Welcome, future engineers, to a deep dive into one of the most powerful theories in chemical bonding: the
Molecular Orbital Theory (MOT). While theories like Valence Bond Theory (VBT) helped us understand many aspects of bonding, MOT takes us a step further, providing a more accurate and comprehensive picture, especially for predicting magnetic properties and explaining the existence (or non-existence) of certain molecules.
Let's begin our journey!
### 1. Introduction: Why Molecular Orbital Theory?
Remember VBT? It told us that atomic orbitals (AOs) overlap to form bonds, and electrons in these overlapping orbitals are localized between the two bonded atoms. It successfully explained bond formation, geometry, and hybridization. However, VBT has its limitations:
* It couldn't explain why oxygen (Oβ) is
paramagnetic (attracted to a magnetic field), even though all its electrons appear to be paired according to its Lewis structure and VBT hybridization.
* It struggles to explain the existence of molecules like Bβ or the non-existence of Heβ.
* It doesn't provide a direct way to calculate bond orders other than through the number of shared electron pairs.
This is where MOT steps in, offering a more refined perspective.
### 2. The Core Idea: Atomic Orbitals Combine to Form Molecular Orbitals
The fundamental premise of MOT is simple yet profound:
When atoms come together to form a molecule, their atomic orbitals no longer remain individual entities. Instead, they combine (or mix) to form new, larger orbitals that belong to the entire molecule. These new orbitals are called
Molecular Orbitals (MOs).
Think of it like this: When two musicians (atoms) play their individual instruments (atomic orbitals), their sounds combine to create a new, larger sound (molecular orbital) that encompasses the entire musical piece (molecule).
#### 2.1. Linear Combination of Atomic Orbitals (LCAO) Method
The most common approach to forming molecular orbitals is the
Linear Combination of Atomic Orbitals (LCAO) method. This method states that molecular orbitals are formed by adding or subtracting the wave functions of atomic orbitals.
Let's consider two atoms, A and B, each with an atomic orbital represented by wave functions $psi_A$ and $psi_B$. When these two AOs combine, they can do so in two ways:
1.
Constructive Interference (Addition): If the wave functions are in phase, they add up, leading to an increased electron probability density between the nuclei. This forms a
Bonding Molecular Orbital (BMO).
* $Psi_{BMO} = Psi_A + Psi_B$
*
Characteristics of BMOs:
* Lower energy than the original atomic orbitals.
* Increased electron density between nuclei, promoting attraction.
* Stabilizes the molecule.
2.
Destructive Interference (Subtraction): If the wave functions are out of phase, they cancel each other out in the region between the nuclei, leading to a decreased electron probability density (even a node, where probability is zero). This forms an
Antibonding Molecular Orbital (ABMO).
* $Psi_{ABMO} = Psi_A - Psi_B$
*
Characteristics of ABMOs:
* Higher energy than the original atomic orbitals.
* Decreased electron density (or a node) between nuclei, causing repulsion.
* Destabilizes the molecule.
Formation of Molecular Orbitals from Atomic Orbitals |
|---|
Atomic Orbital (AO) |
Combination Type |
Molecular Orbital (MO) |
|---|
$psi_A$ |
Constructive (+ overlap) |
Bonding MO (BMO) - $Psi_{BMO}$ (Lower Energy, Stabilizing) |
$psi_B$ |
Destructive (- overlap) |
Antibonding MO (ABMO) - $Psi_{ABMO}$ (Higher Energy, Destabilizing) |
Energy Profile:
Imagine the energy of the original atomic orbitals as a baseline. When they combine:
* The BMO forms at a lower energy level than the original AOs.
* The ABMO forms at a higher energy level than the original AOs.
* The energy lowering of the BMO is often slightly less than the energy raising of the ABMO, but for simplicity, we often draw them symmetrically.
Key takeaway: For every two atomic orbitals that combine, two molecular orbitals are formed: one bonding and one antibonding. The number of MOs formed always equals the number of AOs combined.
### 3. Types of Molecular Orbitals: Sigma ($sigma$) and Pi ($pi$)
Just like atomic orbitals (s, p, d, f) have specific shapes and symmetries, molecular orbitals also have distinct types based on how the atomic orbitals overlap.
