Hopping Probabilities in Tight Binding Model

[R Van Camp]

We recently covered the tight binding model. I have a question from an early lecture regarding sign conventions for the hopping parameter t(T). It was explained that t>0 due to MO theory. I agree and understand why; orbital overlap. It was then stated that t_nn<0; that is, t_nn=-t.

I am not following why t_nn<0. It makes some sense that this is true in view of the increase in charge delocalization associated with any mode of hopping. Ultimately, I am looking to understand the physical significance of the sign of t_nn, t_nnn, etc.

 

[DrDu]

What is t_nn?

 

[R Van Camp]

The subscript 'nn' indicates nearest neighbor hopping; 'nnn' next nearest neighbor hopping. We use Kittel but the lectures on the Tight Binding Model came from Ashcroft's text. Distinctions in notation exist between the two texts. I think Ashcroft was used for this subject but it is a more thorough text.

 

[DrDu]

Well, the sign can be chosen at will to a certain extent. E.g. if you take two 1s orbitals in the hydrogen molecule as a basis, you can change the sign of the off diagonal element (but not of the diagonal element) by changing the sign of one of the two basis functions.
For equal atoms, this is probably not a problem and probably also not for different atoms, e.g. you could fix that e.g. the lobe of a p_x orbital pointing in +x direction should always be positive.
Having said this, there are some semi-empirical rules which are often used to express the t's in terms of the overlapp integrals. See. e.g.:
http://en.wikipedia.org/wiki/Extended_Huckel_method

Nevertheless, even for some canonical choice of the basis functions, the overlapp integrals (and t) may be either positive or negative. For two hydrogen 1s orbitals it will be positive, while for two p_x orbitals it will be negative with the above convention when the two atoms are both on the x-axis ( forming a sigma bond then) and positive if they are both on the y-axis forming a pi bond then).