pi-Complexes I: Alkenes, Alkynes & eta3-Allyl Complexes

Line-by-Line Solution:

(a) Formal Oxidation State and Electron Count:

1. Ligand charges: Three chloride ligands ($-1$ each) and one neutral ethylene ligand ($L$, formal charge $0$).
2. The complex anion is $[\text{PtCl}_3(\text{C}_2\text{H}_4)]^-$ with net charge $q = -1$.

1. Platinum is in Group 10 ($n_v = 10$):

1. Total valence electron count ($VEC$):

• $\text{Pt}(\text{II})$: 8 electrons
• Three $\text{Cl}^-$: $3 \times 2 = 6$ electrons
• One $\eta^2-\text{C}_2\text{H}_4$: 2 electrons

• A classic 16-electron square planar $d^8$ transition metal complex.

(b) Orientation of the $C=C$ Axis Perpendicular to the Square Plane:

1. In a square planar $d^8$ complex in the $xy$-plane, the empty metal orbital accepting the $\sigma$-dative pair from the alkene $\pi$-orbital is a $5d/6s/6p$ hybrid pointing along an in-plane coordination vector (say, along the $x$-axis).
2. For $\pi$-backbonding, the metal must utilize a filled $d$-orbital directed toward the alkene $\pi^*$ LUMO.
3. If the $C=C$ bond lies in the $xy$-plane (parallel):

• The metal orbital of matching $\pi$-symmetry would have to be $d_{xy}$.
• However, $d_{xy}$ is directed between the four in-plane ligands, experiencing strong steric and electrostatic repulsion with the cis chlorides.

1. If the $C=C$ bond is oriented perpendicular to the plane (along the $z$-axis):

• The alkene $\pi^*$ LUMO lies in the $xz$-plane.
• It overlaps with the filled metal $5d_{xz}$ orbital, which projects above and below the coordination plane into empty space.
• This orientation provides maximum orbital overlap for $\pi$-backbonding and minimizes steric clash with the adjacent cis chloride ligands.

(c) Bending Back of the Hydrogen Atoms:

1. In uncoordinated ethylene, the carbons are $sp^2$ hybridized with $120^\circ$ planar geometry.
2. As metal-to-ligand $\pi$-backdonation populates the $\pi^*$ LUMO, electron density between the two carbon nuclei drops, while electron density in the $Pt-C$ bonding region increases.
3. The carbon atoms rehybridize from $sp^2$ toward $sp^3$.
4. To attain tetrahedral-like geometry around each carbon atom, the four hydrogen substituents bend away from the platinum atom.
5. In Zeise's salt, the observed dihedral bend-back angle is $\alpha = 32.5^\circ$, corresponding to intermediate character between planar ethylene ($0^\circ$) and fully $sp^3$ metallacyclopropane ($109.5^\circ - 90^\circ \approx 54^\circ$).

Line-by-Line Solution:

(a) Hapticity and Electron Counting:

• $\eta^1-\text{Allyl}$ (monohapto):
• Bound through a single $M-\text{C}$ $\sigma$-bond.
• Acts as a 1-electron donor in the neutral model ($X$-type) or 2-electron donor as allyl anion ($ ext{C}_3 ext{H}_5^-$).
• The remaining two carbons form an uncoordinated pendant $C=C$ double bond.
• $\eta^3-\text{Allyl}$ (trihapto):
• Bound through all three contiguous carbon atoms.
• Acts as a 3-electron donor in the neutral model ($LX$-type) or 4-electron donor as allyl anion.

(b) Low-Temperature Static $^1\text{H}$ NMR Spectral Prediction:

1. For the $\eta^1-\text{Allyl}$ Isomer ($M-\text{CH}_2-\text{CH}=\text{CH}_2$):

• Contains a localized $\sigma$-alkyl methylene group and a vinyl group.
• Shows three sets of protons:
• Methylene protons ($-\text{CH}_2-M$): $2\text{H}$ at $\delta \approx 2.0-2.5\text{ ppm}$
• Internal vinyl proton ($-\text{CH}=$): $1\text{H}$ at $\delta \approx 5.8-6.2\text{ ppm}$
• Terminal vinyl protons ($=\text{CH}_2$): $2\text{H}$ (split into cis/trans) at $\delta \approx 4.8-5.2\text{ ppm}$
• Total: 3 distinct signals (or 4 if terminal vinyl protons are diastereotopic) with integration ratio $2 : 1 : 2$.

