GW Theory

The GW approximation is a many-body perturbation theory method for calculating quasiparticle energies. This page provides a theoretical overview of the methods implemented in QuasiX.

Introduction

Quasiparticle energies describe the energies required to add or remove an electron from a many-body system. Unlike Kohn-Sham DFT eigenvalues (which are auxiliary quantities), quasiparticle energies have direct physical meaning:

  • Ionization Potential (IP): Energy to remove an electron (negative of HOMO QP energy)

  • Electron Affinity (EA): Energy released when adding an electron (negative of LUMO QP energy)

  • Fundamental Gap: IP - EA = HOMO - LUMO (QP gap)

The GW approximation provides a systematic approach to computing these quantities with typical errors of 0.2-0.5 eV compared to experiment.

Many-Body Perturbation Theory

Starting Point: Green’s Function

The one-particle Green’s function \(G(\mathbf{r}, \mathbf{r}'; \omega)\) contains information about the single-particle excitation spectrum. Its poles correspond to quasiparticle energies.

The Green’s function satisfies the Dyson equation:

\[G = G_0 + G_0 \Sigma G\]

where \(G_0\) is the non-interacting Green’s function and \(\Sigma\) is the self-energy.

The GW Approximation

In the GW approximation, the self-energy is:

\[\Sigma(\mathbf{r}, \mathbf{r}'; \omega) = \frac{i}{2\pi} \int d\omega' \, G(\mathbf{r}, \mathbf{r}'; \omega + \omega') W(\mathbf{r}, \mathbf{r}'; \omega')\]

where:

  • \(G\) is the Green’s function

  • \(W = \varepsilon^{-1} v\) is the screened Coulomb interaction

  • \(v\) is the bare Coulomb interaction

  • \(\varepsilon\) is the dielectric function

This is the first-order term in the screened interaction expansion, hence “GW”.

Quasiparticle Equation

The central equation of GW calculations is the quasiparticle equation:

\[\varepsilon_n^{\text{QP}} = \varepsilon_n^{\text{MF}} + \langle \psi_n | \Sigma(\varepsilon_n^{\text{QP}}) - V_{xc} | \psi_n \rangle\]

where:

  • \(\varepsilon_n^{\text{QP}}\) is the quasiparticle energy

  • \(\varepsilon_n^{\text{MF}}\) is the mean-field (DFT/HF) energy

  • \(\Sigma\) is the GW self-energy

  • \(V_{xc}\) is the exchange-correlation potential from DFT (or exchange for HF)

  • \(\psi_n\) are the mean-field orbitals

Self-Energy Components

The self-energy naturally splits into exchange and correlation:

\[\Sigma = \Sigma^x + \Sigma^c\]

Exchange Self-Energy (exact exchange):

\[\Sigma^x_{nm} = -\sum_i^{\text{occ}} (ni|mi)\]

This is independent of frequency and gives the Hartree-Fock exchange energy when \(n = m\).

Correlation Self-Energy:

\[\Sigma^c_n(\omega) = \frac{i}{2\pi} \int d\omega' \, G_n(\omega + \omega') \left[ W(\omega') - v \right]\]

The correlation self-energy accounts for screening effects beyond mean-field theory.

G0W0: One-Shot GW

In G0W0 (also called “one-shot GW”), the Green’s function and screened interaction are constructed from the starting mean-field orbitals and evaluated once.

Workflow

  1. Mean-field calculation: Obtain orbitals \(\{\psi_n\}\) and energies \(\{\varepsilon_n\}\) from DFT or HF

  2. Polarizability: Construct the non-interacting polarizability

    \[P_0(\omega) = -i \int dt \, e^{i\omega t} G_0(t) G_0(-t)\]

  3. Dielectric function: Compute the RPA dielectric function

    \[\varepsilon(\omega) = 1 - v P_0(\omega)\]

  4. Screened interaction: Invert to get the screened Coulomb

    \[W(\omega) = \varepsilon^{-1}(\omega) v\]

  5. Self-energy: Evaluate exchange and correlation self-energies

  6. QP energies: Solve the quasiparticle equation

Linearized QP Equation

Direct solution of the QP equation requires iteration since \(\Sigma\) depends on \(\varepsilon_n^{\text{QP}}\). For efficiency, QuasiX uses the linearized form:

\[\varepsilon_n^{\text{QP}} \approx \varepsilon_n^{\text{MF}} + Z_n \text{Re}\langle \psi_n | \Sigma(\varepsilon_n^{\text{MF}}) - V_{xc} | \psi_n \rangle\]

where the renormalization factor is:

\[Z_n = \left( 1 - \frac{\partial \text{Re}\Sigma^c_n}{\partial \omega} \bigg|_{\omega=\varepsilon_n^{\text{MF}}} \right)^{-1}\]

The Z-factor is typically 0.7-0.9 and accounts for the energy dependence of the self-energy.

