Character Table: a collection of the characters of all irreducible
representations of the group. The sum of the squares of the
characters of all rows and columns of the character table give the
order of the group (or dimension of the group, or total numbe
Note: 4 Classes
- Sums of squares of all rows and
columns = 4
s, z
1
1
1
1
- x, y, and z all transform with
different symmetries
1
1
-1
-1
- s-orbitals on carbon transform
totally symmetric
1
-1
1
-1
x
- p-orbitals on carbon transform
as x, y, z
1
-1
-1
1
Symmetry Elements: operators that operate on a molecule having
the symmetry of the group that leaves the molecule structurally
unchanged except for possible interchange of equivalent elements.
Common symmetry elements are;
the identity element
rotation ab
OK. Lets learn some new things;
Molecular Motions:
3 Translations
xyz
3 Rotations
about center of mass around inertial axes
3n 6 Vibrations about center of mass
These arise because every atom in the molecule has 3 degrees of
freedom. The xyz coordinates
Rotations are tricky more in a minute
Vibrations are given by transformation of normal mode displacements
center of mass
C
H
C
C
H
Symmetric
Stretch
H
H
H
Asymmetric
Stretch
All 3 of these vibrations conserve the center of mass.
H
Bend
CHEM 4641 Physical Chemistry I
Chapter 15 (32) Recommended Problems, continued
Chapter 15 (32) Problem 15 Vibrational contribution to the constant-volume heat capacity of HCl.
Equation 40 is convenient for calculating vibrational heat capacity.
Figure 3.3. The Lennard-Jones interaction potential, as in Equation (3.17). The energy between an
atom pair, , is expressed in terms of the Lennard-Jones coefficient ; similarly, the pair distance is
in terms of the parameter . The simple form includes bo
expansion,
, where
and are summation coefficients.
is torsional angle,
is a summation index,
On the other hand, the nonbonded interactions apply to atoms that are not connected by
bonds, either within the same molecule or between two different molecules.
where an object lies in space. For microscopic objects, like atoms and electrons, these
distributions look much broader by comparison.
It is beyond the scope of our inquiry to enter a deep discussion about the origins of
the wavefunction and quantum theor
Chapter 3
Energy and how the microscopic world works
3.1
Quantum theory
To conceptualize the molecular origins of thermodynamic equilibrium, we must first
understand the fundamental principles by which molecules interact. So, how does the
world really wor
for all atoms
(3.13)
Alternatively, using Equation (3.12),
(3.14)
When an atomic system evolves according to Newtons laws, its total energy is given by a
classical Hamiltonian, in analogy with the quantum Hamiltonian operator. This is simply a
sum of the
Figure S3: MALDI-MS of PDI-OH. Comparison of PDI-OH in its initial state and post annealing (120
o
C, 16hrs) demonstrates the loss of a methyl group.
S12
Figure S4: EPR raw spectra of as-cast and annealed (a) PDI-OH and (b) PDI-NMe2. Asymmetry in the
PDI-NMe2 spectra is attributed to spin anisotropy within the sample in solid-state.
S13
Figure S7: XPS comparison (Nitrogen edge) of PDI-NMe2 after varied annealing times (as cast, 20
minutes, 4 hours, and 16 hours at 120 oC). Charge carrier concentrations were quantified with EPR for
corresponding samples.
S15
Figure S5: UV/VIS/NIR in solid-state. Comparison of the solid-state spectra for PDI-OH, PDI-I, PDINMe2, and PDI-C6 in the as-cast state and the annealed states. Doping is observed in PDI-OH and PDINMe2 upon film casting. The signal intensity in the PDI po
Figure S9: 1H NMR spectra recorded in CDCl3 (500 MHz). Comparison of PDI-NMe2 neat and post
annealing (120 oC, 16hrs) suggests that longterm thermal treatment results in chemical transformation of
PDI-NMe2.
S17
Quantum effects are dominant at very small scales, typically Angstroms (s) or less.
By quantum effects, we mean that the probability distribution implied by the wavefunction
is not sharply peaked, and uncertainties in the positions and velocities of the a
Here is how we obtain
;
Changes no directions therefore +1 always
Flips all atoms over, -1 ( U D and D U )
Leaves all directions unchanged, +1 ( U U and D D )
Reverses direction for all atoms, -1 ( U D and D U )
etc. for Rx, Rz
When you get more atoms it
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