Chemistry involves studying a vast number of compounds. Scientists continuously discover new ones, arrange facts about them, and develop theories to explain their properties and formation. Chemical bonding is the fundamental concept that explains why atoms combine and how these combinations result in molecules and other chemical species with distinct characteristics.

Most elements, except for the noble gases, do not exist as individual atoms in nature under normal conditions. Instead, atoms group together to form molecules or ions. The attractive force that holds these constituent particles (atoms, ions, etc.) together in various chemical species is called a chemical bond.

Understanding chemical bonding helps answer fundamental questions:

• Why do atoms combine?
• Why do atoms combine in specific ratios or arrangements?
• Why are some combinations possible while others are not?
• Why do molecules have definite shapes?

Various theories have been proposed over time to explain chemical bonding, including the Kössel-Lewis approach, Valence Shell Electron Pair Repulsion (VSEPR) theory, Valence Bond (VB) theory, and Molecular Orbital (MO) theory. These theories have evolved alongside our understanding of atomic structure, electronic configurations, and the periodic table. The tendency of systems to attain lower energy and greater stability is a driving force behind chemical bond formation.

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Kössel-Lewis Approach To Chemical Bonding

Early attempts to explain chemical bonding in terms of electrons were made by W. Kössel and G.N. Lewis independently in 1916. Their approach was significant because it provided a logical explanation for chemical valence based on the observed inertness and stability of noble gases.

Lewis conceptualised an atom as having a positively charged central 'Kernel' (containing the nucleus and inner electrons) and an outer shell capable of holding up to eight electrons. He envisioned these eight outer electrons occupying the corners of a cube around the kernel. A complete set of eight electrons in the outer shell (an octet) represented a highly stable electron arrangement, similar to that of noble gases.

Lewis proposed that atoms form chemical bonds to achieve this stable octet configuration in their outermost shell. This could occur through the transfer of electrons from one atom to another (forming ionic bonds) or by the sharing of electrons between atoms (forming covalent bonds).

Lewis Symbols: To represent the valence electrons (electrons in the outermost shell involved in bonding), Lewis introduced simple notations called Lewis symbols. These symbols show the element's chemical symbol surrounded by dots, where each dot represents a valence electron.

The number of dots directly indicates the number of valence electrons. This number is useful for predicting an element's common valence or group valence, which is typically equal to the number of valence electrons or 8 minus the number of valence electrons (for p-block elements aiming for an octet).

Kössel's contributions focused on the formation of ionic bonds, noting that:

• Highly electronegative halogens (Group 17) and highly electropositive alkali metals (Group 1) are adjacent to noble gases in the periodic table.
• Halogens gain electrons to form negative ions, achieving a stable noble gas configuration.
• Alkali metals lose electrons to form positive ions, also achieving a stable noble gas configuration.
• Noble gases (except He with a duplet, $1s^2$) have a stable octet configuration ($ns^2np^6$).
• The positive and negative ions formed are held together by strong electrostatic attraction, forming an electrovalent bond (ionic bond). The magnitude of the charge on the ion is called its electrovalence.

Example: Formation of NaCl

Na ($[Ne]3s^1$) loses 1 electron to become $\text{Na}^+$ ([Ne]).

Cl ($[Ne]3s^23p^5$) gains 1 electron to become $\text{Cl}^-$ ($[Ne]3s^23p^6$ or [Ar]).

$\text{Na}^+ + \text{Cl}^- \rightarrow \text{NaCl}$ (held by electrostatic attraction).

Kössel's ideas provided a framework for understanding ionic compound formation based on electron transfer to achieve noble gas configurations.

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Octet Rule

Based on the observations that noble gases are very stable and that many atoms form bonds to achieve eight electrons in their valence shell, Kössel and Lewis proposed the electronic theory of chemical bonding (1916). According to this theory, atoms combine by either transferring valence electrons (ionic bonding) or sharing valence electrons (covalent bonding) so that each atom achieves an octet of electrons in its valence shell. This fundamental principle is known as the Octet Rule.

For hydrogen, the goal is to achieve a duplet ($1s^2$ configuration like Helium) rather than an octet.

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Covalent Bond

Irving Langmuir (1919) refined Lewis's concept by proposing that the stable octet is achieved through the sharing of electron pairs between atoms, introducing the term covalent bond. He also moved away from the rigid cubical arrangement of electrons.

Example: Formation of Cl$_2$ molecule.

Each Cl atom has 7 valence electrons ($[Ne]3s^23p^5$). It needs 1 more electron to complete its octet (like Argon, $[Ne]3s^23p^6$). When two Cl atoms approach, they can share a pair of electrons. Each atom contributes one electron to the shared pair.

This shared pair of electrons is counted towards the valence shell of both atoms. Thus, each Cl atom in Cl$_2$ effectively has 8 valence electrons, satisfying the octet rule.

A single shared electron pair forms a single covalent bond, often represented by a dash (e.g., Cl-Cl).

Atoms can share more than one electron pair to form multiple bonds:

• Double bond: Sharing of two electron pairs (e.g., in O$_2$, CO$_2$, C$_2$H$_4$).
• Triple bond: Sharing of three electron pairs (e.g., in N$_2$, C$_2$H$_2$).

In covalent bond formation:

• Bonds are formed by sharing electron pairs.
• Each bonded atom usually contributes at least one electron to the shared pair.
• Combining atoms (except H) generally achieve a noble gas octet (or duplet for H).

Examples of covalent bonds in other molecules:

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Lewis Representation Of Simple Molecules (The Lewis Structures)

Lewis dot structures are diagrams that show the arrangement of valence electrons in a molecule or ion, specifically the shared (bonding) pairs and unshared (lone) pairs. Drawing Lewis structures helps visualise bonding and electron distribution. Here are the general steps:

1. Calculate Total Valence Electrons: Sum the valence electrons of all atoms in the species. For ions, add electrons for negative charges or subtract electrons for positive charges.
2. Determine Skeletal Structure: Identify the central atom (usually the least electronegative, except for H) and connect it to the other atoms with single bonds.
3. Distribute Remaining Electrons: Place remaining valence electrons as lone pairs around the terminal atoms first to complete their octets (or duplets for H).
4. Complete Octet on Central Atom: If the central atom does not have an octet after placing lone pairs on terminal atoms, move lone pairs from terminal atoms to form multiple bonds (double or triple) with the central atom.

Molecule/Ion	Lewis Structure
H$_2$	H : H or H—H
O$_2$	:O : : O: or :O=O:
F$_2$	:F : :F: or :F—F:
H$_2$O	.. :O: .. H H .. or H—O—H ..
CH$_4$	H H : C : H H or H | H—C—H | H
CO$_2$	:Ö::C::Ö: or :Ö=C=Ö:
CO$_3^{2-}$	.. [ :Ö: ]$^{2-}$ | .. .. C — :Ö: :Ö: .. .. or equivalent resonance structures

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Answer:

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Answer:

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Formal Charge

Lewis structures provide a useful model but are not always a perfect representation, especially for polyatomic ions where the charge is distributed. To help evaluate different possible Lewis structures for the same species and to track electron distribution, the concept of formal charge is used. The formal charge on an atom in a Lewis structure is a hypothetical charge assigned to that atom based on a specific set of rules (counting bonding electrons as split equally between bonded atoms).

Formal Charge (F.C.) on an atom = (Total number of valence electrons in the free atom) - (Total number of non-bonding (lone pair) electrons) - $\frac{1}{2}$ (Total number of bonding (shared) electrons).

Example: Ozone molecule (O$_3$). Lewis structure can be drawn with one single and one double O-O bond, and lone pairs.

