Formation of Molecular Orbitals — Explained

Detailed Explanation

The formation of molecular orbitals (MOs) is a cornerstone of understanding chemical bonding beyond the simpler Valence Bond Theory. It provides a more accurate and comprehensive picture of electron distribution and energy states within a molecule. At its heart, Molecular Orbital Theory (MOT) treats a molecule as a single entity where electrons are delocalized over the entire molecular framework, rather than being confined to individual atoms or specific bonds.

Conceptual Foundation: The Quantum Mechanical Basis

Every electron in an atom or molecule is described by a wave function, $Psi$, which is a solution to the Schrödinger equation. In the context of MO formation, we consider the interaction of atomic orbitals (AOs), which are essentially wave functions describing electron behavior around individual nuclei.

When two atoms approach each other to form a bond, their AOs overlap. According to the Linear Combination of Atomic Orbitals (LCAO) approximation, the wave function of a molecular orbital ($Psi_{MO}$) can be expressed as a linear combination (sum or difference) of the atomic orbital wave functions ($Psi_A$ and $Psi_B$) of the participating atoms A and B.

Mathematically, for a diatomic molecule AB, the two possible combinations are:

1
Bonding Molecular Orbital (BMO): — $Psi_{BMO} = c_A Psi_A + c_B Psi_B$
2
Antibonding Molecular Orbital (ABMO): — $Psi_{ABMO} = c_A Psi_A - c_B Psi_B$

Here, $c_A$ and $c_B$ are coefficients that indicate the contribution of each atomic orbital to the molecular orbital. For homonuclear diatomic molecules (e.g., $H_2$, $O_2$), $c_A = c_B$, meaning both AOs contribute equally. For heteronuclear diatomic molecules (e.g., CO, HF), the coefficients differ, reflecting the unequal sharing of electrons due to electronegativity differences.

Key Principles Governing MO Formation

For effective combination of atomic orbitals to form molecular orbitals, three fundamental conditions must be met:

1
Comparable Energy: — Atomic orbitals must have similar energy levels. For instance, a 1s orbital of one atom can combine effectively with a 1s orbital of another atom, but not with a 2s or 2p orbital, because the energy difference would be too large to allow significant overlap and mixing. This ensures that the resulting MOs are energetically favorable.
2
Proper Symmetry: — The atomic orbitals must have the correct symmetry with respect to the internuclear axis to allow for effective overlap. Orbitals with different symmetries cannot combine to form molecular orbitals. For example, a 2s orbital can overlap with a 2p$_z$ orbital (if the z-axis is the internuclear axis) to form a $sigma$ MO, but not with a 2p$_x$ or 2p$_y$ orbital, which would result in zero net overlap.
3
Maximum Overlap: — The atomic orbitals must overlap to a significant extent. The greater the overlap between the interacting AOs, the stronger the resulting bond and the more stable (lower energy) the bonding MO, and the more unstable (higher energy) the antibonding MO.

Types of Molecular Orbitals: Sigma ($sigma$) and Pi ($pi$)

Molecular orbitals are classified based on their symmetry around the internuclear axis:

Sigma ($sigma$) Molecular Orbitals: — These are formed by the head-on (axial) overlap of atomic orbitals. They are cylindrically symmetrical around the internuclear axis. Examples include:

* s-s overlap: Two s orbitals combine head-on (e.g., in $H_2$). This forms a $sigma_{1s}$ (bonding) and $sigma^{*}_{1s}$ (antibonding) MO. * s-p overlap: An s orbital and a p orbital (specifically, the p orbital oriented along the internuclear axis, usually p$_z$) combine head-on (e.

g., in HF). This forms $sigma_{sp}$ and $sigma^{*}_{sp}$ MOs. * p-p overlap: Two p orbitals oriented along the internuclear axis (p$_z$-p$_z$) combine head-on. This forms $sigma_{2p}$ and $sigma^{*}_{2p}$ MOs.

Pi ($pi$) Molecular Orbitals: — These are formed by the sideways (lateral) overlap of atomic orbitals. They have a nodal plane that contains the internuclear axis. Examples include:

* p-p overlap: Two parallel p orbitals (e.g., p$_x$-p$_x$ or p$_y$-p$_y$) combine sideways. This forms $pi_{2p}$ (bonding) and $pi^{*}_{2p}$ (antibonding) MOs. There are two sets of degenerate $pi$ MOs (e.g., $pi_{2p_x}$ and $pi_{2p_y}$) and two sets of degenerate $pi^*$ MOs.

Energy Level Diagrams and Electron Filling Rules

Once MOs are formed, they are arranged in order of increasing energy, similar to atomic orbitals. Electrons are then filled into these MOs following three fundamental rules:

1
Aufbau Principle: — Electrons occupy the lowest energy MOs first.
2
Pauli Exclusion Principle: — Each molecular orbital can hold a maximum of two electrons, and these electrons must have opposite spins.
3
Hund's Rule of Maximum Multiplicity: — For degenerate molecular orbitals (orbitals of the same energy), electrons will first occupy each orbital singly with parallel spins before any orbital is doubly occupied.

