Foundations — MO diagrams of H₂, He₂, N₂, O₂, F₂, NO, CO — bond order, magnetism

This page builds every symbol the parent note leans on, in the order they must be understood. Nothing here assumes you've seen chemistry notation before.

1. What is an electron "wave" ($\psi$)?

Plain words: an electron in an atom is not a tiny ball at a fixed spot. It behaves like a smeared-out cloud — a fuzzy region where the electron might be found. The Greek letter $\psi$ (say "psi") is just the name we give to the mathematical shape of that cloud.

The picture: a soft ball of density around a nucleus (a dot). Denser where the electron spends more time.

Why the topic needs it: every molecular orbital is built by combining these atomic cloud-shapes. You cannot combine what you cannot name — so $\psi$ comes first.

A wave has a sign (+ or −) just like the crest and trough of a water wave — that sign is the key to everything that follows.

2. Atomic orbitals: $1s$, $2s$, $2p$

Plain words: electrons around an atom live in labelled "shelves." The number is the shell (how far out, roughly how much energy). The letter is the shape:

• $s$ = a round ball (no direction).
• $p$ = a dumbbell, two lobes pointing along one axis. There are three of them: $2p_x$, $2p_y$, $2p_z$ — one along each direction in space.

The picture: $1s$ small ball, $2s$ bigger ball, and three dumbbells at right angles.

Why the topic needs it: H₂ is built from $1s$ orbitals; the second-period molecules (N₂, O₂, F₂…) are built from $2s$ and $2p$ orbitals. The subscripts $x,y,z$ are what let us tell $\sigma$ bonds from $\pi$ bonds later.

Full detail on how these combine lives in Molecular Orbital Theory (LCAO basics).

3. Adding and subtracting waves — the "+" and "−" in ${\psi }_A\pm {\psi }_B$

Plain words: bring two atomic clouds close. Their waves overlap. Two things can happen:

• In phase (add, ${\psi }_A+{\psi }_B$): two crests meet → the cloud piles up between the nuclei. Electrons sitting there pull both nuclei inward — glue.
• Out of phase (subtract, ${\psi }_A-{\psi }_B$): a crest meets a trough → they cancel between the nuclei, leaving a node (empty gap). Electrons here sit outside, pulling the nuclei apart — anti-glue.

Why "+" and "−" and not multiply or divide? Because waves superpose — physically, amplitudes add. Two input waves give exactly two independent combinations (sum and difference). That is why two atomic orbitals always make two molecular orbitals.

4. The symbols $\sigma$, $\pi$, and the asterisk $\ast$

Plain words — the shape of the overlap:

• $\sigma$ (sigma): the overlap sits directly along the line joining the two nuclei — head-on. Round, symmetric about that axis. Comes from $s+s$, or from $p_z$ orbitals pointing at each other.
• $\pi$ (pi): the overlap sits sideways, above and below the nuclear axis. Comes from $p_x$ or $p_y$ dumbbells lying parallel.

The asterisk $\ast$ (star): just a tag meaning antibonding — the subtract-version with a node between the nuclei. So $\sigma$ = bonding head-on, ${\sigma }^{\ast }$ = antibonding head-on; $\pi$ = bonding sideways, ${\pi }^{\ast }$ = antibonding sideways.

Why the topic needs it: the whole energy diagram is a stack of these labels. Reading "${\pi }_{2p}^{\ast 2}$" means "two electrons in the sideways antibonding orbital built from $2p$" — and that is what makes O₂ magnetic.

5. Filling the rooms: $N_b$, $N_a$, and three rules

Plain words: once the orbitals are stacked by energy, we drop electrons in like water filling shelves lowest-first. Three seating rules (detailed in Hund's Rule and Pauli Exclusion):

1. Aufbau — fill lowest-energy orbital first.
2. Pauli — at most two electrons per orbital, and they must have opposite spins (spin = a tiny built-in magnetism, drawn as ↑ or ↓).
3. Hund — if two orbitals are equal energy (degenerate), put one electron in each with parallel spins before doubling up.

Now the two counting symbols:

• $N_b$ = total electrons sitting in bonding orbitals ($\sigma$, $\pi$).
• $N_a$ = total electrons sitting in antibonding orbitals (${\sigma }^{\ast }$, ${\pi }^{\ast }$).

Why the topic needs it: bond order is built purely from $N_b$ and $N_a$. Get the seating wrong and both the bond order and the magnetism come out wrong.

6. Bond order — why divide by 2

Plain words: one chemical bond = one pair of shared electrons. Each bonding electron gives $+\frac{1}{2}$ of a bond; each antibonding electron cancels $\frac{1}{2}$ of a bond. Net bonds:

Why the $-$: antibonding electrons undo bonding ones (the node overrides the glue), so we subtract them. Why the $÷2$: because a full bond is a pair — two electrons. Ten bonding minus four antibonding is six net bonding electrons, which is three pairs = a triple bond (that's N₂).

This links to Bond Order, Bond Length, Bond Energy correlation: higher bond order → shorter, stronger bond.

7. Magnetism: paramagnetic vs diamagnetic

Plain words: each electron is a tiny magnet (spin ↑ or ↓). When two share an orbital they point opposite and their magnetism cancels. But if any orbital holds a single, unpaired electron, its magnetism doesn't cancel — the whole molecule is pulled toward a magnet.

• Paramagnetic = has ≥1 unpaired electron → attracted to a magnet.
• Diamagnetic = all electrons paired → weakly repelled, no attraction.

Why the topic needs it: this is MO theory's famous triumph — it predicts that O₂ (two unpaired electrons in the ${\pi }^{\ast }$ pair) sticks to a magnet, which Lewis structures get wrong. Deep dive: Paramagnetism and Diamagnetism.

8. Why the orbital order flips — $s\text{–}p$ mixing

Plain words: for the early second-period atoms (up to nitrogen, Z ≤ 7), the $2s$ and $2p$ energies are close together, so the ${\sigma }_{2s}$ and ${\sigma }_{2p}$ orbitals "repel" each other and swap places — the ${\sigma }_{2p}$ gets shoved above the ${\pi }_{2p}$. By oxygen and fluorine (Z ≥ 8) the $2s$–$2p$ gap is wide, mixing is weak, and the natural order (${\sigma }_{2p}$ below ${\pi }_{2p}$) returns.

Why the topic needs it: using the wrong order for O₂ or F₂ hands you the wrong magnetism. This single fact is the trickiest gate in the whole topic — full treatment in s–p mixing and orbital energy ordering.

9. Two extra ideas the parent leans on

• Isoelectronic = "same number of electrons." CO and N₂ both have 14 → identical MO filling → identical bond order 3. See Isoelectronic species (CO, N₂, CN⁻, NO⁺).
• VBT vs MO — the older Valence Bond Theory draws shared pairs but fails on O₂'s magnetism; MO theory succeeds. Contrast in Valence Bond Theory vs MO Theory.

Prerequisite map

Equipment checklist

Connections

• Molecular Orbital Theory (LCAO basics)
• Hund's Rule and Pauli Exclusion
• Valence Bond Theory vs MO Theory
• Paramagnetism and Diamagnetism
• Bond Order, Bond Length, Bond Energy correlation
• Isoelectronic species (CO, N₂, CN⁻, NO⁺)
• s–p mixing and orbital energy ordering