Let's calculate formal charges for the numbered atoms in the Lewis structure O = $\overset{+}{\text{O}}$ - $\overset{-}{\text{O}}$ (with 4 lone pairs on O=, 1 lone pair on central O, 6 lone pairs on O-):

• Central O (atom 1): Valence e$^-$ = 6. Lone pair e$^-$ = 2. Bonding e$^-$ = 6 (double bond + single bond). F.C. = $6 - 2 - \frac{1}{2}(6) = 6 - 2 - 3 = +1$.
• Terminal O (atom 2, double bond): Valence e$^-$ = 6. Lone pair e$^-$ = 4. Bonding e$^-$ = 4 (double bond). F.C. = $6 - 4 - \frac{1}{2}(4) = 6 - 4 - 2 = 0$.
• Terminal O (atom 3, single bond): Valence e$^-$ = 6. Lone pair e$^-$ = 6. Bonding e$^-$ = 2 (single bond). F.C. = $6 - 6 - \frac{1}{2}(2) = 6 - 6 - 1 = -1$.

Formal charges on O$_3$ are +1, 0, and -1. The net charge of the molecule (0) is the sum of formal charges ($+1 + 0 - 1 = 0$).

Formal charges are not the actual charges but help:

• To determine the most plausible Lewis structure when multiple are possible. Structures with smaller formal charges (closer to zero) and with negative formal charges on more electronegative atoms are generally more stable and represent the molecule better.
• In visualising electron distribution based on an equal sharing assumption for shared pairs.

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Limitations Of The Octet Rule

While the octet rule is a very useful guideline, especially for elements in the second period, it has significant limitations and exceptions:

1. Incomplete Octet of the Central Atom: Some molecules have a central atom surrounded by fewer than eight valence electrons. This is common for elements from Groups 1, 2, and 13, which have fewer than four valence electrons. Examples:
   • LiCl (Li has 2 valence e$^-$)
   • BeH$_2$ (Be has 4 valence e$^-$)
   • BCl$_3$ (B has 6 valence e$^-$)
   • AlCl$_3$ (Al has 6 valence e$^-$)
   These are electron-deficient molecules.
2. Odd-electron Molecules: Molecules with an odd total number of valence electrons cannot possibly satisfy the octet rule for all atoms. Examples:
   • Nitric oxide (NO): 5 (N) + 6 (O) = 11 valence electrons.
   • Nitrogen dioxide (NO$_2$): 5 (N) + 2$\times$6 (O) = 17 valence electrons.
3. Expanded Octet: Elements in the third period and beyond have available d orbitals (e.g., 3d, 4d, etc.) in addition to s and p orbitals. These d orbitals can participate in bonding, allowing the central atom to accommodate more than eight valence electrons in its valence shell. This is known as an expanded octet or hypervalency. Examples:
   • PF$_5$ (P has 10 valence e$^-$)
   • SF$_6$ (S has 12 valence e$^-$)
   • H$_2$SO$_4$ (S can have 12 valence e$^-$ in one common Lewis structure)
   • Many coordination compounds.
   (Note: Sulphur can also form compounds obeying the octet rule, like SCl$_2$, where S has 8 valence e$^-$).

Other drawbacks of the octet theory:

• It is based on the inertness of noble gases, but some noble gases (like Xe and Kr) do form compounds (e.g., XeF$_2$, KrF$_2$, XeOF$_4$), showing they are not entirely inert.
• The theory does not provide any information about the shape of molecules.
• It does not explain the relative stability of molecules in terms of energy.

Ionic Or Electrovalent Bond

As discussed in the Kössel-Lewis approach, an ionic bond is formed by the complete transfer of one or more valence electrons from one atom (typically a metal with low ionization enthalpy) to another atom (typically a non-metal with high negative electron gain enthalpy), resulting in the formation of oppositely charged ions. These ions are then held together by strong electrostatic forces of attraction.

Formation of ionic compound (MX):

• Metal atom M loses electron(s) to form a cation $\text{M}^+$: $\text{M(g)} \rightarrow \text{M}^+\text{(g)} + \text{e}^-$ ($\Delta_i H$, Ionization enthalpy, always endothermic).
• Non-metal atom X gains electron(s) to form an anion $\text{X}^-$: $\text{X(g)} + \text{e}^- \rightarrow \text{X}^-\text{(g)}$ ($\Delta_{eg} H$, Electron gain enthalpy, can be exothermic or endothermic).
• The gaseous ions $\text{M}^+$ and $\text{X}^-$ combine to form a solid ionic compound MX(s): $\text{M}^+\text{(g)} + \text{X}^-\text{(g)} \rightarrow \text{MX(s)}$ (This is a highly exothermic process due to the formation of the crystal lattice).

Ionic bonds are favoured between elements with low ionization enthalpy (easily form cations) and elements with high negative electron gain enthalpy (easily form anions). Cations are usually derived from metallic elements (Groups 1, 2, etc.), and anions are usually derived from non-metallic elements (Groups 16, 17). The ammonium ion ($\text{NH}_4^+$), formed from non-metals, is an exception that forms ionic compounds.

Ionic compounds in the solid state exist as a repeating, ordered three-dimensional arrangement of cations and anions called a crystal lattice. This structure is held together by strong coulombic (electrostatic) forces. The crystal structure is determined by factors like the relative sizes of the ions and their charges.

Even if the energy required for ion formation in the gas phase (sum of ionization enthalpy and electron gain enthalpy) is positive, the overall formation of the ionic solid is often highly exothermic because of the large amount of energy released when the crystal lattice is formed. This energy stabilises the ionic compound.

Example: NaCl formation. $\Delta_i H_1$ (Na) = +495.8 kJ/mol. $\Delta_{eg} H$ (Cl) = -348.7 kJ/mol. Sum = +147.1 kJ/mol (endothermic ion formation). However, the Lattice Enthalpy for NaCl is -788 kJ/mol (energy released when ions form the solid). The overall process $\text{Na(g)} + \text{Cl(g)} \rightarrow \text{NaCl(s)}$ is exothermic and favourable.

The stability of an ionic compound is primarily determined by its lattice enthalpy, not just by the ions achieving noble gas configurations in isolation.

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Lattice Enthalpy

The Lattice Enthalpy of an ionic solid is the energy required to completely separate one mole of the solid ionic compound into its constituent gaseous ions, infinitely far apart.

$\text{NaCl(s)} \rightarrow \text{Na}^+\text{(g)} + \text{Cl}^-\text{(g)}$; Lattice Enthalpy = +788 kJ mol$^{-1}$.

Lattice enthalpy is influenced by the charges on the ions (higher charges lead to stronger attraction and higher lattice enthalpy) and the sizes of the ions (smaller ions can get closer, leading to stronger attraction and higher lattice enthalpy). Crystal structure also plays a role.

A higher lattice enthalpy indicates stronger electrostatic attraction between ions in the solid and thus a more stable ionic compound.

Bond Parameters

Chemical bonds can be described and quantified using various parameters that reflect their strength, length, orientation, and polarity. These parameters provide valuable information about molecular structure and properties.

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Bond Length

Bond length is defined as the equilibrium distance between the nuclei of two atoms that are chemically bonded in a molecule. This distance is the point where the net forces of attraction and repulsion between the atoms are balanced and the potential energy is at a minimum.

Bond lengths are experimental values determined using techniques like spectroscopy, X-ray diffraction (for solids), and electron diffraction (for gases).

For a covalent bond between two atoms A and B, the bond length ($R$) is approximately the sum of their covalent radii ($r_A + r_B$). The covalent radius is considered the radius of an atom's core that is in contact with the core of an adjacent bonded atom.

For identical atoms bonded together, the covalent radius is half the bond length (e.g., covalent radius of Cl = 198 pm / 2 = 99 pm in Cl$_2$).

Van der Waals radius is a different measure of atomic size, representing the overall size including the valence shell in a non-bonded state (half the distance between two non-bonded atoms in adjacent molecules). Van der Waals radii are always larger than covalent radii for the same atom.

Bond Type	Covalent Bond Length (pm)
O–H	96
C–H	107
N–O	136
C–O	143
C–N	143
C–C	154
C=O	121
N=O	122
C=C	133
C=N	138
C$\equiv$N	116
C$\equiv$C	120

Bond lengths for specific common molecules:

Molecule	Bond Length (pm)
H$_2$ (H – H)	74
F$_2$ (F – F)	144
Cl$_2$ (Cl – Cl)	199
Br$_2$ (Br – Br)	228
I$_2$ (I – I)	267
N$_2$ (N $\equiv$ N)	109
O$_2$ (O = O)	121
HF (H – F)	92
HCl (H – Cl)	127
HBr (H – Br)	141
HI (H – I)	160

Bond lengths are influenced by factors like atomic size (larger atoms form longer bonds), and bond multiplicity (multiple bonds are shorter than single bonds between the same atoms).

