Formation of Molecular Orbitals — Revision Notes

⚡ 30-Second Revision

LCAO Principle: — MOs formed by linear combination of AOs ($Psi_{MO} = Psi_A pm Psi_B$).
Conditions for Overlap: — Comparable energy, proper symmetry, maximum overlap.
Bonding MOs (BMOs): — Constructive overlap, lower energy, increased electron density between nuclei, stabilizing.
Antibonding MOs (ABMOs): — Destructive overlap, higher energy, nodal plane between nuclei, destabilizing.
Types of MOs: — $sigma$ (head-on overlap), $pi$ (sideways overlap).
Electron Filling Rules: — Aufbau, Pauli, Hund's.
MO Energy Order (up to 14e-): — $sigma_{1s} < sigma^{*}_{1s} < sigma_{2s} < sigma^{*}_{2s} < pi_{2p} < sigma_{2p} < pi^{*}_{2p} < sigma^{*}_{2p}$ (s-p mixing).
MO Energy Order (> 14e-): — $sigma_{1s} < sigma^{*}_{1s} < sigma_{2s} < sigma^{*}_{2s} < sigma_{2p} < pi_{2p} < pi^{*}_{2p} < sigma^{*}_{2p}$ (no s-p mixing).
Bond Order (BO): — $BO = \frac{1}{2} (N_b - N_a)$.
Magnetic Properties: — Diamagnetic (all paired e-), Paramagnetic (unpaired e-).

2-Minute Revision

Molecular Orbital Theory (MOT) explains bonding by combining atomic orbitals (AOs) into molecular orbitals (MOs) that span the entire molecule. This is done via the Linear Combination of Atomic Orbitals (LCAO) approximation.

When AOs combine, they can do so constructively (in-phase) to form lower-energy bonding MOs (BMOs), which increase electron density between nuclei and stabilize the molecule. Alternatively, they can combine destructively (out-of-phase) to form higher-energy antibonding MOs (ABMOs), which have a nodal plane between nuclei and destabilize the molecule.

For effective combination, AOs must have similar energies, appropriate symmetry, and significant overlap. MOs are classified as sigma ($sigma$) from head-on overlap or pi ($pi$) from sideways overlap.

Electrons fill these MOs following the Aufbau principle, Pauli exclusion principle, and Hund's rule. The specific energy order of MOs varies depending on the number of electrons, particularly due to s-p mixing for lighter elements.

Key applications include calculating bond order ($BO = \frac{1}{2} (N_b - N_a)$) to predict stability and determining magnetic properties (paramagnetic if unpaired electrons, diamagnetic if all paired).

5-Minute Revision

The formation of molecular orbitals (MOs) is a central concept in understanding chemical bonding, offering a more complete picture than simpler theories. It begins with the Linear Combination of Atomic Orbitals (LCAO) approximation, where atomic orbitals (AOs) from different atoms combine to form new, molecule-wide orbitals.

This combination can be either constructive (in-phase addition of wave functions), leading to a bonding molecular orbital (BMO), or destructive (out-of-phase subtraction), resulting in an antibonding molecular orbital (ABMO).

BMOs are characterized by increased electron density between the nuclei, which lowers the system's energy and stabilizes the molecule. ABMOs, conversely, have a region of zero electron density (a nodal plane) between the nuclei, making them higher in energy and destabilizing.

For AOs to combine effectively, they must have comparable energies, suitable symmetry, and sufficient overlap. For example, two 1s orbitals combine to form $sigma_{1s}$ and $sigma^{*}_{1s}$ MOs, while two parallel 2p$_x$ orbitals form $pi_{2p_x}$ and $pi^{*}_{2p_x}$ MOs.

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

The energy ordering of MOs is crucial: for molecules with up to 14 electrons (e.g., $N_2$), s-p mixing causes $pi_{2p}$ MOs to be lower in energy than $sigma_{2p}$ MOs. For molecules with more than 14 electrons (e.

g., $O_2$), $sigma_{2p}$ MOs are lower than $pi_{2p}$ MOs.

Example: For $N_2$ (14 electrons), the configuration is $(sigma_{1s})^2 (sigma^{*}_{1s})^2 (sigma_{2s})^2 (sigma^{*}_{2s})^2 (pi_{2p_x})^2 (pi_{2p_y})^2 (sigma_{2p_z})^2$. Here, $N_b = 10$, $N_a = 4$. Bond Order = $rac{1}{2}(10-4) = 3$. All electrons are paired, so $N_2$ is diamagnetic.