#### 3.1. Sigma ($sigma$) Molecular Orbitals
Sigma MOs are formed by the
head-on (axial) overlap of atomic orbitals.
*
s-s overlap: Two s-orbitals combine axially.
* Constructive: Forms
$sigma_{1s}$ (bonding MO).
* Destructive: Forms
$sigma^*_{1s}$ (antibonding MO).
*
s-p overlap: An s-orbital and a p-orbital combine axially (only if the p-orbital's lobe points towards the s-orbital).
* Constructive: Forms
$sigma_{s-p}$.
* Destructive: Forms
$sigma^*_{s-p}$.
*
p-p head-on overlap: Two p-orbitals overlap along the internuclear axis (e.g., two pz orbitals if the z-axis is the internuclear axis).
* Constructive: Forms
$sigma_{2pz}$.
* Destructive: Forms
$sigma^*_{2pz}$.
Characteristic of $sigma$ MOs: Electron density is symmetrical around the internuclear axis.
#### 3.2. Pi ($pi$) Molecular Orbitals
Pi MOs are formed by the
side-on (lateral) overlap of atomic orbitals. This typically occurs between p-orbitals that are perpendicular to the internuclear axis (e.g., px-px or py-py overlap, if z is the internuclear axis).
*
p-p side-on overlap:
* Constructive: Forms
$pi_{2px}$ and
$pi_{2py}$ (bonding MOs). These are degenerate (have the same energy).
* Destructive: Forms
$pi^*_{2px}$ and
$pi^*_{2py}$ (antibonding MOs). These are also degenerate.
Characteristic of $pi$ MOs: Electron density is concentrated above and below (or to the sides of) the internuclear axis, with a nodal plane containing the internuclear axis.
### 4. Energy Level Diagrams for Homonuclear Diatomic Molecules
To fill electrons into molecular orbitals, we follow the same rules as for atomic orbitals:
1.
Aufbau Principle: Orbitals are filled in order of increasing energy.
2.
Pauli Exclusion Principle: Each molecular orbital can hold a maximum of two electrons with opposite spins.
3.
Hund's Rule of Maximum Multiplicity: If degenerate MOs are available, electrons will first singly occupy each orbital with parallel spins before pairing up.
The energy order of molecular orbitals is crucial and depends on the specific atoms involved. For homonuclear diatomic molecules (like Hβ, Oβ, Nβ), we have two general schemes:
#### 4.1. MO Energy Order for Diatomics with Z β€ 7 (e.g., Liβ, Beβ, Bβ, Cβ, Nβ)
For these lighter elements, there is significant mixing between 2s and 2p atomic orbitals (called
s-p mixing or orbital hybridization between AOs). This interaction pushes the energy of the $sigma_{2p_z}$ orbital *above* the $pi_{2p_x}$ and $pi_{2p_y}$ orbitals.
The energy order is:
$sigma_{1s} < sigma^*_{1s} < sigma_{2s} < sigma^*_{2s} < (pi_{2p_x} = pi_{2p_y}) < sigma_{2p_z} < (pi^*_{2p_x} = pi^*_{2p_y}) < sigma^*_{2p_z}$
#### 4.2. MO Energy Order for Diatomics with Z > 7 (e.g., Oβ, Fβ, Neβ)
For heavier elements, the energy difference between 2s and 2p atomic orbitals is larger, so s-p mixing is negligible. In this case, the $sigma_{2p_z}$ orbital remains *below* the $pi_{2p_x}$ and $pi_{2p_y}$ orbitals.
The energy order is:
$sigma_{1s} < sigma^*_{1s} < sigma_{2s} < sigma^*_{2s} < sigma_{2p_z} < (pi_{2p_x} = pi_{2p_y}) < (pi^*_{2p_x} = pi^*_{2p_y}) < sigma^*_{2p_z}$
JEE Focus: It is absolutely critical to remember these two energy orders and when to apply them. A common mistake is using the wrong order, leading to incorrect predictions of magnetic behavior and bond order.
