Download 2015 Ch112 – problem set 5 Due: Thursday, Nov

2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
Problem 1 (2 points)
1. Consider the MoMoCr heterotrimetallic complex shown below (Berry, et. al. Inorganica
Chimica Acta 2015, p. 241). Metal-metal bonds are not drawn. The ligand framework distorts
this structure so that it is in the C4 point group, as seen by looking down the CrMoMo axis. For
this problem assume C4v symmetry.
a. Assign an average oxidation state for the metal centers and total number of d-electrons in this
molecule:
N
N
Cl
N
Mo
N
Mo
N
N
N
N
N
N
Cl
Cr
N
N
N
N
Looking Down The
Mo-Mo-Cr axis:
N
N
N =
Cr
NN
N
N
N
NN N
N
Mo2+, Mo2+, Cr2+
12 electrons
N
N
N
N
b. Draw a qualitative MO diagram for the Mo-Mo interaction in the C4v point group, using the
ten Mo d-based orbitals as your basis set. Ignore Cr for this part. The z axis is along the M-M
vector.
i. Assign Mulliken symbols
ii. Label each orbital with the type of interaction (nb, σ, σ*, etc)
iii. Fill in the diagram with electrons assuming the average oxidation state as above. Predict the
spin state of this fragment.
iv. Determine the Mo-Mo bond order
δ∗
dx2-y2, B1
dx2-y2, B1
δ
σ∗
π∗
Mo-Mo quadruple
bond.
δ∗
dxy, B2
dxy, B2
dz2, A1
dxz, dyz, E
dz2, A1
δ
dxz, dyz, E
π
σ
1
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
c. Starting from your answer to part b, draw the qualitative MO diagram for the Mo-Mo-Cr
interaction. Consider whether atomic orbitals/molecular orbitals will mix significantly based not
only on their symmetry, but also spatial overlap. Assume that the Mo & Cr based atomic orbitals
are approximately equal in energy. Note that the crystallographic Mo-Mo bond distance is 2.098
Å and the Mo-Cr bond distance is 2.689 Å.
i. Assign Mulliken symbols
ii. Label each orbital with the type of interaction (nb, σ, σ*, etc)
iii. Fill in the diagram with electrons
iv. Predict the Mo-Cr and Mo-Mo bond orders.
Mo-Mo: 3.5
Cr-Mo: 0.5
The labels in the center reflect the character of the interactions between Mo and Cr. The very
long Cr-Mo distance compared to Mo-Mo indicate a much weaker interaction with the first row
transition metal. Computationally, the bond Mo-Mo order is similar to the one observed in Mo2
compounds (quadruple), although the Mo-Cr interaction is non-zero.
d. Draw the Mo-Mo-Cr molecular orbitals that correspond to the interactions between the three
metals that have σ and π symmetry with respect to the z axis. Comment on their bonding / nonbonding / antibonding character.
2
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
e. Magnetic susceptibility experiments on the Mo-Mo-Cr trimetallic are consistent with a quintet
ground state (S = 2). Rationalize the experimental result and discuss in the context of potential
lower spin electronic configurations.
The MO diagram drawn above is consistent with this picture, in which there are four unpaired
electrons. Cr is effectively an isolated monometallic center, except for the dz2 σ-interaction.
These non-bonding orbitals are all close enough in energy for the electrons to overcome spinpairing energy. Stronger field ligands on Cr or a stronger interaction with the Mo center could
lead to a lower spin complex.
f. How do you expect the MO diagram to change for an analogous Mo-Mo-Mo complex of D4h
symmetry? Predict the bond orders of the two Mo-Mo interactions.
The overall M-M bond order of the compound will be the same, assuming that
the terminal Mo centers lead to equivalent interactions with the central one. The
four M-M bonds will be shared equally over both M(central)-M(terminal)
interactions for a total bond order of two each. No antibonding orbitals will be
populated.
3
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
Problem 2 (3 points)
y
x'
y'
L
L
L
L
M
L
x
L
L
L
(view along the M-M vector)
In class, the electronic structure of bimetallic complexes of the form [L4M−ML4] in D4h
symmetry was analyzed. Consider here the staggered conformer. The z axis is perpendicular to
the plane of the page. Using the coordinate system as given above (x,y for the metal in the front
and xꞌ,yꞌ for the metal in the back), answer the following questions:
1. What is the point group of [L4M−ML4]?
D4d
2. Considering σ interactions only, provide a qualitative d orbital splitting diagram for the
square planar [ML4] fragment. Label each orbital with its d orbital parentage. Label each
orbital with the type of M−L interaction (nb, σ, σ*)
x2-y2
σ∗
z2
σ∗
xy, xz, yz
nb
3. For the basis set consisting of the dx2-y2, dxy, dx2-y2ꞌ, dxyꞌ orbitals determine which ones are
related by symmetry and express them as a sum of irreducible representations. Provide the
normalized wavefunction for each SALC. Show your work.
D4d
E 2S8 2C4 2S83 C2 4C2ꞌ 4σd
Γ(dx2-y2, dxy, dx2-y2ꞌ, dxyꞌ) 4 0
−4
0
4
0
0
Γ(dx2-y2, dxy, dx2-y2ꞌ, dxyꞌ) = 2E2
4
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
ψ1(E2) = dx2-y2; ψ2(E2) = dx2-y2ꞌ; ψ3(E2) = dxy; ψ4(E2) = dxyꞌ
dx2-y2 and dxy or dx2-y2ꞌ and dxyꞌ are not related by symmetry. The two E2 sets are,
respectively:
1a) dxy and 1b) dxyꞌ
2a) dx2-y2 and 2b) dx2-y2ꞌ
4. For the basis set consisting of the dxz, dyz, dxzꞌ, dyzꞌ orbitals determine which ones are related
by symmetry and express them as a sum of irreducible representations. Provide the
normalized wavefunction for each SALC. Show your work.
D4d
E 2S8 2C4 2S83 C2 4C2ꞌ 4σd
Γ(dxz, dyz, dxzꞌ, dyzꞌ) 4 0
0
0
−4
0
0
Γ(dxz, dyz, dxzꞌ, dyzꞌ) = E1 + E3
D4d E
S8
S87
S83
S85
C2
1/2
1/2
1/2
E1 2
2
2
−2
−21/2 −2
dxz dxz dyzꞌ −dxzꞌ dxzꞌ −dyzꞌ −dxz
dyz dyz −dxzꞌ −dyzꞌ dyzꞌ dxzꞌ − dyz
ψ1(E1) = 2 dxz + 21/2 dyzꞌ − 21/2 dxzꞌ
ψ2(E1) = 2 dyz − 21/2 dyzꞌ − 21/2 dxzꞌ
D4d E
S8
S87 S83 S85
C2
1/2
1/2
1/2
1/2
E3 2 −2
−2
2
2
−2
dxz dxz dyzꞌ −dxzꞌ dxzꞌ −dyzꞌ −dxz
dyz dyz −dxzꞌ −dyzꞌ dyzꞌ dxzꞌ − dyz
ψ3(E3) = 2 dxz − 21/2 dyzꞌ + 21/2 dxzꞌ
ψ4(E3) = 2 dyz + 21/2 dyzꞌ + 21/2 dxzꞌ
5. Starting from the d-orbital splitting diagram of the two [ML4] fragments from part 2, sketch a
qualitatively correct molecular orbital diagram for [L4M−ML4]. Label each molecular orbital
with the type of M−M interaction (σ, σ* etc.). Draw the δ bonding interactions.
5
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
σ∗
dx2-y2
δ∗
dx2-y2'
π∗
dz2'
dz2
dxy', dyz', dxz'
dxy, dyz, dxz
δ
π
σ
6. Assume that the donors L correspond to monoanionic L2 chelating ligands and M = Mo to
generate a fragment related to the complex analyzed in problem 1 part b, but of different
symmetry. What is the Mo−Mo bond order under the symmetry of problem 2? What is the
spin state? Discuss the difference in bond order between the Mo2 compounds of different
symmetry, and provide an example of an organic molecule that undergoes a change in bond
order upon rotation around a C−C bond.
