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Fundamental Atomic Units
Symbol and Value
Atomic Units
Quantity
Name
Mass
electron rest mass
in
me=1
e=1
SI value
9.109 3826(16)10-31 kg
1.602 176 53(14)10-19 C
Charge
electron charge
angular momentum
Reduced Planck's constant
h /( 2 ) 1
1.054 5
Multielectron Atoms
What is different with many electrons?
Repulsion between electrons new terms in Hamiltonian
Concept of electron pairing
Indistinguishability of electrons permutations of wavefunctions
At
ele tract
nu ctro ion
A
cl e t t r n o f
u
ele
Computational Quantum Chemistry
DFT
Ab Initio
Walter Kohn (left) and John
Pople in Stockholm, 2001, were
awarded Nobel Prizes for their
theoretical work
John Pople 1925-2004, Nobel
Prize in 1998 for his
development of computational
methods in quantum chem
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Mechanism
A
B
k1
k2
MathcadKinetics_A_B_C.xmcd
B
Runge-Kutta approach is used to solve the
mechanism: A->B->C, where the first step has
rate constant k1, and the second, k2.
C
1
k1 1.3410 1
k2 1.3910 1
[A]0
x, is a vector containing
the initial
oo
Schrodinger Equation
Erwin Schrdinger, Nobel Prize in
1933 for discovery of new and
productive forms of atomic theory
Postulate 5
time dependent
energy operator is the
Hamiltonian operator
kinetic potential
energy energy
Operators
Classical Quantities
Chemical Kinetics
Mechanisms
Chemical
Kinetics
Reactions Happen Through
Collisions
In a chemical reaction, bonds are
broken and new bonds are formed.
Molecules can only react if they collide
with each other.
Chemical
Kinetics
Furthermore, molecules must c
Two Spectroscopic Models
Harmonic Oscillator and Rigid Rotor
Molecular Vibrations
E. Bright Wilson (1908-1992).
wrote Introduction to Quantum
Mechanics with Linus Paulong
and Molecular Vibrations: The
Theory of Infrared and Raman
Vibrational Spectra with
4)
Script:-
Result
5)
Script:-
H=[1 2; 3 4]
[C,E]=eigs(H)
Result
So C is giving us the eigenvectors and the elements on the diagonal of matrix E are the eigenvalues.
6) Script:-
Result
7)
CHEM 4300, Homework 1
Intro, Postulates, Complex Numbers, MATLAB, WebMO
Due 5 PM Friday, Sept. 4th
1. Do problem 4-14 on commutators (which are important for Postulate 2).
2. Do problem A-3 on complex numbers.
3. Do problem 3-3 on eigenvalues and eigenfun
CHEM 4300, Homework 2
Particle-in-a-Box, DVR
Due 5 PM Friday, Sept. 11th, 2015.
1. Do problems 3-20 and 3-21 on the uncertainty principle for the particle-in-a-box
problem. What is the value of and how does it compare to the uncertainty relation
of textbo
CHEM 4300, Homework 3
Harmonic Oscillator
Due 5 PM Friday, Sept. 18th, 2015.
1. Do problems 3-23 and 3-24 for a particle in a 3D box.
2. Do problem 5-22 on the zero point motions of some diatomic molecules.
3. Harmonic oscillator wavefunctions are given i
CHEM 4300, Homework 4
Rigid Rotor and Harmonic Oscillator
Due 5 PM Friday, Oct. 2nd, 2015.
1. Do problem 5-35 on the bond distance of the CO diatomic molecule from rotational
spectroscopy.
2. Do problem 5-31 which starts from the result of problem 5-30 an
CHEM 4300, Coe, Homework 5 Hydrogen Atom, Mostly
Due 5 PM Friday, Oct. 9th, 2015.
You are encouraged to use Matlab or Mathcad (or any other computational program, like
Mathematica) to do these problems.
1. Do problem 6-25 on the expectation values of radi
CHEM 4300, Coe, Homework 6 Approximate Methods
Due 5 PM Monday, Oct. 19th, 2015
1. Do problem 7-3 using the trial function 1/(1 + 2 )2 to calculate the ground state of
a general harmonic oscillator (which is a Gaussian) using the variational theorem.
