CHEM443 Physical Chemistry 1
Midterm Exam 1
October 7, 2011
Instructions: There are 5 questions for a total of 100 points. Please make sure to present your
solutions in a clear, logical, and legible manner. Indicate your final answer(s) where appropriate.
two different (non-orthogonal) orbitals ( - x') and ( + x'). This picture is closely
related to the so-called generalized valence bond (GVB) model that W. A. Goddard and his
co-workers have developed (see, for example, W. A. Goddard and L. B. Harding, Ann
3
1
1
E
E(X) + E(Y)
1/2 [E(X) + E(Y) + E(Y) + E(X:)]
E(Y) + E(X:)
1
R
Configuration correlation diagram for a homonuclear case in which heterolytic bond
cleavage is energetically favored.
It is essential to realize that the energies H of the CSFs do not
The above strong CI mixing of 1 and 1* as R qualitatively alters the
configuration correlation diagrams shown above. Descriptions of the resulting valence
singlet and triplet states are given below for homonuclear situations in which covalent
products lie
2
2
2
1e 2e
2
1
1e 1o
2
1
Even
1
Odd
2
1
1e 1o 2e
1
3
Odd
1
1e 2e 1o
1
Even
2
2
1e 1o
2
1e 2e
2
This SCD predicts that the thermal (i.e., on the ground electronic surface)
disrotatory rearrangement of 1,3-butadiene to produce cyclobutene will experience a
Section 3 Exercises, Problems, and Solutions
Review Exercises
1. For the given orbital occupations (configurations) of the following systems, determine
all possible states (all possible allowed combinations of spin and space states). There is no
need to f
*
E
sY
sX
R
Re
Energies of the bonding and antibonding * orbitals as functions of internuclear distance.
Here, X is more electronegative than Y.
For the homonuclear case, as R approaches , the energies of the g and u
orbitals become degenerate. Moreover,
1.
b. Forming SALC-AOs from the C and H atomic orbitals would generate the
following:
H
H
C
C
H
H
H1s + H1s = g = a1
H1s - H 1s = u = b2
H
H
H
H
C
C
H
H
C2s = a1
C
C
H
H
C2p = a1
C2p = b2
z
C2p = b1
y
x
The bonding, nonbonding, and antibonding orbitals of
Orbital-correlation diagram for the reaction C + H2 -> CH 2 (bent)
u(b2)
2px(b1)
2py(b2)
2s(a1)
g(a1)
b2(antibonding)
a1(antibonding)
2pz(a1)
b1(2p)
a1(non-bonding)
b2(bonding)
a1(bonding)
CH2 (bent)
C + H2
1.
d. - e. It is necessary to determine how the
1.
b. Remember that when coupling non-equivalent linear molecule angular momenta,
one simple adds the individual Lz values and vector couples the electron spin. So, in this
case (1 u12 u1), we have ML values of 1+1, 1-1, -1+1, and -1-1 (2, 0, 0, and -2).
7.
1
iv. 1D(ML=0,M S =0) = ( 2|p0p0| + |p 1p-1| + |p -1p1|)
6
Evaluating <1D(ML=0,M S =0)|H|1D(ML=0,M S =0)> we note that all the Slater Condon matrix
elements generated are the same as those evaluated in part iii. (the signs for the
wavefunction componen
CHEM-443, Fall 2013, Section 010
Final Exam
Student Name_
December 9, 2013
Directions: Please answer each question to the best of your ability. Make sure your response is legible, precise,
includes relevant dimensional units (where appropriate), logically
CHEM-443, Fall 2013, Section 010
Quiz 1
Student Name_
August 30, 2013
Directions: Please answer each question to the best of your ability. Make sure your response is legible, precise,
includes relevant dimensional units (where appropriate), logically pres
In constructing the CCD, one must trace the energies of all four of the above CSFs
(actually there are more because the singlet and triplet excited CSFs must be treated
independently) along the proposed reaction path. In doing so, one must realize that th
is the triplet CSF. In each case, the spin quantum number S, its z-axis projection MS , and
the quantum number are given in the conventional 2S+1 (MS) notation.
2. Orbital, CSF, and State Correlation Diagrams
i. Orbital Diagrams
The two orbitals of the co
1. Orbitals, Configurations and States
The resultant family of six electronic states can be described in terms of the six
configuration state functions (CSFs) that arise when one occupies the pair of bonding
and antibonding * molecular orbitals with two
Although, in principle, it is possible for some fraction of the events to follow the 1Odd
surface beyond this second intersection and to thus lead to 1Odd product molecules that
might fluoresce, quenching is known to be rapid in most polyatomic molecules;
to produce acetylene and 1D carbon is known to occur with an activation energy equal to
the reaction endothermicity. However, when triplet carbene decomposes to acetylene and
ground-state (triplet) carbon, the activation energy exceeds this reaction's end
Even for the relatively simple two-electron, two-orbital single-bond interactions
between a pair of atoms or functional groups, the correlations among energy-ordered
molecular states and energy-ordered asymptotic states is complex enough to warrant
consid
1
E
1
3
E(Y) + E(X:)
1/2 [E(X) + E(Y) + E(Y) + E(X:)]
E(X) + E(Y)
1
R
Re
Configuration correlation diagram for homonuclear case in which homolytic bond cleavage
is energetically favored.
When heterolytic bond cleavage is favored, the configuration energ
III. Various Types of Configuration Mixing
A. Essential CI
The above examples of the use of CCD's show that, as motion takes place along the
proposed reaction path, geometries may be encountered at which it is essential to describe
the electronic wavefunc
1
E
3
1
E(X) + E(Y:)
E(X) + E(Y)
E(X:) + E(Y)
1
R
State correlation diagram for heteronuclear case in which heterolytic
bond cleavage to one product is energetically favored but homolytic
cleavage lies below the second heterolytic asymptote.
are substituted into the Slater determinant definitions of the CSFs. Here C and C* are the
normalization constants. The parameter z is 1.0 in the homonuclear case and deviates from
1.0 in relation to the sx and sy orbital energy difference (if sx lies bel