= Covalent bond
{wiki}
A <covalent bond> joins <chemical atoms> through shared <Electrons>. In the familiar two-centre description, one <bond pair> contributes to a connection between two <atomic nuclei>. Sharing means that the <Electrons> belong to the combined arrangement and interact with both <atomic nuclei>; it does not mean that identifiable dots shuttle along drawn paths. The https://goldbook.iupac.org/terms/view/C01384[IUPAC Gold Book, covalent bond]{ref} emphasizes the concentration of <Electrons> between <atomic nuclei> and the resulting attraction and characteristic separation. The pair picture is especially useful for simple <molecules>, but is not a universal description of every <covalent bond>.
A <Lewis structure> turns this picture into practical bookkeeping. Count the <valence electrons>, indicate which <chemical atoms> are connected, and allocate two <Electrons> to each line. A <bond pair> is shared; a <lone pair> is assigned to one <chemical atom> without connecting it to another. A double line represents two <bond pairs>. Dots and lines record allocations, rather than positions, trajectories, or a measured three-dimensional arrangement. https://openstax.org/books/chemistry-2e/pages/7-3-lewis-symbols-and-structures[Flowers et al., Chemistry 2e, Lewis symbols and structures]{ref}
For <molecular hydrogen>, $\mathrm H_2$, each neutral <hydrogen> <chemical atom> supplies one <valence electron>. Thus $1+1=2$ <valence electrons> make one <bond pair>, represented by $\mathrm{H-H}$, with no <lone pairs>. Each <hydrogen> counts the same shared pair when checking its two-electron arrangement. The <molecule> still contains two <Electrons>, not four: counting a pair around both endpoints does not create extra <Electrons>.
For <water>, $\mathrm H_2\mathrm O$, neutral <oxygen> supplies six <valence electrons> and the two <hydrogen> <chemical atoms> supply one each: $6+1+1=8$. Two O–H connections consume four <Electrons> as two <bond pairs>. The remaining $8-4=4$ <Electrons> form two <lone pairs> on <oxygen>. Around <oxygen>, the two <bond pairs> and two <lone pairs> count as eight <Electrons>; each <hydrogen> counts two. The whole <water> <molecule> has ten <Electrons>, because two additional <oxygen> <Electrons> are omitted from this <valence electron> drawing.
For <ammonia>, $\mathrm{NH}_3$, the total is $5+3\times1=8$ <valence electrons>: five from <nitrogen> and three from <hydrogen>. Three N–H connections use six <Electrons> as three <bond pairs>, leaving two <Electrons> as one <lone pair> on <nitrogen>. Thus <water> and <ammonia> have the same total of eight <valence electrons> but different allocations: two plus two pairs versus three plus one. This small calculation already distinguishes their connectivity without a detailed <quantum mechanics> calculation.
\Image[/lewis-electron-pair-counts.png]
{title=<Lewis structures> count <bond pairs> and <lone pairs> in <molecular hydrogen>, <water>, <ammonia>, and <molecular oxygen>; schematic layouts do not specify molecular shapes}
For <molecular oxygen>, $\mathrm O_2$, $6+6=12$ <valence electrons> give the conventional $\mathrm{O=O}$ <Lewis structure>. Its two <bond pairs> use four <Electrons>, leaving eight as four <lone pairs>, two on each <oxygen>. Each <oxygen> consequently counts eight <Electrons> locally. However, actual <molecular oxygen> has two unpaired <Electrons> and exhibits <paramagnetism>: it is attracted into a stronger <magnetic field>. The all-paired <Lewis structure> cannot capture this observation. <Molecular orbital theory> describes the unpaired <Electrons>; the successful counting diagram alone is insufficient. https://openstax.org/books/chemistry-2e/pages/8-4-molecular-orbital-theory[Flowers et al., Chemistry 2e, molecular orbital theory]{ref}
Why does sharing produce a stable separation? For <molecular hydrogen>, the electronic arrangement responds as the <atomic nuclei> approach. Their interactions can lower the <energy>, but pushing them too close raises it. A curve of <potential energy> against separation therefore has a <local minimum>, rather than falling forever. Attraction and repulsion balance there. This is a statement about the combined system, not a claim that stationary <Electrons> sit between otherwise motionless <atomic nuclei>. <Chemistry 2e by Paul Flowers and others/7.2 Covalent Bonding>{ref}
The following <Morse potential> makes those features calculable. Set $x=r/r_e$, where $r$ is separation and $r_e$ its equilibrium value, and let $D$ be the well depth. With the self-chosen parameter $a=2$, define the <dimensionless quantities>
$$
u(x)=\frac{V(r)}{D}=\left(1-e^{-2(x-1)}\right)^2-1,
\qquad f(x)=\frac{F(r)r_e}{D}=-\frac{du}{dx}=4\left(e^{-4(x-1)}-e^{-2(x-1)}\right).
$$
Here $F=-dV/dr$ is the net separating <force>, obtained from the <derivative> of the <potential energy>. Positive $f$ pushes the <chemical atoms> apart; negative $f$ draws them together. These are illustrative curves with $r_e=D=1$ in chosen units, not measured <molecular hydrogen> data or fitted parameters.
\Image[/covalent-bond-energy-model.png]
{title=Illustrative <dimensionless> <Morse potential> and its <force>: a stable <local minimum>, short-distance repulsion, and separation toward zero <potential energy>}
At $x=1$, $u=-1$ and $f=0$. The <second derivative> is $u''(1)=8>0$, confirming a <local minimum>. At $x=0.5$, $u=(1-e)^2-1\approx1.9525$; at $x=1.5$, $u=(1-e^{-1})^2-1\approx-0.6004$. As $x$ increases without bound, $u$ approaches zero from below. Thus moving from the well bottom to separated <chemical atoms> requires an increase of $D$ in this model's <potential energy>. An actual <quantum mechanics> dissociation threshold also depends on the initial motion of the <atomic nuclei>, so the well depth alone is not every measured dissociation <energy>.
The <force> panel separates $f$ into a positive term $4e^{-4(x-1)}$ and a negative term $-4e^{-2(x-1)}$. At $x=1$ they are $+4$ and $-4$, cancelling exactly; nearer separations favour the positive term. These are algebraic components of the chosen <Morse potential>, not separately measured microscopic <forces>. Their usefulness is to show how a balance arises from the slope of one <energy> curve.
The beauty of the pair picture is its economy: a few marks summarize connectivity and constrain what a proposed <molecule> can contain. Its historical anchor is Gilbert N. Lewis's 1916 paper https://doi.org/10.1021/ja02261a002[The Atom and the Molecule, Journal of the American Chemical Society 38, 762–785]{ref}, also available as an https://zenodo.org/records/1429068[archived original paper]{ref}. <Lewis structures> remain useful because they answer counting questions quickly; <quantum mechanics> supplies the richer account when <energy>, shape, or unpaired <Electrons> matter.
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