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Bond Angle

In molecules containing three or more atoms, the atoms are arranged in specific orientations in space. A bond angle is defined as the angle between the lines representing the orbitals containing bonding electron pairs around the central atom. It is measured in degrees and can be determined experimentally by spectroscopic methods.

Bond angles help describe the spatial arrangement of atoms and thus the shape of the molecule. For example, in the water molecule (H$_2$O), the bond angle between the two O-H bonds is approximately 104.5$^\circ$.

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Bond Enthalpy

Bond enthalpy (or bond dissociation enthalpy) is the energy required to break one mole of a specific type of bond between two atoms in the gaseous state. It is a measure of the strength of the chemical bond.

Units are typically kilojoules per mole (kJ mol$^{-1}$).

Example: Bond enthalpy of H–H bond in H$_2$.

$\text{H}_2\text{(g)} \rightarrow 2\text{H(g)}$; $\Delta_a H^\ominus$ = +435.8 kJ mol$^{-1}$.

The superscript '$\ominus$' indicates standard state, and '$a$' indicates atomization.

For molecules with multiple bonds, breaking the bond requires more energy:

• $\text{O}_2\text{(g)}$ (O=O) $\rightarrow 2\text{O(g)}$; $\Delta_a H^\ominus$ = +498 kJ mol$^{-1}$.
• $\text{N}_2\text{(g)}$ (N$\equiv$N) $\rightarrow 2\text{N(g)}$; $\Delta_a H^\ominus$ = +946.0 kJ mol$^{-1}$.

Higher bond enthalpy corresponds to a stronger bond.

For polyatomic molecules, bond enthalpies are more complex. For example, in H$_2$O, the energy to break the first O-H bond is different from the energy to break the second O-H bond in the OH radical:

• $\text{H}_2\text{O(g)} \rightarrow \text{H(g)} + \text{OH(g)}$; $\Delta_a H_1^\ominus$ = +502 kJ mol$^{-1}$.
• $\text{OH(g)} \rightarrow \text{H(g)} + \text{O(g)}$; $\Delta_a H_2^\ominus$ = +427 kJ mol$^{-1}$.

The difference is due to the change in the chemical environment after the first bond is broken. For polyatomic molecules, an average bond enthalpy is used, which is the total bond dissociation enthalpy divided by the number of bonds broken. For H$_2$O, the average O-H bond enthalpy is $(502 + 427) / 2 = 464.5$ kJ mol$^{-1}$.

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Bond Order

In the Lewis description, the bond order between two atoms in a molecule is the number of covalent bonds (shared electron pairs) between them.

• Single bond: Bond order = 1 (e.g., H–H in H$_2$, Cl–Cl in Cl$_2$).
• Double bond: Bond order = 2 (e.g., O=O in O$_2$, C=C in C$_2$H$_4$).
• Triple bond: Bond order = 3 (e.g., N$\equiv$N in N$_2$, C$\equiv$O in CO).

Isoelectronic species (those with the same number of electrons) often have the same bond order (e.g., N$_2$, CO, $\text{NO}^+$ all have bond order 3; $\text{F}_2$ and $\text{O}_2^{2-}$ have bond order 1).

There is a general relationship between bond order, bond enthalpy, and bond length: As bond order increases, bond enthalpy increases (stronger bond), and bond length decreases (shorter bond).

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Resonance Structures

Sometimes, a single Lewis structure cannot accurately represent a molecule's properties, particularly its bond lengths, which may be experimentally determined to be intermediate between single, double, or triple bond lengths. This situation is addressed by the concept of resonance.

Example: Ozone (O$_3$). Possible Lewis structures show one O=O double bond and one O–O single bond. Normal bond lengths are 121 pm (double) and 148 pm (single). However, experimental measurements show *both* O–O bond lengths in O$_3$ are identical at 128 pm, intermediate between single and double bonds.

Resonance proposes that when a single Lewis structure is inadequate, the molecule's true structure is a resonance hybrid of multiple contributing structures, called canonical forms or resonance structures. These canonical forms have the same atomic connectivity but differ in the arrangement of electrons (specifically, the placement of multiple bonds and lone pairs). The resonance hybrid is the weighted average of these canonical forms and is the true representation of the molecule.

The resonance hybrid is represented by drawing the canonical forms separated by a double-headed arrow ($\leftrightarrow$). Structure III for O$_3$ is the resonance hybrid, showing delocalised electrons and intermediate bond lengths.

Examples of species exhibiting resonance include carbonate ion ($\text{CO}_3^{2-}$) and carbon dioxide (CO$_2$).

Answer:

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Key points about resonance:

• Resonance stabilizes the molecule. The resonance hybrid is more stable and has lower energy than any of its individual canonical forms.
• Resonance leads to averaged bond characteristics (lengths, energies) across the bonds involved in resonance.

Misconceptions about resonance:

• Canonical forms are theoretical constructs and do not actually exist.
• The molecule does not rapidly switch or oscillate between different canonical forms.
• Resonance is not an equilibrium between different structures (unlike tautomerism).
• The molecule has only one real structure, the resonance hybrid, which cannot be fully depicted by a single Lewis structure.

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Polarity Of Bonds

Chemical bonds are rarely purely ionic or purely covalent. Most bonds have some degree of both characters.

• Nonpolar Covalent Bond: Forms between two identical atoms (e.g., H$_2$, O$_2$, Cl$_2$). The shared electron pair is equally attracted by both nuclei and is located symmetrically between them. There is no separation of charge.
• Polar Covalent Bond: Forms between two different atoms with different electronegativities (e.g., HF, H$_2$O). The more electronegative atom attracts the shared electron pair more strongly, causing it to be shifted closer to that atom. This creates partial positive ($\delta+$) and partial negative ($\delta-$) charges on the bonded atoms.

This separation of charge creates a bond dipole. A molecule containing polar bonds may have a net dipole moment ($\mu$), which is a measure of the molecule's overall polarity. Dipole moment is a vector quantity defined as the product of the magnitude of the charge separation ($Q$) and the distance ($r$) between the centres of positive and negative charge ($\mu = Q \times r$). It is typically measured in Debye (D) units, where 1 D $\approx 3.33564 \times 10^{-30}$ C m.

In chemistry, a bond dipole is often represented by a crossed arrow pointing from the positive end to the negative end ($\longrightarrow$). For a molecule with multiple polar bonds, the net dipole moment is the vector sum of the individual bond dipoles. Molecular geometry is crucial in determining the net dipole moment.

Example: CO$_2$ vs H$_2$O

• CO$_2$ is a linear molecule (O=C=O). Although the C=O bonds are polar (O is more electronegative), the two bond dipoles are equal in magnitude and point in opposite directions, cancelling each other out. Net dipole moment = 0. CO$_2$ is a nonpolar molecule.
• H$_2$O has a bent structure (H-O-H angle $\approx 104.5^\circ$). The two O-H bonds are polar (O is more electronegative). Because of the bent shape, the two bond dipoles do not cancel. Their vector sum results in a significant net dipole moment ($\approx 1.85$ D). H$_2$O is a polar molecule.

Example: NH$_3$ vs NF$_3$

Both NH$_3$ and NF$_3$ have pyramidal shapes with a lone pair on the nitrogen atom. The N-H bonds are polar (N is slightly more electronegative than H). The N-F bonds are highly polar (F is much more electronegative than N).

In NH$_3$, the bond dipoles of the three N-H bonds add up, and the dipole contributed by the lone pair on N is in the same general direction, resulting in a large net dipole moment ($\approx 1.47$ D or $4.90 \times 10^{-30}$ C m).

