Molecular Orbital Theory

Molecular Orbital Theory (MOT) emerged as a powerful quantum mechanical approach to describe chemical bonding, addressing several limitations of the Valence Bond Theory (VBT). While VBT successfully explains the geometry of many molecules and the concept of localized bonds, it often falls short in explaining phenomena like the paramagnetism of dioxygen ($O_2$) or the existence of species like $H_2^+$.

MOT provides a more comprehensive picture by treating electrons as delocalized over the entire molecular framework, rather than confined to specific bonds between two atoms.

Conceptual Foundation: Limitations of VBT and the Need for MOT

Valence Bond Theory, with its emphasis on the overlap of atomic orbitals to form localized electron-pair bonds, is intuitive and widely used. However, it struggles with:

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Magnetic Properties: — VBT predicts $O_2$ to be diamagnetic (all electrons paired), but experimentally, $O_2$ is paramagnetic (contains unpaired electrons). MOT correctly predicts this.
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Delocalization: — While resonance structures in VBT attempt to describe delocalization, MOT inherently accounts for it by forming molecular orbitals that span the entire molecule.
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Existence of certain species: — VBT has difficulty explaining the stability of species like $H_2^+$ (one electron) or $He_2$ (which doesn't exist). MOT provides a clear explanation based on bond order.

MOT postulates that when atoms combine to form a molecule, their atomic orbitals (AOs) combine to form an equivalent number of molecular orbitals (MOs). These MOs are polycentric, meaning they are associated with all the nuclei in the molecule, unlike AOs which are monocentric.

Key Principles and Laws of MOT:

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Linear Combination of Atomic Orbitals (LCAO): — This is the cornerstone of MOT. It states that molecular orbitals are formed by the linear combination (addition or subtraction) of atomic orbital wave functions. For two atoms A and B, combining their atomic orbitals $\psi_A$ and $\psi_B$ leads to two molecular orbitals:

* Bonding Molecular Orbital (BMO): Formed by the constructive interference (addition) of atomic orbital wave functions. It has lower energy than the original AOs, increased electron density between the nuclei, and stabilizes the molecule.

Represented as $\psi_{BMO} = \psi_A + \psi_B$. * Antibonding Molecular Orbital (ABMO): Formed by the destructive interference (subtraction) of atomic orbital wave functions. It has higher energy than the original AOs, a nodal plane between the nuclei (zero electron density), and destabilizes the molecule.

Represented as $\psi_{ABMO} = \psi_A - \psi_B$.

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Conditions for Combination of Atomic Orbitals: — For AOs to combine effectively to form MOs, three conditions must be met:

* Comparable Energies: The combining AOs must have similar energies. For example, a 1s orbital of one atom can combine with a 1s orbital of another, but not effectively with a 2s orbital (unless the atoms are very different in electronegativity, leading to some mixing).

* Proper Symmetry: The AOs must have the same symmetry with respect to the molecular axis. For instance, an s orbital can combine with another s orbital or a $p_z$ orbital (if z is the internuclear axis), but not with a $p_x$ or $p_y$ orbital to form a sigma bond.

* Maximum Overlap: The AOs must overlap to a significant extent. Greater overlap leads to stronger bonds and more stable MOs.

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Types of Molecular Orbitals: — Based on the symmetry around the internuclear axis, MOs are classified as:

* **Sigma ($\sigma$) MOs:** Formed by the head-on or axial overlap of AOs (s-s, s-$p_z$, $p_z$-$p_z$). Electron density is cylindrically symmetrical around the internuclear axis. * **Pi ($\pi$) MOs:** Formed by the lateral or sideways overlap of AOs ($p_x$-$p_x$, $p_y$-$p_y$). Electron density is concentrated above and below the internuclear axis, with a nodal plane containing the internuclear axis.

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Energy Level Diagrams: — The relative energies of MOs are crucial for filling electrons. For homonuclear diatomic molecules, the general order of MO energies depends on whether there is s-p mixing or not.

* **Without s-p mixing (for $O_2$, $F_2$, and heavier elements):** $\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < \sigma 2p_z < (\pi 2p_x = \pi 2p_y) < (\pi^* 2p_x = \pi^* 2p_y) < \sigma^* 2p_z$ * **With s-p mixing (for $B_2$, $C_2$, $N_2$, and lighter elements):** $\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < (\pi 2p_x = \pi 2p_y) < \sigma 2p_z < (\pi^* 2p_x = \pi^* 2p_y) < \sigma^* 2p_z$ The s-p mixing occurs when the energy difference between 2s and 2p atomic orbitals is small enough for them to interact, leading to a change in the relative order of $\sigma 2p_z$ and $\pi 2p$ orbitals.

