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In bent molecular geometry, two neighbours and their central chemical atom form a noncollinear triangle, with an interior bond angle strictly between $0^{\circ }$ and $180^{\circ }$. Water, ${\mathrm{H}}_2\mathrm{O}$, is the standard example. The NIST CCCBDB experimental water entry, internal coordinates, citing Hoy and Bunker (1979) gives the equilibrium H–O–H bond angle as $104.4776^{\circ }$ with quoted uncertainty $0.0019^{\circ }$, usually rounded to $104.5^{\circ }$.

The oxygen in water has two bonding electron domains and two lone pairs. Its ideal electron pair geometry is tetrahedral, while the three atomic nuclei give bent molecular geometry. Three points always lie in a plane; this does not make the four-domain scaffold planar. Nor does the word bent by itself determine an electron domain count: the same atomic shape can arise from other domain arrangements.

In an illustrative model with two equal bond-dipole contributions of magnitude $p$ separated by bond angle $\theta$, their sum of vectors points along the bisector and has magnitude $2p\cos (\theta /2)$. At $180^{\circ }$ it vanishes; at $104.4776^{\circ }$ it is approximately $1.2247p$. This ratio is a geometric calculation, not a prediction of water's measured electric dipole moment. The entire electron density, including contributions represented by lone pairs, matters. Water is a polar molecule; chemical polarity helps explain its interactions with other polar molecules. See McMurry, Organic Chemistry, §2.2: water, lone-pair contributions, and polarity.

A bond dipole represents the separation of positive and negative electric charge associated with a chemical bond. In a neutral two-point charge model put $+q$ at ${\mathbf{r}}_A$ and $-q$ at ${\mathbf{r}}_B$, with $q>0$. Its physical electric dipole moment isIf $B$ is the negative end, the physical electric dipole moment points from $B$ to $A$. The customary crossed-tail chemical arrow points towards the negative, more electronegative end, from $A$ to $B$. It therefore has the opposite direction. State the convention before doing vector addition; use physical electric dipole moments in the interaction potential energy $U=-\mathbf{p}\cdot \mathbf{E}$, where $\mathbf{E}$ is the electric field.^[ref]

For a fixed neutral molecule, its complete electric dipole moment follows from all atomic nuclei and the electron density, not just point charges assigned to sites. If $n(\mathbf{r})$ is the expected number of electrons per unit volume, the corresponding charge density is ${\rho }_e(\mathbf{r})=-en(\mathbf{r})$; the nuclear terms contribute separately. Partitioning that distribution into bond dipoles is a model. Lone pairs and electric polarization within a chemical atom can contribute as well. A measured molecular electric dipole moment cannot uniquely recover every individual bond dipole or partial atomic charge.

A heteronuclear diatomic molecule consisting of carbon and sulfur. Its permanent electric dipole moment permits an electric-dipole rotational spectrum. The linear rigid rotor model relates its rotational constant to its reduced mass and bond length.

Chemical polarity concerns unequal distribution of electron density in a chemical bond or molecule. A polar molecule has a permanent electric dipole moment; the effects of individual bonds must be combined with their spatial arrangement. In a symmetric linear carbon dioxide molecule, opposing bond-dipole contributions cancel. In water, its bent molecular geometry prevents that cancellation, giving a permanent electric dipole moment.^[ref]

The debye is a unit of electric dipole moment: $1\mathrm{D}\approx 3.33564\times 10^{-30}\mathrm{C}\mathrm{m}$.^[ref]

A dielectric is a material in which an electric field produces electric polarization while transport of electric charge is sufficiently small for the intended use. An insulating solid in a capacitor is an example; real dielectrics can still have dielectric loss and nonzero electrical conductivity.

The important distinction is between rearranging bound electric charge and transporting mobile charge carriers through the specimen. Bound electric charges can move a little within a chemical atom or molecule, or an existing electric dipole can turn. Their averaged electric dipole moments produce electric polarization even if there is negligible steady electric current between the electrodes. A changing electric polarization does give a bound current; the absence of steady conduction does not mean that the material is electrically unresponsive.^[ref]

For uniform electric polarization $\mathbf{P}=P\hat{\mathbf{x}}$ in a rectangular slab, polarization density gives zero bound volume charge density in the interior and opposite bound surface charge densities $-P,+P$ on the left and right faces. The electric charge has been redistributed, not created. More generally, the divergence theorem makes the cancellation explicit:Here $\mathbf{n}$ is the outward normal vector on the material boundary. This accounts for both bound volume charge density and bound surface charge density; omitting either can falsely suggest that an initially neutral dielectric acquired net electric charge.

