Methane CH4 molecule. Intensity is proportional to ∫ |Ψ(r)|2dz.
Use your fingers or mouse to control the model. Use "zoom" (e.g. mouse wheel) to adjust intensity. Canvas is matched to your browser window.

Valence bond theory

Valence bond theory proposes that covalent bonds form when the atomic orbitals of two atoms overlap. The combined, overlapping orbitals, contain a total of two electrons with opposed spins.

Atomic (Hydrogen) orbitals

Hydrogen wave functions are
    Ψ1s ~ 2 e -r,     Ψ2s ~ (2 - r) e -r/2,     Ψ2px ~ x e -r/2.




σ bonds

Sigma (σ) bonds arise from the 'end-on' overlap between adjacent orbitals. This leads to a region of high electron density along the inter-nuclear axis.

H2 molecule

unbonding orbital

1s-2px bond

HF molecule

px-px bond

F2 molecule

Hybrid orbitals

Hybridization of atomic orbitals provides for molecular shapes that cannot be accommodated by using s, p, and d orbitals.



This hybridization allows carbon to form four rather than two covalent bonds and to orient them so as to minimize the e- pair repulsions (i.e., tetrahedrally).

s-sp3 bond





1. The fragment shader shader-fs-calc for every (x, y) point (pixel) renders integrated intensity into 64x64 texture. Then this texture is mapped on the whole canvas. For complex molecule orbitals are rendered in sequence.
2. Laguerre integration formula for integral of the type o f(z)e-zdz is used (25.4.45 "Handbook of Mathematical Functions" under the direction of M.Abramowitz and I.Stegun)
    o g(z)dz = ∑i wi e zi g(zi )
where wi e zi, zi are tabulated values for i = 1, 9. Integration may be not very accurate (but simple, fast and funny :)


[1] Lecture 18 - Theories of Covalent Bonding by Warren Gallagher
[2] Atomic orbital viewer by Andy Brummer
Open Source molecule viewers     updated 3 May 2012