🔗 Chemical Bonding & Molecular Orbital Theory

Chemical bonding governs molecular geometry, bond order, magnetic properties, and intermolecular forces. Advanced problems test Molecular Orbital (MOT) energy levels and mixing ($sp$-mixing vs non-$sp$-mixing), $\pi$-back-bonding bond length shifts, isoelectronic geometries (VSEPR), and $3c-2e$ banana bonding.

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1. 📐 VSEPR Theory & Steric Number Geometries

$$\text{Steric Number (SN)}=(\text{Number of }\sigma \text{-bonds})+(\text{Number of Lone Pairs on Central Atom})$$

Steric No.	Hybridization	Bond Pairs	Lone Pairs	Molecular Geometry	Ideal Bond Angle	Classic Examples
2	$sp$	2	0	Linear	${180}^{\circ }$	$BeC{l}_2,C{O}_2,HCN,I_3^{-}$
3	$s{p}^2$	3	0	Trigonal Planar	${120}^{\circ }$	$B{F}_3,S{O}_3,N{O}_3^{-}$
3	$s{p}^2$	2	1	Bent / V-shaped	$<{120}^{\circ }$ (${119}^{\circ }$)	$S{O}_2,O_3,N{O}_2^{-}$
4	$s{p}^3$	4	0	Tetrahedral	${109.5}^{\circ }$	$C{H}_4,N{H}_4^{+},S{O}_4^{2-}$
4	$s{p}^3$	3	1	Trigonal Pyramidal	$<{109.5}^{\circ }$ (${107}^{\circ }$)	$N{H}_3,PC{l}_3,H_3{O}^{+}$
4	$s{p}^3$	2	2	Bent / Angular	$<{109.5}^{\circ }$ (${104.5}^{\circ }$)	$H_{2}O,O{F}_2,SC{l}_2$
5	$s{p}^{3}d$	5	0	Trigonal Bipyramidal (TBP)	${120}^{\circ },{90}^{\circ }$	$PC{l}_5,P{F}_5$
5	$s{p}^{3}d$	4	1	See-Saw	$<{120}^{\circ },<{90}^{\circ }$	$S{F}_4,TeC{l}_4$
5	$s{p}^{3}d$	3	2	T-Shaped	$<{90}^{\circ }$ (${87.5}^{\circ }$)	$Cl{F}_3,Br{F}_3$
5	$s{p}^{3}d$	2	3	Linear	${180}^{\circ }$	$Xe{F}_2,I_3^{-},[IC{l}_2{]}^{-}$
6	$s{p}^3{d}^2$	6	0	Octahedral	${90}^{\circ }$	$S{F}_6,[P{F}_6{]}^{-}$
6	$s{p}^3{d}^2$	5	1	Square Pyramidal	$<{90}^{\circ }$	$Br{F}_5,I{F}_5,XeO{F}_4$
6	$s{p}^3{d}^2$	4	2	Square Planar	${90}^{\circ }$	$Xe{F}_4,[IC{l}_4{]}^{-}$
7	$s{p}^3{d}^3$	7	0	Pentagonal Bipyramidal	${72}^{\circ },{90}^{\circ }$	$I{F}_7$
7	$s{p}^3{d}^3$	6	1	Distorted Octahedral	Variable	$Xe{F}_6$

Bent's Rule

More electronegative substituents prefer hybrid orbitals having less $s$-character (axial positions in TBP), whereas lone pairs and electropositive groups prefer hybrid orbitals with more $s$-character (equatorial positions in TBP).

