Diatomic moleculei

💼 Case: Consider a diatomic molecule of identical ions at position $x=0$ and $x=a$ with a free electron. We assume the situation is in 1 dimension for simplicity

• The Schrödinger equation for the system is

  $$ \hat H \psi(x)=E\psi(x) \quad \text{where} \quad \hat H =\frac{\hat p^2}{2m}+W(x)+W(x-a) $$

  where $W(x)$ is the attractive potential due to the ions

  🗒️ Note: if the ions are far apart then the electron will stay close to one

• Since we expect the electron to be close to one of the atom we can use linear combinations of atom orbitals $\phi_n(x)$

  $$ \left [ \frac{ p^2}{2m}+W(x) \right ]\phi_n(x) = E_n \phi_n(x) $$

  🗒️ Note: there is a translational symmetry at $x=a$ ie

  $$ \left [ \frac{ p^2}{2m}+W(x-a) \right ]\phi_n(x-a) = E_n \phi_n(x-a) $$

  is the same equation

• We can guess a trial wavefunction using the fact that the lowest-lying state should resemble the ground state of the two ions

  $$ \psi (x) \simeq c_0 \phi_0 (x) + c_1 \phi_0(x-a) $$

  🗒️ Note: We assume that the electron wavefunction is approximately the superposition of the two atomic orbitals

• Sub in our trial solution into the TISE and integrate over $x$ between $-\infin$ and $+ \infin$ we get

  $$ c_0(E_0-E)-c_1 t=0 \quad \text{with} \quad -t =\int^{+\infin}_{-\infin} \, \text d x\,\phi^*_0 (x) W(x) \phi_0 (x-a)
  $$

• Expand for the complete derivation (not examinable)

🗒️ Notes:

• $t$ is the hopping amplitude. If the two ions were at $a\to \infin$ then $t\to 0$ ie the two wavefunctions $\phi_0^*(x)$ and $\phi_0(x-a)$ have no overlap. Thus the solution is $E=E_0$ and the electron is in the ground state of an isolated atom

• The equation is symmetric ie both of the following are true, where $t$ is the same

  $$ \begin{aligned} c_0(E_0-E)-c_1 t&=0 \\ c_1(E_0-E)-c_0 t&=0 \end{aligned} $$

  where $|c_0|^2+|c_1|^2=1$ for the electron to be somewhere at all times

• We can visualise the solution to this equation:

  

🗂️ Recap of everything:

We consider a diatomic molecule with identical ions in 1D

1. When we bring 2 ions together the electron wavefunction will have a hoping amplitude $t$ which represents the overlap of the wavefunctions (orange circle)
2. Since there is an overlap in the wavefunction it means the electron can jump from one ion to the other even though it is unlikely
3. Thus it makes sense that the equations are symmetric because it can either be a hop from the $x=0$ to $x=a$ ion or from the $a$ to the $0$ ion

Solutions

• For $c_1=c_0$ we get $E=E_0 - t$ which is the ground state and is symmetric expressed as

  $$ \psi_{s,a}=\frac{\phi_0(x)+ \phi_0(x-a)}{\sqrt 2} $$

• For $c_1=-c_0$ we get $E=E_0 + t$ which is the first excited state and is antisymmetric

  $$ \psi_{s,a}=\frac{\phi_0(x)- \phi_0(x-a)}{\sqrt 2} $$

If the 2 atoms were different they would have 2 different atomic energies $E_0$ and $E_1$ which would give