AAMPLITUDE
Wavefunction
h = cₛφₛ + cₚφₚ
Positive hNegative h
Two signs of the same orbital. Color is wavefunction sign, not electric charge.
SIGNED AMPLITUDE · h
A laboratory for orbital character
Shape comes
from interference.
Change the balance of s and p character. See how adding wavefunctions creates directionality—and why adding their densities gives a different picture. Then explore a separate 3s/3p/3d basis.
THE EXPERIMENT
One basis.
Three ways to look.
h = √f · 2s + σ√(1 − f) · 2p
f is the s fraction; σ is the relative sign.01
Mix one orbital
A coefficient scan, not a time evolution.
50% s / 50% p
Pure p · 0% sPure s · 100% s
Character is the squared coefficient: f = |cₛ|². These labels describe one orbital. Explore whole sets below.
h = 0.7071 · 2s + 0.7071 · 2p_z
BCOHERENT
Hybrid density
ρ = |cₛφₛ + cₚφₚ|²
Nonnegative |h|²
Square the combined wavefunction. The shape includes constructive and destructive interference.
HYBRID DENSITY · |h|²
CINCOHERENT
Density-only mixture
ρₘᵢₓ = f|φₛ|² + (1 − f)|φₚ|²
Nonnegative ρₘᵢₓ
Add the component densities. No interference term, so the two p directions remain equivalent.
DENSITY MIXTURE · ρₘᵢₓ
Drag any 3D view to rotate all views; use its arrow keys to rotate or + / − to zoom. The three comparison views share one spatial scale. A and B use the same boundary: |h| = √cutoff and |h|² = cutoff.
Cross-sections through the nucleus. Horizontal z, vertical x; y = 0. Each panel spans −10 to +10 a₀. Density colors share a fixed square-root scale; these are slices, not projections.
CHECK A POINT
Find the interference.
Point to a cross-section, or focus it and use the arrow keys. The same coordinate is sampled in all three views.
Hybrid density
=Density mixture
+Interference term
The cross term can be positive or negative. The total probability density stays nonnegative.
TRY THIS
At 50% s character, reverse the relative sign. Predict which images change before you switch. Then try pure s or pure p: what happens to the interference term?
Reveal the explanation
Reversing only the p contribution reflects the hybrid density along the selected p axis. The density-only mixture is unchanged because it contains squared component amplitudes. At either pure endpoint, one coefficient vanishes and the interference term is zero everywhere.
02
Build the complete set
Four basis orbitals in. Four basis orbitals out.
Positive amplitudeNegative amplitudeUnused p orbitals: wireframe
Select an individual orbital to inspect overlapping lobes. The 2s basis function retains its radial node; hybrid nodes change through interference. A cutaway can reveal interior surfaces.
The coefficients
Each row is one normalized orbital in the basis (2s, 2pₓ, 2pᵧ, 2p_z). Values below are rounded for display.
| Orbital | 2s | 2pₓ | 2pᵧ | 2p_z |
|---|
WHAT STAYS THE SAME?A new basis.
A new basis.
The same space.
The four orbitals remain orthonormal. At every position, summing their squared amplitudes gives the same result as summing the original four basis densities:
Σᵢ |hᵢ|² = |2s|² + |2pₓ|² + |2pᵧ|² + |2p_z|²
Here hᵢ includes any unused p orbitals. This unweighted identity is not a calculation of a molecule’s electron density; electron occupations and a molecular wavefunction would also be needed.
03
Bring d orbitals into the basis
Idealized constructions, with the full basis accounted for.
Explore square-planar, trigonal-bipyramidal, and octahedral coefficient sets. This section uses a separate, normalized 3s / 3p / 3d hydrogenic basis at Z = 1. Its ruler and density cutoffs differ from the 2s/2p views above.
A construction is not a bonding diagnosis.
These combinations illustrate basis changes. A molecular geometry alone does not establish d-orbital participation. In particular, sp³d and sp³d² labels are not evidence that main-group molecules such as PF₅ or SF₆ bond through substantial d hybridization. For real transition-metal compounds, metal–ligand interactions, orbital energies, and occupations must be considered, for example with molecular-orbital or ligand-field models.
Preparing analytic cross-sections for the selected orbital.
AAMPLITUDE
Wavefunction
h = √fₛ S + √fₚ P + √fd D
Positive hNegative h
Combine signed amplitudes. Color shows wavefunction sign, not electric charge.
SIGNED AMPLITUDE · h
BCOHERENT
Hybrid density
ρ = |√fₛ S + √fₚ P + √fd D|²
Nonnegative |h|²
Square the combined wavefunction. Interference changes the density. A and B share exactly the same boundary.
HYBRID DENSITY · |h|²
CINCOHERENT
Density-only mixture
ρₘᵢₓ = fₛ|S|² + fₚ|P|² + fd|D|²
Nonnegative ρₘᵢₓ
Add the s-, p-, and d-component densities. Remove interference between these groups; keep the chosen shape within each group.
DENSITY MIXTURE · ρₘᵢₓ
The slices sample the analytic functions, independently of the 3D mesh.
Preparing the three d-section views.
Each panel overlays separate orbital surfaces, not their summed density or wavefunction. Unused orbitals appear as wireframes.
Drag any view or use its arrow keys to rotate all views; + / − zoom. Each d-section panel has the same 5 a₀ scale bar. A shows h = ±√τ; B and C show density = τ. This cutoff is not an enclosed-probability percentage.
CHECK THE SAME POINT
Resolve the interference.
Point to any slice or tap it. With a slice focused, arrow keys move 0.5 a₀; Shift + arrows move 2 a₀; Home returns to the nucleus. Changing the plane also returns the probe to the nucleus.