Example: For $O_2$ (16 electrons), the configuration is $(sigma_{1s})^2 (sigma^{*}_{1s})^2 (sigma_{2s})^2 (sigma^{*}_{2s})^2 (sigma_{2p_z})^2 (pi_{2p_x})^2 (pi_{2p_y})^2 (pi^{*}_{2p_x})^1 (pi^{*}_{2p_y})^1$.

Here, $N_b = 10$, $N_a = 6$. Bond Order = $rac{1}{2}(10-6) = 2$. It has two unpaired electrons, so $O_2$ is paramagnetic. This correctly explains oxygen's magnetic behavior, a major success of MOT. The bond order calculation and magnetic property prediction are frequently tested in NEET.

Prelims Revision Notes

Formation of Molecular Orbitals (MOs) - NEET Revision Notes

1. LCAO Approximation:

Molecular orbitals are formed by the Linear Combination of Atomic Orbitals (LCAO).
$Psi_{MO} = Psi_A pm Psi_B$, where $Psi_A$ and $Psi_B$ are atomic orbital wave functions.
Constructive Interference ($Psi_A + Psi_B$): — Forms Bonding Molecular Orbitals (BMOs).

* Lower energy than parent AOs. * Increased electron density between nuclei. * Stabilizes the molecule.

Destructive Interference ($Psi_A - Psi_B$): — Forms Antibonding Molecular Orbitals (ABMOs).

* Higher energy than parent AOs. * Nodal plane (zero electron density) between nuclei. * Destabilizes the molecule.

2. Conditions for Effective AO Combination:

Comparable Energy: — AOs must have similar energy levels (e.g., 1s with 1s, 2p with 2p).
Proper Symmetry: — AOs must have the correct orientation and symmetry for effective overlap (e.g., 2s with 2p$_z$, 2p$_x$ with 2p$_x$).
Maximum Overlap: — Greater overlap leads to stronger bonds and larger energy splitting between BMOs and ABMOs.

3. Types of Molecular Orbitals:

Sigma ($sigma$) MOs: — Formed by head-on (axial) overlap (e.g., s-s, s-p$_z$, p$_z$-p$_z$). Cylindrically symmetrical.
Pi ($pi$) MOs: — Formed by sideways (lateral) overlap (e.g., p$_x$-p$_x$, p$_y$-p$_y$). Have a nodal plane containing the internuclear axis.

4. Rules for Filling MOs:

Aufbau Principle: — Fill lowest energy MOs first.
Pauli Exclusion Principle: — Max 2 electrons per MO, with opposite spins.
Hund's Rule: — For degenerate MOs, fill singly with parallel spins before pairing.

5. MO Energy Level Diagrams (Homodiatomic Molecules):

**For total electrons $le 14$ (e.g., $H_2, B_2, C_2, N_2$):**

$sigma_{1s} < sigma^{*}_{1s} < sigma_{2s} < sigma^{*}_{2s} < pi_{2p_x} = pi_{2p_y} < sigma_{2p_z} < pi^{*}_{2p_x} = pi^{*}_{2p_y} < sigma^{*}_{2p_z}$ (due to s-p mixing).

**For total electrons $> 14$ (e.g., $O_2, F_2, Ne_2$):**

$sigma_{1s} < sigma^{*}_{1s} < sigma_{2s} < sigma^{*}_{2s} < sigma_{2p_z} < pi_{2p_x} = pi_{2p_y} < pi^{*}_{2p_x} = pi^{*}_{2p_y} < sigma^{*}_{2p_z}$ (reduced s-p mixing).

6. Key Applications:

Bond Order (BO): — $BO = \frac{1}{2} (N_b - N_a)$, where $N_b$ = electrons in BMOs, $N_a$ = electrons in ABMOs.

* $BO > 0$: Stable molecule. * $BO = 0$: Unstable, molecule does not exist (e.g., $He_2, Ne_2$). * Higher BO $implies$ greater stability, shorter bond length, higher bond energy.

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., $N_2, F_2$).

Important Note: Always count total electrons first to decide the correct MO energy order.

Vyyuha Quick Recall

To remember the MO energy order for $le 14$ electrons: 'Sigma Star Sigma Star Pi Pi Sigma Pi Star Pi Star Sigma Star' (for 1s, 2s, 2p orbitals respectively, omitting 1s and 2s for brevity in the mnemonic, focusing on 2p: 'Pi Pi Sigma, Pi Star Pi Star Sigma Star'). For $> 14$ electrons, just swap $pi_{2p}$ and $sigma_{2p}$ positions: 'Sigma Pi Pi, Pi Star Pi Star Sigma Star' (for 2p orbitals).