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Hartree-Fock Approximation

Overview

The Hartree-Fock (HF) method is a mean-field approximation that treats electron-electron interactions through static effective potentials. Rather than solving the full many-body problem, HF replaces the two-body interaction with an effective single-particle potential constructed self-consistently from the electron density. The total HF self-energy decomposes into two contributions: the Hartree (direct) term and the Fock (exchange) term.

Hartree Self-Energy

The Hartree term represents the classical electrostatic potential felt by an electron due to the charge density of other electrons. It is given by:

\[ \Sigma^{H\, pr}_{ij\sigma}(\mathbf{k}) = \sum_{\sigma'} \sum_{q} \sum_{kl} V^{p\sigma;q\sigma'}_{ijkl}(\mathbf{k}=0) \times \frac{1}{N_\mathbf{k}} \sum_{\mathbf{k}'} n^{q}_{lk\sigma'}(\mathbf{k}') \delta_{pr} \]

where \(N_\mathbf{k}\) is the number of k-points in the Brillouin zone, \(n^{q}_{lk\sigma'}(\mathbf{k}')\) represents the momentum-resolved density, and the interaction \(V^{p\sigma;q\sigma'}_{ijkl}(\mathbf{k}=0)\) is evaluated at zero momentum transfer. This term is diagonal in the basis-site index (\(\delta_{pr}\)) and captures the average electrostatic repulsion.

In QAssemble, the SigmaHartree class (child of FLatStc) computes this quantity. It accepts the density matrix \(n\) and the bare interaction \(V\) as inputs and returns \(\Sigma^H\) as an FLatStc object.

Fock Self-Energy

The Fock term accounts for exchange interactions arising from the antisymmetry of the fermionic wavefunction, providing a non-local correction:

\[ \Sigma^{F\, p,q}_{ij\sigma}(\mathbf{k}) = -\sum_{\mathbf{R}} \sum_{kl} n^{q,p}_{lk\sigma}(\mathbf{R}) \times V^{p\sigma;q\sigma'}_{ijkl}(\mathbf{R}) \delta_{\sigma\sigma'} e^{i\mathbf{k}\cdot\mathbf{R}} \]

where \(n^{q,p}_{lk\sigma}(\mathbf{R})\) is the density matrix in real space connecting sites separated by lattice vector \(\mathbf{R}\), and \(\delta_{\sigma\sigma'}\) ensures that exchange occurs only between electrons of the same spin. This term is generally non-diagonal in both real space and orbital indices, and is responsible for phenomena like exchange splitting and magnetic ordering.

In QAssemble, the SigmaFock class (child of FLatStc) accepts \(n\) and \(V\) as inputs and returns \(\Sigma^F\) as an FLatStc object.

HF Hamiltonian and Self-Consistent Field Procedure

The total HF self-energy combines both contributions:

\[ \Sigma^{HF} = \Sigma^H + \Sigma^F \]

The effective single-particle HF Hamiltonian is then:

\[ H_{HF}(\mathbf{k}) = H_0(\mathbf{k}) + \Sigma^{HF}(\mathbf{k}) - \mu \hat{I} \]

where \(\mu\) is the chemical potential of the system and \(H_0\) is the non-interacting Hamiltonian constructed from hopping amplitudes and on-site energies.

Since both \(\Sigma^H\) and \(\Sigma^F\) depend on the density matrix \(n\), which is itself determined by \(H_{HF}\), the problem must be solved self-consistently. The SCF procedure proceeds as follows:

  1. Initialize: Construct the non-interacting Hamiltonian \(H_0\) from hopping amplitudes \(t\) and on-site energies \(\epsilon\) (NIHamiltonian class). Construct the bare Coulomb interaction \(V\) (VBare class).

  2. Compute self-energies: Given the current density matrix \(n\), evaluate \(\Sigma^H\) (SigmaHartree) and \(\Sigma^F\) (SigmaFock).

  3. Update Hamiltonian: Form \(H = H_0 + \Sigma^H + \Sigma^F\) in the Hamiltonian class.

  4. Adjust chemical potential: Search for \(\mu\) such that the target electron filling \(N_e\) is satisfied.

  5. Compute density: Evaluate the Fermi-Dirac distribution \(n = 1/(e^{\beta(H-\mu)} + 1)\) to obtain the updated density matrix.

  6. Check convergence: Compare the new density matrix with the previous iteration. If converged, stop; otherwise, apply density mixing (OccMixing) and return to step 2.

The Hamiltonian class acts as the central hub for this workflow, aggregating the self-energy contributions and managing the chemical potential search via CalMu0, SearchMu, and UpdateMu.

Bare Coulomb Interaction

The VBare class constructs the bare Coulomb interaction \(V\) from user-specified parameters. Interactions are handled in both local and non-local forms:

  • Local interactions: Specified through Slater or Kanamori parameterizations, with support for transformations between the two. The VLoc class in BLocStc provides SlaterKanamori, SlaterParameter, and KanamoriParameter methods.
  • Non-local interactions: Specified either explicitly through site-to-site couplings \(V_{ij}\) or generated from model potentials:
    • Ohno: \(V(r) = U / \sqrt{1 + (Ur/e^2)^2}\), interpolating between on-site \(U\) and long-range \(e^2/r\) Coulomb behavior.
    • JTH: Adopts the same functional form as Ohno but allows a consistent on-site Coulomb value across equivalent orbitals.

References

  • D. R. Hartree & W. Hartree, Proc. R. Soc. A 150, 9-33 (1935).
  • J. C. Slater, Phys. Rev. 32, 339 (1928); 35, 210 (1930); 81, 385 (1951).
  • C. Froese Fischer, Comput. Phys. Commun. 43, 355 (1987).
  • J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, 1960).
  • J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).
  • D. Van Der Marel & G. A. Sawatzky, Phys. Rev. B 37, 10674 (1988).
  • H. U. R. Strand, Phys. Rev. B 90, 155108 (2014).
  • K. Ohno, Theor. Chim. Acta 2, 219 (1964).