Signed h: a₀−3/2
Hybrid density = density mixture + interference
Hybrid density · ρ
Density mixture · ρₘᵢₓ
Interference · Δρ
s–p · 2AₛAₚ
s–d · 2AₛAd
p–d · 2AₚAd
All density and interference values are in a₀⁻³. Aₛ, Aₚ, and Ad are the coefficient-weighted component amplitudes: √fₛ S, √fₚ P, and √fd D. The three terms sum to Δρ before display rounding. Probe values evaluate the analytic functions at the marked point; color saturation does not clip these numbers.
WHAT IS BEING MIXED?
Three components. Three cross terms.
S, P, and D are the fixed, normalized components shown in the character controls; fₛ + fₚ + fd = 1. The mixture represents an incoherent mixture of these component states. It is not another rendering of the same pure hybrid state.
ρ − ρₘᵢₓ = 2√(fₛfₚ) SP + 2√(fₛfd) SD + 2√(fₚfd) PD
The functions here are real. The difference can be positive or negative locally; both densities stay nonnegative and integrate to one for a selected normalized orbital.
TRY THIS
Choose an individual orbital and move any character slider to 100%. B and C now agree everywhere. This also holds for a pure p or d component that combines several basis functions.
Within P or D, the selected basis functions still interfere. C removes only the interference between s, p, and d components.
READ THE CONSTRUCTION
Four equivalent hybrids
Directions describe the symmetry of the coefficient construction. They are not predicted bond angles or guaranteed lobe maxima at every radius: the 3s and 3p radial factors change sign.
KEEP THE WHOLE SPACE
Nine in. Nine out.
Including all unused orbitals gives an orthonormal nine-function basis. Its unweighted sum of squared amplitudes equals that of the original 3s, three 3p, and five 3d functions.
This identity describes a basis transformation, not nine electrons or a molecule’s occupied orbitals.
Inspect all nine coefficient rows
The nine rows belong to the selected reference set.
Columns use normalized 3s, 3p, and real 3d functions. Character is the sum of squared coefficients within each s, p, or d group. Values here are rounded; calculations retain full precision.
| Orbital | 3s | 3pₓ | 3pᵧ | 3p_z | 3dₓᵧ | 3dₓz | 3dᵧz | 3dₓ²₋ᵧ² | 3d_z² |
|---|
Model, radial nodes, and chemical interpretation
The nine functions are real hydrogenic n = 3 states at Z = 1 in the nonrelativistic, fixed-nucleus model. They are mathematically degenerate in this model; that does not establish equal orbital energies in a many-electron atom or molecule. This section does not model the 3d/4s/4p shell ordering of a transition metal or distinguish inner- and outer-orbital bonding schemes.
The 3s basis function has two radial nodes, 3p has one at r = 6 a₀, and 3d has none. These radial factors are retained, so the full surfaces can differ markedly from textbook angular-lobe cartoons. Hybrid nodes change by interference. The finite triangle mesh can miss very small features near a node or cutoff.
The sp³d and sp³d² coefficients follow the constructions tabulated in Wannier90’s official documentation. The display combines those coefficients with the hydrogenic n = 3 radial functions; it is not a Wannier-function calculation. The square-planar dsp² rows use the orthonormal combination of s, pₓ, pᵧ, and dₓ²₋ᵧ².
For the distinction between hypercoordination and d-hybrid bonding, see Noury, Silvi, and Gillespie, “Chemical Bonding in Hypervalent Molecules: Is the Octet Rule Relevant?”. Adding d-type basis functions in a calculation is also not, by itself, evidence for substantial occupied atomic d-orbital participation.
04 / MODEL & LIMITS
Know what you’re looking at.
These are mathematical combinations of normalized hydrogenic orbitals. They illustrate character and interference, not a computed bonding orbital of carbon or a reaction trajectory.
2s/2p basis functions and units
Fixed nucleus, one electron, Z = 1; nonrelativistic Coulomb model. In coordinates measured in a₀, with r = √(x² + y² + z²) and C = 1/(4√(2π)):
φ₂ₛ = C(2 − r)e−r/2
φ₂ₚₓ = Cxe−r/2, and likewise for y and z.
Wavefunctions have units a₀−3/2; densities have units a₀−3. The pure 2s radial node is at r = 2 a₀. Hybridization changes the zero set through interference.
What the surfaces and colors mean
The 3D surfaces use the selected density cutoff. They do not enclose a prescribed percentage of probability and are not hard orbital edges. The three comparison panels use the same cutoff and spatial scale. The larger complete-set view has its own visible scale bar. In A, the same boundary is split by positive and negative wavefunction sign. B and C show nonnegative densities. The finite triangle mesh approximates curved boundaries and can miss very small features. The slices and probe evaluate the analytic functions independently of the mesh.
Slice density colors use √ρ on one fixed scale. The signed-amplitude slice uses a fixed diverging scale. The cutaway affects only the 3D rendering, not the orbital or the full cross-sections.
Single-orbital composition and complete sets
The continuous slider changes one orbital’s coefficients. The complete-set presets construct orthonormal bases with fixed compositions. Four equivalent orthogonal s/p hybrids have 25% s character each; varying all four equally at fixed tetrahedral directions would generally break orthogonality.
The label sp² describes a hybrid constructed within an s-plus-two-p subspace. Its p component can be expressed as a single oriented p function. A single sp²-character orbital alone does not specify a trigonal set.
Sources and browser use
- Oregon State: hydrogen radial functions and spherical harmonics
- TU Braunschweig: normalized hybrid-orbital constructions
- Related prior art: Falstad’s orbital viewer
- Related prior art: The Hybridization Explorer
This display runs in your browser with bundled code. It sends no measurements, has no analytics, and does not save your settings. External reference links open only when selected. Three.js is distributed under its included MIT license.