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Filling of Molecular Orbitals: — Electrons are filled into MOs according to:

* Aufbau Principle: MOs are filled in increasing order of energy. * Pauli Exclusion Principle: Each MO can hold a maximum of two electrons with opposite spins. * Hund's Rule of Maximum Multiplicity: For degenerate MOs (of the same energy), electrons are first filled singly with parallel spins before pairing up.

Derivations (Qualitative LCAO for $H_2$ and $H_2^+$):

Consider two hydrogen atoms, $H_A$ and $H_B$, each with a 1s atomic orbital ($\psi_{1sA}$ and $\psi_{1sB}$). When they approach each other, their 1s AOs combine to form two MOs:

Bonding MO ($\sigma 1s$): — $\psi_{BMO} = \psi_{1sA} + \psi_{1sB}$. This results in increased electron density between the nuclei, leading to attraction and stabilization.
**Antibonding MO ($\sigma^* 1s$):** $\psi_{ABMO} = \psi_{1sA} - \psi_{1sB}$. This results in a nodal plane between the nuclei, reducing electron density and leading to repulsion and destabilization.

For $H_2^+$ (1 electron): The single electron occupies the $\sigma 1s$ BMO. Bond order = $(1-0)/2 = 0.5$. It exists. For $H_2$ (2 electrons): Both electrons occupy the $\sigma 1s$ BMO with opposite spins. Bond order = $(2-0)/2 = 1$. It exists and is stable. For $He_2$ (4 electrons): Two electrons go into $\sigma 1s$ and two into $\sigma^* 1s$. Bond order = $(2-2)/2 = 0$. Hence, $He_2$ does not exist as a stable molecule.

Real-World Applications and NEET-Specific Angle:

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Bond Order: — A crucial concept derived from MOT. It is defined as half the difference between the number of electrons in bonding MOs ($N_b$) and antibonding MOs ($N_a$).

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Magnetic Properties: — MOT accurately predicts whether a molecule is paramagnetic or diamagnetic.

* Paramagnetic: Molecules with one or more unpaired electrons in their MOs are attracted to a magnetic field (e.g., $O_2$, $B_2$). * Diamagnetic: Molecules with all electrons paired in their MOs are repelled by a magnetic field (e.

g., $N_2$, $F_2$). The classic example is $O_2$. Its MO configuration is $\sigma 1s^2 \sigma^* 1s^2 \sigma 2s^2 \sigma^* 2s^2 \sigma 2p_z^2 (\pi 2p_x^2 = \pi 2p_y^2) (\pi^* 2p_x^1 = \pi^* 2p_y^1)$. The presence of two unpaired electrons in the degenerate $\pi^* 2p$ antibonding orbitals explains its paramagnetism, a significant success of MOT over VBT.

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Stability and Bond Length: — Directly related to bond order. Higher bond order means stronger attraction between nuclei, leading to greater stability and shorter bond lengths. For example, $N_2$ has a bond order of 3 (very stable, short bond), while $O_2$ has a bond order of 2, and $F_2$ has a bond order of 1.

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Heteronuclear Diatomic Molecules: — For molecules like CO, NO, HF, the AOs of the more electronegative atom will have lower energy. The MOs will be polarized, with bonding MOs having a greater contribution from the more electronegative atom's AOs and antibonding MOs having a greater contribution from the less electronegative atom's AOs.

Common Misconceptions:

Atomic vs. Molecular Orbitals: — Students often confuse the two. AOs belong to individual atoms; MOs belong to the entire molecule.
Number of Orbitals: — The number of MOs formed is always equal to the number of combining AOs, not just the bonding ones.
Energy Order: — Incorrectly applying the s-p mixing rule. Remember, for $B_2, C_2, N_2$, the $\pi 2p$ orbitals are lower in energy than $\sigma 2p_z$. For $O_2, F_2$, it's the reverse.
Hund's Rule: — Forgetting to fill degenerate orbitals singly before pairing electrons, especially in $\pi$ and $\pi^*$ orbitals.

MOT provides a robust framework for understanding the electronic structure and properties of molecules, particularly diatomic species, and is a frequently tested concept in NEET UG.