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Helium Atom Variational Method (2D): Effective Nuclear Charge

Interactive 2D variational-method simulator for the helium atom: drag the effective nuclear charge Z' of a trial 1s wavefunction, watch a cross-section of the two-electron density cloud shrink or swell in a pannable/zoomable canvas, and find the electron-shielding minimum of the ground-state energy E(Z') live.

Chemistry & Materials2DAdvanced60 FPS📱 Mobile-adapted⇄ 3D version

This 2D sibling applies the same quantum-chemistry variational method to helium as the 3D version, but renders the electron probability density as a flat cross-section you can pan and zoom instead of an orbiting point cloud. Dragging Z' rescales the density disc live and moves a marker along the exact closed-form energy curve E(Z') = Z'² − 2ZZ' + (5/8)Z', with kinetic, nuclear-attraction and electron-repulsion terms broken out live. An auto-minimize button runs real gradient descent toward the textbook shielding result Z'_eff = 1.6875, the same variational logic (in miniature) that self-consistent-field methods like Hartree-Fock and DFT use on real molecules.

⚙ Under the hood

Drag the effective nuclear charge Z' of a variational trial wavefunction for helium and watch a 2D cross-section of the two-electron density cloud rescale live while a marker moves along the exact energy curve E(Z'), converging to the textbook electron-shielding minimum at Z'=1.6875. Pan and zoom the cloud freely.

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install