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In quantum mechanics, electrons around an atom are described by wave functions, which represent the probability distribution of finding an electron in a certain region around the nucleus. It means that electrons do not have precise positions in an atom, making it hard to determine a specific boundary for the atom. An atomic radius is the distance from the nucleus to the outermost electron shell. However, due to the probabilistic nature of the electron positions, it becomes challenging to define the precise atomic radius.

The electrons in many-electron atoms are distributed in different orbitals with various shapes and energies. The distribution depends on the principle quantum number, azimuthal quantum number, and magnetic quantum number. Due to the distinct electron distribution, there is no well-defined boundary where the electron density drops to zero. Thus, defining a precise atomic radius becomes difficult as it relies heavily on the electron distribution in atomic orbitals.

When nonbonded atoms come close to each other, the closest approach of their nuclear centers is determined by the balance between attractive and repulsive forces: 1. Attractive Forces: These forces primarily come from the van der Waals interactions. These interactions arise due to the temporary fluctuations in the electron distribution around the atom, causing temporary regions of positive and negative charge. The induced dipoles attract neighboring atoms, leading to weak attractive forces. 2. Repulsive Forces: When atoms get very close, the outer electron clouds start to overlap, which leads to repulsive forces as a result of the Pauli exclusion principle. This principle states that no two electrons can have the same set of quantum numbers. Thus, when electron clouds overlap, the electrons in the overlapping region experience repulsion due to the requirement that they occupy different quantum states. The closest approach of nuclear centers depends on the balance between these two forces. When the attractive and repulsive forces are equal, the atoms are in a stable position, and the distance between their nuclear centers is called the van der Waals radius.