In NF$_3$, the bond dipoles of the three N-F bonds point away from N towards F atoms. The dipole contributed by the lone pair on N points in the opposite direction to the resultant of the N-F bond dipoles. The lone pair dipole partially cancels the bond dipoles, resulting in a smaller net dipole moment ($\approx 0.23$ D or $0.8 \times 10^{-30}$ C m).

This illustrates how molecular geometry significantly affects the overall polarity (dipole moment) of a molecule.

Even purely ionic bonds have a degree of partial covalent character. This is explained by Fajan's rules. A cation can distort the electron cloud of an anion (polarise the anion), pulling some electron density into the space between the nuclei. This sharing of electron density contributes covalent character to the ionic bond. Factors favouring covalent character in an ionic bond:

• Small size of the cation.
• Large size of the anion.
• High charge on the cation.
• Cations with $(n-1)d^{1-10}ns^0$ electron configuration (transition metals) are more polarising than those with noble gas ($ns^2np^6$) configuration (alkali/alkaline earth metals) of similar size and charge.

In summary, bond polarity (dipole moment) provides a measure of the charge separation within a bond or molecule, reflecting the interplay of electronegativity differences and molecular geometry. Most real chemical bonds exist on a spectrum between ideal ionic and ideal covalent.

The Valence Shell Electron Pair Repulsion (Vsepr) Theory

The Lewis concept helps draw structures but doesn't explain molecular shapes. The Valence Shell Electron Pair Repulsion (VSEPR) theory provides a simple model to predict the geometry of covalent molecules. Proposed by Sidgwick and Powell (1940) and refined by Nyholm and Gillespie (1957), it's based on the idea that electron pairs in the valence shell of a central atom repel each other and arrange themselves to minimise this repulsion.

Key postulates of VSEPR theory:

1. The shape of a molecule is determined by the total number of electron pairs (both bonding and non-bonding/lone pairs) around the central atom.
2. These electron pairs repel each other because they are negatively charged.
3. Electron pairs orient themselves in space to be as far apart as possible, thus minimising repulsion and achieving maximum stability.
4. The valence shell can be considered a sphere, with electron pairs localising on its surface at maximum distances from one another.
5. Multiple bonds (double or triple) are treated as a single "super" electron pair for predicting geometry, although they cause greater repulsion than single bonds.
6. If a molecule has resonance structures, the VSEPR model applies to any of these structures.

The strength of repulsion between electron pairs decreases in the order:

Lone pair - Lone pair (lp-lp) > Lone pair - Bond pair (lp-bp) > Bond pair - Bond pair (bp-bp)

Lone pairs occupy more space around the central atom than bonding pairs because they are attracted by only one nucleus, while bonding pairs are shared between two. This difference in spatial requirement leads to greater repulsion by lone pairs, influencing bond angles and causing deviations from ideal geometries.

VSEPR theory allows the prediction of molecular geometry by considering the number of bonding pairs (bp) and lone pairs (lp) around the central atom (A) in an $\text{AB}_x\text{E}_y$ type molecule (where x is bp, y is lp). Molecules can be broadly categorised into those with no lone pairs on the central atom and those with one or more lone pairs.

Geometries for central atoms with no lone pairs (only bonding pairs, AB$_x$ type): Electron pairs are arranged to maximise distance.

No. of electron pairs (bp + lp)	Molecular geometry	Examples
2	Linear	BeCl$_2$, HgCl$_2$
3	Trigonal planar	BF$_3$, AlCl$_3$, CO$_3^{2-}$
4	Tetrahedral	CH$_4$, SiCl$_4$, $\text{NH}_4^+$
5	Trigonal bipyramidal	PCl$_5$, PF$_5$
6	Octahedral	SF$_6$, $\text{SiF}_6^{2-}$

Shapes of molecules with one or more lone pairs on the central atom (AB$_x$E$_y$ type): Lone pairs occupy positions that minimise repulsion, often distorting the geometry predicted solely by the total number of electron pairs.

No. of electron pairs (bp + lp)	No. of bonding pairs (bp)	No. of lone pairs (lp)	Arrangement of electron pairs	Molecular geometry	Examples
3	2	1	Trigonal planar	Bent (V-shape)	SO$_2$, O$_3$
4	3	1	Tetrahedral	Trigonal pyramidal	NH$_3$, PCl$_3$
4	2	2	Tetrahedral	Bent (V-shape)	H$_2$O, SCl$_2$
5	4	1	Trigonal bipyramidal	See-saw	SF$_4$
5	3	2	Trigonal bipyramidal	T-shape	ClF$_3$
5	2	3	Trigonal bipyramidal	Linear	XeF$_2$
6	5	1	Octahedral	Square pyramidal	BrF$_5$
6	4	2	Octahedral	Square planar	XeF$_4$

Lone pairs tend to occupy positions that provide more space, like equatorial positions in a trigonal bipyramid, to minimise 90$^\circ$ repulsions with other electron pairs.

Summary of shapes and reasons for common distorted geometries involving lone pairs:

Molecule type	No. of bonding pairs	No. of lone pairs	Arrangement of electrons	Shape	Reason for the shape acquired
AB$_2$E	2	1	Trigonal planar	Bent	Total pairs = 3 (2 bp, 1 lp). Electron arrangement is trigonal planar. Lone pair-bond pair repulsion is > bond pair-bond pair repulsion, reducing the bond angle from ideal 120$^\circ$.
AB$_3$E	3	1	Tetrahedral	Trigonal pyramidal	Total pairs = 4 (3 bp, 1 lp). Electron arrangement is tetrahedral. Lone pair-bond pair repulsion is > bond pair-bond pair repulsion, reducing bond angles from ideal 109.5$^\circ$.
AB$_2$E$_2$	2	2	Tetrahedral	Bent	Total pairs = 4 (2 bp, 2 lp). Electron arrangement is tetrahedral. Lone pair-lone pair repulsion is > lone pair-bond pair repulsion > bond pair-bond pair repulsion, reducing bond angle further from ideal 109.5$^\circ$.
AB$_4$E	4	1	Trigonal bipyramidal	See-saw	Total pairs = 5 (4 bp, 1 lp). Electron arrangement is trigonal bipyramidal. Lone pair prefers equatorial position to minimise 90$^\circ$ repulsions, distorting the shape.
AB$_3$E$_2$	3	2	Trigonal bipyramidal	T-shape	Total pairs = 5 (3 bp, 2 lp). Electron arrangement is trigonal bipyramidal. Lone pairs occupy equatorial positions (to minimise lp-lp and lp-bp repulsions), leaving bonding pairs in a T-shape.
AB$_2$E$_3$	2	3	Trigonal bipyramidal	Linear	Total pairs = 5 (2 bp, 3 lp). Electron arrangement is trigonal bipyramidal. Lone pairs occupy all three equatorial positions, leaving bonding pairs at the axial positions, resulting in a linear molecule.
AB$_5$E	5	1	Octahedral	Square pyramidal	Total pairs = 6 (5 bp, 1 lp). Electron arrangement is octahedral. Lone pair occupies one position, distorting the base of the pyramid.
AB$_4$E$_2$	4	2	Octahedral	Square planar	Total pairs = 6 (4 bp, 2 lp). Electron arrangement is octahedral. Lone pairs occupy opposite positions to minimise lp-lp repulsion, leaving bonding pairs in a square planar arrangement.

VSEPR theory is successful in predicting shapes for a wide range of molecules, particularly those of p-block elements, and accounts for subtle distortions in bond angles caused by lone pairs. However, it is a qualitative model and doesn't provide a theoretical basis for the electron pair repulsions themselves.

Valence Bond Theory

The Lewis approach and VSEPR theory are useful for describing bonding and predicting geometry but do not explain how chemical bonds are formed or why they have specific strengths and directional properties. To address these aspects, theories based on quantum mechanics were developed, including the Valence Bond (VB) theory.

Introduced by Heitler and London (1927) and further developed by Pauling, VB theory explains bond formation in terms of the overlap of atomic orbitals. It requires knowledge of atomic orbitals, electronic configurations, and the principles of quantum mechanics.

