Molecular Orbital Theory (MOT) — LCAO, bonding - antibonding orbitals: Foundations

This page assumes nothing. Before you can read the parent note MOT parent topic, you must own every symbol it uses. We build each one from a picture, in an order where each idea leans on the one before it.

1. What is a wave? (the picture behind everything)

Before we talk about electrons, look at the simplest wave: a wiggle that goes up and down as you move along a line.

Figure — Molecular Orbital Theory (MOT) — LCAO, bonding - antibonding orbitals

Why does the topic need this? Because in MOT an electron is described by a wave. The single most important thing a wave can do is meet another wave and either reinforce it or cancel it. Hold on to the two pictures in the figure — the whole theory is those two pictures wearing a lab coat.

2. The symbol $ψ$ (psi) — the wavefunction

The parent note is full of $ψ$ . Let's earn it.

You'll see $ψ_{A}$ and $ψ_{B}$ in the parent:

You need Atomic Orbitals and Quantum Numbers to know what these atomic waves look like (s is a ball, p is a dumbbell). We use those shapes below.

3. The symbol $∣ ψ ∣^{2}$ — where the electron actually is

Here is a subtle but crucial jump the parent note makes without slowing down.

The little $^{2}$ as a superscript (e.g. $ψ_{A}^{2}$ ) means "squared". The same superscript notation later counts electrons: $σ_{1 s}^{2}$ means "2 electrons in that orbital". Same symbol, two jobs — watch the context.

4. Adding, subtracting, and the $\pm$ sign

The parent's master equation is: $ψ_{M O} = c_{A} ψ_{A} \pm c_{B} ψ_{B}$

Let's decode every piece:

Symbol	Plain words	Picture
$ψ_{M O}$	the new molecular orbital wave	a wave spread over both atoms
$c_{A}, c_{B}$	mixing coefficients: "how much" of each atom's wave to use	dial knobs setting each wave's loudness
$+$	in-phase (crests add)	constructive → glue
$-$	out-of-phase (crest meets trough)	destructive → node

5. The node — the star of "antibonding"

6. Overlap and the integral sign $\int$

The parent's condition 2 says orbitals need "significant overlap", and defines the overlap integral $S = \int ψ_{A} ψ_{B} d τ$ . Let's not let that $\int$ scare anyone.

7. Greek letters σ and π — labels for bond shape

Full shape details live in Sigma and Pi Bonds. For MOT you only need: σ = on the line, π = off the line, star = antibonding.

8. Filling the orbitals — the rules you already know

Once we have molecular orbitals, we drop electrons in exactly like atomic orbitals. That machinery is Aufbau Principle, Pauli Exclusion, Hund's Rule:

Aufbau: fill the lowest-energy orbital first (water fills the lowest basin).
Pauli: at most 2 electrons per orbital, and they must spin opposite.
Hund: given equal-energy ("degenerate") orbitals, put one electron in each first, all same spin, before pairing up.

9. Bond order symbols $N_{b}$ , $N_{a}$

The parent's payoff formula is $B.O. = \frac{N _{b} - N _{a}}{2}$ .

You'll use this in Bond Order, Bond Length and Bond Energy. Higher bond order → shorter, stronger bond.

10. How it all fits together

Compare this map with Valence Bond Theory, which never builds the "subtract" (antibonding) branch — that missing branch is exactly why VBT can't explain O₂.

Equipment checklist

Test yourself — you're ready for the parent note only if you can answer each without peeking:

I can say what a crest, trough, and "in phase" mean

Crest = wave above middle (positive), trough = below (negative), in phase = crests line up.

I know what

ψ

stands for

The wavefunction — the wave describing an electron; positive, negative, or zero at each point.

I know why we use

∣ ψ ∣^{2}

and not

ψ

∣ ψ ∣^{2}

is probability density (must be positive); squaring a sum of waves creates the

2 ψ_{A} ψ_{B}

glue term.

I can read

ψ_{A} \pm c ψ_{B}

Combine atom A's and atom B's waves, once by adding (in phase) and once by subtracting (out of phase).

I know what a node is and why it means antibonding

A point of zero density; a node between nuclei removes glue, so nuclei repel → higher energy.

I can explain the overlap integral

S

in words

A sum over all space of

ψ_{A} ψ_{B}

; measures how much two waves share the same region; zero if symmetry cancels it.

I can tell σ from π

σ is symmetric about the bond axis (head-on), π has lobes off the axis with the axis as a node (sideways); star = antibonding.

I know Aufbau, Pauli, Hund

Fill lowest first; max 2 opposite-spin electrons per orbital; singly fill equal-energy orbitals before pairing.

I can state and read the bond order formula

(N_{b} - N_{a}) /2

;

N_{b}

= bonding electrons,

N_{a}

= antibonding electrons; divide by 2 because a bond is a pair.

I know what "paramagnetic" requires

At least one unpaired electron, so the molecule is drawn toward a magnet.

Recall One-line summary to lock it in

An electron is a wave ( $ψ$ ); waves add (glue = bonding) or subtract (node = antibonding); square to see density; fill with Aufbau/Pauli/Hund; count glue minus push, halve it → bond order. That's the whole toolkit.