An electrical insulator between capacitor plates supports a useful combination: appreciable electric polarization, little leakage electric current, and tolerance of the intended electric field. This lets a small parallel-plate capacitor store separated electric charge and recoverable capacitor energy. The comparison in dielectric capacitor response explains why specifying fixed electric charge or fixed electric potential difference changes the answer. High relative permittivity alone does not establish good insulation or a large usable energy density; dielectric loss and dielectric breakdown matter as well.

Reduction of an effective electric dipole moment by the opposing local field of neighboring dipoles.

A permanent electric dipole moment gives evidence of separated positive and negative electric charge in an overall neutral molecule. This is a more direct probe of chemical polarity than an electronegativity table. For hydrogen chloride, NIST lists about $1.109$ debye for the lowest vibrational state and $1.093$ debye for the equilibrium value. They describe different averaging conventions, not two contradictory measurements. The NIST WebBook traces the vibrational value $1.1085$ debye to molecular-beam electric-resonance measurements in 1970.^[ref][ref]

How does molecular beam electric resonance reveal an electric dipole moment? A source and apertures form a beam of molecules in vacuum. A first nonuniform electric field, A, selects quantum states through their different Stark effects. In a central region, C, electrodes supply a nearly uniform electric field and applied radiation drives transitions at selected frequencies. A second nonuniform electric field, B, analyzes the outgoing quantum states; a detector measures the rate of arriving molecules. In a transmission-loss arrangement, resonant radiation transfers some molecules to a quantum state rejected by B, so the measured arrival rate dips. The observable is a change in detected molecules as the radiation frequency is scanned, rather than light absorption itself. This general apparatus description is documented in a later experimental thesis; it is not a reconstruction of the exact 1970 HCl instrument.^[ref]

The inference comes from how transition frequencies change with the known electric field. The interaction $-\mathbf{\mu }\cdot \mathbf{E}$ couples the electric dipole moment to the electric field; fitting the quantum mechanics of the relevant rotational states extracts its magnitude. A free rotating molecule need not show a first-order shift proportional to $|\mathbf{E}|$: the weak-field Stark effect can be quadratic. The diagram uses a simplified rigid rotor, with $B$ an energy constant, to illustrate this behavior. It omits the fine details of actual spectra.

Useful controls include scans with the radiation off to measure the baseline arrival rate, repeated scans at several calibrated electric fields, and agreement between independently assigned transitions. These are methodological checks, not claims about undocumented steps of the 1970 experiment. Field calibration and state assignment limit the inferred electric dipole moment. The published 1970 abstract confirms analysis of HCl and DCl beam-resonance spectra and combination with infrared spectroscopy intensities to obtain a distance-dependent electric dipole moment; its full apparatus and calibration record were unavailable here.^[ref]

Even an accurately measured electric dipole moment is not an absolute partial atomic charge. Replacing a hydrogen chloride molecule by two point electric charges $\pm q$ separated by bond length $r$ gives $|\mu |=|q|r$. Using $1.109$ debye and $r=1.27455\times 10^{-10}\mathrm{m}$ gives $|q|\approx 0.181$ times the elementary electric charge. This is an effective two-point model, combining a vibrationally averaged electric dipole moment with an equilibrium bond length, not a unique partition of the actual electron density or a measured “18.1 percent ionic bond”.^[ref]

For a molecule with several chemical bonds, molecular geometry can cancel opposing contributions to the total electric dipole moment. Carbon dioxide illustrates why an overall nonpolar molecule need not have unpolarized individual chemical bonds. An electronegativity comparison helps anticipate local bond polarization; the measured molecular electric dipole moment tests the entire electric charge distribution.