### 5. Understanding Bond Order: The Quantitative Measure of Bonding
One of the most powerful outcomes of MOT is the ability to calculate
Bond Order (BO), which directly correlates with the stability and strength of a chemical bond.
The bond order is defined as:
$$ ext{Bond Order (BO)} = frac{1}{2} ( ext{Number of electrons in BMOs} - ext{Number of electrons in ABMOs}) $$
$$ mathbf{BO = frac{1}{2} (N_b - N_a)} $$
Where:
* $mathbf{N_b}$ = total number of electrons in bonding molecular orbitals.
* $mathbf{N_a}$ = total number of electrons in antibonding molecular orbitals.
#### 5.1. Significance of Bond Order:
*
Positive Bond Order (BO > 0): Indicates that the molecule is stable and can exist. A higher positive bond order implies greater stability and bond strength.
*
Zero or Negative Bond Order (BO β€ 0): Indicates that the molecule is unstable and unlikely to exist.
*
Bond Length: Inversely proportional to bond order. Higher bond order means stronger attraction, thus shorter bond length.
*
Bond Energy: Directly proportional to bond order. Higher bond order means a stronger bond, requiring more energy to break.
### 6. Applying MOT: Step-by-Step Examples
Let's apply MOT to some simple homonuclear diatomic molecules.
#### Example 1: Hydrogen Molecule (Hβ)
*
Total electrons: 1 (from H) + 1 (from H) = 2 electrons.
*
MO diagram: Each H atom has a 1s orbital. These combine to form $sigma_{1s}$ and $sigma^*_{1s}$.
*
Electron configuration: $(sigma_{1s})^2$
*
$N_b$ = 2 (2 electrons in $sigma_{1s}$)
*
$N_a$ = 0
*
Bond Order (BO) = Β½ (2 - 0) = 1
* This indicates a stable single bond, consistent with VBT.
*
Magnetic Character: All electrons are paired, so Hβ is
diamagnetic.
#### Example 2: Helium Molecule (Heβ)
*
Total electrons: 2 (from He) + 2 (from He) = 4 electrons.
*
MO diagram: Two 1s orbitals combine to form $sigma_{1s}$ and $sigma^*_{1s}$.
*
Electron configuration: $(sigma_{1s})^2 (sigma^*_{1s})^2$
*
$N_b$ = 2
*
$N_a$ = 2
*
Bond Order (BO) = Β½ (2 - 2) = 0
* Since the bond order is zero, Heβ is unstable and does not exist. This is a significant success of MOT!
*
Magnetic Character: All electrons are paired, so if it existed, it would be diamagnetic.
#### Example 3: Nitrogen Molecule (Nβ)
*
Total electrons: 7 (from N) + 7 (from N) = 14 electrons.
*
Applicable MO order: Z β€ 7, so s-p mixing occurs.
$sigma_{1s} < sigma^*_{1s} < sigma_{2s} < sigma^*_{2s} < (pi_{2p_x} = pi_{2p_y}) < sigma_{2p_z} < ...$
*
Electron configuration:
$(sigma_{1s})^2 (sigma^*_{1s})^2 (sigma_{2s})^2 (sigma^*_{2s})^2 (pi_{2p_x})^2 (pi_{2p_y})^2 (sigma_{2p_z})^2$
*
$N_b$ = 2 (in $sigma_{1s}$) + 2 (in $sigma_{2s}$) + 4 (in $pi_{2p}$) + 2 (in $sigma_{2p_z}$) = 10
(Note: Inner shell electrons in 1s orbitals usually cancel out in terms of bond order, but we count them for complete electron configuration).
*
$N_a$ = 2 (in $sigma^*_{1s}$) + 2 (in $sigma^*_{2s}$) = 4
*
Bond Order (BO) = Β½ (10 - 4) = 3
* This aligns perfectly with VBT and the Lewis structure (Nβ‘N), indicating a very stable triple bond.
*
Magnetic Character: All electrons are paired, so Nβ is
diamagnetic.
#### Example 4: Oxygen Molecule (Oβ)
*
Total electrons: 8 (from O) + 8 (from O) = 16 electrons.
*
Applicable MO order: Z > 7, so
NO s-p mixing.
