= Molecular dipole cancellation
{title2=$\mathbf p_{\rm mol}=\sum_b\mathbf p_b$}
<Molecular dipole cancellation> occurs when nonzero contributions to a <molecule>'s <electric dipole moment> cancel under <vector addition>. In a <bond dipole> model, use $\mathbf p_{\rm mol}=\sum_b\mathbf p_b$; adding magnitudes instead discards the <molecular geometry>. A symmetric <linear molecule> with two equivalent opposed <bond dipoles> has $\mathbf p_1=p_0\widehat{\mathbf x}$ and $\mathbf p_2=-p_0\widehat{\mathbf x}$, hence zero total <electric dipole moment> even though each <chemical bond> is polarized. <Carbon dioxide> provides the familiar molecular example; <water>'s <bent molecular geometry> permits a nonzero total.<Chemistry 2e>[§7.6, Figure 7.27 and accompanying paragraphs: cancellation in carbon dioxide, noncancellation in water]{ref}
For an original geometry calculation, give a fictitious three-site model $AB_2$ a central effective <electric charge> $+2q$ at the origin and two terminal <electric charges> $-q$ at
$$
\mathbf r_1=r\bigl(-\sin(\theta/2),\cos(\theta/2)\bigr),\qquad
\mathbf r_2=r\bigl(\sin(\theta/2),\cos(\theta/2)\bigr),\qquad 0<\theta\leq\pi.
$$
Here $r$ is the stipulated <bond length>, $\theta$ the <angle> between the two connections, and $q>0$ is independent of any <electronegativity> scale. Equivalently, split the central $+2q$ into two colocated $+q$ contributions, one for each neutral <bond dipole>. With $p_0=qr$, the physical <electric dipole moments> point from the terminal sites towards the central site:
$$
\begin{aligned}
\mathbf p_1&=p_0\bigl(\sin(\theta/2),-\cos(\theta/2)\bigr),\\
\mathbf p_2&=p_0\bigl(-\sin(\theta/2),-\cos(\theta/2)\bigr),\\
\mathbf p_{\rm mol}&=\mathbf p_1+\mathbf p_2
=\bigl(0,-2p_0\cos(\theta/2)\bigr).
\end{aligned}
$$
The sideways components cancel; the components along the <angle bisector> add. Thus $|\mathbf p_{\rm mol}|=2p_0\cos(\theta/2)$. At $\theta=180^\circ$ the result is zero; at the stipulated $\theta=120^\circ$ it is exactly $p_0$. This is an angle example for a model, not the measured angle or <electric dipole moment> of <water>.
\Image[/molecular-dipole-components-and-cancellation.png]
{title=<Vector addition> of physical <bond dipoles> in a stipulated neutral three-site model: cancellation at $180^\circ$, a bisector resultant at $120^\circ$}
{id=image-molecular-dipole-components-and-cancellation}
{description=Central effective <electric charge> $+2q$, terminal <electric charges> $-q$, and equal <bond lengths> $r$. Arrows point from negative to positive <electric charge>. The lower panels translate the <bond dipoles> to a common origin and show their components; they are vectors, not electron trajectories or extra bonds.}
Opposition alone is insufficient: a <linear molecule> with unequal opposed <bond dipoles> of magnitudes $p_1$ and $p_2$ retains magnitude $|p_1-p_2|$. Conversely, symmetry can enforce exact cancellation beyond a particular <bond dipole> partition. If the complete <charge density> invariant under <point reflection> about the origin satisfies $\rho(\mathbf r)=\rho(-\mathbf r)$, contributions to $\int\mathbf r\rho(\mathbf r)\,d^3r$ cancel in pairs. That <proof> uses the whole <electric charge> distribution, including <lone pairs> and <atomic nuclei>. It explains why molecular <symmetry> is stronger evidence for zero permanent <electric dipole moment> than a rule based only on tabulated <electronegativity> differences.
A permanent molecular <electric dipole moment> is specified relative to the <molecule>'s orientation. An ensemble with random orientations can have zero mean laboratory <vector> while each <molecule> is polar. That orientational average differs from internal <molecular dipole cancellation>. The <Stark effect> and <molecular beam electric resonance> probe molecular <electric dipole moments> through state-dependent responses to an <electric field>.
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