Formation of the Hydrogen molecule (H$_2$): Consider two hydrogen atoms, A and B, approaching each other. Each has a nucleus (N$_A$, N$_B$) and an electron (e$_A$, e$_B$) in a 1s atomic orbital.

As the atoms come closer, new forces arise:

• Attractive Forces: Between the nucleus of one atom and the electron of the other ($\text{N}_A - \text{e}_B$ and $\text{N}_B - \text{e}_A$). Also, between the nucleus and its own electron ($\text{N}_A - \text{e}_A$ and $\text{N}_B - \text{e}_B$), which exist even when atoms are far apart.
• Repulsive Forces: Between the electrons of the two atoms ($\text{e}_A - \text{e}_B$) and between the nuclei of the two atoms ($\text{N}_A - \text{N}_B$).

Bond formation occurs if the magnitude of the new attractive forces is greater than that of the new repulsive forces. As the atoms approach, the potential energy of the system decreases due to net attraction. This continues until a minimum energy state is reached at a specific internuclear distance. At this distance, the attractive and repulsive forces balance, and a stable covalent bond is formed. This minimum energy corresponds to the bond enthalpy, and the internuclear distance at this minimum is the bond length.

The potential energy curve for H$_2$ shows a minimum at an internuclear distance of 74 pm, corresponding to the bond length. The energy released upon bond formation is 435.8 kJ/mol, which is the bond enthalpy.

Bond formation is thus an exothermic process, resulting in a more stable molecule than the isolated atoms.

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Orbital Overlap Concept

The decrease in potential energy during bond formation is explained by the overlap of atomic orbitals. When atoms approach the equilibrium distance, their atomic orbitals containing valence electrons undergo partial interpenetration or merging. This overlap allows the pairing of electrons with opposite spins, forming a covalent bond. The region of overlap has increased electron density, which attracts both nuclei, holding them together.

According to the orbital overlap concept, a covalent bond is formed by the pairing of valence electrons with opposite spins through the partial overlap of atomic orbitals.

The strength of a covalent bond is directly related to the extent of overlap: Greater the overlap, stronger the bond.

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Directional Properties Of Bonds

Covalent bonds have directional properties because atomic orbitals, except s orbitals, are oriented in specific directions in space (e.g., p orbitals along axes, d orbitals in specific spatial arrangements). The maximum overlap occurs when orbitals overlap in a specific direction, leading to defined bond angles and molecular shapes.

Simple overlap of ground state atomic orbitals does not fully explain the observed geometries of polyatomic molecules (like the 109.5$^\circ$ tetrahedral angle in CH$_4$ or the 107$^\circ$ pyramidal angle in NH$_3$). This limitation led to the development of the concept of hybridisation within VB theory.

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Overlapping Of Atomic Orbitals

Overlap of atomic orbitals is crucial for covalent bond formation. The overlap can be positive, negative, or zero, depending on the phase (sign) and orientation of the overlapping parts of the atomic orbitals.

• Positive Overlap (Constructive Interference): Occurs when lobes of atomic orbitals with the same sign overlap. This increases electron density in the overlap region and leads to bond formation.
• Negative Overlap (Destructive Interference): Occurs when lobes of atomic orbitals with opposite signs overlap. This decreases electron density between nuclei and does not lead to bond formation.
• Zero Overlap: Occurs when orbitals are oriented such that there is no net overlap (e.g., an s orbital overlapping with the side lobes of a p$_x$ or p$_y$ orbital if the bond axis is z).

Effective bond formation requires positive overlap of atomic orbitals with appropriate symmetry along the internuclear axis.

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Types Of Overlapping And Nature Of Covalent Bonds

Based on the mode of overlap, covalent bonds are classified into two main types:

1. Sigma ($\sigma$) Bond:
   • Formed by end-to-end (head-on or axial) overlap of atomic orbitals along the internuclear axis.
   • Types of $\sigma$ overlap:
     • s-s overlap: Between two s orbitals.
     • s-p overlap: Between an s orbital and a p orbital along the axis.
     • p-p overlap: Between two p orbitals along the axis.
   • Electron density is concentrated symmetrically along the internuclear axis.
   • A single bond is always a $\sigma$ bond.
2. Pi ($\pi$) Bond:
   • Formed by sidewise (lateral) overlap of atomic orbitals whose axes are parallel to each other and perpendicular to the internuclear axis. Typically involves p or d orbitals.
   • Electron density is concentrated in two regions, one above and one below the internuclear axis, with a nodal plane along the axis.
   • Multiple bonds (double or triple) consist of one $\sigma$ bond and one or two $\pi$ bonds. A double bond has one $\sigma$ and one $\pi$ bond. A triple bond has one $\sigma$ and two $\pi$ bonds.

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Strength Of Sigma And Pi Bonds

Bond strength is related to the extent of orbital overlap. $\sigma$ overlap is generally more extensive than $\pi$ overlap because the orbitals overlap directly along the internuclear axis. Therefore, $\sigma$ bonds are generally stronger than $\pi$ bonds.

However, multiple bonds (a combination of $\sigma$ and $\pi$ bonds) are stronger than single bonds ($\sigma$ only) between the same atoms, and triple bonds are stronger than double bonds. The total strength of a multiple bond is the sum of the strengths of the individual $\sigma$ and $\pi$ bonds.

Hybridisation

To explain the observed definite shapes and bond angles in polyatomic molecules that cannot be accounted for by simple atomic orbital overlap, Linus Pauling introduced the concept of hybridisation. Hybridisation is the process of intermixing atomic orbitals of slightly different energies belonging to the same atom to form a new set of equivalent orbitals called hybrid orbitals.

These hybrid orbitals are then used for bond formation and are more effective in forming stable bonds than pure atomic orbitals. The type of hybridisation on the central atom dictates the molecule's geometry.

Salient features of hybridisation:

1. The number of hybrid orbitals formed equals the number of atomic orbitals that hybridise.
2. Hybrid orbitals are equivalent in energy and shape.
3. Hybrid orbitals are more effective than pure atomic orbitals in forming stable covalent bonds.
4. Hybrid orbitals are oriented in specific directions in space to minimise electron pair repulsion, which determines the molecular geometry.

Important conditions for hybridisation:

1. Only orbitals in the valence shell participate in hybridisation.
2. The atomic orbitals undergoing hybridisation must have comparable energies.
3. Promoting an electron to a higher energy orbital is not always necessary before hybridisation.
4. Both filled and half-filled valence orbitals can participate in hybridisation.

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Types Of Hybridisation

Different combinations of s, p, and d orbitals can hybridise, resulting in various types of hybridisation and corresponding geometries.

(I) sp Hybridisation:

• Involves the mixing of one s orbital and one p orbital to form two equivalent sp hybrid orbitals.
• The two sp hybrid orbitals are oriented linearly, pointing in opposite directions (180$^\circ$ apart).
• Each sp hybrid orbital has 50% s character and 50% p character.
• This type of hybridisation is also called diagonal hybridisation, leading to linear geometry.

Example: BeCl$_2$. Ground state Be: $1s^22s^2$. Excited state: $1s^22s^12p^1$. One 2s and one 2p orbital hybridise to form two sp hybrid orbitals. These sp orbitals overlap axially with the 2p orbitals of two Cl atoms to form two Be-Cl $\sigma$ bonds. The molecule has linear geometry with a 180$^\circ$ Cl-Be-Cl bond angle.

(II) sp² Hybridisation:

• Involves the mixing of one s orbital and two p orbitals to form three equivalent sp² hybrid orbitals.
• The three sp² hybrid orbitals are oriented in a trigonal planar arrangement, at angles of 120$^\circ$ to each other.
• Each sp² hybrid orbital has 33.3% s character and 66.7% p character.

Example: BCl$_3$. Ground state B: $1s^22s^22p^1$. Excited state: $1s^22s^12p^2$. One 2s and two 2p orbitals hybridise to form three sp² hybrid orbitals. These sp² orbitals overlap with the 2p orbitals of three Cl atoms to form three B-Cl $\sigma$ bonds. The molecule has trigonal planar geometry with 120$^\circ$ Cl-B-Cl bond angles.

