Hydrogenic basis · coherent superposition

Orbital Superposition Studio

Start with the quantum rules that make orbital mixing meaningful, then build a linear combination such as 1s + 2p, 2px − 2py, or 2s + 0.5*3dz2. The app evaluates the signed wavefunction ψ and visualizes interference. Bare p means pz; bare d means dz².

Before mixing orbitals

Pauli constrains occupation, not superposition.

A single electron may occupy any normalized superposition of basis orbitals. The exclusion principle enters when several electrons are assigned spin-orbitals. For the orbital-box diagrams, antisymmetry argument and filling examples, use the dedicated primer.

Read the Pauli primer →

Build one state

Expression

Syntax: signed sums with optional scalar coefficients. Examples: 1s + 2p; 2px - 2py; 2s + .5*3dz2.

Add basis terms

Surface level Half-width a₀ Grid resolution

Python parity: extent = 18 a₀, grid = 64³, level = 8% of max|ψ|.

Interpretation. This is a linear combination of normalized hydrogenic atomic orbitals at one center. Calling the result a “hybrid orbital” is appropriate only when the chosen basis and coefficients represent a hybridization model. In particular, 1s + 2p is a superposition, not a conventional valence hybrid.

3D signed level surface

Drag to rotate · wheel to zoom · orange ψ > 0 · blue ψ < 0

x–z slice at y = 0

Brightness tracks |ψ|; phase is encoded by sign. The same extent and grid are used as the 3D field.

positive phasenegative phaseatomic units: Z = 1, a₀ = 1

Preset library

Save states built above, then add any saved state into the current expression. Presets are stored locally in this browser.

Local moments · exchange model

d¹ orbital lattice and magnetic order

This section separates orbital shape from magnetic order. Each rendered site carries one illustrative local 3d basis function and one spin-½ moment. The spin pattern is an input to a nearest-neighbour Heisenberg toy model, not a prediction from the orbital picture alone.

Local d¹ orbital basis function

A real d basis function is shown at each site. A real crystal-field eigenstate may instead be a linear combination of these functions.

Lattice Rendered cluster Spin texture Exchange J · meV

J < 0 favours parallelJ > 0 favours antiparallel

Weight bond brightness by orbital-direction overlap proxy

This proxy only visualizes lobe orientation. It does not calculate a hopping integral or determine J.

Local spinS = ½

Spin-only μeff≈ 1.73 μB

Sites shown—

NN bonds shown—

Mean sᵢsⱼ—

Evis / site—

H = J Σ<i,j> Sᵢ·Sⱼ Sᵢ = (1/2) sᵢ ẑ, sᵢ ∈ {−1,+1} Evis = (J/4) Σ<i,j> sᵢsⱼ

Drag to rotate · wheel to zoom · orange/blue show orbital phase · arrows show the assumed local spin direction. Nearest-neighbour bond thickness can encode a geometric orbital-direction proxy.