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What is a covalent bond, quantum mechanically? How does it hold the two atoms together, and at one point can you qualify the electron as being shated between two atoms, versus feeling attractive forces?

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It's rather chemistry, and matter of descriptive word. For me, if one solves the Hartree-Fock-Roothaan equation and gets lower energy of molecule than atoms (attractive), and find the most crucial part of eigenfunction $\psi_i=c_{ia}\phi_a+c_{ib} \phi_b$ (take diatomics for instance, a,b are two atomic orbitals), if $c_{ia} \approx c_{ib}$, it's a covalent bond. –  user26143 Jun 7 '14 at 9:02

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