(III) sp³ Hybridisation:

• Involves the mixing of one s orbital and three p orbitals to form four equivalent sp³ hybrid orbitals.
• The four sp³ hybrid orbitals are directed towards the corners of a tetrahedron, with angles of 109.5$^\circ$ between them.
• Each sp³ hybrid orbital has 25% s character and 75% p character.

Example: CH$_4$. Ground state C: $1s^22s^22p^2$. Excited state: $1s^22s^12p^3$. One 2s and three 2p orbitals hybridise to form four sp³ hybrid orbitals. These sp³ orbitals overlap with the 1s orbitals of four H atoms to form four C-H $\sigma$ bonds. The molecule has tetrahedral geometry with 109.5$^\circ$ H-C-H bond angles.

sp³ hybridisation can also explain the shapes of molecules with lone pairs, such as NH$_3$ and H$_2$O. The total number of electron pairs (bonding + lone pairs) determines the hybridisation.

Example: NH$_3$. Central atom N (Group 15). Valence electrons: 5. Three H atoms form three N-H bonds. So, 3 bonding pairs. Remaining electrons: 5 - 3 = 2. This forms one lone pair on N. Total electron pairs = 3 bp + 1 lp = 4. Nitrogen undergoes sp³ hybridisation (four sp³ orbitals arranged tetrahedrally). Three sp³ orbitals form $\sigma$ bonds with H atoms. One sp³ orbital contains the lone pair. The molecular shape is determined by the position of the atoms, which is pyramidal. The bond angle (H-N-H) is slightly less than 109.5$^\circ$ (approx. 107$^\circ$) due to lp-bp repulsion being greater than bp-bp repulsion.

Example: H$_2$O. Central atom O (Group 16). Valence electrons: 6. Two H atoms form two O-H bonds. So, 2 bonding pairs. Remaining electrons: 6 - 2 = 4. This forms two lone pairs on O. Total electron pairs = 2 bp + 2 lp = 4. Oxygen undergoes sp³ hybridisation. Two sp³ orbitals form $\sigma$ bonds with H atoms. Two sp³ orbitals contain the two lone pairs. The molecular shape is determined by the position of the atoms, which is bent or angular. The bond angle (H-O-H) is less than 109.5$^\circ$ (approx. 104.5$^\circ$) due to greater lp-lp and lp-bp repulsions compared to bp-bp repulsion.

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Other Examples Of Sp³, Sp² And Sp Hybridisation

Hybridisation can also explain bonding in molecules with multiple bonds.

• sp³ in C$_2$H$_6$ (Ethane): Each carbon atom is bonded to four other atoms (one C and three H). Each C undergoes sp³ hybridisation. One sp³ hybrid orbital from each C overlaps axially to form a C-C $\sigma$ bond. The remaining three sp³ orbitals on each C overlap with H 1s orbitals to form C-H $\sigma$ bonds. C-C bond length is 154 pm, C-H bond length is 109 pm.
• sp² in C$_2$H$_4$ (Ethene): Each carbon atom is bonded to three other atoms (one C and two H). Each C undergoes sp² hybridisation. One sp² orbital from each C overlaps axially to form a C-C $\sigma$ bond. The other two sp² orbitals on each C overlap with H 1s orbitals to form C-H $\sigma$ bonds. The unhybridised p orbital on each C atom (perpendicular to the sp² plane) overlaps sidewise to form a C-C $\pi$ bond. The molecule is planar. The C=C bond consists of one $\sigma$ and one $\pi$ bond, with a bond length of 134 pm. C-H $\sigma$ bonds have a length of 108 pm. H-C-H angle is $\approx$ 117.6$^\circ$, H-C-C angle is $\approx$ 121$^\circ$.
• sp in C$_2$H$_2$ (Ethyne): Each carbon atom is bonded to two other atoms (one C and one H). Each C undergoes sp hybridisation. One sp orbital from each C overlaps axially to form a C-C $\sigma$ bond. The other sp orbital on each C overlaps with H 1s orbital to form a C-H $\sigma$ bond. The two unhybridised p orbitals on each C (perpendicular to the sp axis and each other) overlap sidewise to form two C-C $\pi$ bonds. The molecule is linear. The C$\equiv$C bond consists of one $\sigma$ and two $\pi$ bonds, with a bond length of 120 pm. C-H $\sigma$ bonds have a length of 106 pm. The H-C-C angle is 180$^\circ$.

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Hybridisation Of Elements Involving D Orbitals

Elements in the third period and beyond can use their d orbitals for hybridisation in addition to s and p orbitals, provided the d orbitals have comparable energy to the s and p orbitals involved. Hybridisation schemes involving d orbitals explain the geometries of molecules where the central atom has an expanded octet.

Shape of molecules/ions	Hybridisation type	Atomic orbitals involved	Examples
Square planar	dsp$^2$	d + s + p(2)	[Ni(CN)$_4$]$^{2-}$, [Pt(Cl)$_4$]$^{2-}$
Trigonal bipyramidal	sp$^3$d	s + p(3) + d	PF$_5$, PCl$_5$
Square pyramidal	sp$^3$d$^2$	s + p(3) + d(2)	BrF$_5$
Octahedral	sp$^3$d$^2$ or d$^2$sp$^3$	s + p(3) + d(2) or d(2) + s + p(3)	SF$_6$, [CrF$_6$]$^{3-}$, [Co(NH$_3$)$_6$]$^{3+}$

Example: PCl$_5$. Central atom P (Group 15). Ground state electron configuration: $[Ne]3s^23p^3$. To form 5 bonds, P needs 5 unpaired electrons. In the excited state, one 3s electron is promoted to a vacant 3d orbital: $[Ne]3s^13p^33d^1$. One 3s, three 3p, and one 3d orbital hybridise to form five sp³d hybrid orbitals. These orbitals are directed towards the corners of a trigonal bipyramid.

The five sp³d hybrid orbitals of P overlap with Cl orbitals to form five P-Cl $\sigma$ bonds. Three equatorial bonds lie in a plane at 120$^\circ$ to each other. Two axial bonds are perpendicular to the equatorial plane (90$^\circ$). Axial bonds experience greater repulsion from equatorial bonds, making them slightly longer and weaker than equatorial bonds, contributing to PCl$_5$'s reactivity.

Example: SF$_6$. Central atom S (Group 16). Ground state: $[Ne]3s^23p^4$. To form 6 bonds, S needs 6 unpaired electrons. In the excited state, two electrons (one from 3s, one from 3p) are promoted to two vacant 3d orbitals: $[Ne]3s^13p^33d^2$. One 3s, three 3p, and two 3d orbitals hybridise to form six sp³d² hybrid orbitals. These are directed towards the corners of a regular octahedron (all bond angles 90$^\circ$).

The six sp³d² hybrid orbitals of S overlap with F orbitals to form six S-F $\sigma$ bonds. The molecule has perfect octahedral geometry.

Molecular Orbital Theory

Molecular Orbital (MO) theory, developed by F. Hund and R.S. Mulliken (1932), provides an alternative description of bonding that treats the molecule as a whole, rather than focusing on individual atoms and their overlapping orbitals.

Salient features of MO theory:

1. Electrons in a molecule occupy molecular orbitals (MOs), which are associated with the entire molecule, similar to how electrons in an atom occupy atomic orbitals (AOs) associated with a single nucleus. MOs are polycentric (influenced by multiple nuclei), while AOs are monocentric.
2. MOs are formed by the combination of atomic orbitals from different atoms in the molecule. Only AOs of comparable energy and appropriate symmetry can combine.
3. The number of MOs formed is equal to the number of combining AOs. When two AOs combine, they form two MOs: a bonding molecular orbital (BMO) and an antibonding molecular orbital (ABMO).
4. BMOs have lower energy and are more stable than the original AOs. ABMOs have higher energy and are less stable than the original AOs. Electrons in BMOs stabilise the molecule, while electrons in ABMOs destabilise it.
5. Molecular orbitals describe the electron probability distribution around the group of nuclei in a molecule.
6. Electrons fill MOs according to the Aufbau principle (increasing energy order), Pauli Exclusion Principle (max 2 electrons per MO with opposite spins), and Hund's rule (degenerate MOs filled singly with parallel spins before pairing).

