Hartree-Fock and Starting Points

The previous page defined the finite-basis Hamiltonian in terms of the one-electron matrix $H_0$, the overlap matrix $S$, and the bare Coulomb integrals $U$. The next object needed before GF2 or GW is a mean-field description of the same Hamiltonian. In the Green weak-coupling literature, that role is usually played by Hartree-Fock (HF), density functional theory (DFT), or a density matrix imported from one of these calculations.¹, ², ³

This page introduces Hartree-Fock as a controlled first approximation to the interacting-electron problem, explains the Hartree and exchange terms that form the Fock matrix, and connects the Fock contribution to the static self-energy $\Sigma^\infty$. We still avoid the time- and frequency-dependent Green’s-function machinery; the only self-energy quantity used here is $\Sigma^\infty$, the frequency-independent part that behaves like an effective one-particle potential.

Mean-field idea

The exact many-electron wave function depends on the coordinates of all electrons at once. Hartree-Fock replaces it by a single Slater determinant, which is the antisymmetrized product of one-particle orbitals. This is the simplest wave function that obeys the Pauli principle. It cannot describe all correlation effects, because each electron moves in an average field produced by the others, but it does include exchange exactly within that one-determinant picture.

In a finite basis, the Hartree-Fock solution is represented by a one-particle density matrix $P$. For a closed-shell molecule in a spatial-orbital basis, $P_{ij}$ roughly measures how much the occupied molecular orbitals overlap basis functions $i$ and $j$. In a spin-orbital or periodic Bloch basis the same object carries spin and possibly $k$ labels. This density matrix is central: once $P$ is known, the electron-electron interaction can be contracted into an effective one-electron matrix.

Hartree and exchange

The Coulomb interaction contributes two static mean-field terms. The Hartree term is the classical electrostatic potential generated by the electron density. It is repulsive: an electron in orbital $i$ feels the average charge of all other occupied orbitals. In periodic GW notation this term is often denoted $J$.²

The exchange term is different. It has no classical electrostatic analogue; it comes from antisymmetry of the Slater determinant. Exchanging two same-spin electrons changes the sign of the many-electron wave function, and that sign change produces an effective nonlocal potential. In the periodic GW notation used in the literature, this contribution is denoted $K$ and enters with a minus sign in the usual Coulomb-integral convention.²

Together these terms form the Hartree-Fock contribution to the one-particle problem. A common molecular closed-shell convention, used for example in the Legendre-spectral GF2 paper, writes the Hartree-Fock self-energy as $\Sigma^{\mathrm{HF}}_{ij}=\sum_{kl}P_{kl}(v_{ijkl}-v_{ilkj}/2)$, where $v_{ijkl}$ is the electron-electron Coulomb integral and $P$ is the density matrix.¹ The first part of the parenthesis is the direct Hartree contribution. The second part is the exchange contribution. The factor $1/2$ is a convention-dependent consequence of using spatial orbitals with spin summed into $P$; in a spin-orbital notation the same physics is distributed differently among the spin sums.

The Fock matrix is then $F=h+\Sigma^{\mathrm{HF}}$, where $h$ is the one-electron part containing the kinetic energy and the electron-nuclear or pseudopotential terms.¹ In the notation used elsewhere in these docs, $h$ is the same object as $H_0$ unless spin-orbit or other one-body structure makes a more explicit index notation useful.

Self-consistency in Hartree-Fock

Hartree-Fock is self-consistent because the Fock matrix depends on the density matrix, while the density matrix is obtained from the orbitals that diagonalize the Fock matrix. The calculation therefore alternates between two operations.

First, build $F$ from the current density matrix $P$. Second, solve the generalized one-particle problem $FC=SC\varepsilon$ in a non-orthogonal basis, or $FC=C\varepsilon$ in an orthonormal basis, to obtain new molecular orbitals or Bloch orbitals. Occupying those orbitals gives a new $P$. The cycle is repeated until the density, energy, and Fock matrix stop changing within chosen tolerances.

This procedure is often called the self-consistent field (SCF) method. The word “field” means the average field generated by the electrons themselves. At convergence, each orbital is optimal for the average field created by all occupied orbitals, and that average field is consistent with the orbitals that produced it.