2. For the $\eta^3-\text{Allyl}$ Isomer:

• Exhibits $C_s$ mirror plane symmetry passing through the central carbon and the metal atom.
• Shows three distinct signals:
• Central proton ($H_c$ at C2): $1\text{H}$ multiplet at $\delta \approx 4.8-5.5\text{ ppm}$
• Two syn-protons ($H_s$ at C1, C3): $2\text{H}$ doublet at $\delta \approx 3.8-4.2\text{ ppm}$
• Two anti-protons ($H_a$ at C1, C3): $2\text{H}$ doublet at $\delta \approx 2.8-3.2\text{ ppm}$
• Total: Exactly 3 signals with integration ratio $1 : 2 : 2$.

(c) Thermodynamic Preference for $\text{Pd}(\text{II})$:

• In $[\text{Pd}(\eta^1-\text{C}_3\text{H}_5)(\text{PPh}_3)\text{Cl}]$, palladium is 3-coordinate and possesses only:

• In $[\text{Pd}(\eta^3-\text{C}_3\text{H}_5)(\text{PPh}_3)\text{Cl}]$, the $\eta^3$-allyl donates 4 electrons (ionic model):

• 16 valence electrons represents the closed-shell, thermodynamically stable configuration for square planar $d^8$ $\text{Pd}(\text{II})$.
• Therefore, the $\eta^3-\text{allyl}$ isomer is overwhelmingly favored thermodynamically by $>60\text{ kJ/mol}$.

Line-by-Line Solution:

(a) $[(\eta^5-\text{C}_5\text{H}_5)(\eta^6-\text{C}_6\text{H}_6)\text{Fe}]^+$:

• Ligands present: $\eta^5-\text{cyclopentadienyl}$ (odd, closed) and $\eta^6-\text{benzene}$ (even, closed).
• Apply Rule 1: Nucleophiles attack even polyenes preferentially over odd polyenes:

• Attack occurs exclusively on the $\eta^6-\text{benzene}$ ring, converting it into an uncharged neutral $\eta^5-\text{cyclohexadienyl}$ complex:

(b) $[(\eta^5-\text{C}_5\text{H}_5)(\eta^4-\text{C}_4\text{H}_6)\text{Fe}(\text{CO})]^+$:

• Ligands present: $\eta^5-\text{Cp}$ (odd, closed) and $\eta^4-\text{butadiene}$ (even, open).
• Apply Rule 1: Attack occurs at the even polyene ($\eta^4-\text{butadiene}$) rather than the odd polyene ($\eta^5-\text{Cp}$).
• Apply Rule 3: For even, open polyenes, attack occurs preferentially at the terminal carbon (C1 or C4):

• Attack delivers a neutral $\eta^3-\text{allyl}$ complex: $(\eta^5-\text{Cp})\text{Fe}(\text{CO})(\eta^3-\text{CH}_2\text{CHCHCH}_2\text{OMe})$.

(c) $[(\eta^5-\text{C}_5\text{H}_5)\text{Mo}(\text{CO})_2(\eta^3-\text{C}_3\text{H}_5)]^+$:

• Both polyene ligands are odd: $\eta^5-\text{Cp}$ (odd, closed) and $\eta^3-\text{allyl}$ (odd, open).
• Apply Rule 2: Between ligands of comparable parity, attack occurs at the open polyene rather than the closed polyene:

• Apply Rule 3: Attack occurs at a terminal carbon of the allyl ligand, generating a neutral $\eta^2-\text{alkene}$ complex:

Line-by-Line Solution:

(a) Rate Constant at Coalescence ($k_c$): For an uncoupled two-site exchange with equal population probabilities:

Given $\Delta \nu = 48\text{ Hz}$:

(b) Eyring Activation Parameters ($\Delta H^\ddagger$ and $\Delta S^\ddagger$): Given:

• $T_1 = -20^\circ\text{C} = 253.15\text{ K}$, with $k_1 = 18\text{ s}^{-1}$.
• $T_c = +10^\circ\text{C} = 283.15\text{ K}$, with $k_c = 106.6\text{ s}^{-1}$.

From the linear Eyring formulation:

Let $Y = \ln(k/T)$ and $X = 1/T$:

• At $T_1 = 253.15\text{ K}$:

• At $T_c = 283.15\text{ K}$:

1. Compute Slope:

1. Compute Activation Enthalpy:

1. Compute Activation Entropy:

From $\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger$ at $T_c = 283.15\text{ K}$:

• Physical Interpretation: The negative activation entropy ($\Delta S^\ddagger \approx -99\text{ J/(mol}\cdot\text{K)}$) indicates an ordered transition state where solvent and ancillary phosphine ethyl groups become restricted during the $90^\circ$ rotation.