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Formation Of Molecular Orbitals Linear Combination Of Atomic Orbitals (LCAO)

The Linear Combination of Atomic Orbitals (LCAO) is an approximate method used to describe the formation of molecular orbitals. Atomic orbitals can be represented by wave functions ($\psi_A, \psi_B$). When two AOs combine, their wave functions can interact constructively (adding) or destructively (subtracting), forming two MOs.

Mathematically, the combination of two AOs ($\psi_A$ and $\psi_B$) from two atoms A and B forms two molecular orbitals:

• $\psi_{\text{bonding}} = \psi_A + \psi_B$ (constructive interference) - forms a bonding molecular orbital (BMO), lower energy.
• $\psi_{\text{antibonding}} = \psi_A - \psi_B$ (destructive interference) - forms an antibonding molecular orbital (ABMO), higher energy.

In a BMO, the electron density is increased in the region between the nuclei, attracting the nuclei and holding them together. In an ABMO, there is decreased electron density between the nuclei, often with a nodal plane where density is zero, leading to repulsion between nuclei and destabilisation.

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Conditions For The Combination Of Atomic Orbitals

For atomic orbitals to combine effectively and form molecular orbitals, the following conditions must be met:

1. Comparable Energies: The combining AOs must have similar energies. For homonuclear diatomic molecules (same atoms), AOs from the same shell or subshell combine (e.g., 1s with 1s, 2p with 2p). For heteronuclear diatomic molecules (different atoms), AOs of similar energies can combine, even if from different shells (e.g., H 1s might combine with O 2p).
2. Same Symmetry: The combining AOs must have the same symmetry with respect to the molecular axis. By convention, the z-axis is usually taken as the molecular axis. A $\text{p}_z$ orbital can combine with another $\text{p}_z$ orbital, but a $\text{p}_z$ cannot combine with a $\text{p}_x$ or $\text{p}_y$ orbital due to different symmetry relative to the z-axis.
3. Maximum Overlap: The AOs must overlap to a significant extent. Greater overlap leads to stronger bonding and antibonding MOs.

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Types Of Molecular Orbitals

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

1. Sigma ($\sigma$) Molecular Orbitals:
   • Symmetrical around the internuclear axis.
   • Formed by head-on overlap of AOs (s-s, s-p$_z$, p$_z$-p$_z$ if z is the axis).
   • Designated as $\sigma$ (bonding) and $\sigma^*$ (antibonding).
2. Pi ($\pi$) Molecular Orbitals:
   • Not symmetrical around the internuclear axis; electron density is concentrated above and below the axis.
   • Formed by sidewise overlap of AOs (p$_x$-p$_x$, p$_y$-p$_y$).
   • Designated as $\pi$ (bonding) and $\pi^*$ (antibonding). A $\pi$ BMO has two lobes (above and below the axis), while a $\pi^*$ ABMO has a nodal plane between the nuclei.

Delta ($\delta$) MOs are formed from d orbital overlap but are not discussed in detail here.

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Energy Level Diagram For Molecular Orbitals

The relative energies of the molecular orbitals determine the order in which electrons fill them. A diagram showing the energy levels of MOs relative to the original AOs is called a molecular orbital energy level diagram.

For two combining atoms, their AOs (e.g., 1s, 2s, 2p) split into corresponding BMOs (lower energy) and ABMOs (higher energy).

The order of increasing energy for homonuclear diatomic molecules of second row elements (Li$_2$ to Ne$_2$) varies slightly depending on the element. The order is generally determined experimentally or by calculations.

For $\text{O}_2$ and $\text{F}_2$ (and heavier):

$\sigma_{1s} < \sigma^*_{1s} < \sigma_{2s} < \sigma^*_{2s} < \sigma_{2p_z} < (\pi_{2p_x} = \pi_{2p_y}) < (\pi^*_{2p_x} = \pi^*_{2p_y}) < \sigma^*_{2p_z}$

For $\text{Li}_2, \text{Be}_2, \text{B}_2, \text{C}_2, \text{N}_2$ (and lighter):

$\sigma_{1s} < \sigma^*_{1s} < \sigma_{2s} < \sigma^*_{2s} < (\pi_{2p_x} = \pi_{2p_y}) < \sigma_{2p_z} < (\pi^*_{2p_x} = \pi^*_{2p_y}) < \sigma^*_{2p_z}$

The key difference is the relative energy order of $\sigma_{2p_z}$ and $\pi_{2p_x}, \pi_{2p_y}$ MOs. For lighter molecules (up to N$_2$), the $\pi_{2p}$ MOs are lower in energy than the $\sigma_{2p_z}$ MO. For heavier molecules (O$_2$, F$_2$), the $\sigma_{2p_z}$ MO is lower in energy than the $\pi_{2p}$ MOs.

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Electronic Configuration And Molecular Behaviour

Once the relative energy levels of MOs are known, electrons are filled into these MOs according to the Aufbau principle, Pauli Exclusion Principle, and Hund's rule. The distribution of electrons in MOs is the molecular electronic configuration.

This configuration provides insights into several molecular properties:

1. Stability of Molecules: A molecule is stable if the number of electrons in bonding molecular orbitals ($N_b$) is greater than the number of electrons in antibonding molecular orbitals ($N_a$). If $N_b \le N_a$, the molecule is unstable and does not exist.
2. Bond Order: Defined as half the difference between the number of electrons in bonding and antibonding orbitals:

   Bond Order (b.o.) = $\frac{1}{2}(N_b - N_a)$

   A positive bond order ($N_b > N_a$) indicates a stable molecule. A zero or negative bond order indicates an unstable molecule. The calculated bond order often corresponds to the number of bonds (single=1, double=2, triple=3).
3. Nature of the bond: The calculated bond order can predict the type of bond (single, double, triple).
4. Bond-length: Bond length is inversely related to bond order. Higher bond order implies a stronger bond and shorter bond length.
5. Magnetic nature: Molecules with all electrons paired in MOs are diamagnetic (repelled by a magnetic field). Molecules with one or more unpaired electrons in MOs are paramagnetic (attracted by a magnetic field). MO theory can successfully predict paramagnetism, even for molecules where Lewis structures might suggest otherwise (e.g., O$_2$).

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Bonding In Some Homonuclear Diatomic Molecules

Let's apply MO theory to some simple diatomic molecules of second-row elements.