The dynamic electric polarizability relates the induced electric dipole moment at a driving frequency to the applied electric field at that frequency in linear response theory. A complex coefficient can encode both a dispersive phase and absorption. For a charged one-dimensional quantum harmonic oscillator with charge magnitude $e$, mass $m$, and natural frequency ${\omega }_0$, the undamped response isOne derivation uses the oscillator electric-dipole selection rule and first-order time-dependent perturbation theory. The same result follows from the classical Lorentz oscillator model: $m\ddot{z}+m{\omega }_0^{2}z=-eE$ and $d=-ez$ imply $\ddot{d}+{\omega }_0^{2}d=e^{2}E/m$. At exact resonance there is no bounded periodic response without damping; a finite switch-on produces secular growth rather than a finite steady-state value of the pole expression.

A finite electric dipole consists of equal and opposite electric charges separated by a nonzero displacement. It has charge neutrality but a nonzero electric dipole moment $\mathbf{p}=q\mathbf{d}$. An ideal point electric dipole is the mathematical limit $d=|\mathbf{d}|\to 0$ with $\mathbf{p}$ fixed, so $q$ must increase as $1/d$. Shrinking the separation while holding $q$ fixed instead makes the electric dipole moment vanish. The ideal electric dipole describes an exterior electric field; it does not reproduce the two near-charge singularities or the internal structure of a molecule. ^[ref].

Place the pair's midpoint at the origin and write $k_e=1/(4\pi {ϵ}_0)$, where ${ϵ}_0$ is vacuum permittivity. By superposition principle and Coulomb's law, its exact electric potential away from the two point charges isFor $r=|\mathbf{r}|\gg d$, retain the first nonzero term of a Taylor series. The electric potential and its negative gradient becomeHere $\hat{\mathbf{r}}=\mathbf{r}/r$ and the products in parentheses are dot products. These are exact exterior formulas for the ideal electric dipole and distant approximations for a finite pair. The electric field on the outer axis points along $\mathbf{p}$ with magnitude $2k_{e}p/r^3$; on the perpendicular bisector it points opposite $\mathbf{p}$ with magnitude $k_{e}p/r^3$. This directional structure is what a scalar total electric charge misses. ^[ref].

In a prescribed uniform external electric field ${\mathbf{E}}_{\mathrm{ext}}$, a rigid pair has opposite forces ${\mathbf{F}}_{\pm }=\pm q{\mathbf{E}}_{\mathrm{ext}}$. Its net force is zero, but summing their moments about the midpoint gives the torqueThe cross product makes the torque perpendicular to both the electric dipole moment and the external electric field. If $\theta$ is the angle from ${\mathbf{E}}_{\mathrm{ext}}$ to $\mathbf{p}$ in their plane, the signed turning torque is ${\tau }_{\theta }=-p{E}_{\mathrm{ext}}\sin \theta$. It tends to reduce a small positive $\theta$. ^[ref].

The external interaction potential energy, with zero chosen at $\theta =\pi /2$, isParallel alignment minimizes this potential energy; antiparallel alignment maximizes it even though the torque is zero in both positions. This omits the pair's constant internal binding potential energy and assumes a fixed permanent electric dipole moment. ^[ref].

A nonuniform external electric field can exert a net force on a neutral electric dipole. Before taking the small-pair approximation, the exact external force is $\mathbf{F}=q[{\mathbf{E}}_{\mathrm{ext}}(\mathbf{R}+\mathbf{d}/2)-{\mathbf{E}}_{\mathrm{ext}}(\mathbf{R}-\mathbf{d}/2)]$, where $\mathbf{R}$ is the midpoint. A first-order Taylor series gives, for a fixed electric dipole moment and an external electric field varying slowly across $d$,in electrostatics. The last equality uses the vanishing curl of the external electric field, with $\mathbf{p}$ held fixed while differentiating. Thus uniform external electric fields turn a misaligned electric dipole, whereas gradients can also move its midpoint. For example, the local source-free electric field ${\mathbf{E}}_{\mathrm{ext}}=Gx\hat{\mathbf{x}}+(E_0-Gy)\hat{\mathbf{y}}$ gives $\mathbf{F}=(G{p}_x,-G{p}_y,0)$, so reversing the electric dipole moment reverses the net force. This stipulated local model has zero divergence and zero curl.

A motivating experiment is Filsinger and colleagues' study of electrostatic deflection of polar molecules. Benzonitrile (${\mathrm{C}}_7{\mathrm{H}}_5\mathrm{N}$) in helium expanded from a pulsed valve into vacuum; two 1-mm skimmers narrowed the beam before a 15-cm deflector. A grounded trough and a high-voltage rod produced a vertical electric field gradient. Scanning a focused laser vertically downstream and recording benzonitrile ions from photoionization measured the beam's relative density. With voltage off, the profile spanned about 2 mm; applying voltage shifted and broadened it upwards. At 10 kV, many molecules lay outside the original profile. ^[ref].

