Quantum Mechanics of Matter
Types of Matter
- Materials:
- single atoms,
- molecules,
- crystalline materials,
- amorphous materials
- Materials can consist of many atoms
Classical description of matter:
- Continuum approximation: volumes with properties
- Electric/heat conductivity
- Elasticity, mass, etc.
How can we describe matter with Quantum mechanics?
Matter: Single Atoms
- Materials:
- single atoms,
- molecules,
- crystalline materials,
- amorphous materials
- Materials consist of many atoms
Matter: Multiple Atoms
- Materials:
- single atoms,
- molecules,
- crystalline materials,
- amorphous materials
- Materials can consist of many atoms
- Describing electrons in matter:
- (Relative) positions of the nuclei
- Positions of the electrons
- Interactions between all nuclei and all electrons
Matter: Multiple Atoms
- Materials:
- single atoms,
- molecules,
- crystalline materials,
- amorphous materials
- Materials can consist of many atoms
- Describing electrons in matter:
- (Relative) positions of the nuclei
- Positions of the electrons
- Interactions between all nuclei and all electrons
Matter: Molecules
- Materials:
- single atoms,
- molecules,
- crystalline materials,
- amorphous materials
- Covalent bonds
- Electronic interactions
- Configuration in space
- Vibrations
Molecules consist of multiple atoms:
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Matter: Large molecules
- Materials:
- single atoms,
- molecules,
- crystalline materials,
- amorphous materials
- Covalent bonds
- Electronic interactions
- Configuration in space
- Vibrations
Covalent bonds
- Superposition of orbitals: \(sp^2\)-hybridization
- Orbitals in the plane
- s-orbital is now symmetric with p-orbitals
- Improved orbital overlap/access to all electrons
- \(\sigma\)-bonds: Strong overlap between \(sp^2\)-hybridized orbitals
- \(\pi\)-bonds: Less overlap between \(p_z\)-orbitals but also in the plane
Tight-binding / LCAO
- Tight-binding method assumes electrons close to nuclei
- Wave functions are Linear Combinations of Atomic Orbitals (LCAO)
- Example of a molecule with two atoms:
\[
\hat{H} = \frac{\hat{p}^2}{2m} + \hat{V}_1 + \hat{V}_2
\]
Eigenstates atomic orbitals:
\[
\begin{aligned}
(\frac{\hat{p}^2}{2m} + \hat{V}_1) | 1 \rangle &= \varepsilon_0 \,| 1 \rangle\\
(\frac{\hat{p}^2}{2m} + \hat{V}_2) | 2 \rangle &= \varepsilon_0 \,| 2 \rangle\\
\end{aligned}
\]
Tight-binding / LCAO
Assume Linear Combination of Atomic Orbitals:
\[
|\psi\rangle = \phi_1\, |1\rangle + \phi_2\, |2\rangle
\]
with \(\phi_j\) amplitude of jth atom orbital and \(\langle 1| 2\rangle = 0\)
\[
\begin{aligned}
\hat{H} |\psi\rangle &= E |\psi\rangle = \phi_1 E |1\rangle + \phi_2 E |2\rangle\\
\end{aligned}
\]
Multiply to the left with bra’s \(\langle 1 |\) and \(\langle 2 |\)
\[
\begin{aligned}
E \phi_1 = \phi_1 \langle 1|\hat{H}|1\rangle + \phi_2 \langle 1|\hat{H}|2\rangle\\
E \phi_2 = \phi_2 \langle 2|\hat{H}|1\rangle + \phi_2 \langle 2|\hat{H}|2\rangle\\
\end{aligned}
\]
This gives us a new Hamiltonian
\[
\Longrightarrow
\begin{pmatrix}
\langle 1|\hat{H}|1\rangle & \langle 1|\hat{H}|2\rangle\\
\langle 2|\hat{H}|1\rangle & \langle 2|\hat{H}|2\rangle\\
\end{pmatrix}
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
=
E
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
\]
Tight-binding / LCAO
\[
\Longrightarrow
\begin{pmatrix}
\langle 1|\hat{H}|1\rangle & \langle 1|\hat{H}|2\rangle\\
\langle 2|\hat{H}|1\rangle & \langle 2|\hat{H}|2\rangle\\
\end{pmatrix}
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
=
E
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
\]
- The diagonal elements \(\langle j|\hat{H}|j\rangle = E_0\) represent the onsite energy
- The off-diagonal elements \(\langle 1|\hat{H}|2\rangle = -t\) represent the hopping term
\[
\Longrightarrow
\begin{pmatrix}
E_0 & -t\\
-t & E_0\\
\end{pmatrix}
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
=
E
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
\]
- This eigenvalue equation has eigenvalues/eigenstates:
\[
E_\pm = E_0 \mp t, \quad \qquad \psi_\pm = \frac{1}{\sqrt{2}}(| 1\rangle \pm | 2 \rangle )
\]
Tight-binding
\[
\begin{pmatrix}
E_0 & -t\\
-t & E_0\\
\end{pmatrix}
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
=
E
\begin{pmatrix} \phi_1 \\ \phi_2
\end{pmatrix}
\]
\[
E_\pm = E_0 \mp t
\]
\[
\psi_\pm = \frac{1}{\sqrt{2}}(| 1\rangle \pm | 2 \rangle )
\]