$sigma_{1s} < sigma^*_{1s} < sigma_{2s} < sigma^*_{2s} < sigma_{2p_z} < (pi_{2p_x} = pi_{2p_y}) < (pi^*_{2p_x} = pi^*_{2p_y}) < sigma^*_{2p_z}$
*
Electron configuration:
$(sigma_{1s})^2 (sigma^*_{1s})^2 (sigma_{2s})^2 (sigma^*_{2s})^2 (sigma_{2p_z})^2 (pi_{2p_x})^2 (pi_{2p_y})^2 (pi^*_{2p_x})^1 (pi^*_{2p_y})^1$
(Note: Hund's rule applied to the degenerate $pi^*$ orbitals)
*
$N_b$ = 2 (in $sigma_{1s}$) + 2 (in $sigma_{2s}$) + 2 (in $sigma_{2p_z}$) + 4 (in $pi_{2p}$) = 10
*
$N_a$ = 2 (in $sigma^*_{1s}$) + 2 (in $sigma^*_{2s}$) + 1 (in $pi^*_{2p_x}$) + 1 (in $pi^*_{2p_y}$) = 6
*
Bond Order (BO) = Β½ (10 - 6) = 2
* This indicates a double bond (O=O), consistent with VBT.
*
Magnetic Character: Crucially, there are
two unpaired electrons in the $pi^*_{2p}$ orbitals. Therefore, Oβ is
paramagnetic. This is a monumental success of MOT, explaining a property that VBT failed to predict!
#### Example 5: Fluorine Molecule (Fβ)
*
Total electrons: 9 (from F) + 9 (from F) = 18 electrons.
*
Applicable MO order: Z > 7, so
NO s-p mixing.
*
Electron configuration:
$(sigma_{1s})^2 (sigma^*_{1s})^2 (sigma_{2s})^2 (sigma^*_{2s})^2 (sigma_{2p_z})^2 (pi_{2p_x})^2 (pi_{2p_y})^2 (pi^*_{2p_x})^2 (pi^*_{2p_y})^2$
*
$N_b$ = 2 (in $sigma_{1s}$) + 2 (in $sigma_{2s}$) + 2 (in $sigma_{2p_z}$) + 4 (in $pi_{2p}$) = 10
*
$N_a$ = 2 (in $sigma^*_{1s}$) + 2 (in $sigma^*_{2s}$) + 2 (in $pi^*_{2p_x}$) + 2 (in $pi^*_{2p_y}$) = 8
*
Bond Order (BO) = Β½ (10 - 8) = 1
* This indicates a stable single bond (F-F), consistent with VBT.
*
Magnetic Character: All electrons are paired, so Fβ is
diamagnetic.
### 7. JEE Advanced Insights and Summary
*
Understanding s-p mixing is paramount. This small detail fundamentally changes the MO energy diagram for lighter elements and directly impacts the prediction of magnetic properties and bond order.
*
MOT is superior to VBT for magnetic properties. It correctly predicts the paramagnetism of Oβ and other species with unpaired electrons in MOs.
*
Fractional Bond Orders: MOT can explain molecules or ions with fractional bond orders (e.g., OββΊ, Oββ», OβΒ²β»), which VBT finds difficult. For example, Oββ» (superoxide ion) has 17 electrons. Its configuration would be ...$(pi_{2p_x})^2 (pi_{2p_y})^2 (pi^*_{2p_x})^2 (pi^*_{2p_y})^1$. $N_b=10$, $N_a=7$. BO = Β½(10-7) = 1.5. This indicates a bond stronger than a single bond but weaker than a double bond.
*
Stability of ions: By calculating bond orders, we can compare the stability of various diatomic ions (e.g., NββΊ, Nββ», NβΒ²β»).
* Always draw the MO diagram, fill electrons, then calculate bond order and determine magnetic character.
Molecular Orbital Theory offers a robust framework for understanding the intricacies of chemical bonding, moving beyond localized electron pairs to a model where electrons occupy orbitals spanning the entire molecule. Its ability to predict magnetic behavior and fractional bond orders makes it an indispensable tool in advanced chemistry.