The specific energy order of MOs varies depending on the atoms involved. For homonuclear diatomic molecules, there are two common energy orderings:

**For $H_2$ to $N_2$ (up to 14 electrons):**

$sigma_{1s} < sigma^{*}_{1s} < sigma_{2s} < sigma^{*}_{2s} < pi_{2p_x} = pi_{2p_y} < sigma_{2p_z} < pi^{*}_{2p_x} = pi^{*}_{2p_y} < sigma^{*}_{2p_z}$ (Note: $pi_{2p}$ are lower in energy than $sigma_{2p}$ due to s-p mixing)

**For $O_2$, $F_2$, $Ne_2$ (more than 14 electrons):**

$sigma_{1s} < sigma^{*}_{1s} < sigma_{2s} < sigma^{*}_{2s} < sigma_{2p_z} < pi_{2p_x} = pi_{2p_y} < pi^{*}_{2p_x} = pi^{*}_{2p_y} < sigma^{*}_{2p_z}$ (Note: $sigma_{2p}$ is lower in energy than $pi_{2p}$ due to reduced s-p mixing)

Derivations (Qualitative Understanding)

Let's consider the simplest case: the formation of MOs from 1s atomic orbitals in $H_2$.

Constructive Overlap: — When two 1s orbitals (spherical, positive phase) approach each other along the internuclear axis, their wave functions add up. This leads to an increased electron density between the nuclei. The resulting MO, $sigma_{1s}$, is lower in energy than the original 1s AOs and is cylindrically symmetrical. It has no nodal plane perpendicular to the internuclear axis.

Destructive Overlap: — When two 1s orbitals approach each other out of phase (one positive, one negative phase, or one inverted), their wave functions subtract. This creates a region of zero electron density (a nodal plane) between the nuclei. The resulting MO, $sigma^{*}_{1s}$, is higher in energy than the original 1s AOs and is also cylindrically symmetrical. The nodal plane lies perpendicular to the internuclear axis.

Similarly, for p orbitals:

Head-on (axial) overlap of p$_z$ orbitals: — If the internuclear axis is defined as the z-axis, two p$_z$ orbitals (each with two lobes, one positive, one negative phase) can overlap head-on. Constructive overlap leads to a $sigma_{2p_z}$ MO, with increased electron density along the axis. Destructive overlap leads to a $sigma^{*}_{2p_z}$ MO, with a nodal plane between the nuclei.

Sideways (lateral) overlap of p$_x$ or p$_y$ orbitals: — Two p$_x$ orbitals (or p$_y$ orbitals) can overlap sideways. Constructive overlap leads to a $pi_{2p_x}$ MO, with electron density above and below the internuclear axis. Destructive overlap leads to a $pi^{*}_{2p_x}$ MO, with a nodal plane perpendicular to the internuclear axis, in addition to the nodal plane containing the internuclear axis inherent to p orbitals.

Real-World Applications and Molecular Properties

MOT successfully explains several molecular properties:

1
Bond Order: — Calculated as $ext{Bond Order} = \frac{1}{2} (\text{Number of electrons in BMOs} - \text{Number of electrons in ABMOs})$. A positive bond order indicates a stable molecule. A bond order of zero suggests the molecule does not exist (e.g., $He_2$). Higher bond order generally means stronger and shorter bonds.
2
Magnetic Properties: — Molecules with all electrons paired in their MOs are diamagnetic (repelled by a magnetic field). Molecules with one or more unpaired electrons are paramagnetic (attracted to a magnetic field). MOT correctly predicts the paramagnetism of $O_2$, which Valence Bond Theory fails to explain.
3
Stability: — Molecules with more electrons in bonding MOs than in antibonding MOs are stable. The energy difference between BMOs and ABMOs also contributes to stability.
4
Electronic Spectra: — The energy differences between MOs correspond to electronic transitions, which can be observed in UV-Vis spectroscopy.

Common Misconceptions

Confusing AOs with MOs: — Students often think MOs are just AOs stuck together. MOs are entirely new orbitals spanning the whole molecule.
Incorrect Energy Ordering: — Especially for $N_2$ vs. $O_2$ type molecules, the s-p mixing effect changes the relative energies of $sigma_{2p}$ and $pi_{2p}$ MOs. This is a frequent source of error.
Ignoring Symmetry: — Forgetting that AOs must have compatible symmetry to combine effectively. Not all overlaps lead to bonding.
Overlooking Nodal Planes: — Not understanding that antibonding MOs always have at least one more nodal plane than their corresponding bonding MOs, specifically between the nuclei.

NEET-Specific Angle

For NEET, the focus is primarily on:

Diatomic Molecules: — Understanding MO diagrams for homonuclear (e.g., $H_2, N_2, O_2, F_2$) and simple heteronuclear (e.g., CO, NO) diatomic species.
Calculating Bond Order: — A very common question type. Students must be able to write the MO electronic configuration and apply the bond order formula.
Predicting Magnetic Properties: — Determining if a molecule or ion is paramagnetic or diamagnetic based on its MO configuration.
Comparing Stability: — Using bond order to compare the relative stability of different species (e.g., $O_2, O_2^+, O_2^-$).
Identifying Nodal Planes: — Understanding the presence and location of nodal planes in bonding and antibonding MOs.
Energy Level Diagrams: — Being able to draw or interpret simplified MO energy level diagrams for common diatomic molecules.