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Hi everyone, I'm trying to fit the Tight Binding molecule for a more complicated system, so I'm first trying to understand it for a simpler one, graphene. I've read several guides but they're all confusing me.

Right now, I'm trying to understand the graphene example on this site. My biggest confusion seems to be about how since graphene is a honeycomb lattice, it has two sublattices, or, two atoms per unit cell (as does my more complicated system, so I need to understand this well).

The guide says that each carbon atom has one ##2p_z## valence orbital, so the ##\phi_{2p_{z1}}## orbital is centered at one of the atoms in the primitive cell and ##\phi_{2p_{z2}}## is centered at the other. The primitive lattice vectors are ##\vec a_1## and ##\vec a_2##.

So, going over every primitive cell in the lattice (though of course we're going to cut it off at nearest neighbors), the total wave function is:

##\psi_{\vec{k}}\left(\vec{r}\right)=\frac{1}{\sqrt{N}}\sum\limits_{h,j}e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2\right) \right)

##

I don't understand why we do this, though. I understand that this isn't a real Bravais lattice, it's a lattice with a basis, but why does that matter if the atoms are all the same and we're just going to look at a few nearest neighbors?

Anyway, then they do this little trick (using orthogonality of the functions) to get some equations that can be solved:

##\begin{array}{a} \langle\phi_{\text{2p}_{z1}}|\hat{H}|\psi_{k}\rangle = E\langle\phi_{\text{2p}_{z1}}|\psi_{k}\rangle , \\ \langle\phi_{\text{2p}_{z2}}|\hat{H}|\psi_{k}\rangle = E\langle\phi_{\text{2p}_{z2}}|\psi_{k}\rangle . \end{array}##

And here's where my confusion starts. They get this as the result of those two lines (and continue on to get the determinant and such), keeping only on-site and nearest neighbor terms:

##\begin{array}{a} \epsilon c_1 -tc_2\left(1+e^{-i\vec{k}\cdot\vec{a_1}} + e^{-i\vec{k}\cdot\vec{a_2}}\right) = Ec_1 ,\\ \epsilon c_2 -tc_1\left(1+e^{i\vec{k}\cdot\vec{a_1}} + e^{i\vec{k}\cdot\vec{a_2}}\right) = Ec_2. \end{array}##

(where ##\epsilon = \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z1}}\left(\vec{r}\right)\rangle## and ##t = - \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z1}}\left(\vec{r}-\vec{a}_1\right)\rangle##)

First of all, this seems to have only used ##\vec a_1## and ##\vec a_2## for the nearest neighbor terms...but looking at the lattice diagram above, doesn't any atom in one of the sublattices have 6? For example, if you look at the middle C2 atom in that diagram, there are 6 other equidistant C2 atoms.

The answer to that might explain this as well, but if I do the algebra out (even just using ##\vec a_1## and ##\vec a_2## for nearest neighbors the way they seem to) on the left hand sides to try and get to those last equations, I get:

##\psi_{\vec{k}}\left(\vec{r}\right)=e^0 \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}\right) \right) +

\\

e^{i\left(\vec{k}\cdot\vec{a}_1\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-\vec{a}_1\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-\vec{a}_1\right) \right) +

\\

e^{i\left(\vec{k}\cdot\vec{a}_2\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-\vec{a}_2\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-\vec{a}_2\right) \right)##

So

##\langle\phi_{\text{2p}_{z1}}|\hat{H}|\psi_{k}\rangle = c_1 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z1}}(\vec r)\rangle + c_2 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z2}}(\vec r)\rangle

\\

+ e^{i\left(\vec{k}\cdot\vec{a}_1\right)}(c_1 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z1}}(\vec r - \vec {a}_1)\rangle + c_2 \langle\phi_{\text{2p}_{z1}}(\vec r )|\hat{H}|\phi_{\text{2p}_{z2}}(\vec r- \vec {a}_1)\rangle)

\\

+ e^{i\left(\vec{k}\cdot\vec{a}_2\right)}(c_1 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z1}}(\vec r - \vec {a}_2)\rangle + c_2 \langle\phi_{\text{2p}_{z1}}(\vec r )|\hat{H}|\phi_{\text{2p}_{z2}}(\vec r- \vec {a}_2)\rangle)##

Which certainly has the terms they have, but a few more that they seem to have dropped. Unless I'm understanding this really horribly, they seemed to have dropped all the terms with a ##|\phi_{\text{2p}_{z1}}\rangle## ket except the on-site term.

What's going on? I'm so confused...

Thank you!

Right now, I'm trying to understand the graphene example on this site. My biggest confusion seems to be about how since graphene is a honeycomb lattice, it has two sublattices, or, two atoms per unit cell (as does my more complicated system, so I need to understand this well).