Line-by-Line Solution:

(a) Stereochemistry of the Palladium-Allyl Intermediate:

1. Starting material: $(R,E)$-1,3-diphenylallyl acetate. The acetate leaving group ($-\text{OAc}$) resides on one defined face of the allylic plane.
2. In the oxidative addition step, the palladium(0) catalyst coordinates the double bond and performs a nucleophilic displacement on the acetate-bearing carbon.
3. This displacement occurs via an outer-sphere inversion pathway: palladium attacks from the face opposite to the leaving acetate group:

1. The resulting cationic $[(\eta^3-\text{1,3-diphenylallyl})\text{Pd}(\text{chiraphos})]^+]$ intermediate has the palladium atom located exclusively on the face opposite to the original acetate.

(b) Absolute Stereochemistry with Soft Nucleophile (Dimethyl Malonate):

1. Dimethyl sodiomalonate is a stabilized carbanion ($\text{p}K_a \approx 13$, soft nucleophile).
2. Soft nucleophiles attack coordinated $\eta^3$-allyl ligands via an outer-sphere mechanism:

• The malonate anion attacks the allylic carbon directly from the solution side, on the face opposite to the palladium atom.
• This nucleophilic addition proceeds with inversion of configuration at carbon:

3. Summing the two elementary steps:

1. The product retains the original $(R)$ absolute configuration: $(R,E)$-dimethyl 2-(1,3-diphenylallyl)malonate.

(c) Stereochemical Divergence with Hard Nucleophile ($Me\text{MgBr}$):

1. Methylmagnesium bromide is a hard, localized carbanion.
2. Hard nucleophiles cannot perform outer-sphere attack on the external face of the allyl ligand.
3. Instead, $Me\text{MgBr}$ attacks the electropositive palladium metal center directly via transmetallation:

1. The methyl group then undergoes intramolecular reductive elimination onto the allyl carbon from the same face as the palladium atom (retention in the elimination step).
2. Net Stereochemical Trajectory with Hard Nucleophile:

• Thus, the reaction with $Me\text{MgBr}$ yields the opposite $(S)$ enantiomer!

Line-by-Line Solution:

(a) Ligand Classifications and Oxidation States:

• $\text{CO}$ is a neutral 2-electron $L$-ligand (formal charge $0$).
• Diethyldithiocarbamate $\text{S}_2\text{CNEt}_2^-$ is a bidentate monoanionic ligand ($L X$-type, donating 4 electrons per ligand, charge $-1$).
• Two dithiocarbamate ligands carry a total charge of $-2$.
• Overall complex charge $q = 0$.
• Tungsten is in Group 6 ($n_v = 6$).
• Model 1: Alkyne as 2-Electron Donor ($L$-type, charge $0$):

• Model 2: Alkyne as 4-Electron Donor ($L_2$-type or $C^2-$ metallacyclopropene, charge $0$ or $-2$):
• Under covalent CBC model: alkyne is $L_2$, formal charge is $0$. Metal oxidation state remains $+2$ ($d^4$).

(b) Total Valence Electron Count ($VEC$):

• Under the 2-Electron Donor Hypothesis:
• Tungsten (Group 6): 6 electrons
• CO: 2 electrons
• Two dithiocarbamates: $2 \times 3\text{e}$ (neutral model: $S^\bullet + S: = 3\text{e}$ each) $= 6$ electrons
• Alkyne (2e): 2 electrons

Under this model, the tungsten center is sub-18e (16 electrons), leaving an empty valence orbital.

• Under the 4-Electron Donor Hypothesis:
• Both orthogonal $\pi$-systems ($\pi_\parallel$ and $\pi_\perp$) donate into tungsten:

• Conclusion: To achieve electronic saturation and satisfy the 18-electron rule, the alkyne must act as a 4-electron donor ($L_2$).

