Chapter 5. The Particle in the Box and the Real World
Real particles in boxes:
Conjugated molecules with alternating single and double bonds
Butadiene: CH2-CH=CH-CH2
The electrons are delocalized, excitation energy:
5h 2
5(6.626 1034 Js) 2
E23
8mL2 8(9.1
Chapter 11. Quantum states for many-electron atoms
and atomic spectroscopy
Atomic spectra
H atom:
Not all transitions are allowed, due to conservation of
angular momentum (photon has 1 of angular momentum).
Selection rules:
l 1, ml 0,1.
Example:
1s 2s, fo
Chapter 12. The Chemical Bond in Diatomic
Molecules
I. Born-Oppenheimer approximation
Because of the small mass of an electron, it can respond
instantaneously to change of nuclear coordinates.
(r, R) (r : R) (R)
Substituting to H E and ignoring the action
Chapter 15. Computational Chemistry
Hartree-Fock theory
(1,2,.N ) Pij (1) (2). ( N )
where Pij is the projection operator to antisymmetrize the
Hartree wavefunction. is the molecular spin-orbital with its
spatial part as LCAO:
(i) cAO
A set of coupled li
Chapter 13. Molecular Structure and Energy Levels
for Polyatomic Molecules
Valence bond (VB) theory
A localized view: A bond is formed with e-pairs between
bonding atoms.
Valence shell electron pair repulsion (VSEPR) model:
ligands and lone pairs repel ea
Chapter 8. The vibrational and rotational
spectroscopy of diatomic molecules
General features
Interaction of electromagnetic field with atoms/molecules
Spectral range:
~
Wave number (frequency )
Wavelength
Radio
MW
rot.
IR
vib.
VIS
UV
elec. (valence)
X-r
Chapter 7. A Quantum Mechanical Model for the
Vibration and Rotation of Molecules
Harmonic oscillator
Hookes law:
F kx , x is displacement
Harmonic potential:
1
V ( x) Fdx kx 2
2
k is force constant:
k
d 2V
dx
2
(curvature of V at equilibrium)
x 0
Newtons
Chapter 2. The Schrdinger Equation
Classical waves
xt
1 ( x, t ) A sin[2 ( ) ] A sin(kx t )
T
2 ( x, t ) B cos(kx t )
3 ( x, t ) C[sin(kx t ) i cos(kx t )] Cei ( kxt )
Wave equation:
d 2 4 2
2 0
dx 2
(standing wave)
Recall h / p and p 2 [ E V ( x)]/ 2m
Chapter 1. From Classical to Quantum Mechanics
Classical Mechanics (Newton): It describes the motion of a
classical particle (discrete object).
F ma
dp
,
dt
p = m = m
dx
dt
F: force (N)
a: acceleration ( a d / dt d 2 x / dt 2 )
p: momentum (kg m/s)
x: po
Chapter 3. The Quantum Mechanical Postulates
Postulates 1: A quantum state is completely specified by a wave
function, which is the solution of the Schrdinger equation.
Born's interpretation: The probability of finding the system between x
2
and x+dx is p
Chapter 4. Using Quantum Mechanics on Simple
Systems
Free particle Hamiltonian:
2 d 2
H
2m dx 2
(V=0)
Schrdinger equation:
2 d 2
E
2m dx 2
Solutions:
1/ 2
( x) e ikx ,
2mE
k 2
(k )2
E
,
2m
p k
Particle in a 1D box:
V ( x) 0, 0 x L
, x 0, and x L
Sc
Chapter 6. Commuting and Noncommuting Operators
Order of operations might be important in QM!
Two operators commute if the commutator is zero
[1, 2 ] 12 21 0
p and T commute:
p2
p3 p3
p2
p2
f ( x)
pf ( x)
p, f ( x ) p
f ( x) 0
2m
2m
2m
2m 2m
whe
Chapter 16. Molecular Symmetry
I. Symmetry
Elements
Operations
axis
mirror plane
inversion center
.
rotation about an axis
reflection thru a plane
inversion thru a center
Five symmetry elements and corresponding operations:
i. Doing nothing, identity E
ii