This attraction is much weaker than covalent bonds but drops slower with increasing distance.
The dipole attraction is much weaker than the covalent bonds but drops slower with increasing distance.
??? question "How does the strength of a covalent bond scale with distance?"
The strength of the bond is determined by the interatomic hopping integral $-t = \langle 1 | H | 2 \rangle$. Since the wavefunction of a bound electron typically decays exponentially, so does the overlap integral.
Although the Van der Waals force is weak, it is the only force when there are no chemically active electrons or when the atoms are too far apart to form covalent bonds.
Therefore, there are materials where Van der Waals interactions are the dominant interactions.
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The solutions to the eigenvalue problem are phonon modes.
### Electrons
Let us now look at the electronic behaviour of 3 atoms on a chain.
We just looked at how a chain of atoms moves.
Let us now look at how the electrons of those atoms behave.
To that end, we consider a 3 atom chain without any motion.
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Figure having 3 atoms with a hopping -t
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Because the three atoms are identical, the onsite energy is the same on all atoms $\langle 1|H|1 \rangle = \langle2|H|2 \rangle = \langle3|H|3 \rangle = E_0$.
Furthermore, we assume hopping only between the nearest neighbors and assume that it is real valued: $\langle1|H|2\rangle = \langle2|H|3\rangle = -t$.
This also holds for its complex conjugate.
We also assume that the orbitals are orthogonal to eachother.
Just as we did in the previous lecture, we use the Schrödinger equation $H |\psi\rangle = E |\psi\rangle$ to set up a system of equations:
