linguistic:voice_change, quality:corruption
Modern quantum mechanical models

Schrödinger's wave model for hydrogen replaced Bohr's model, with its neat, clearly defined circular orbits. The modern model of the atom describes the positions of electrons in an atom in terms of probabilities. An electron can potentially be found at any distance from the nucleus, but, depending on its energy level and angular momentum, exists more frequently in certain regions around the nucleus than others; this pattern is referred to as its atomic orbital. The orbitals come in a variety of shapes— sphere, dumbbell, torus, etc.—with the nucleus in the middle. The shapes of atomic orbitals are found by solving the Schrödinger equation. Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the hydrogen atom and the hydrogen molecular ion. Beginning with the helium atom—which contains just two electrons—numerical methods are used to solve the Schrödinger equation.