Jellium Model and Hartree-Fock Approximation for Electron Gas Energy

Hello! Welcome back to our journey into diagrammatic many-body theory.

In our last lesson, we calculated the first-order self-energy, ${\Sigma }_{\text{HF}}$, for the Jellium model and used it in Dyson's equation to find the renormalized quasiparticle energies, ${ϵ}_k={ϵ}_k^0+{\Sigma }_F(k)$. We also saw that this approximation, while insightful, leads to the unphysical prediction of a vanishing density of states at the Fermi level due to its inadequate treatment of the long-range Coulomb interaction.

Today, we will use the Hartree-Fock (HF) approximation to address a different, but equally fundamental, question: what is the total ground-state energy, $E_0$, of the electron gas? This calculation is a classic milestone in condensed matter physics. It will allow us to assess whether interactions lead to a stable, bound system and will introduce the concept of the "exchange hole," a beautiful physical picture that explains cohesion in this model.

1. Total Energy vs. Quasiparticle Energies

A common pitfall is to assume that the total ground-state energy is simply the sum of all occupied quasiparticle energies, $E_0\ne {\sum }_{|k|<k_F}{ϵ}_k$. This is incorrect because the quasiparticle energy ${ϵ}_k$ already includes the interaction potential from all other particles. Summing these up would effectively count each pair-interaction twice.

The correct way to find the ground-state energy is to compute the expectation value of the full Hamiltonian $\hat{H}$ in the non-interacting ground state $|{\Phi }_0⟩$ (the Fermi sea of plane waves):

$$E_0=⟨{\Phi }_0|\hat{H}|{\Phi }_0⟩=⟨{\Phi }_0|\hat{T}+\hat{V}|{\Phi }_0⟩$$

The relationship between the total energy and the single-particle energies can be shown to be (for the HF approximation):

$$E_0=\sum _{|k|<k_F,\sigma }\left({ϵ}_k^0+\frac{1}{2}{\Sigma }_F(k)\right)=⟨\hat{T}{⟩}_0+\frac{1}{2}\sum _{|k|<k_F,\sigma }{\Sigma }_F(k)$$

The factor of 1/2 is precisely the correction to avoid double-counting the interaction energy. Our task is to calculate these two terms.

2. The Jellium Model Hamiltonian

Let's first formally define the Jellium model. It consists of $N$ electrons moving in a volume $\Omega$, with a uniform positive background charge to ensure overall neutrality.

A crucial simplification occurs in the Jellium model. The repulsive energy of the uniform positive background, $⟨{\hat{H}}_b⟩$, and the attractive energy between the electrons and the background, $⟨{\hat{H}}_{el-b}⟩$, are exactly cancelled by the direct (or Hartree) part of the electron-electron interaction. You can see this cancellation mentioned at the end of part (b) of "Exercise 2" in the resource LINK.

This leaves us with only two contributions to the total energy:

1. The kinetic energy of the electrons.
2. The exchange energy from the Fock term of the electron-electron interaction.

3. Calculating the Ground-State Energy

We will now calculate these two terms. The full derivation is a fantastic exercise in applying the concepts we've learned.

3.1. Kinetic Energy $⟨T⟩$

The average kinetic energy is the sum of $\frac{{\hslash }^2{k}^2}{2m}$ over all occupied states in the Fermi sea. In the thermodynamic limit ($\Omega \to \infty$), we replace the sum with an integral:

$$⟨T⟩=\sum _{|k|<k_F,\sigma }\frac{{\hslash }^2{k}^2}{2m}=2\frac{\Omega }{(2\pi {)}^3}{\int }_{|k|<k_F}{d}^{3}k\frac{{\hslash }^2{k}^2}{2m}$$

The factor of 2 is for spin degeneracy. Performing the integral in spherical coordinates gives:

$$⟨T⟩=2\frac{\Omega }{(2\pi {)}^3}(4\pi ){\int }_0^{k_F}dk{k}^2\left(\frac{{\hslash }^2{k}^2}{2m}\right)=\frac{\Omega {\hslash }^2}{10{\pi }^{2}m}{k}_F^5$$

Since the electron number is $N=\frac{\Omega k_F^3}{3{\pi }^2}$, the kinetic energy per particle is:

$$\frac{⟨T⟩}{N}=\frac{3{\hslash }^2}{10m}{k}_F^2$$

3.2. Exchange Energy $⟨V_{\text{exch}}⟩$

As we established, the exchange energy is $\frac{1}{2}{\sum }_{|k|<k_F,\sigma }{\Sigma }_F(k)$. From the last lesson, we know that ${\Sigma }_F(k)=-{\int }_{|k^{\prime }|<k_F}\frac{d^3{k}^{\prime }}{(2\pi {)}^3}V(\mathbf{k}-{\mathbf{k}}^{\prime })$. For the Coulomb interaction $V(q)=\frac{4\pi e^2}{q^2}$, this integral can be solved analytically.

