2.4 Molecular Orbital Theory and Magnetochemistry of Diatomic Molecules

Molecular Orbital Theory and Magnetochemistry of Diatomic Molecules

Molecular Orbital (MO) Theory provides an advanced quantum mechanical description of chemical bonding where valence electrons delocalize over the entire molecular framework rather than remaining localized between paired atoms. This section covers the Linear Combination of Atomic Orbitals (LCAO) approximation, orbital symmetry and parity, $s-p$ mixing crossover, paramagnetism, and Frontier Molecular Orbital (FMO) theory tested on the PRC Chemist Licensure Examination.

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1. Principles of Linear Combination of Atomic Orbitals (LCAO-MO)

In the LCAO approximation, a molecular orbital $\psi_{MO}$ is formulated as a linear combination of atomic basis functions $\phi_i$:

$\psi_{MO} = c_A \phi_A + c_B \phi_B$

where $c_A$ and $c_B$ are weighting coefficients representing the orbital contribution of each atom. The total electron probability density is given by squaring the wavefunction:

$\lvert\psi_{MO}\rvert^2 = c_A^2 \phi_A^2 + c_B^2 \phi_B^2 + 2 c_A c_B \phi_A \phi_B$

• The terms $c_A^2 \phi_A^2$ and $c_B^2 \phi_B^2$ represent electron densities associated with isolated atomic orbitals.
• The cross-term $2 c_A c_B \phi_A \phi_B$ represents quantum mechanical interference:
  • When wavefunctions combine in-phase ($+$), constructive interference increases electron probability density in the internuclear region, shielding the nuclei and lowering the potential energy $\implies$ Bonding Molecular Orbital (lower in energy than parent atomic orbitals).
  • When wavefunctions combine out-of-phase ($-$), destructive interference produces a nodal plane between the nuclei where $\psi = 0$, exposing the nuclei to mutual electrostatic repulsion $\implies$ Antibonding Molecular Orbital (designated with $*$, higher in energy than parent atomic orbitals).

The Three Cardinal Rules of Effective Orbital Overlap

1. Energy Compatibility: Atomic orbitals must have comparable energies (typically $\Delta E \le 12-14\text{ eV}$). If two atomic orbitals differ greatly in energy, they mix weakly and remain effectively nonbonding.
2. Symmetry Compatibility: Atomic orbitals must possess identical spatial symmetry with respect to the internuclear axis ($z$). The overlap integral $S_{AB}$ must be non-zero: $S_{AB} = \int \phi_A \phi_B \, d\tau \neq 0$ For example, an $s$ orbital has zero net overlap with a $p_x$ or $p_y$ orbital oriented perpendicular to the bond axis because positive and negative phase overlaps cancel identically ($S = 0$).
3. Spatial Proximity: Orbitals must be sufficiently close in space for their radial wavefunctions to overlap significantly.

Symmetry Classification and Parity ($g$ vs $u$)

• $\sigma$ (Sigma) Orbitals: Cylindrically symmetric about the internuclear axis. Rotation by any arbitrary angle around the bond axis leaves the wavefunction unchanged. Formed by coaxial overlap ($s+s, s+p_z, p_z+p_z$).
• $\pi$ (Pi) Orbitals: Possess a nodal plane containing the internuclear axis. A $180^\circ$ rotation around the bond axis reverses the sign of the wavefunction. Formed by parallel sideways overlap ($p_x+p_x, p_y+p_y$).
• Parity / Inversion Symmetry: In centrosymmetric (homonuclear) diatomics, orbitals are classified under the inversion operation $\hat{i}$ through the molecular midpoint:
  • Gerade ($g$): Symmetric upon inversion: $\psi(-x, -y, -z) = +\psi(x, y, z)$.
    • $\sigma$ bonding orbitals are $\sigma_g$.
    • $\pi^*$ antibonding orbitals are $\pi_g^*$.
  • Ungerade ($u$): Antisymmetric upon inversion: $\psi(-x, -y, -z) = -\psi(x, y, z)$.
    • $\sigma^*$ antibonding orbitals are $\sigma_u^*$.
    • $\pi$ bonding orbitals are $\pi_u$.

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2. Homonuclear Diatomics and the $s-p$ Mixing Crossover

Bond Order Formulation

The bond order quantifies the net number of covalent bonds: $\text{Bond Order} = \frac{1}{2}(N_b - N_a)$ where $N_b$ is the number of electrons in bonding molecular orbitals and $N_a$ is the number of electrons in antibonding molecular orbitals. A positive bond order indicates thermodynamic stability relative to separated atoms; higher bond orders correlate with shorter internuclear bond distances and higher bond dissociation energies.