2. D
CHEM 4300, Coe, Homework 7 Multielectron Atoms, Bonding and Diatomics
Due 5 PM Friday, Oct. 30th, 2015.
1. Do problem 8-11 (using the results from 8-10) concerning the relation between
orbital energies in helium atom and the total electronic energy.
2. De
Molecules and Spectroscopy
Linus Pauling, 1901-1994,
Nobel Prize in 1954 for his
research into the nature of the
chemical bond and its
application to the elucidation of
the structure of complex
structures. proposed the alpha
helix as the basic structure o
11/20/2013
Chemical Kinetics
glow sticks
chemiluminescence reaction
hot
cold
Svante Arrhenius, Nobel Prize
in 1903 for his electrolytic
theory of dissociations,
temperature dependence of
rate constants
Equilibrium
is a balance of forward and backward rat
Free Particle
x
V(x)=0
solve Schrdinger eq.
Hamiltonian is kinetic+potential
As with particle-in-a-box
Schrdinger eq. is
but no boundaries
Write in standard form
General solution
( x) = Ae ikx + Be ikx = Aei
it would seem that
2 mE x /
+ Be i
2 mE x /
Fundamental Atomic Units
Symbol and Value
Atomic Units
in
Quantity
Name
SI value
Mass
electron rest mass
me=1
9.109 3826(16)10-31 kg
Charge
electron charge
e=1
1.602 176 53(14)10-19 C
angular momentum
Reduced Planck's constant
= h /( 2 ) = 1
1.054 571 68
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Postulate 1
Postulate 2
Wavefunction ( x )
probability of system being at x
( x) * ( x)dx
Operators for properties come from commutator rules
,
,
uncertainty principle
Postulate 3
0
,
0
Operator Eigenvalue equations
Postulate 4
average v
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Postulate 1
The state of a quantum-mechanical system is completely specified by a function ( x )
that depends upon the coordinates of the particle. All possible information about the
system can be derived from ( x ). This function, called the wa
9/16/2013
Free Particle
x
V(x)=0
solve Schrdinger eq.
.
+
+0
Hamiltonian is
kinetic+potential
As with particle-in-a-box
2
Schrdinger eq. is
)
but no boundaries
Write in standard form
2
0
General solution
( x) Aeikx Be ikx Aei
2 mE x /
Be i
2 mE x /
8/21/2013
Introduction to Quantum Chemistry
Quantum Mechanics provides basis for
Lasers and the Optical Revolution
Atomic Clock and Measurement of Time (GPS)
Nuclear Magnetic Resonance and
Medical Diagnostics
Spectroscopic Diagnostics for our
World an
Textbook Postulates of QM
Postulate 1
The state of a quantum-mechanical system is completely specified by a function ( x )
that depends upon the coordinates of the particle. All possible information about the
system can be derived from ( x ). This functio
Discrete Variable Representation (DVR)
When theoreticians
write theory that
experimentalists can
program with one page,
you have something
powerful!
enter Potential here, you
will mess with this part
matrix form of Hamiltonian
H i , j = * H i d
j
basis se
9/27/2013
Raising and Lowering Operators for the Harmonic Oscillator
Lowering Operator
Raising Operator
) and momentum (
Text problems redefined position (
) operators as
and . Here we leave them as they were originally defined.
We know that
1
2
1
2
9/30/2013
Raising and Lowering Operators for the Harmonic Oscillator
Lowering Operator
Raising Operator
Text problems redefined position (
) and momentum (
) operators as
and . Here we leave them as they were originally defined.
We know that
1
2
1
2
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Douglas Hartree, 18971958, pioneered the
use of computers in
research
Vladimir Fock, 18981974, generalized
Hartrees eqs. for
interchange of electrons
The Hartree-Fock calculation scheme is a self-consistent iterative
procedure to calculate the s
10/16/2013
What is different with many electrons?
Repulsion between electrons new terms in Hamiltonian
Concept of electron pairing
Indistinguishability of electrons permutations of wavefunctions
-e
r1
r12
Z=+2e
helium
r2
-e
Internal Electronic Hamiltonia