Homework Week 2: Due Friday, October 17
Chemistry 110A 2014
Professor McCurdy
1. Properties of wave functions. Show by explicit integration that the particle-in-a-box
wave functions, which are given by,
2 $ n x '
n (x) =
sin&
n = 1,2,3,
)
L % L (
satisfy
Ch17 The Boltzmann Factor and Partition Function1(132)
I. The Boltzmann factor
Energy
Probability of
residing in state Ej
j+1
Partition function
(Normalization factor)
.
.
.
Probability = Pj = [exp(-Ej /kT)]/Q
For N particles in state Ej
Energy level dep
Ch16 The Properties of Gases1(119)
I. Ideal Gas Equation
Ideal gas-equation of state
or
Extensive properties:
e.g.: P, V
All gases obey ideal gas at a sufficiently low density
where
Intensive properties:
e.g.: density, T, and
Pressure Unit:
Gravitational
Hour Exam I (10-22-2008)
1.
The four tetrahedral hybrid orbitals [SP3(1),SP3(2), SP3(3), and SP3(3)] are
shown the figure. Fill in the coefficients in the parentheses below for the 2s, 2px,
2py, and 2pz atomic orbitals such that SP3(1),SP3(2), SP3(3), and
Hour Exam II
1. (20 points) Find the best match for each item in the left column from among
those in the right column and its letter on the blank provided.
1. Harmonic oscillator selection rules
_
A. 0
v = 0, 2
2. Centrifugal distortion correction
_
B. -e
CHEMISTRY 110B-FINAL EXAM
NAME: _
Last,
First
ID# _
Question #
Score
I (20 points)
II (20 points)
III (20 points)
IV (20 points
V (20 points)
VI (20 points)
VII (20 points)
VIII (20 points)
IX (40 pts)
Total: (200 pts)
Useful Constants:
h = 6.626x10-34 J.
The theme I have for this problem solution set is getting used to picking out the salient features of
a solutions so that you can generalize and reuse the work you do.
To this end, Dirac notation is previewed as a shorthand. Prof. McCurdy will give a more
1
Find the values of the following commutators for the hydrogen atom Hamiltonian. Only very little algebra is required, because you may use the values of any of the
angular momentum commutators we wrote in class without deriving them.
(a) H, Lx (b
Chem 110A Equations for Exams
x
1D:
p x = i
3D:
Operators
2
2 2
KE =
H =
+V ( x )
2m x 2
2m x 2
2 2
H =
+V ( x, y, z)
2m
2
p = -i
2
2
2
2 = + +
3D volume element: dx dy dz = r 2 sin dr d d
2
2
2
x y z
1
1
1
2
sin +
2 = 2 r 2 + 2
r r r r sin
r 2 s
CHE 110B
2012
HOMEWORK 1 - SOLUTIONS
1. Express the Hamiltonian operator for a hydrogen molecule in atomic units.
10 points
The Hamiltonian operator for a hydrogen molecule is given by SI units by equation 9.2.
Let me = e = = 40 = 1 to convert SI units to
Symmetry of molecular systems
Symmetry elements: plane, axis and point (center of inversion)
Symmetry operations:
1. rotation about a proper axis. For example, C2 means rotation about 3600/2 (1800),
C3 rotation about 3600/3 (1200), where n is an order of
Ch15 Laser Spectroscopy and Photochemistry8(112)
IV. Population Inversion in a Three-Level System
At equilibrium:
Ch15 Laser Spectroscopy and Photochemistry10(114)
Example:
Radiant power
Av. Radiant power
Ch15 Laser Spectroscopy and Photochemistry9(113)
V
Ch14 Nuclear Magnetic Resonance Spectroscopy1(83)
I.
Intrinsic Nuclear Spin Angular Momenta
Intrinsic spin for electrons
Intrinsic nuclear spin
Spin associated with a magnetic
dipole moment
The properties of some common nuclei used in NMR experiments:
Nuc
Chemistry .110A
Midterm 2 Exam
May 21, 2014
Prof. P.B. Kelly
NAME ALA/
Values of same Physical Constants . _ _ '
-._._._
Constant Symbol ._ I .Vglue - I _ _
W
1 15 Atomic mass constant mu I _ - [.660 "5402 x 10- kg _
' Avogadro conspant - NA- -. 6.022 136
Ch13 Molecular Spectroscopy1(52)
I. Electromagnetic Spectrum-Molecular Structure
A. Spectroscopy: Interactions of electromagnetic radiation with atoms and molecules
E E f E f h hc /
eV 23.06kcal / mol 8065.54cm 1
E ~ 1
v (cm 1 )
hc
eV
12398.4
; cm 1 29
Ch10: General LCAO-MO Method (16)
I. Convert Differential Equation into problem in linear Algebra (p.249-256, McQ. & Simon)
A. Seek solution of
B. Have a known (complete) set of function
(r )
Expand
Cj
j
i
(r )
j
[
C. Insert in differential equation
Cj
j
Ch10 continued (6)
II. B. Planar sp2 hybrids (e.g., BCl3, triangular planar geometry)
1) From previous example for linear geometry, we see that the direction of the hybrid
orbital is formed by the vector like character of the p-orbitals. This gives us a q
Chapter 10: Bonding in Polyatomic Molecules (1)
I. Solving the molecular electronic structure problem with the Born-Oppenheimer
approximation
A. Born-Oppenheimer approach
Seek solution for
(r , R) (r , R)
E (r , R)
where r stands for the electron coordina
Ch10: LCAO (22)
0
E
0
0
E
Secular determinant:
0
0
E
0
0
E
x,
Energy
E
Thus
x 1
1 x 1
1
x 1
1 x
-1.618
0
-0.618
Expansion of determinant gives:
+0.618
x4 - 3x2 + 1 =0; set y=x2,
we have y2 3y +1 =0
y
x
E=
( 3)
3
( 3) 2
2
5
2
4
3
5
2
1.618; 0.618
1.618 and
Ch11: Computational Quantum Chemistry-1(25)
I. Ab initio Quantum Chemistry
A. The wavefunction for a closed-shell molecule with N (even) electrons is:
(1, 2,3,., N )
1
N!
1 (1) (1)
1 (2) (2)
1 (1) (1)
1 (2) (2)
.
.
.
.
.
.
.
.
N / 2 (1) (1)
N / 2
Ch129(42)
IV. Character Table Most Important Property of Point Group
For the application of group theory: only the SUMs of the diagonal elements of the
(irreducible representation) matrices (=TRACEs=Characters) are needed.
The character table of the C2v p
Ch12: Application of Group Theory in Molecular Problems 1(34)
I.
Symmetry consideration simplifies molecular problems
can often provide enough simplification that the problem may be understood on the
symmetry analysis alone.
A. Block diagonalizes matrix r
Ch13 Molecular Spectroscopy5(56)
J = -1
Line spacing
increases with J
J=0
J = +1
Line spacing
decreases with J
Rotational-vibrational spectrum of the 0 1 vibrational transition of HBr
C. Vibration-rotation interaction
The equilibrium distance Re slightly
Ch13 Molecular Spectroscopy20(71)
H. Vibations of Polyatomic Molecules: Normal Coordinates
Degrees of freedom for a polyatomic molecule with N atoms
Total number of degrees of freedom:
3N
Linear Molecules:
Translation
3
Rotation
2
Vibrational
3N - 5
Nonli