The Mechanism of $s-p$ Mixing (Orbital Crossover)

In second-period homonuclear diatomics, atomic $2s$ and $2p_z$ orbitals both form molecular orbitals of $\Sigma_g^+$ symmetry ($\,\sigma_{2s}\,\text{and}\,\sigma_{2p_z}\,$, respectively). Quantum mechanics dictates that orbitals sharing the same symmetry interact and repel one another if their energy gap is small:

1. Without $s-p$ Mixing ($O_2, F_2, Ne_2$): In Oxygen and Fluorine, the high effective nuclear charge binds $2s$ electrons very tightly, creating a large energy separation between $2s$ and $2p$ ($\Delta E_{2s-2p} = 16.5\text{ eV}$ in O, $21.6\text{ eV}$ in F). Minimal mixing occurs. The standard energy sequence is: $\sigma_{2s} (1\sigma_g) < \sigma_{2s}^* (1\sigma_u^*) < \mathbf{\sigma_{2p_z} (2\sigma_g)} < \mathbf{\pi_{2p_x} = \pi_{2p_y} (1\pi_u)} < \pi_{2p_x}^* = \pi_{2p_y}^* (1\pi_g^*) < \sigma_{2p_z}^* (2\sigma_u^*)$

2. With $s-p$ Mixing ($Li_2, Be_2, B_2, C_2, N_2$): In lighter elements, lower nuclear charge yields a small $2s-2p$ energy gap ($\Delta E_{2s-2p} = 5.7\text{ eV}$ in B, $8.8\text{ eV}$ in C, $12.4\text{ eV}$ in N). Strong quantum mixing occurs between the $1\sigma_g$ and $2\sigma_g$ orbitals: the lower $1\sigma_g$ is stabilized further down, while the upper $2\sigma_g$ ($\sigma_{2p_z}$) is pushed upward above the degenerate $1\pi_u$ orbitals! The modified energy sequence for $B_2, C_2, N_2$ is: $\sigma_{2s} (1\sigma_g) < \sigma_{2s}^* (1\sigma_u^*) < \mathbf{\pi_{2p_x} = \pi_{2p_y} (1\pi_u)} < \mathbf{\sigma_{2p_z} (2\sigma_g)} < \pi_{2p_x}^* = \pi_{2p_y}^* (1\pi_g^*) < \sigma_{2p_z}^* (2\sigma_u^*)$

Systematic Period 2 Homonuclear Diatomic Series

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3. Magnetochemistry and the Triplet Ground State of $O_2$

Paramagnetism vs Diamagnetism

• Diamagnetism: All electrons in the molecule are paired. When placed in an external magnetic field gradient, paired electrons generate an induced orbital magnetic moment that opposes the applied field, causing weak expulsion.
• Paramagnetism: The molecule contains one or more unpaired electrons. The intrinsic electron spin produces a permanent magnetic dipole moment that aligns parallel to the external field, drawing the substance strongly into the magnetic field.

Spin-Only Magnetic Moment Formula

The quantitative magnetic moment due to electron spin is given by:

$\mu_{so} = \sqrt{n(n+2)} \, \mu_B$

where $n$ is the number of unpaired electrons and $\mu_B$ is the Bohr Magneton ($\mu_B = \frac{e\hbar}{2m_e} = 9.274 \times 10^{-24}\text{ J/T}$). Measurements using a Gouy balance or SQUID magnetometer determine $\mu_{so}$ directly.

The Triumphant Explanation of $O_2$ Paramagnetism

Under classical Lewis bonding, dioxygen is written as $:\ddot{O}=\ddot{O}:$, predicting an octet-complete, diamagnetic molecule with zero unpaired electrons. However, liquid oxygen is strongly paramagnetic and clings suspended between magnetic pole pieces.

