Helium Atom Variational Method: Effective Nuclear Charge

This simulator applies the quantum-chemistry variational method to the simplest multi-electron atom, helium. Instead of the true, mathematically intractable two-electron Schrödinger equation, it uses a trial wavefunction of two hydrogen-like 1s orbitals sharing one adjustable effective nuclear charge Z'. Dragging Z' rescales a live 3D point cloud of the electron probability density 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.