= Molecular orbital theory
{wiki}
<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 https://ocw.mit.edu/courses/5-61-physical-chemistry-fall-2017/ef9da1132ff6233d732a1d6115099467_MIT5_61F17_lec24.pdf[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 $1s$ <atomic orbitals> of the <hydrogen atom> by two <Gaussian functions> centred at $x=\pm a$, with <dimensionless> $x$ and $a>0$:
$$
\phi_A(x)=\pi^{-1/4}e^{-(x+a)^2/2},\qquad
\phi_B(x)=\pi^{-1/4}e^{-(x-a)^2/2},\qquad
S=\int_{-\infty}^{\infty}\phi_A\phi_B\,dx=e^{-a^2}.
$$
The normalized sum and difference are $\psi_\pm=(\phi_A\pm\phi_B)/\sqrt{2(1\pm S)}$, following the same <linear combination of atomic orbitals> algebra. At $a=1$, the <orbital overlap> is $S=e^{-1}\approx0.367879$. At the midpoint, $\phi_A(0)=\phi_B(0)\approx0.455581$, giving
$$
\psi_+(0)\approx0.550879,\qquad |\psi_+(0)|^2\approx0.303468,\qquad
\psi_-(0)=|\psi_-(0)|^2=0.
$$
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>.
\Image[/molecular-orbital-interference.png]
{title=Normalized one-dimensional <Gaussian function> toy: signed <probability amplitudes> add or cancel before their squares give <probability densities>}
{description=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.}
Now fill the two $1s$-derived <molecular orbitals> in the minimal model. The lower <bonding molecular orbital> is labelled $\sigma_g$; the upper <antibonding molecular orbital> is $\sigma_u^*$. 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 $\sigma_g$ first. The subscripts $g$ and $u$ denote <even> and <odd> behaviour under inversion through the midpoint; the star marks an <antibonding molecular orbital>. See https://openstax.org/books/chemistry-2e/pages/8-4-molecular-orbital-theory[Flowers et al., Chemistry 2e, 8.4: occupancy and bond order] and https://ocw.mit.edu/courses/5-61-physical-chemistry-fall-2017/e9f9413400be87f6a0a512e757d52e3c_MIT5_61F17_lec25.pdf[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>, $\mathrm H_2$: two <Electrons> give $(\sigma_g)^2$ and $b=(2-0)/2=1$.
* The <hydrogen> molecular <ion> $\mathrm H_2^+$: one <Electron> gives $(\sigma_g)^1$ and $b=(1-0)/2=1/2$.
* Two neutral <helium> <chemical atoms>, $\mathrm{He}_2$: four <Electrons> give $(\sigma_g)^2(\sigma_u^*)^2$ and $b=(2-2)/2=0$ in this minimal $1s$ model.
\Image[/molecular-orbital-occupancy.png]
{title=Schematic <molecular orbital> occupancy for $\mathrm H_2$, $\mathrm H_2^+$, and $\mathrm{He}_2$: <bond orders> one, one half, and zero}
{description=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.}
The zero <bond order> says this minimal model gives no net <covalent bond> for $\mathrm{He}_2$. 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 https://www.atom.uni-frankfurt.de/publications/files/ZellerPNAS-2016.pdf[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 https://scholarworks.gvsu.edu/chm_articles/6/[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 https://openstax.org/books/chemistry-2e/pages/8-4-molecular-orbital-theory[Chemistry 2e, 8.4: the molecular orbital description of oxygen].
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