The guide says that each carbon atom has one ##2p_z## valence orbital, so the ##\phi_{2p_{z1}}## orbital is centered at one of the atoms in the primitive cell and ##\phi_{2p_{z2}}## is centered at the other. The primitive lattice vectors are ##\vec a_1## and ##\vec a_2##.

So, going over every primitive cell in the lattice (though of course we're going to cut it off at nearest neighbors), the total wave function is:

##\psi_{\vec{k}}\left(\vec{r}\right)=\frac{1}{\sqrt{N}}\sum\limits_{h,j}e^{i\left(h\vec{k}\cdot\vec{a}_1 + j\vec{k}\cdot\vec{a}_2\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-h\vec{a}_1-j\vec{a}_2\right) \right)

##

I don't understand why we do this, though. I understand that this isn't a real Bravais lattice, it's a lattice with a basis, but why does that matter if the atoms are all the same and we're just going to look at a few nearest neighbors?

Anyway, then they do this little trick (using orthogonality of the functions) to get some equations that can be solved:

##\begin{array}{a} \langle\phi_{\text{2p}_{z1}}|\hat{H}|\psi_{k}\rangle = E\langle\phi_{\text{2p}_{z1}}|\psi_{k}\rangle , \\ \langle\phi_{\text{2p}_{z2}}|\hat{H}|\psi_{k}\rangle = E\langle\phi_{\text{2p}_{z2}}|\psi_{k}\rangle . \end{array}##

And here's where my confusion starts. They get this as the result of those two lines (and continue on to get the determinant and such), keeping only on-site and nearest neighbor terms:

##\begin{array}{a} \epsilon c_1 -tc_2\left(1+e^{-i\vec{k}\cdot\vec{a_1}} + e^{-i\vec{k}\cdot\vec{a_2}}\right) = Ec_1 ,\\ \epsilon c_2 -tc_1\left(1+e^{i\vec{k}\cdot\vec{a_1}} + e^{i\vec{k}\cdot\vec{a_2}}\right) = Ec_2. \end{array}##

(where ##\epsilon = \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z1}}\left(\vec{r}\right)\rangle## and ##t = - \langle\phi_{\text{2p}_{z1}}\left(\vec{r}\right)|\hat{H}|\phi_{\text{2p}_{z1}}\left(\vec{r}-\vec{a}_1\right)\rangle##)

First of all, this seems to have only used ##\vec a_1## and ##\vec a_2## for the nearest neighbor terms...but looking at the lattice diagram above, doesn't any atom in one of the sublattices have 6? For example, if you look at the middle C2 atom in that diagram, there are 6 other equidistant C2 atoms.

The answer to that might explain this as well, but if I do the algebra out (even just using ##\vec a_1## and ##\vec a_2## for nearest neighbors the way they seem to) on the left hand sides to try and get to those last equations, I get:

##\psi_{\vec{k}}\left(\vec{r}\right)=e^0 \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}\right) \right) +

\\

e^{i\left(\vec{k}\cdot\vec{a}_1\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-\vec{a}_1\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-\vec{a}_1\right) \right) +

\\

e^{i\left(\vec{k}\cdot\vec{a}_2\right)} \left( c_1\phi_{\text{2p}_{z1}}\left(\vec{r}-\vec{a}_2\right) + c_2 \phi_{\text{2p}_{z2}}\left(\vec{r}-\vec{a}_2\right) \right)##

So

##\langle\phi_{\text{2p}_{z1}}|\hat{H}|\psi_{k}\rangle = c_1 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z1}}(\vec r)\rangle + c_2 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z2}}(\vec r)\rangle

\\

+ e^{i\left(\vec{k}\cdot\vec{a}_1\right)}(c_1 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z1}}(\vec r - \vec {a}_1)\rangle + c_2 \langle\phi_{\text{2p}_{z1}}(\vec r )|\hat{H}|\phi_{\text{2p}_{z2}}(\vec r- \vec {a}_1)\rangle)

\\

+ e^{i\left(\vec{k}\cdot\vec{a}_2\right)}(c_1 \langle\phi_{\text{2p}_{z1}}(\vec r)|\hat{H}|\phi_{\text{2p}_{z1}}(\vec r - \vec {a}_2)\rangle + c_2 \langle\phi_{\text{2p}_{z1}}(\vec r )|\hat{H}|\phi_{\text{2p}_{z2}}(\vec r- \vec {a}_2)\rangle)##

Which certainly has the terms they have, but a few more that they seem to have dropped. Unless I'm understanding this really horribly, they seemed to have dropped all the terms with a ##|\phi_{\text{2p}_{z1}}\rangle## ket except the on-site term.

What's going on? I'm so confused...

Thank you!

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