Slater-Koster parameter

Question

When we use Slater-Koster parameter for tight binding model, we represent hopping integral using direction cosine (l,m,n) and bond intergral $V$.

For example, the energy between s and px orbital is expressed as

$$ E_{s,x} = l V_{sp\sigma} $$

where $l$ is the direction from position of $s$ to $px$ orbital. But for $V_{sp\sigma}$, we have two choices of intergral, below figure (1) and (2). Their values are same, but sign is different. So which one is used when calculating $V_{sp\sigma}$?

Answer 1

You can deduce this by looking at the pictures.

On the left the energy is lower than on the right, because the +-parts repel each other. If you assume $V_{sp\sigma}$ to be positive it is calculated for the right. Then $-V_{sp\sigma}$ applies to the left.

This argument only holds for an isolated molecule (as in your picture). For an crystal you have to extend this thinking. Most tight-binding codes use this parametrization: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.54.4519