Theories of Covalent Bonding and Molecular Shapes

Electronic Configurations and Magnetic Properties

The study of covalent bonding begins with an understanding of electronic configurations and the resulting magnetic properties of atoms and molecules. For instance, the electronic configuration of elements involves the distribution of electrons in orbitals such as $1s$, $2s$, $2p$, $3s$, $3p$, and $3d$. Specific sub-orbitals for the d-shell include $d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x^2-y^2}$, and $d_{z^2}$.

Paramagnetism is determined by the presence of unpaired electrons. The magnetic moment ($\mu$) can be calculated using the formula $\mu = \sqrt{n(n+2)}$, where $n$ represents the number of unpaired electrons. For an atom with 3 unpaired electrons ($n=3$), the calculation is $\mu = \sqrt{3(3+2)} = \sqrt{15}$, which is approximately $3.9\,BM$ (Bohr Magnetons). Understanding these properties allows for comparisons of paramagnetic strength between different ions, such as $Ni^{2+}$ and others.

Valence Bond Theory (VBT)

Valence Bond Theory (VBT) is a foundational concept in the study of covalent bonding. It was first presented by Heitler and London in 1927 and was later extended by Linus Pauling. This theory provides essential explanations for the formation and breaking of covalent bonds, bond energy levels, bond lengths, the distance between atoms, and types of covalent bonds, as well as the overall shapes of molecules.

According to VBT, a covalent bond is formed by the overlapping of half-filled atomic orbitals. A single covalent bond is formed by the overlapping of exactly two such atomic orbitals. During bond formation, heat energy is released as the system moves to a more stable state. The electrons within these overlapped orbitals must have opposite spins. The number of bonds an atom can form is directly proportional to the number of half-filled orbitals it contains in its valence shell. Furthermore, only orbitals with the same symmetry can overlap to form a stable covalent bond.

Examples of VBT in practice include the formation of Hydrogen ($H_2$), where two $1s^1$ orbitals overlap ($s-s$ overlapping). In the formation of Hydrochloric Acid ($HCl$), the $1s^1$ orbital of Hydrogen overlaps with the $3p_z^1$ orbital of Chlorine ($s-p$ overlapping). In the case of Fluorine ($F_2$), the bond results from the overlapping of two $2p_z^1$ orbitals ($p-p$ overlapping).

Characteristics of Sigma ($\sigma$) and Pi ($\pi$) Bonds

Covalent bonds are categorized into Sigma ($\sigma$) and Pi ($\pi$) bonds based on the nature of orbital overlapping.

A Sigma ($\sigma$) bond is formed by the head-on or "front-to-front" linear (axial) overlapping of atomic orbitals. This bond is significantly stronger and possesses higher Bond Energy (B.E.) than a pi bond. It can be formed by the overlapping of $s-s$ orbitals, $s-p$ orbitals, or $p-p$ orbitals, and can involve both hybridized and unhybridized orbitals. The sigma bond is always the first bond formed between two atoms. The electron density is at its maximum at the internuclear axis, and because of this axial symmetry, it allows for the free rotation of the bonded atoms. Sigma bonds are generally less reactive and do not involve resonance.

A Pi ($\pi$) bond is formed by the lateral or "side-by-side" (parallel) overlapping of atomic orbitals. It is weaker than a sigma bond and has a lower Bond Energy. These bonds are typically formed only by the $p-p$ overlapping of unhybridized orbitals. The pi bond represents the second or third bond formed between atoms (after the initial sigma bond). Its electron density is concentrated above and below the internuclear axis, which restricts the rotation of the atoms. Pi bonds are more reactive than sigma bonds and are the primary bonds involved in resonance (delocalization).

Resonance and Bond Relationships

Resonance involves the delocalization of only $\pi$ bonds within a molecule. This process is possible because rotation is restricted in $\pi$ bonds. There are critical relationships established between bond properties: Bond strength is directly proportional to Bond Energy (B.E.), and both are inversely proportional to the reactivity of the molecule. This implies that a bond with higher energy is harder to break and therefore less chemically reactive.

Valence Shell Electron Pair Repulsion (VSEPR) Theory

VSEPR Theory was presented by Sidgwick and Powell and later extended by Nyholm and Gillespie. This theory is used to explain the shapes of polyatomic molecules like $H_2O$, $NH_3$, and $BF_3$. In any molecule, there are two types of electron pairs: Bond Pairs ($B.P.$), which are shared pairs of valence electrons, and Lone Pairs ($L.P.$), which are unshared pairs.

These electron pairs repel each other, and the geometry of the molecule is determined by the need to occupy positions that minimize these repulsions and maximize stability. The order of repulsive strength is: L.P-L.P > L.P-B.P > B.P-B.P. Lone pairs occupy more space than bond pairs because they are under the electrostatic influence of only one nucleus (the central atom), whereas bond pairs are shared between two nuclei. Diatomic molecules are inherently linear, but for polyatomic molecules, the shape depends on the arrangement of both bond and lone pairs.