(c) $^{13}\text{C}$ NMR Chemical Shift Prediction:

• For free alkynes, $sp$-hybridized carbons resonate at $\delta = 70 - 90\text{ ppm}$.
• In a 2-electron alkyne complex, carbons shift downfield to $\delta = 110 - 150\text{ ppm}$.
• In a 4-electron donor alkyne complex, both $\pi$-orbitals donate heavily into the metal, while the metal backdonates into $\pi^*$. The $C-C$ bond order approaches a single bond, and the carbons experience extreme downfield deshielding:

This downfield shift (in the range typical of carbenes and metal-alkylidynes) is the definitive spectroscopic signature of a 4-electron donating alkyne.

Line-by-Line Solution:

1. Perturbation Energy for Nucleophilic Attack: According to Klopman-Salem frontier molecular orbital theory, the stabilization energy $\Delta E$ upon interaction between a nucleophile ($ ext{Nu}$) and a coordinated polyene complex is:

where the second term represents the frontier orbital charge-transfer interaction.

2. Frontier Orbital Energy of Coordinated Polyenes: Consider the Hückel $\pi$-orbital levels of even ($2n$) versus odd ($2n+1$) polyenes:

• Even Closed Polyenes (e.g., Benzene, $2n = 6$):
• Ground-state neutral benzene has 6 electrons completely filling the three bonding MOs ($a_{1g}, e_{1g}$).
• Coordination to a transition metal in an 18-electron complex involves donation from $a_{1g}$ and $e_{1g}$ into empty metal orbitals, and backdonation from metal $d$ into the degenerate $e_{2u}$ LUMO.
• However, because the complex bears a formal positive charge ($+m$), the entire orbital manifold is depressed to low energy.
• The lowest unoccupied molecular orbital (LUMO) of the complex has substantial amplitude localized on the ligand carbon atoms with a low orbital energy $E_\text{LUMO}$, producing a small denominator $E_\text{LUMO} - E_\text{HOMO}(\text{Nu})$ and a large orbital interaction $\Delta E$.
• Odd Polyenes (e.g., Cyclopentadienyl $Cp$, $2n+1 = 5$):
• The cyclopentadienyl ligand is formally a $6\pi$ aromatic system as $Cp^-$ ($e_{1g}$ completely filled).
• Its LUMO is $e_{2u}^*$, which lies at an exceptionally high energy level ($> +3\text{ eV}$ higher than benzene $e_{2u}$).
• Even upon coordination to a cationic metal center, the $Cp$ $e_{2u}^*$ orbital remains at a prohibitive energy level.
• Instead, the LUMO of the complex is primarily metal-centered ($d_{z^2}^$ or $e_g^$), with negligible atomic orbital coefficients $c_i$ on the cyclopentadienyl carbon atoms ($c_i \approx 0$).

3. Comparison of Atomic Orbital Coefficients ($c_i$):

• In even polyene complexes ($\eta^6-\text{benzene}$, $\eta^4-\text{diene}$), the LUMO has large coefficients at the carbon atoms ($|c_i| \approx 0.4 - 0.6$).
• In odd polyene complexes ($\eta^5-\text{Cp}$), the LUMO is metal-localized ($c_\text{carbon} \approx 0.05$).
• Consequently:

• Therefore, nucleophilic attack on even polyenes is favored both electrostatically and by frontier orbital overlap by factors exceeding $10^5$, establishing the theoretical foundation of DGM Rule 1.

Line-by-Line Solution:

(a) Kinetic Derivation of $\pi-\sigma-\pi$ Exchange with Added Base $L$:

1. Let the starting 16-electron complex be $\text{Pd}_{\eta^3}$.
2. Step 1: Associative Attack of Base $L$:

Incoming base $L$ coordinates to palladium, inducing an $\eta^3 \to \eta^1$ hapticity ring slip:

3. Step 2: Carbon-Carbon Single Bond Rotation:

In the 16-electron $\sigma$-allyl intermediate, rotation around the $C_\alpha-C_\beta$ bond occurs with rate constant $k_\text{rot}$:

4. Step 3: Dissociation of $L$ and Re-coordination:

Applying the steady-state approximation to the intermediate $[\text{Pd}_{\eta^1}-L]$:

The observed rate of exchange of syn/anti protons is:

Thus, the apparent pseudo-first-order rate constant is:

(b) Saturation Kinetic Regimes:

1. At Low Base Concentration or Fast Rotation ($k_{-1} \gg k_\text{rot}$):

The exchange rate depends linearly on $[L]$ (first-order kinetics).

2. At High Base Concentration ($k_\text{rot} \gg k_{-1}$ or rapid pre-equilibrium):

When coordination of $L$ is quantitative, the rate-determining step becomes the intrinsic single-bond rotation $k_\text{rot}$:

The exchange rate becomes independent of $[L]$ (zero-order saturation).