Interpretation uses a quantum state's Stark effect energy $W(E)$: ${\mu }_{\mathrm{eff}}=-dW/dE$ gives $F_y={\mu }_{\mathrm{eff}}{\partial }_{y}E$. States whose energy falls with increasing electric field seek stronger fields. The voltage-off profile controls the mechanical aperture. Inferring state populations needs calculated trajectories, speed distributions and a rotational temperature model; profile displacement alone does not measure a permanent electric dipole moment uniquely. This is an electric Stark effect experiment, with photoionization occurring after deflection. ^[ref].

For the symmetric finite pair, the first omitted electric multipole expansion term after the electric dipole is third order, so typical relative errors scale as $(d/r{)}^2$. This stronger error estimate need not hold for an arbitrary neutral molecule with a nonzero electric quadrupole moment. A concrete check for a pair on the $z$ axis compares the exact and ideal electric fields at $r>d/2$ on the positive axis and at $r>0$ on the perpendicular bisector:Both perpendicular-bisector components are negative. At $r=5d$, the ratios are approximately $1.0203$ and $0.9852$: errors of $+2.03\%$ and $-1.48\%$ relative to the ideal electric field. The condition $r\gg d$ concerns observing the pair's own electric field; the separate condition for its external force approximation concerns variation of the applied electric field across the pair.

An ideal electric dipole at height $a$ above a grounded plane has an image electric dipole moment with reversed tangential component and unchanged normal component. If $\theta$ is its angle to the normal, its conductor-response potential energy is the displayed expression. The factor one half in $U=-\mathbf{p}\cdot {\mathbf{E}}_{\mathrm{im}}/2$ accounts for induced surface charge. Therefore its downward force magnitude is $3p^2(1+{\cos }^2\theta )/(64\pi {ϵ}_0{a}^4)$, and its orienting torque follows from $-{\partial }_{\theta }U$. The two normal orientations are stable against tilting; the parallel orientation is unstable. The net induced plane electric charge is zero for a neutral source, although its local surface charge density is nonzero.

For a localized charge distribution in an external field varying little across its extent, expand the external scalar potential about a reference point $\mathbf{R}$. The constant term couples to total charge; the first spatially varying term gives the electric-dipole interaction $-\mathbf{d}\cdot \mathbf{E}(\mathbf{R},t)$, where $\mathbf{d}$ is the electric dipole moment. Higher spatial multipoles are neglected.

For a passing charge at minimum distance $b$, the atomic extent must be small compared with $b$. For a plane electromagnetic wave, the analogous long-wavelength condition is $ka\ll 1$, as in the electric-dipole approximation for radiation. A short interaction time and a spatially uniform field across the atom are different assumptions; neither implies the other.

At distances large compared with the separation of a neutral charge pair, its electric field has the displayed leading form in terms of the electric dipole moment $\mathbf{p}$. Measure the polar angle from $\mathbf{p}$. The radial and angular components are $E_r=2p\cos \theta /(4\pi {ϵ}_0{r}^3)$ and $E_{\theta }=p\sin \theta /(4\pi {ϵ}_0{r}^3)$, giving $|\mathbf{E}|=p\sqrt{1+3{\cos }^2\theta }/(4\pi {ϵ}_0{r}^3)$. At fixed radius the polar field magnitude is twice the equatorial one. This leading neutral-pair field falls as $r^{-3}$, faster than the $r^{-2}$ electric field of an isolated point charge.

The electric dipole moment measures the first spatial moment of signed electric charge. For localized point charges $q_i$ at positions ${\mathbf{r}}_i$, or an integrable charge density $\rho (\mathbf{r})$ with a finite first moment, define^[ref].

The electric dipole moment is a vector with SI units $\mathrm{C}\mathrm{m}$: coulombs times length in metres. Opposite electric charges can cancel in $Q$ while contributing in the same direction to $\mathbf{p}$. For a neutral pair with positive electric charge $q>0$ at ${\mathbf{r}}_{+}$ and negative electric charge $-q$ at ${\mathbf{r}}_{-}$,Thus the electric dipole moment points from negative to positive electric charge, not along the attraction between the two point charges. ^[ref].

