Foundations — Valence Bond Theory (VBT) — hybridization (sp, sp², sp³, sp³d, sp³d²)

This page assumes nothing. The parent note leans on one master formula — a "steric number" built from bonds and lone pairs — but those words are meaningless until you can picture each ingredient. So we build every symbol from the ground up, in an order where each one leans only on the ones before it, and only assemble the master formula in §7 once every piece exists.

1. The atom and its "shells" of electrons

• Picture: the nucleus is a dot; electrons live in nested shells like layers of an onion. Only the outer layer reaches out to touch other atoms.
• Why the topic needs it: every hybridization calculation starts with $V$ (e.g. carbon $V=4$, oxygen $V=6$). If you can't find $V$ you can't start.

Read $V$ aloud as "how many outer electrons this atom brought to the party."

2. Atomic orbitals: the shapes electrons live in

• Why the subscript letters? $p_x$ just means "the dumbbell lying along the $x$ axis." The letter after $p$ names the direction — nothing scarier.
• Why the topic needs it: hybridization mixes these exact shapes. The name "$s{p}^3$" literally means "one $s$ ball blended with three $p$ dumbbells." You must see the ingredients before you can read the recipe.

See Atomic Orbitals (s, p, d shapes) for the full gallery.

3. The superscript number: reading "$s{p}^3$", "$s{p}^2$", "$sp$", "$s{p}^{3}d$"

• The raised number is not a power here (we are not multiplying); it is a tally of ingredients, and it applies to whichever letter it sits on — the $3$ on $p$ counts p orbitals, the $2$ on $d$ counts d orbitals. This is a common trap.
• Golden rule: number of hybrids = number of atomic orbitals mixed. Two in, two out. Six in, six out. Nothing is created or destroyed.

4. Bonds: σ (sigma) and π (pi)

• Picture: a σ bond is two arrows meeting tip-to-tip; a π bond is two dumbbells lying parallel and touching along their sides.
• The crucial fact for this topic: every single bond is one σ. A double bond is one σ + one π. A triple bond is one σ + two π. In every case there is exactly one σ bond between the two atoms; the extras are all π.
• Why the topic needs it: the steric number counts only σ bonds (see §7). The π bonds come from leftover unhybridized p orbitals and are ignored when finding the shape. Miss this and $C{O}_2$ looks bent instead of linear.

Full detail lives in Sigma and Pi Bonds.

5. Lone pairs

• Picture: two electrons parked in one of the hybrid "hands," pointing outward but grabbing nothing.
• Why the topic needs it: lone pairs still take up a hybrid orbital and still push other pairs away. They must be counted, or you get the wrong shape (water would look linear).

We work out how many lone pairs exist using the formula in §6.

6. The counting formula $\text{LP}=\frac{V-B}{2}$

Reading it as a story:

• Start with $V$ electrons.
• Subtract $B$, the ones spent reaching neighbours → what's left is unbonded electrons.
• Divide by $2$ because electrons pair up → number of pairs left over = lone pairs.

7. The steric number $\text{SN}$ — the master symbol

Now every ingredient exists — σ bonds (§4) and lone pairs (§5, §6) — so we can finally assemble the parent note's master formula.

• Picture: count every "direction the central atom must point a hand" — one for each neighbour it holds, one for each resting lone pair. That total is SN.
• Why divide bonds into σ only? From §4, a double bond is 1 σ + 1 π; only the σ needs a hybrid orbital (the π uses leftover p). So a double bond adds 1 to SN, not 2.
• Why the topic needs it: SN is the single number that unlocks the whole table below. Everything downstream — hybridization, geometry, angle — is read straight off SN.

8. Angle notation and the "balloons" picture

• Picture: tie $\text{SN}$ balloons at one knot — they shove apart to the most spread-out arrangement. That is the geometry, and the angle between neighbouring balloons is the bond angle.
• Why the topic needs it: shapes and angles are the whole payoff of hybridization. The degree symbol $°$ just means "part of a full turn," where a full circle is $360°$ and a straight line is $180°$.

The refinement of why angles shrink (lone pairs push harder) is handled by VSEPR Theory and Bond Angle and Bent's Rule. The deeper energy-based alternative to VBT is Molecular Orbital Theory (MOT).

9. The dot product $\vec{A}\cdot \vec{B}$ (only for the angle derivation)

The parent note derives $109.5°$ using $\cos \theta =\frac{\vec{A}\cdot \vec{B}}{|\vec{A}||\vec{B}|}$. Here is that tool from zero.

• Why this tool and not another? The dot product is the one operation that turns two arrows directly into the angle between them, via $\cos \theta =\frac{\vec{A}\cdot \vec{B}}{|\vec{A}||\vec{B}|}.$ No triangle-drawing needed — feed in coordinates, out comes $\cos \theta$.
• What ${\cos }^{-1}$ does: $\cos \theta$ gives a ratio; ${\cos }^{-1}$ (arccos) asks the reverse question — "which angle has this cosine?" — and hands back the angle.

Prerequisite map

Equipment checklist