MO theory resolves this discrepancy effortlessly: Dioxygen possesses 12 valence electrons. Filling the molecular orbitals in order of increasing energy: $\sigma_{2s}^2 \; \sigma_{2s}^{*2} \; \sigma_{2p_z}^2 \; \pi_{2p}^4 \; (\pi_{2p_x}^{*1} = \pi_{2p_y}^{*1})$ Ten electrons fill the bonding and core-like antibonding orbitals. The remaining two electrons must enter the degenerate $\pi_{2p}^*$ antibonding orbitals. By Hund's rule, they occupy separate degenerate orbitals with parallel spins ($S = \frac{1}{2} + \frac{1}{2} = 1$, spin multiplicity $2S+1 = 3$, forming a triplet ground state ($^3\Sigma_g^-$)): $\mu_{so}(O_2) = \sqrt{2(2+2)} = \sqrt{8} \approx 2.83\text{ BM}$

Dioxygen Redox Series: Bond Order, Length, and Magnetism

Stepwise reduction or oxidation of $O_2$ systematically alters the population of the antibonding $\pi^*$ orbitals:

Thermodynamic Deduction: Removing electrons from antibonding $\pi^*$ orbitals ($O_2 \rightarrow O_2^+$) increases bond order, shortens bond length, and raises the vibrational stretching frequency. Adding electrons into $\pi^*$ ($O_2 \rightarrow O_2^- \rightarrow O_2^{2-}$) weakens and lengthens the $O-O$ bond.

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4. Heteronuclear Diatomics and Frontier Molecular Orbital (FMO) Theory

In heteronuclear diatomics ($A-B$), unequal electronegativities cause the atomic orbitals of the more electronegative atom to lie lower in energy. Consequently:

• Bonding MOs are lower in energy and receive greater mathematical contribution ($c_B > c_A$) from the more electronegative atom.
• Antibonding MOs are higher in energy and receive greater mathematical contribution ($c_A^* > c_B^*$) from the less electronegative atom.

Carbon Monoxide ($CO$)

Carbon Monoxide has 10 valence electrons ($4$ from C, $6$ from O) and is isoelectronic with $N_2$:

• Atomic orbital energies: $C(2s) = -19.4\text{ eV}, C(2p) = -10.7\text{ eV}$; $O(2s) = -32.4\text{ eV}, O(2p) = -15.9\text{ eV}$.
• Valence configuration: $(1\sigma)^2 (2\sigma)^2 (1\pi)^4 (3\sigma)^2$
• HOMO ($3\sigma$): The Highest Occupied Molecular Orbital is a weakly antibonding / nonbonding $\sigma$ orbital. Crucially, the $3\sigma$ HOMO is polarized heavily toward Carbon ($c_C \gg c_O$), with a large lone-pair lobe protruding outward from the carbon nucleus into space!
• LUMO ($2\pi^*$): The Lowest Unoccupied Molecular Orbital consists of empty degenerate $\pi^*$ orbitals that are also polarized predominantly on Carbon ($c_C \gg c_O$).

The Biochemical & Catalytic Consequence: Although Oxygen is far more electronegative, $CO$ binds transition metals exclusively via Carbon! Carbon monoxide acts as a classical $\sigma$-donor (donating its carbon-centered HOMO $3\sigma$ electron density into empty metal $d$-orbitals) and a powerful $\pi$-acceptor (accepting electron density from filled metal $d$-orbitals into its empty carbon-centered LUMO $2\pi^*$). This synergistic bonding mechanism (Dewar-Chatt-Duncanson model) explains why $CO$ binds hemoglobin $200\times$ more tightly than $O_2$, causing lethal carbon monoxide poisoning.

Nitric Oxide ($NO$)

Nitric oxide has 11 valence electrons; ground-state configuration: $(1\sigma)^2 (2\sigma)^2 (1\pi)^4 (3\sigma)^2 (2\pi^*)^1$

• Bond order = $\frac{8 - 3}{2} = 2.5$.
• The unpaired electron resides in an antibonding $2\pi^*$ orbital polarized primarily on nitrogen.
• Oxidation to the nitrosonium ion ($NO^+$) removes this antibonding electron, increasing the bond order to $3.0$, contracting the bond from $1.15\text{ \AA}$ to $1.06\text{ \AA}$, and shifting the vibrational frequency from $1876\text{ cm}^{-1}$ to $2300\text{ cm}^{-1}$.

Hydrogen Fluoride ($HF$)

• $H(1s)$ lies at $-13.6\text{ eV}$. $F(2s)$ lies at $-40.2\text{ eV}$ (too low to interact; forms nonbonding $1\sigma$).
• $F(2p_z)$ lies at $-18.6\text{ eV}$ and shares $\sigma$ symmetry with $H(1s)$, combining to form a bonding $2\sigma$ and antibonding $3\sigma^*$ orbital.
• $F(2p_x)$ and $F(2p_y)$ have $\pi$ symmetry and zero overlap with $H(1s)$ ($S=0$), remaining completely nonbonding lone pairs ($1\pi$).
• Bond order = $1.0$; the MO description directly mirrors the three nonbonding lone pairs on fluorine in the Lewis structure.