(c) Orbital Correlation Diagram ($\eta^3 \to \eta^1$):

• In $\eta^3-\text{allyl}$, the three carbon $p$-orbitals form $\psi_1, \psi_2, \psi_3$. Both $\psi_1$ (symmetric) and $\psi_2$ (antisymmetric) overlap with metal $d$-orbitals, donating 4 electrons.
• As one terminal carbon $C_3$ pulls away:
• $\psi_1$ and $\psi_2$ re-hybridize into a localized $C_1-M$ $\sigma$-bonding orbital and a localized $C_2=C_3$ $\pi$-bonding orbital.
• The metal $d$-orbital previously bonded to $C_3$ becomes vacant, receiving the electron pair from the incoming base $L$.
• The $C_1-C_2$ bond becomes a pure $\sigma$-single bond, allowing barrier-free rotation ($E_a \approx 25-35\text{ kJ/mol}$) before reverse slip regenerates $\eta^3$ with exchanged proton environments.

Line-by-Line Solution:

(a) Steric vs. Electronic Regiocontrol:

1. Steric Factor:

• C1 bears a methyl substituent; C3 bears only hydrogen atoms.
• Bulky nucleophiles or sterically encumbered diphosphine ligands favor attack at the less hindered, unsubstituted terminus C3, yielding the linear product ($E$-alkene).

2. Electronic Ground-State Trans-Influence:

• The methyl group at C1 is an electron-releasing inductive substituent ($+I$), which stabilizes the partial positive charge developing in the transition state.
• Furthermore, according to the trans-influence, the carbon terminus trans to the stronger trans-influence ligand experiences greater $Pd-C$ bond lengthening and higher carbocationic character, directing nucleophilic attack to that site.

(b) Regiochemical Outcome with Unsymmetrical $P_1-P_2$ Ligand:

• $P_1$ is a strong $\sigma$-donor (e.g., $\text{PMe}_3$, alkylphosphine).
• $P_2$ is a strong $\pi$-acceptor (e.g., $\text{P(OPh)}_3$, phosphite).

1. In square planar $[(\eta^3-\text{allyl})\text{Pd}(P_1)(P_2)]^+$, one allyl terminus is trans to $P_1$, and the other is trans to $P_2$.
2. The strong $\sigma$-donor $P_1$ exerts a large trans-influence: it directs electron density into the metal, weakening and lengthening the trans $Pd-\text{C}$ bond.
3. However, the strong $\pi$-acceptor $P_2$ withdraws electron density from the metal. The allyl carbon trans to $P_2$ receives significantly less backdonation, rendering it more electrophilic (more carbocationic).
4. Nucleophilic attack by soft carbanions (dimethyl malonate) occurs preferentially at the allyl terminus trans to the stronger $\pi$-acceptor ($P_2$).
5. By designing ligands where the more sterically accessible or substituted carbon aligns trans to the $\pi$-acceptor, the reaction can be steered selectively toward either the linear or branched product with $>95:5$ regiocontrol.

(c) Mathematical Derivation of Enantiomeric Excess ($ee$): In an asymmetric catalytic reaction where two enantiomeric pathways proceed through transition states $TS_R$ and $TS_S$ with activation free energies $\Delta G_R^\ddagger$ and $\Delta G_S^\ddagger$:

1. According to transition-state theory:

1. The enantiomeric ratio ($er$) is:

where $\Delta\Delta G^\ddagger = \Delta G_S^\ddagger - \Delta G_R^\ddagger > 0$ (assuming $R$ is favored).

1. The enantiomeric excess ($ee$) is defined as:

1. Substituting $er = \exp\left(\frac{\Delta\Delta G^\ddagger}{RT}\right)$:

• Numerical Benchmark at $298\text{ K}$:
• For $90\%\ ee$: $er = 19:1 \implies \Delta\Delta G^\ddagger = RT \ln(19) = (8.314)(298)(2.944) = 7.3\text{ kJ/mol}$ ($1.74\text{ kcal/mol}$).
• For $99\%\ ee$: $er = 199:1 \implies \Delta\Delta G^\ddagger = RT \ln(199) = 13.1\text{ kJ/mol}$ ($3.13\text{ kcal/mol}$).

A difference of just $3\text{ kcal/mol}$ in transition state free energy delivers near-perfect enantioselectivity!