1D problem

Each atom has an atomic orbital $\phi_m(\vec{r})$ with $m=\lambda$ for the adsorbate and $m,=0,...,N-1$ the substrate. We can write the total system wavefunction (molecular orbital) $\psi_k$ as an LCAO, where $k$ indicates different energy levels of the molecular orbitals:

Or in bra-ket notation:

where $c_{mk}=\langle m|k \rangle$.

Each molecular orbital satisfies the 1-electron Schrödinger equation

By inserting $|k\rangle$ and multiplying by $\langle n|$, we get the Schrödinger equation in terms of matrix elements:

If we assume orthonormal atomic orbitals, $\langle n|m \rangle= \delta_{nm}$. We also write the Hamiltonian in the tight-binding approximation: $$ H =

which transforms our Schrödinger equation into

This equation has three ‘boundary conditions’:

• For $n=0$: $(E_k-\alpha_s)c_{0k} = \beta c_{1k}+\beta_a c_{\lambda k}$

• For $n=\lambda$: $(E_k-\alpha_a) c_{\lambda k} = \beta_a c_{0k}$

• For $n=N$: $c_{Nk}=0$

Now we take the Ansatz $c_{n\pm \ell,k}=e^{\pm i\ell\theta_k} c_{nk}$. Actually, this ‘plane wave Ansatz’ is a common Ansatz for a wave function in periodic systems, and corresponds to Bloch’s theorem. The book provides a derivation of this Ansatz from scratch. By inserting this Ansatz, the Schrödinger equation can then be written as

This defines the energy band. Hence, $\beta$ is related to the band width.

We can also express the LCAO coefficient at any point in terms of the plane waves and the coefficient at $n=0$ (actually not entirely clear to me how using positive and negative waves should be motivated here),

or by re-expressing some parameters in terms of others

Now we apply the boundary conditions. $c_{Nk}=0$ gives

and with the cosine/sine sum rule this yields

which implies that $A=c_{0k}$.

The other two boundary conditions can also be applied. Through some algebra (including some cosine/sine sum rules), one gets an ‘eigenvalue equation’:

where in the derivation process the following parameters were defined:

• the chemisorption parameter $z_a = (\alpha-\alpha_a)/\beta$

• the surface parameter $z_s=(\alpha-\alpha_s)/\beta$

• the chemisorption (adbond) parameter $\eta=\beta_a/\beta$.

Solutions¶

Let’s consider solutions for $\eta=0$. Then the solutions $\theta_k$ to the eigenvalue problem satisfy either

(if $-2\leq z_a\leq 2$), or

The first equation gives one solution and the second one gives $N$ solutions.

The $N$ solutions are visible in the graph below, where the purple lines are two possible values of $-z_s$ (1 and -1),

Play with the graph on Desmos

But when $-z_s$ becomes larger than 1 or smaller than -1, a solution disappears -- at least it does on the real number line. We can consider solutions for complex $\theta_k = \xi_k + i\mu_k$, where $\mu_k>0$ (it does not make sense to include $\mu_k=0$, because then we get real solutions; I think the reason to not include negative $\mu_k$ is because it will give duplicate solutions). We can then write $X_k=\cos\xi_k \cosh \mu_k - i\sin \xi_k \sinh \mu_k$. However, the energy must be real, so $\sin\xi_k=0$, which gives $\xi_k=j\pi,j=0,1,2,...$. Consequently $\cos \xi_k$ is either 1 or -1, so $\xi_k=0$ and $\xi_k=\pi$ already describe the complete space of solutions, higher $j$ just gives duplicates.

So overall we get two possible additional solutions by considering complex $\theta_k$,

The positive one lies above the bulk band, and we call them P-states. The negative one lies below the bulk band, and we call them N-states.

In conclusion, there are $N+1$ atoms, each with one electron, so there are $N+1$ molecular orbital states, i.e. $N+1$ solutions for $\theta_k$:

• $N-1$ real solutions in the bulk band.

• 1 solution for the adatom

• 1 solution for the surface The latter two may be real (in bulk band), or complex (P-states and N-states).

We now show that bulk band states are delocalized and P/N-states are localized. Find the corresponding graph on Desmos.

Bulk band states¶

Oscillates in $n$ with constant period. So delocalized.

P-states¶

Large $N$ and if $\theta_k$ has a positive imaginary part $(\mu_k>0)$:

If $\theta_k=i\mu_k$:

which is a decaying wavefunction.

Inserting the large-$N$ approximation for the wavefunction in the ‘eigenvalue equation’, we get

By setting $x=e^\mu$ we can solve this equation graphically; we get a function

and we find the intersections with

The graphical solution is shown below:

Find the Desmos graph here.

So for these particular parameters there are 2 localized P-states.

N-states¶

If $\theta_k=\pi+i\mu_k$:

Decaying but oscillatory wave-function.

By inserting the large-$N$ approximation for the N-state wavefunction in the eigenvalue equation, we get an algebraic equation

which can be solved as before.

The $N$-states appear at different parameter values than $P$-states. They can even coexist. Check out Desmos!