Active Sets and Molecular Geometries

The "Active Set" of electron pairs refers to the number of electron pairs that determine the geometry of the central atom. The following table details the relationship between active sets, bond pairs, lone pairs, and the resulting molecular geometry:

• 2 Active Sets (2 B.P., 0 L.P.): Result in a Linear geometry with a bond angle of $180^{\circ}$ and $sp$ hybridization (e.g., $BeCl_2$, $CO_2$).

• 3 Active Sets (3 B.P., 0 L.P.): Result in a Trigonal Planar geometry with a bond angle of $120^{\circ}$ and $sp^2$ hybridization (e.g., $BF_3$, $BH_3$).

• 3 Active Sets (2 B.P., 1 L.P.): Result in a Bent or V-shaped geometry with a bond angle less than $120^{\circ}$ and $sp^2$ hybridization (e.g., $SO_2$, $NO_2^-$).

• 4 Active Sets (4 B.P., 0 L.P.): Result in a Tetrahedral geometry with a bond angle of $109.5^{\circ}$ and $sp^3$ hybridization (e.g., $CH_4$, $CCl_4$).

• 4 Active Sets (3 B.P., 1 L.P.): Result in a Trigonal Pyramidal geometry with a bond angle of approximately $107.5^{\circ}$ and $sp^3$ hybridization (e.g., $NH_3$, $PH_3$, $NF_3$).

• 4 Active Sets (2 B.P., 2 L.P.): Result in a Bent or V-shaped geometry with a bond angle of approximately $104.5^{\circ}$ and $sp^3$ hybridization (e.g., $H_2O$, $H_2S$, $SCl_2$).

• 5 Active Sets (5 B.P., 0 L.P.): Result in a Trigonal Bipyramidal geometry with bond angles of $90^{\circ}$ and $120^{\circ}$ and $sp^3d$ hybridization (e.g., $PCl_5$, $PF_5$).

• 6 Active Sets (6 B.P., 0 L.P.): Result in an Octahedral geometry with bond angles of $90^{\circ}$ and $sp^3d^2$ hybridization (e.g., $SF_6$).

• 7 Active Sets (7 B.P., 0 L.P.): Result in a Pentagonal Bipyramidal geometry with bond angles of $72^{\circ}$ and $90^{\circ}$ and $sp^3d^3$ hybridization (e.g., $IF_7$).

Hybridization Theory (Hybrid Orbital Theory)

Hybridization Theory, presented by Linus Pauling, was developed to address the limitations of VBT. Specifically, VBT failed to explain the observed valency of Carbon (4), Boron (3), and Beryllium (2) in their compounds, the paramagnetic nature of Oxygen ($O_2$), and the delocalization of $\pi$ electrons.

Hybridization is the process in which atomic orbitals of slightly different energies are mixed together to form new equivalent orbitals, known as hybrid orbitals, which have identical shapes and energies. This process occurs within the central atom of polyatomic molecules.

In $sp^3$ hybridization, one s-orbital and three p-orbitals ($p_x, p_y, p_z$) intermix to produce four new $sp^3$ hybrid orbitals. For example, in Carbon ($C$), the ground state configuration is $1s^2, 2s^2, 2p_x^1, 2p_y^1$, showing a valency of 2. In the excited state, one electron from the $2s$ orbital promotes to the $2p_z$ orbital, yielding a configuration of $1s^2, 2s^1, 2p_x^1, 2p_y^1, 2p_z^1$. This provides four half-filled orbitals (valency of 4) that go through hybridization to form a Tetrahedral model with $109.5^{\circ}$ angles, as seen in Methane ($CH_4$).

Questions & Discussion

Question: In which molecule does $s-p$ overlapping take place? Response: Correct identification would be Hydrogen Iodide ($H-I$), where the $1s$ orbital of Hydrogen overlaps with a $p$ orbital of Iodine.

Question: Calculate the ratio of sigma to pi bonds in cyanide ($C \equiv N$). Response: In a triple bond such as $C \equiv N$, there is 1 sigma bond and 2 pi bonds. Therefore, the ratio is $1:2$.

Question: What is the relationship between S-character and P-character in bond angles? Response: Bond angles are directly proportional to S-character and inversely proportional to P-character. For instance, $sp$ hybridization (50% S) results in $180^{\circ}$, while $sp^3$ (25% S) results in $109.5^{\circ}$.

Question: Which molecule has the least bond angle among $H_2O$, $NH_3$, $CH_4$, and $H_2S$? Response: $H_2S$ typically shows a significantly smaller bond angle (approaching $92^{\circ}$) compared to the others due to the larger size and lower electronegativity of Sulfur compared to Oxygen, Nitrogen, or Carbon.

Question: Select a pair of orbitals that can form a covalent bond from: a) $2p_x^1, 2p_x^1$; b) $4s^1, 4p_x^1$; c) $3p_y^1, 3p_y^1$; d) All of these. Response: The correct answer is d) All of these, as all pairs consist of half-filled orbitals capable of overlapping.