Relation to $\Sigma^\infty$

In GF2 and GW calculations, the self-energy is separated into a static part and a dynamical part. The static part is called $\Sigma^\infty$. It is the part that does not depend on time or frequency, and it is the piece that survives as the high-frequency limit of the self-energy. In the implementations discussed in this literature, this static part is the Hartree-Fock self-energy built from the current density matrix.¹, ²

For this reason, the same idea appears under several names. In molecular GF2 discussions one often sees $F=h+\Sigma^{\mathrm{HF}}$. In fully self-consistent periodic GW, the static contribution is written as $\Sigma^\infty=J+K$, with $J$ the Hartree term and $K$ the exchange term.² These notations refer to the same physical layer: the instantaneous Coulomb and exchange response of the electrons to the density matrix.

This connection is useful operationally. A weak-coupling calculation does not discard Hartree-Fock theory when it goes beyond it. Instead, it keeps the static Hartree-Fock-like term as $\Sigma^\infty$ and adds a dynamical correlation contribution in the later GF2 or GW equations. The static part is cheap compared with the dynamical part; in the periodic GW implementation, the exchange build is the expensive part of $\Sigma^\infty$, but the dynamical contribution is typically still much more costly overall.²

Density functional starting points

Density functional theory is another common way to obtain a mean-field-like starting point. Kohn-Sham DFT also solves an effective one-particle problem, but the effective potential is built from the electron density through an exchange-correlation functional rather than from exact Hartree-Fock exchange alone. In practice, DFT provides orbitals, orbital energies, and a density matrix.

Several DFT functionals appear repeatedly in the literature. The periodic self-consistent GW implementation was tested from LDA, PBE, and HF starting solutions for silicon, and the final fully self-consistent GW spectra were found to converge consistently to the same result in that case.² PBE is also used as a standard generalized-gradient DFT reference in relativistic solid calculations and with GTH-PBE pseudopotentials.³ PBE0, a hybrid functional that mixes exact exchange with PBE-based exchange and correlation, also appears as a mean-field comparison in relativistic molecular benchmarks.³

For GF2 or fully self-consistent GW, a DFT calculation should be viewed primarily as a way of choosing an initial density matrix and orbital representation. The DFT exchange-correlation potential is not the same object as the later GF2 or GW correlation treatment. Once the weak-coupling calculation is started, the static part $\Sigma^\infty$ is built from the Coulomb integrals and the current density matrix, while the later correlation approximation is supplied by the GF2 or GW equations.

Thus HF, LDA, PBE, and PBE0 do not define different Hamiltonians if the same $H_0$, $S$, and $U$ are used. They define different starting points for solving that Hamiltonian. The difference enters through the initial density matrix: HF tends to produce densities and orbital energies shaped by exact exchange and no dynamical correlation; semilocal DFT functionals such as LDA and PBE include approximate exchange-correlation effects through the density; hybrid functionals such as PBE0 mix a fraction of exact exchange with a DFT exchange-correlation functional.

Why the starting point still matters

In a fully self-consistent calculation, the formal goal is to remove the dependence on the initial mean-field reference. This is one of the motivations for self-consistent GW: unlike one-shot GW, it is not supposed to retain an arbitrary dependence on whether the calculation began from HF, PBE, PBE0, or another reference.²

The starting point nevertheless matters in practice. It can strongly affect convergence speed and stability. It can also matter in systems with multiple self-consistent solutions, where an initial density matrix may guide the iteration toward a metastable state.², ⁴ This is why Green workflows typically prepare the Hamiltonian data together with a reasonable starting density matrix from a mean-field calculation before the GF2 or GW iteration begins.

The conceptual hierarchy is therefore simple. The Hamiltonian page defines $H_0$, $S$, and $U$. Hartree-Fock and DFT provide practical one-particle starting points and density matrices. The static Hartree-Fock contribution built from that density matrix is the $\Sigma^\infty$ term used later. Only after these ingredients are in place do GF2 and GW introduce the dynamical many-body corrections.