inert shell orbitals
# basic sets for
valance shell orbitals
• Some common types
3-21G
3-21+G
6-21G
6-31G*
etc.
3-21G*
3-21+G*
6-31G
6-31+G*
John A. Pople (1925-2004)
Nobel Prize in Chemistry
(1998)
Hartree–fock Method (HF)
Self-consistent Field (SCF)
• Self-consistent Field (SCF)
Directly solve the electronic wavefunction is very difficult
because, for one electron, the distribution of other
electrons we do not know, but it’s necessary to be known if
we want to figure out the electronic wavefunction.
What preferable way is guess an initial condition and
then using a mathematical method (i.e. Iterative Method) to
approach the exact solution gradually.
Hartree–fock Method (HF)
Self-consistent Field (SCF)
• Solution process
Choose a basic sets
There seems that we almost
could find no more lower
energy for the system.
Work many times.
Hartree–fock Method (HF)
• Brief conclusions
1.) If the electronic wavefunction can be expressed as a single
Slater determinant, we can decompose the many-electron
Hamiltonian as the sum of all single-electron Hamiltonian.
 i.e. the electron is independent of others, and the
correlation and exchanging energy of electrons is
neglected.
2.) The electron motion is regarded as on electron under a
mean electric field composed by others.
 but we do not know any information about the
distribution of electrons.
 all we can do is guess the value and optimize it.
Eyring Equation
Henry Eyring (1901-1981)
Eyring Equation
• Transition State Theory
Reaction Coordinate
Eyring Equation
• Derive Eyring eq.
For a reaction
Assume its mechanism:
Pre-equilibrium + Transition state
Eyring Equation
By definition,
In gaseous phase, the equilibrium const. for this reaction can be written as:
concentration
Eyring Equation
Recall,
rate const. of the reaction:
Eyring Equation
•
Eyring Equation
•
Eyring Equation
•
Eyring Equation
•
Eyring Equation
Until now, we had deduced:
Recall,
Eyring eq.
#
Procedure

Fragment

bond 和 modify angle

bond→”OK”
View建構Gaussian03之imput
↓

Lable 欲調整的原子

GaussView儲存的imput檔
k c a l/m o l
tsi4 i8
tsi4 i5
H
tsi2 i4
B
i2
H
C
C
H
H
H
i8
H
C
C
B
H
H
i4
0 .0
H
B
H
C
H
i5
H
H
C
C
H
H
C
B
H
References
• Atkins' Physical Chemistry 9/E, Ch24-4
• Levine I.N. Quantum Chemistry 4/E, Ch10-Ch13
• http://www.iams.sinica.edu.tw/lab/wbtzeng/labtech/term_calch
•
•
•
•
•
•
em.htm, 20120304
http://www.iams.sinica.edu.tw/lab/wbtzeng/labtech/basis_set.h
tm, 20120304
http://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_a
pproximation, 20120303
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1
998/index.html, 20120305
http://www.shodor.org/chemviz/basis/teachers/background.ht
ml, 20120304