1. Hydrogen molecule (H$_2$): Each H atom has 1 electron ($1s^1$). Total electrons = 2. MO configuration: $(\sigma_{1s})^2$. $N_b=2, N_a=0$. Bond order = $\frac{1}{2}(2-0) = 1$. Stable molecule with a single bond. Diamagnetic (all electrons paired). Experimental bond enthalpy = 438 kJ/mol, bond length = 74 pm. Matches theory.
2. Helium molecule (He$_2$): Each He atom has 2 electrons ($1s^2$). Total electrons = 4. MO configuration: $(\sigma_{1s})^2(\sigma^*_{1s})^2$. $N_b=2, N_a=2$. Bond order = $\frac{1}{2}(2-2) = 0$. Bond order is zero, so He$_2$ molecule is unstable and does not exist. Matches observation.
3. Lithium molecule (Li$_2$): Each Li atom has 3 electrons ($1s^22s^1$). Total electrons = 6. Inner shell (K shell) $1s$ AOs form $(\sigma_{1s})^2(\sigma^*_{1s})^2$ (often abbreviated KK). Valence shell $2s$ AOs form $(\sigma_{2s})^2(\sigma^*_{2s})^0$. MO configuration: KK$(\sigma_{2s})^2$. For valence shell: $N_b=2, N_a=0$. Bond order = $\frac{1}{2}(2-0) = 1$. Stable molecule with a single bond. Diamagnetic (all electrons paired). Li$_2$ exists in the gas phase.
4. Carbon molecule (C$_2$): Each C atom has 6 electrons ($1s^22s^22p^2$). Total electrons = 12. MO configuration: KK$(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p_x})^2(\pi_{2p_y})^2$. Note: $\pi$ MOs are lower energy than $\sigma_{2p_z}$ for C$_2$. Valence shell electrons: $2s$ form $(\sigma_{2s})^2(\sigma^*_{2s})^2$. $2p$ form $(\pi_{2p_x})^2(\pi_{2p_y})^2(\sigma_{2p_z})^0(\pi^*_{2p_x})^0(\pi^*_{2p_y})^0(\sigma^*_{2p_z})^0$. MO configuration (valence): $(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p_x})^2(\pi_{2p_y})^2$. $N_b$ (valence) = 2 ($\sigma_{2s}$) + 2 ($\pi_{2p_x}$) + 2 ($\pi_{2p_y}$) = 6. $N_a$ (valence) = 2 ($\sigma^*_{2s}$) = 2. Total $N_b = 2+6=8$, Total $N_a = 2+2=4$. Bond order = $\frac{1}{2}(8-4) = 2$. Stable molecule with a double bond. Diamagnetic (all electrons paired in valence MOs). C$_2$ is detected in the gas phase. Note that the double bond in C$_2$ consists of two $\pi$ bonds, unlike most double bonds which are one $\sigma$ and one $\pi$.
5. Nitrogen molecule (N$_2$): Each N atom has 7 electrons ($1s^22s^22p^3$). Total electrons = 14. MO configuration: KK$(\sigma_{2s})^2(\sigma^*_{2s})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\sigma_{2p_z})^2$. Note: $\pi$ MOs are lower energy than $\sigma_{2p_z}$ for N$_2$. $N_b$ (valence) = 2 ($\sigma_{2s}$) + 2 ($\pi_{2p_x}$) + 2 ($\pi_{2p_y}$) + 2 ($\sigma_{2p_z}$) = 8. $N_a$ (valence) = 2 ($\sigma^*_{2s}$) = 2. Total $N_b = 2+8=10$, Total $N_a = 2+2=4$. Bond order = $\frac{1}{2}(10-4) = 3$. Stable molecule with a triple bond. Diamagnetic (all electrons paired). N$_2$ is a very stable molecule (high bond enthalpy).
6. Oxygen molecule (O$_2$): Each O atom has 8 electrons ($1s^22s^22p^4$). Total electrons = 16. MO configuration: KK$(\sigma_{2s})^2(\sigma^*_{2s})^2(\sigma_{2p_z})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\pi^*_{2p_x})^1(\pi^*_{2p_y})^1$. Note: $\sigma_{2p_z}$ is lower energy than $\pi$ MOs for O$_2$. Hund's rule applies to degenerate $\pi^*$ MOs. $N_b$ (valence) = 2 ($\sigma_{2s}$) + 2 ($\sigma_{2p_z}$) + 2 ($\pi_{2p_x}$) + 2 ($\pi_{2p_y}$) = 8. $N_a$ (valence) = 2 ($\sigma^*_{2s}$) + 1 ($\pi^*_{2p_x}$) + 1 ($\pi^*_{2p_y}$) = 4. Total $N_b = 2+8=10$, Total $N_a = 2+4=6$. Bond order = $\frac{1}{2}(10-6) = 2$. Stable molecule with a double bond. Paramagnetic due to two unpaired electrons in the degenerate $\pi^*_{2p}$ MOs. This prediction is a major success of MO theory, as Lewis theory predicts O$_2$ to be diamagnetic.
7. Fluorine molecule (F$_2$): Each F atom has 9 electrons ($1s^22s^22p^5$). Total electrons = 18. MO configuration: KK$(\sigma_{2s})^2(\sigma^*_{2s})^2(\sigma_{2p_z})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\pi^*_{2p_x})^2(\pi^*_{2p_y})^2$. Note: $\sigma_{2p_z}$ lower energy than $\pi$ MOs for F$_2$. $N_b$ (valence) = 2 ($\sigma_{2s}$) + 2 ($\sigma_{2p_z}$) + 2 ($\pi_{2p_x}$) + 2 ($\pi_{2p_y}$) = 8. $N_a$ (valence) = 2 ($\sigma^*_{2s}$) + 2 ($\pi^*_{2p_x}$) + 2 ($\pi^*_{2p_y}$) = 6. Total $N_b = 2+8=10$, Total $N_a = 2+6=8$. Bond order = $\frac{1}{2}(10-8) = 1$. Stable molecule with a single bond. Diamagnetic (all electrons paired).
8. Neon molecule (Ne$_2$): Each Ne atom has 10 electrons ($1s^22s^22p^6$). Total electrons = 20. MO configuration: KK$(\sigma_{2s})^2(\sigma^*_{2s})^2(\sigma_{2p_z})^2(\pi_{2p_x})^2(\pi_{2p_y})^2(\pi^*_{2p_x})^2(\pi^*_{2p_y})^2(\sigma^*_{2p_z})^2$. Total $N_b = 2+8=10$, Total $N_a = 2+8=10$. Bond order = $\frac{1}{2}(10-10) = 0$. Bond order is zero, so Ne$_2$ molecule is unstable and does not exist.

Hydrogen Bonding

A special type of attractive force exists between molecules when a hydrogen atom is bonded to a highly electronegative atom like Nitrogen (N), Oxygen (O), or Fluorine (F). In such a polar covalent bond (e.g., H—X where X = N, O, or F), the electron density is significantly shifted towards the electronegative atom, leaving the hydrogen atom with a partial positive charge ($\delta+$). This partially positive hydrogen atom is then attracted to the lone pair of electrons on a nearby electronegative atom (N, O, or F) in another molecule (or sometimes within the same molecule).

This attractive force is called a hydrogen bond. It is weaker than a covalent bond but stronger than typical van der Waals forces. Hydrogen bonds are usually represented by dotted lines (--).

Example: Hydrogen fluoride (HF). In liquid or solid HF, hydrogen bonds form between the partially positive H of one molecule and the partially negative F of another:

$\overset{\delta+}{\text{H}} - \overset{\delta-}{\text{F}} \cdots \overset{\delta+}{\text{H}} - \overset{\delta-}{\text{F}} \cdots \overset{\delta+}{\text{H}} - \overset{\delta-}{\text{F}} \cdots$

Here, the dotted lines represent hydrogen bonds.

Hydrogen bonding acts like a bridge, linking atoms from different parts of a structure (molecule or collection of molecules). The magnitude of hydrogen bonding varies with the physical state, being strongest in solids, weaker in liquids, and weakest in gases.

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Cause Of Formation Of Hydrogen Bond

Hydrogen bonding arises from the strong polarity of the bond formed when hydrogen is bonded to a highly electronegative atom (like F, O, or N). The electron pair is pulled significantly towards the electronegative atom (X), creating a large partial positive charge on the hydrogen ($\text{H}^{\delta+}$) and a large partial negative charge on the electronegative atom ($\text{X}^{\delta-}$). This creates a significant electrostatic attraction between the $\text{H}^{\delta+}$ of one molecule and the $\text{X}^{\delta-}$ of another (specifically, the lone pair on X).

The strength of the hydrogen bond depends on the electronegativity of X and the distance between H and X in the hydrogen bond.

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Types Of H-Bonds

Hydrogen bonds are classified into two types:

1. Intermolecular Hydrogen Bond: Formed between two different molecules. The molecules can be of the same substance (e.g., water, alcohol, HF) or different substances. This type of hydrogen bonding leads to association of molecules, influencing physical properties like boiling points and solubility.
2. Intramolecular Hydrogen Bond: Formed when the hydrogen atom is bonded to an electronegative atom and is also hydrogen bonded to another electronegative atom located within the same molecule. This typically occurs in molecules where functional groups capable of hydrogen bonding are positioned close to each other due to the molecule's structure. Intramolecular hydrogen bonding can affect the conformation and properties of the molecule.

Hydrogen bonding plays a crucial role in determining the structure and properties of many substances, including water, proteins, and DNA.

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