# Tight-binding Hamiltonian

I am currently in a computational physics course and am working on a final project involving carbon dimers. The reason this topic is applicable in my class is that once I figure out the physics involved, the problem involves using a lot of the numerical methods I learned in class. I am solid on the numerical methods, but a complete novice in terms of solid-state physics and quantum mechanics. I previously posted this https://www.physicsforums.com/showthread.php?t=361408" in the Homework and Coursework Questions forum, but wasn't getting any replies, so I am rethinking my approach to getting help on this.

Can someone explain to me what an empirical tight-binding Hamiltonian is?

I understand that in classical mechanics the Hamiltonian represents the energy of the system.

In the tight binding model, I assume it still represents the energy, but in what way?

Is the tight-binding hamiltonian the same as the Hamiltonian in the Schrödinger equation?

Why would the tight binding Hamiltonian be represented as a matrix?

In the case above, does empirical imply that the values used in the matrix should be measure?
(for example, the off-diagonal elements are supposed to be described by a set of orthogonal sp3 two-center hopping parameters, $$V_{ss\sigma}$$,
$$V_{sp\sigma}$$, $$V_{pp\sigma}$$, and $$V_{pp\pi}$$, scaled with interatomic separation r as a function of s(r); and the on-site elements are the atomic orbital energies of the corresponding atom.)

Should the Hamiltonian matrix have dimensions 8x8 because carbon is tetravalent and I am attempting to construct a Hamiltonian for C2?

What does off-site element mean? Does it refer to the diagonal elements of the matrix?

Any help would greatly be appreciated.

Also, if anyone here hasn't seen my thread in the coursework forum, but think they might be able to help, please do. Any kind of human-to-human interaction is better than trying to figure this stuff out through journals and wikipedia.

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## Answers and Replies

Tight binding is a method to construct a Hamiltonian for a system starting from the assumption there is a small basis of localized orbitals that will adequately describe the physics you want to capture.

Is the tight-binding hamiltonian the same as the Hamiltonian in the Schrödinger equation?
In tight-binding, you have your hopping integrals:

$$t_{a,b}(\vec{R},\vec{R'}) = \langle a \vec{R} | H | b \vec{R'} \rangle$$

where H is the full Hamiltonian. IIRC the term "empirical tight-binding" implies that the values of t are taken from somewhere else, such as by fitting them to band structure data.

In the tight binding model, I assume it still represents the energy, but in what way?
Diagonalizing this Hamiltonian gives you the energy eigenvalues for the single particle states. If you wanted to compute the total energy of the system, a sum over the eigenvalues of all occupied states would be involved.

Is the tight-binding hamiltonian the same as the Hamiltonian in the Schrödinger equation?

Not in two senses. The true Hamiltonian of a system of electrons includes the quantum Coulomb repulsion, which prevents you from decoupling the Schrodinger equation. So, in doing tight-binding you neglect this term, or assume that it can be handled in a mean-field way, (or at best you assume that the system has long-lived quasiparticles) so you can still treat the electrons as independent particles. The other sense in which it's not the true Hamiltonian of a system is that it's an approximation technique, and one of those approximations is to cut off the basis to a finite, and usually small, number of states.

But it is the same sort of Hamiltonian as would be used in quantum mechanics, so all the usual interpretations for a non-interacting system would apply.

In the case above, does empirical imply that the values used in the matrix should be measure?
The hopping integrals that go into the tight binding Hamiltonian can be interpreted as transition amplitudes between states. You are picking some basis of states, usually atomically motivated, which are not the eigenstates of the system, so the elements of the Hamiltonian represent a amplitude for an electron in one state to "hop" into another state.

Also, a good exercise would be to look at the equation for the hopping integral written above and ask yourself why $$V_{sp\pi}$$ is not allowed.

Should the Hamiltonian matrix have dimensions 8x8 because carbon is tetravalent and I am attempting to construct a Hamiltonian for C2?
Yes, if that is the right number of basis states to give you the accuracy that you want. For carbon that is probably sufficient, since there is usually strong sp^3 hybridization you can't drop the 2s states. For more accuracy you might include the 3d states but that is probably not necessary for a class.

What does off-site element mean? Does it refer to the diagonal elements of the matrix?
I would think that would refer to a hopping element which is between orbitals on different atoms.

Could you recommend books to read on tight binding theory?

I asked a question about the calculaton of tight binding bandstructure in this froum (Tight Binding calculation of band structure), i think the answers are useful for you.
in the tight binding the basis set for expanding the wave function are localized atomic robitals so the basis set dimension is equall to the number of the atomic orbitals.

the answer of all the questions about tight binding find in this book: Solid state Physics GIUSEPPE GROSSO and GIUSEPPE PASTORI PARRAVICINI (pages : 16 & 145)
and of course in this address and references there in : http://en.wikipedia.org/wiki/Tight_binding 