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Bond order 2026-10-13

In the elementary molecular orbital theory count, bond order is half the excess of Electrons in bonding molecular orbitals over those in antibonding molecular orbitals. Thus adding one Electron to a bonding molecular orbital raises

b

by

1/2

, while adding one to an antibonding molecular orbital lowers it by

1/2

. The configurations drawn above give

1

,

1/2

, and

0

. This count summarizes net covalent bond occupancy; it is neither a measured energy nor a universal test for every kind of binding. See Chemistry 2e, 8.4: molecular orbital bond order.

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Molecular orbital theory 2026-10-13

Molecular orbital theory describes the electronic arrangement of a molecule using molecular orbitals: one-Electron spatial wavefunctions extending over the combined atomic nuclei. Filling bonding molecular orbitals can stabilize a chemical bond; filling antibonding molecular orbitals opposes that stabilization. A molecular orbital is a building block of a many-Electron description, rather than the entire many-Electron wavefunction. See Robert W. Field, MIT 5.61, lecture 24, pp. 1–4: molecular orbitals and the LCAO approximation.

First make the interference visible with an exactly normalized one-dimensional toy. Replace the three-dimensional

1 s

atomic orbitals of the hydrogen atom by two Gaussian functions centred at

x = \pm a

, with dimensionless

x

and

a > 0

:

ϕ_{A} (x) = π^{- 1/4} e^{- (x + a)^{2} /2}, ϕ_{B} (x) = π^{- 1/4} e^{- (x - a)^{2} /2}, S = \int_{- \infty}^{\infty} ϕ_{A} ϕ_{B} d x = e^{- a^{2}} .

The normalized sum and difference are

ψ_{\pm} = (ϕ_{A} \pm ϕ_{B}) / 2 (1 \pm S)

, following the same linear combination of atomic orbitals algebra. At

a = 1

, the orbital overlap is

S = e^{- 1} \approx 0.367879

. At the midpoint,

ϕ_{A} (0) = ϕ_{B} (0) \approx 0.455581

ψ_{+} (0) \approx 0.550879, ∣ ψ_{+} (0) ∣^{2} \approx 0.303468, ψ_{-} (0) = ∣ ψ_{-} (0) ∣^{2} = 0.

Both probability densities have total integral one: interference redistributes probability, rather than creating an extra Electron. These are toy wavefunctions, not actual hydrogen atom orbitals or predictions of molecular hydrogen energies.

Figure 1.
Normalized one-dimensional Gaussian function toy: signed probability amplitudes add or cancel before their squares give probability densities
. Centres at $x = - 1$ and $x = 1$ give orbital overlap $S = e^{- 1}$ . The difference has a zero at the midpoint; its negative lobe does not carry positive electric charge. All density curves shown are normalized on the line, rather than in three-dimensional space. Details.

Now fill the two

1 s

-derived molecular orbitals in the minimal model. The lower bonding molecular orbital is labelled

σ_{g}

; the upper antibonding molecular orbital is

σ_{u}^{*}

. The Pauli exclusion principle permits at most two Electrons in each spatial molecular orbital, with opposite spin projections along a chosen axis. For the lowest electron configuration in this model, fill

σ_{g}

first. The subscripts

g

and

u

denote even and odd behaviour under inversion through the midpoint; the star marks an antibonding molecular orbital. See Flowers et al., Chemistry 2e, 8.4: occupancy and bond order and Field, MIT 5.61, lecture 25, pp. 1–3: bonding, antibonding, and symmetry labels.

With

N_{b}

Electrons in bonding molecular orbitals and

N_{a}

in antibonding molecular orbitals, the model's bond order is

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

. The three elementary counts are:

Molecular hydrogen, $H_{2}$ : two Electrons give $(σ_{g})^{2}$ and $b = (2 - 0) /2 = 1$ .
The hydrogen molecular ion $H_{2}^{+}$ : one Electron gives $(σ_{g})^{1}$ and $b = (1 - 0) /2 = 1/2$ .
Two neutral helium chemical atoms, $He_{2}$ : four Electrons give $(σ_{g})^{2} (σ_{u}^{*})^{2}$ and $b = (2 - 2) /2 = 0$ in this minimal $1 s$ model.

Figure 2.
Schematic molecular orbital occupancy for $H_{2}$ , $H_{2}^{+}$ , and $He_{2}$ : bond orders one, one half, and zero
. Each horizontal line is one spatial molecular orbital; paired arrows represent opposite spin projections. Vertical positions show ordering only, with arbitrary gaps rather than measured energies or a common scale across species. Details.

The zero bond order says this minimal model gives no net covalent bond for

He_{2}

. It does not rule out every bound pair of helium chemical atoms: a very weak Van der Waals force can bind a neutral helium dimer, which has been observed experimentally. See Zeller et al., Imaging the He2 quantum halo state using a free electron laser, PNAS 113, 14651–14655 (2016), p. 14651: van der Waals binding and Luo et al., The weakest bond: Experimental observation of helium dimer, Journal of Chemical Physics 98, 3564–3567 (1993). The bond order count does not include that weak binding mechanism.

The useful surprise is that sharing alone does not determine a chemical bond: the relative quantum phase and the occupancy matter together. The same two shapes explain why adding an Electron to a bonding molecular orbital increases the model's bond order, while adding one to an antibonding molecular orbital decreases it. Molecular orbital theory also accounts for the unpaired Electrons and paramagnetism of molecular oxygen, which its all-paired Lewis structure misses. See Chemistry 2e, 8.4: the molecular orbital description of oxygen.

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