Molecular Orbital Theory — Revision Notes

⚡ 30-Second Revision

LCAO Principle: — MOs formed by $psi_A pm psi_B$.
Bonding MO ($sigma, pi$): — Lower energy, increased electron density between nuclei, stabilizing.
**Antibonding MO ($sigma^*, pi^*$):** Higher energy, nodal plane between nuclei, destabilizing.
Energy Order (up to $N_2$): — $sigma 1s < sigma^* 1s < sigma 2s < sigma^* 2s < (pi 2p_x = pi 2p_y) < sigma 2p_z < (pi^* 2p_x = pi^* 2p_y) < sigma^* 2p_z$
Energy Order ($O_2, F_2$): — $sigma 1s < sigma^* 1s < sigma 2s < sigma^* 2s < sigma 2p_z < (pi 2p_x = pi 2p_y) < (pi^* 2p_x = pi^* 2p_y) < sigma^* 2p_z$
Bond Order (BO): — $BO = \frac{1}{2}(N_b - N_a)$.
Stability: — $BO > 0 Rightarrow$ stable; $BO = 0 Rightarrow$ unstable. Higher BO $Rightarrow$ more stable, shorter bond, higher bond energy.
Magnetic Properties: — Unpaired electrons $Rightarrow$ Paramagnetic; All paired electrons $Rightarrow$ Diamagnetic.

2-Minute Revision

Molecular Orbital Theory (MOT) explains chemical bonding by forming molecular orbitals (MOs) from atomic orbitals (AOs) via the Linear Combination of Atomic Orbitals (LCAO) principle. This results in bonding MOs (lower energy, stabilizing) and antibonding MOs (higher energy, destabilizing).

Electrons fill these MOs following Aufbau, Pauli, and Hund's rules. Crucially, the energy order of MOs differs for lighter elements ($B_2, C_2, N_2$) due to s-p mixing, placing $pi 2p$ before $sigma 2p_z$.

For heavier elements ($O_2, F_2$), $sigma 2p_z$ comes before $pi 2p$. The 'bond order' ($BO = \frac{1}{2}(N_b - N_a)$) determines molecular stability, bond strength, and bond length. A positive BO indicates stability, with higher BO meaning shorter and stronger bonds.

MOT also accurately predicts magnetic properties: molecules with unpaired electrons are paramagnetic (attracted to magnetic fields), while those with all paired electrons are diamagnetic (repelled). This explains the paramagnetism of $O_2$, a key success of MOT.

5-Minute Revision

Molecular Orbital Theory (MOT) provides a quantum mechanical description of chemical bonding, where atomic orbitals (AOs) combine to form molecular orbitals (MOs) that are delocalized over the entire molecule.

This combination occurs through the Linear Combination of Atomic Orbitals (LCAO) method, resulting in two types of MOs for every pair of combining AOs: bonding molecular orbitals (BMOs) and antibonding molecular orbitals (ABMOs).

BMOs are formed by constructive interference, leading to increased electron density between nuclei, lower energy, and molecular stabilization. ABMOs are formed by destructive interference, resulting in a nodal plane between nuclei, higher energy, and molecular destabilization.

Electrons are filled into these MOs following the Aufbau principle (lowest energy first), Pauli exclusion principle (max two electrons per orbital with opposite spins), and Hund's rule (maximize unpaired spins in degenerate orbitals).

The energy order of MOs is critical. For homonuclear diatomic molecules with up to 14 electrons (e.g., $B_2, C_2, N_2$), s-p mixing occurs, leading to the order: $sigma 1s < sigma^* 1s < sigma 2s < sigma^* 2s < (pi 2p_x = pi 2p_y) < sigma 2p_z < (pi^* 2p_x = pi^* 2p_y) < sigma^* 2p_z$.

For molecules with more than 14 electrons (e.g., $O_2, F_2$), s-p mixing is negligible, and the order is: $sigma 1s < sigma^* 1s < sigma 2s < sigma^* 2s < sigma 2p_z < (pi 2p_x = pi 2p_y) < (pi^* 2p_x = pi^* 2p_y) < sigma^* 2p_z$.