Both are Mo(II), d4. Strictly, the bond order is 4, but the δ interactions are very weak
because of the energy mismatch of the orbitals that mix (ex: dxyꞌ and dx2-y2). Due to the
weak overlap, the predicted bond order is 3. The compound is expected to be paramagnetic,
with S = 1. The δ interaction would be stronger in the D4h symmetry, and the bond order
would be 4. Also, the D4h conformer is expected to be diamagnetic.
Problem 3 (2 points)
1. In class, the “isolobal analogy” was presented as a way to predict the stability and bonding of
hypothetical organometallic compounds. Roald Hoffman describes it as a “bridge between
organic and inorganic chemistry” (Angew. Chem. Int. Ed., 1982, 21, p. 711). Using the isolobal
analogy, several different metallacyclopropane fragments can be envisioned. For the neutral,
closed shell structures below, fill in the table, and assign the identity of the first-row transition
metal:
6
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
H
H
H
e-s required for
closed shell
e s required from
central atom
identity of atom
H
H
C
M
H
H
geometry
L
H
H
H
H
L
H
H
H
L
L
M
L
H
H
H
L
M
tetrahedral
octahedral
Square planar
Octahedral/piano
stool
8
18
16
18
4
8
10
9
Carbon
Fe
Ni
Co
2. To the right, it is shown that Mn(CO)5 is isolobal
CO
with methyl radical, and [Mn(CO)5]+ is isolobal with
OC
methyl cation. Note that in describing these
OC Mn
fragments, there is no need to consider the most stable
geometry, as they are not isolated as species. You
CO CO
simply need to keep the geometry the same from
fragment to fragment.
+e
Consider the reaction shown below. A Mn-C bond of
Mn(CO)5 (cis to the open site) is homolyzed,
releasing CO+. What is the resultant Mn fragment?
What first-row metal can you swap in for Mn to make
this an isoelectronic neutral compound?
OC
OC
OC
OC
-e
CO
OC
+e
-CO
CO
CO
-e
-
+
+
CO CO
"methyl cation"
still tetrahedral
+
OC
-
Mn
-
Mn
CO CO
OC
methyl radical
CO
Mn
L
H
OC
OC
CO
OC
CO
Cr
CO CO
Fe
CO
3. To extend the analogy, provide the identity of four neutral M(PR3)x fragments (x = 4, 5, or 6)
that are each isolobal to the following organic fragments. Provide the identity of the transition
metal, M.
a. methane
b. methyl radical
c. methylene
d. methyl anion
7
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
R 3P
R 3P
PR3
Cr
R 3P
PR3
PR3PR3
R 3P
PR3
Mn
PR3PR3
R 3P
R 3P
PR3
Fe
R 3P
R 3P
PR3
PR3
Fe
PR3PR3
4. For each of the four metal fragments drawn in part 3, use the isolobal analogy to predict if
each of them might react with an incoming carbon dioxide molecule to generate a product with
the CO2 fragment in bent geometry, and if so, draw the structure of the product of that reaction.
Problem 4 (3 points)
Part A
In class, we considered a free ion with a p2 electronic configuration. Consider now the case in
which two free ions with p1 electronic configuration come together, forming a p1p1 configuration.
The electron does not ‘hop’ from one ion to the other, and the two ions are distinguishable.
1. Determine the total number of possible microstates for the p1p1 configuration.
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2. Determine the ground state term using the shortcut presented in class.