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2. 🌌 Molecular Orbital Theory (MOT) for Diatomics

2.1 Molecular Orbital Energy Level Sequences

1. For species with $\le 14$ electrons ($B_2,C_2,N_2$) — With $2s-2p$ Mixing:$$\sigma 1s<{\sigma }^{\ast }1s<\sigma 2s<{\sigma }^{\ast }2s<(\pi 2p_x=\pi 2p_y)<\sigma \mathbf{2}{\mathbf{p}}_{\mathbf{z}}<({\pi }^{\ast }2p_x={\pi }^{\ast }2p_y)<{\sigma }^{\ast }2p_z$$
2. For species with $>14$ electrons ($O_2,F_2,N{e}_2$) — No $2s-2p$ Mixing:$$\sigma 1s<{\sigma }^{\ast }1s<\sigma 2s<{\sigma }^{\ast }2s<\sigma \mathbf{2}{\mathbf{p}}_{\mathbf{z}}<(\pi 2p_x=\pi 2p_y)<({\pi }^{\ast }2p_x={\pi }^{\ast }2p_y)<{\sigma }^{\ast }2p_z$$

2.2 Bond Order & Magnetic Properties Matrix

$$\text{Bond Order (BO)}\mathbf{=}\frac{{\mathbf{N}}_{\mathbf{b}}\mathbf{-}{\mathbf{N}}_{\mathbf{a}}}{\mathbf{2}}$$

Molecule / Ion	Total Electrons	Configuration (Valence MO)	Bond Order	Magnetic Nature	Stability & Bond Length
$N_2$	14	$\dots (\pi 2p{)}^4(\sigma 2p_z{)}^2$	3.0	Diamagnetic	Shortest bond ($1.10\text{AA}$), highest BDE
$N_2^{+}$	13	$\dots (\pi 2p{)}^4(\sigma 2p_z{)}^1$	2.5	Paramagnetic (1 unpaired)	Bond length: $N_2<N_2^{+}$
$O_2$	16	$\dots (\sigma 2p_z{)}^2(\pi 2p{)}^4({\pi }^{\ast }2p{)}^2$	2.0	Paramagnetic (2 unpaired in ${\pi }^{\ast }$)	Experimental proof of MOT
$O_2^{+}$	15	$\dots ({\pi }^{\ast }2p{)}^1$	2.5	Paramagnetic (1 unpaired)	Higher BO than $O_2$
$O_2^{-}$ (Superoxide)	17	$\dots ({\pi }^{\ast }2p{)}^3$	1.5	Paramagnetic (1 unpaired)	Bond length: $O_2^{+}<O_2<O_2^{-}$
$O_2^{2-}$ (Peroxide)	18	$\dots ({\pi }^{\ast }2p{)}^4$	1.0	Diamagnetic	Longest $O-O$ single bond
$C_2$	12	$\dots (\sigma 2s{)}^2({\sigma }^{\ast }2s{)}^2(\pi 2p{)}^4$	2.0	Diamagnetic	Both bonds are strictly $\pi$-bonds!
$CO$	14	Isoelectronic with $N_2$	3.0	Diamagnetic	Highest bond energy ($1072\text{ kJ/mol}$)
$C{O}^{+}$	13	Highest occupied orbital is non-bonding ${\sigma }^{\ast }$	3.5	Paramagnetic	Anomalous: $C{O}^{+}$ has HIGHER bond order than $CO$!

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3. 🔄 Back-Bonding & Multi-Center Bonds

1. $d\pi -p\pi$ Back-Bonding:
   • In Trisilylamine $(Si{H}_3{)}_{3}N$, the nitrogen lone pair delocalizes into the empty $3d$-orbitals of Silicon.
   • Result: Geometry of $(Si{H}_3{)}_{3}N$ is strictly planar $s{p}^2$, and it is non-basic (unlike pyramidal $s{p}^3$ basic $(C{H}_3{)}_{3}N$).
2. $3c-2e$ Banana Bond in Diborane ($B_2{H}_6$):
   • Contains two terminal $B-H$ $2c-2e$ bonds on each Boron and two bridging $B-H-B$ $3c-2e$ bonds.
   • Bridging $B-H$ bonds are longer ($1.33\text{AA}$) and weaker than terminal bonds ($1.19\text{AA}$).