Key Applications:

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Bond Order (BO): — Calculated as $BO = \frac{1}{2}(N_b - N_a)$, where $N_b$ are bonding electrons and $N_a$ are antibonding electrons. A positive BO indicates a stable molecule. Higher BO implies greater stability, shorter bond length, and higher bond dissociation energy. For example, $N_2$ has BO=3, $O_2$ has BO=2, $F_2$ has BO=1. $He_2$ has BO=0, hence it's unstable.
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Magnetic Properties: — Molecules with unpaired electrons are paramagnetic (attracted to magnetic fields), while those with all paired electrons are diamagnetic (repelled). $O_2$ is famously paramagnetic due to two unpaired electrons in its $pi^* 2p$ orbitals, a prediction uniquely explained by MOT.

Example: Determine the bond order and magnetic nature of $N_2^+$.

Total electrons in $N_2$ = 14. $N_2^+$ has $14-1 = 13$ electrons.
MO configuration (with s-p mixing): $sigma 1s^2 sigma^* 1s^2 sigma 2s^2 sigma^* 2s^2 (pi 2p_x^2 = pi 2p_y^2) sigma 2p_z^1$.
$N_b = 2+2+4+1 = 9$.
$N_a = 2+2 = 4$.
$BO = \frac{1}{2}(9 - 4) = 2.5$.
Since there is one unpaired electron in the $sigma 2p_z$ orbital, $N_2^+$ is paramagnetic.

Prelims Revision Notes

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Core Idea: — Atomic orbitals (AOs) combine to form molecular orbitals (MOs) that span the entire molecule. Electrons are delocalized.
2
LCAO Principle: — MOs are formed by linear combination (addition/subtraction) of AO wave functions. Number of MOs = Number of AOs combined.
3
Types of MOs:

* Bonding MOs (BMOs): Lower energy, increased electron density between nuclei, stabilizing. ($sigma, pi$) * Antibonding MOs (ABMOs): Higher energy, nodal plane between nuclei, destabilizing. ($sigma^*, pi^*$)

1
Conditions for AO Combination: — Comparable energy, proper symmetry, maximum overlap.
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Rules for Electron Filling:

* Aufbau Principle: Fill MOs in increasing order of energy. * Pauli Exclusion Principle: Max 2 electrons per MO, with opposite spins. * Hund's Rule: For degenerate MOs, fill singly with parallel spins first, then pair up.

1
MO Energy Order (Crucial for NEET):

* **For $B_2, C_2, N_2$ (and their ions, total electrons $le 14$):** $sigma 1s < sigma^* 1s < sigma 2s < sigma^* 2s < (pi 2p_x = pi 2p_y) < sigma 2p_z < (pi^* 2p_x = pi^* 2p_y) < sigma^* 2p_z$ (Note: $pi 2p$ are lower than $sigma 2p_z$ due to s-p mixing) * **For $O_2, F_2$ (and their ions, total electrons $> 14$):** $sigma 1s < sigma^* 1s < sigma 2s < sigma^* 2s < sigma 2p_z < (pi 2p_x = pi 2p_y) < (pi^* 2p_x = pi^* 2p_y) < sigma^* 2p_z$ (Note: $sigma 2p_z$ is lower than $pi 2p$ as s-p mixing is negligible)

1
Bond Order (BO): — $BO = \frac{1}{2}(N_b - N_a)$.

* $N_b$: Number of electrons in bonding MOs. * $N_a$: Number of electrons in antibonding MOs. * BO > 0: Stable molecule. BO = 0: Unstable (e.g., $He_2$). * Higher BO $Rightarrow$ greater stability, shorter bond length, higher bond dissociation energy.

1
Magnetic Properties:

* Paramagnetic: Contains one or more unpaired electrons (attracted to magnetic field). E.g., $O_2, B_2, NO$. * Diamagnetic: All electrons are paired (repelled by magnetic field). E.g., $N_2, F_2, CO$.

1
Isoelectronic Species: — Species with the same total number of electrons often have the same bond order and similar magnetic properties (e.g., $N_2$, $CO$, $CN^-$, $NO^+$ all have 14 electrons and BO=3).

Vyyuha Quick Recall

**MO-Diagram Order (for $le 14$ e-):** 'Sigma Star Sigma Star Pi Pi Sigma Pi Star Pi Star Sigma Star'

**MO-Diagram Order (for $> 14$ e-):** 'Sigma Star Sigma Star Sigma Pi Pi Pi Star Pi Star Sigma Star'

(Remember to insert '1s' and '2s' for the first four, then '2p' for the rest. The key difference is the position of $sigma 2p_z$ relative to $pi 2p$.)

Bond Order: Bonding - Antibonding / 2 (BO = (Nb - Na)/2)

Magnetic Properties: Unpaired = Paramagnetic; All Paired = Diamagnetic (UAPD)