3D
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2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
3. Prepare a microstate table and populate with all possible microstates. Example notation:
(1a+,1b+)
ML\Ms
2
0
(1a+,1b+)
(1a+,1b−)(1a−,1b+)
−1
(1a−,1b−)
1
(1a+,0b+)(0a+,1b+)
(1a+,0b−) (1a−,0b+)
(0a+,1b−)(0a−,1b+)
(1a−,0b−)(0a−,1b−)
0
(1a+,−1b+)(0a+,0b+)(−1a+,1b+)
(1a−,−1b−)(0a−,0b−)(−1a−,1b−)
−1
(−1a+,0b+)(0a+,−1b+)
(1a+,−1b−)(1a−,−1b+)
(−1a+,1b−)(−1a−,1b+)
(0a+,0b−)(0a−,0b+)
(−1a+,0b−) (−1a−,0b+)
(0a+,−1b−)(0a−,−1b+)
−2
(−1a+,−1b+)
(−1a+,−1b−)(−1a−,−1b+)
(−1a−,−1b−)
1
(−1a−,0b−)(0a−,−1b−)
4. Reduce the microstate table for the p1p1 configuration to its component free-ion terms.
Include the terms resulting from spin-orbit coupling. Order all terms based on energy strictly
following Hund’s rules.
3D: J = {3, 2, 1}  3D < 3D < 3D
1D: J = {2}  1D
1
2
3
2
3P: J = {2, 1, 0}  3P < 3P < 3P
1P: J = {1}  1P
0
1
2
1
3S: J = {1}  3S
1S: J = {0}  1S
1
0
3D (ground state) < 3P < 3S < 1D < 1P < 1S
Part B
Consider a free ion with a d3 electronic configuration.
1) By inspection, obtain the term symbol (2S+1L) for the ground state.
4F
2) For this ground state, obtain all possible J values and order them from lowest to highest in
energy.
J = {9/2, 7/2, 5/2, 3/2}: 3/2 < 5/2 < 7/2 < 9/2
3) The first excited state for the d3 ion is the 4P state. How many microstates does the 4P state
contain?
S = 3/2  2S+1 = 4
L = 1  2L+1 = 3
(2S+1)(2L+1) = 12 microstates
4) Splitting of terms and orbitals in a chemical environment
9
2015 Ch112 – problem set 5
Due: Thursday, Nov. 19 – before class
Remember to turn in your project abstract with this problem set
Consider an octahedral ligand field on a set of atomic wave functions. The full symmetry of the
octahedron is Oh but we can work with the rotational subgroup O. An atomic orbital can be
represented as follows (radial, angular (θ, φ), and spin wavefunctions):
Ψ = 𝑅𝑅(𝑟𝑟) ∙ Θ(𝜃𝜃) ∙ Φ(𝜙𝜙) ∙ 𝜓𝜓𝑠𝑠
We can ignore ψs assuming negligible spin-orbit coupling. The radial function R has no
directionality, so it can also be ignored. The angular function Θ is invariant with respect to
rotation in the z axis (principal rotation axis), so it can be ignored as well. Working only with Φ,
we show without derivation that the character of the reducible representation under Cα for a basis
set in which the orbital angular momentum is l is given by:
1
sin[�𝑙𝑙 + 2� 𝛼𝛼]
𝜒𝜒(𝛼𝛼) =
sin(𝛼𝛼/2)
For example, for the set of f orbitals (l = 3) :
𝜒𝜒(𝐶𝐶3 ) =
1 2𝜋𝜋
sin[�3 + 2� � 3 �]
In the limit where α = 0, χ(E) = 2l+1.
sin(𝜋𝜋/3)
=
sin(7𝜋𝜋/3)
=1
sin(𝜋𝜋/3)
a) By applying the formula, complete the table below.
O
E
6 C4
3 C2 (= C42)
8 C3
6 C2 ꞌ
S (l = 0)
1
1
1
1
1
P (l = 1)
3
−1
1
−1
0
D (l = 2)
5
1
−1
1
−1
F (l = 3)
7
1
−1
−1
−1
G (l = 4)
9
1
1
1
1
H (l = 5)
11
−1
1
−1
−1
I (l = 6)
13
1
−1
1
1
b) Each row is a reducible representation in the O point group. Represent each row as a sum of
irreducible representations.
S: A1
P: T1
D: E + T2
F: A2 + T1 + T2
G: A1 + E + T1 + T2
H: E + 2T1 + T2
I: A1 + A2 + E + T1 + 2T2
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