Changing the coordinate origin changes the position vectors, not the physical electric charge distribution. If the new origin is at $\mathbf{a}$ relative to the old origin, ${\mathbf{r}}_i^{\prime }={\mathbf{r}}_i-\mathbf{a}$, so direct substitution givesConsequently charge neutrality makes the electric dipole moment independent of the origin. For a system with nonzero total electric charge, always specify the origin: a single point charge at the old origin has $\mathbf{p}=0$ but ${\mathbf{p}}^{\prime }=-Q\mathbf{a}$. This is coordinate bookkeeping, not a newly created electric dipole. The full electric potential stays the same at the same physical point; the terms of its electric multipole expansion and the observation coordinates change together.

As a stipulated numerical example, put $+2\mathrm{nC}$ at $(1,0,0)\mathrm{mm}$ and $-2\mathrm{nC}$ at $(-1,0,0)\mathrm{mm}$. Their total electric charge is zero and $\mathbf{p}=(4\times 10^{-12},0,0)\mathrm{C}\mathrm{m}$. Translating the origin by $(3,-2,0)\mathrm{mm}$ leaves this electric dipole moment unchanged. Adding $+1\mathrm{nC}$ at the old origin instead gives $Q=1\mathrm{nC}$ and ${\mathbf{p}}^{\prime }=(1,2,0)\times 10^{-12}\mathrm{C}\mathrm{m}$. Distinguishing these two cases prevents the mistaken claim that every electric dipole moment is independent of the origin.

The usefulness of the electric dipole moment is compression: many internal electric charges determine one vector governing the leading distant electric potential of a neutral localized object when $\mathbf{p}\ne 0$. ^[ref]. A polar molecule need not literally contain two point charges; its atomic nuclei and electron density collectively give an electric dipole moment. For example, the chemical polarity of water reflects its bent molecular geometry, whereas the symmetric linear molecular geometry of carbon dioxide makes its permanent electric dipole moment vanish. ^[ref]. A collection of such electric dipole moments per unit volume gives electric polarization. ^[ref].

An electric dipole transition is a transition driven, in the long-wavelength approximation, by the electric-dipole interaction $H_{\mathrm{int}}=-\mathbf{d}\cdot \mathbf{E}(t)$. Its leading amplitude is proportional to a matrix element of the electric dipole moment between the initial and final states. A vanishing element gives an electric-dipole selection rule; a nonzero element alone does not ensure a large probability, since the drive frequency, duration and strength also matter. For a charged quantum harmonic oscillator driven along its axis, $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ gives the single-quantum rule $\Delta n=\pm 1$.

The matrix element of the electric dipole moment operator between two quantum states. Electric-dipole transition strength is proportional to its squared magnitude; molecular symmetry can force every Cartesian component to vanish.

For $H=p^2/(2m)+k{x}^2/2-qEx$, completing the square translates the well by $x_0=qE/k$ and shifts every energy by $-q^2{E}^2/(2k)$. The wavefunction of the ground state is translated without changing its width. Consequently the induced electric dipole moment is exactly linear in $E$, even when a low-order perturbative approximation to the state is inaccurate.

Electric polarizability relates an induced electric dipole moment to an applied electric field. In a weak-field model with linearity, ${\mathbf{d}}_{\mathrm{ind}}=\mathbf{\alpha }\mathbf{E}$. A tensor response allows different electric field directions to induce electric dipole moments in different directions. A response that changes with molecular rotation enables rotational Raman spectroscopy.

An electric quadrupole is a charge distribution whose total charge and electric dipole moment vanish while its second charge moment does not. Its potential decays as $r^{-3}$.

The trace-free electric quadrupole moment is a symmetric tensor measuring the second-order term in an electric multipole expansion. For a localized charge distribution its contribution to the far-field electrostatic potential is $Q_{ij}{\hat{r}}_i{\hat{r}}_j/(8\pi {ϵ}_0{r}^3)$, with summation over indices. The moment may be nonzero even when the total electric charge or electric dipole moment is nonzero; it should not be confused with an electric quadrupole whose lower moments vanish. Changing the origin generally changes the tensor when lower moments are nonzero.