The full derivation of both the kinetic and exchange energy is quite involved. The following resource walks through the entire calculation, arriving at the final expression for energy per particle.

3.3. The Final Result

By combining the kinetic and exchange terms and expressing them in terms of the dimensionless density parameter $r_s$, we arrive at the celebrated result for the ground-state energy per particle in the Hartree-Fock approximation:

$$\frac{E_{\text{HF}}}{N}=\left(\frac{2.21}{r_s^2}-\frac{0.916}{r_s}\right)\text{Ry}$$

Here:

• $r_s=r_0/a_0$ is the radius of the volume occupied by one electron, measured in units of the Bohr radius $a_0$. A small $r_s$ means high density.
• 1 Rydberg (Ry) $\approx 13.6$ eV is the natural unit of energy for atomic systems.

The first term, $\propto 1/r_s^2$, is the kinetic energy. The second term, $\propto -1/r_s$, is the exchange energy.

4. Physical Interpretation: Cohesion and the Exchange Hole

What does this result tell us?
The positive kinetic energy term reflects the energy cost of confining electrons into a small volume, as dictated by the uncertainty principle. This term dominates at high densities (small $r_s$) and would cause the system to fly apart.

The negative exchange energy term provides cohesion. Its origin is a purely quantum mechanical effect rooted in the Pauli exclusion principle.

In essence, the Pauli principle forbids two electrons with the same spin from occupying the same position. This effectively creates a "hole" in the distribution of same-spin electrons around any given electron. The electron is therefore surrounded by a region with a net positive charge (from the background jellium), which lowers its electrostatic energy. This attractive interaction is the source of the negative exchange energy and is what holds the electron gas together in this model.

The competition between the repulsive kinetic energy and the attractive exchange energy leads to a stable ground state. We can see this by plotting the energy as a function of $r_s$.

Test your understanding!

In the high-density limit ($r_s\to 0$), which term in the HF energy dominates? In the low-density limit ($r_s\to \infty$), which term dominates? What does this imply about the importance of interactions in these two regimes?

Show answer

• High density ($r_s\to 0$): The kinetic energy term $(+2.21/r_s^2)$ dominates. The system behaves more like a non-interacting free electron gas.
• Low density ($r_s\to \infty$): The exchange energy term $(-0.916/r_s)$ dominates relative to the kinetic energy. This implies that electron-electron interactions (correlations) become increasingly important as the density decreases.

This trend is a general feature of interacting electron systems.

Conclusion

In this lesson, we completed a cornerstone calculation of many-body physics, deriving the ground-state energy of the electron gas within the Hartree-Fock approximation.

Key Takeaways:

• The total ground-state energy is not the sum of quasiparticle energies; it must be calculated from the expectation value of the Hamiltonian, $⟨{\Phi }_0|\hat{H}|{\Phi }_0⟩$.
• For the Jellium model, the direct (Hartree) interactions are cancelled by the uniform background, leaving only kinetic and exchange energy contributions.
• The final result, $E_{\text{HF}}/N=(2.21/r_s^2-0.916/r_s)$ Ry, shows a competition between repulsive kinetic energy and attractive exchange energy.
• The cohesion is provided by the exchange hole, a manifestation of the Pauli principle where an electron is surrounded by a region of depleted same-spin electron density, leading to a net attraction to the positive background.

Preview of the Next Lesson:

While Hartree-Fock gives us a stable ground state, we know it has serious flaws, like the zero density of states at the Fermi level. These flaws stem from its poor treatment of the long-range Coulomb interaction. To fix this, we must go beyond first-order perturbation theory and account for screening. In the next lesson, we will begin this process by calculating the fundamental object that describes the response of the electron gas to a perturbation: the polarization bubble, also known as the Lindhard function. This will be our first step towards the Random Phase Approximation (RPA).