Newton Solver

For difficult cases (e.g., strong satellites), QuasiX also provides a Newton solver that iteratively solves the full QP equation:

\[\varepsilon_n^{(k+1)} = \varepsilon_n^{(k)} - \frac{f(\varepsilon_n^{(k)})}{f'(\varepsilon_n^{(k)})}\]

where \(f(\omega) = \omega - \varepsilon_n^{\text{MF}} - \text{Re}\langle \psi_n | \Sigma(\omega) - V_{xc} | \psi_n \rangle\).

evGW: Eigenvalue Self-Consistent GW

In G0W0, results depend on the starting point (HF, PBE, PBE0, etc.). Eigenvalue self-consistent GW (evGW) reduces this dependence by iterating the QP energies.

Algorithm

  1. Initialize: \(\varepsilon_n^{(0)} = \varepsilon_n^{\text{MF}}\)

  2. For iteration \(k = 1, 2, \ldots\):

    1. Compute \(P_0^{(k)}\) using energies \(\varepsilon_n^{(k-1)}\)

    2. Compute \(W^{(k)}\) from \(P_0^{(k)}\)

    3. Evaluate \(\Sigma^{(k)}\)

    4. Solve QP equation for \(\varepsilon_n^{(k)}\)

    5. Check convergence: \(|\varepsilon_n^{(k)} - \varepsilon_n^{(k-1)}| < \text{tol}\)

  3. Output converged \(\varepsilon_n^{\text{QP}}\)

Note

In evGW, only the eigenvalues are updated; the orbitals remain frozen. This typically converges in 5-15 iterations with tolerance 10^-4 eV.

Starting Point Dependence

Different starting points give different results:

Starting Point Effects (H2O/def2-TZVP)

Method

Starting Point

HOMO (eV)

Starting Point Variation

G0W0

HF

-12.72

0.4 eV

G0W0

PBE

-12.38

(spread)

G0W0

PBE0

-12.57

evGW

HF

-12.55

0.05 eV

evGW

PBE

-12.52

(reduced)

evGW

PBE0

-12.54

evGW significantly reduces starting point dependence.

Density Fitting (RI Approximation)

QuasiX uses the Resolution of Identity (RI) approximation for computational efficiency. Orbital products are expanded in an auxiliary basis:

\[\psi_i(\mathbf{r})\psi_a(\mathbf{r}) \approx \sum_P C_{ia}^P \chi_P(\mathbf{r})\]

where \(\{\chi_P\}\) is the auxiliary basis. The fitting coefficients are:

\[C_{ia}^P = \sum_Q (ia|Q) (Q|P)^{-1/2}\]

This reduces the scaling of four-center integrals \((ia|jb)\) to products of three-center quantities:

\[(ia|jb) \approx \sum_{PQ} C_{ia}^P S_{PQ}^{-1} C_{jb}^Q = \sum_P C_{ia}^P C_{jb}^P\]

Benefit: O(N^5) -> O(N^4) scaling with negligible loss of accuracy (~1 meV).

Frequency Integration

Imaginary Frequency Formulation

The correlation self-energy is evaluated on the imaginary frequency axis where functions are smooth:

\[\Sigma^c(\omega) = \frac{1}{\pi} \int_0^\infty d\omega' \, \text{Im}[W(i\omega')] F(\omega, i\omega')\]

where \(F\) contains Green’s function contributions.

Analytic Continuation

Results on the imaginary axis are analytically continued to the real axis using:

  • Pade approximants (default): Fit rational function to imaginary-axis data

  • Two-pole model: For difficult cases with strong satellite features

Frequency Grids

QuasiX supports two frequency integration schemes:

  1. Minimax grids (default): Optimized for GW, requiring minimal points (32 typical)

  2. Gauss-Legendre: Standard quadrature, requiring more points (64+)

# Minimax (recommended)
gw = G0W0Driver(mf, freq_method='minimax', n_freq=32)

# Gauss-Legendre
gw = G0W0Driver(mf, freq_method='gauss-legendre', n_freq=64)

Implementation Details

QuasiX implements GW using the contour deformation approach:

  1. Contour Deformation: Splits the frequency integral into contributions along the real and imaginary axes

  2. RI-GW: All quantities computed using density-fitted integrals

  3. Parallel Frequency Integration: Frequency points evaluated in parallel using Rayon

  4. SIMD Optimization: Tensor operations use AVX-512 where available

References

  1. Hedin, L. (1965). Phys. Rev. 139, A796. “New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem”

  2. Hybertsen, M. S. & Louie, S. G. (1986). Phys. Rev. B 34, 5390. “Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies”

  3. Aryasetiawan, F. & Gunnarsson, O. (1998). Rep. Prog. Phys. 61, 237. “The GW method”

  4. van Setten, M. J. et al. (2015). J. Chem. Theory Comput. 11, 5665. “GW100: Benchmarking G0W0 for Molecular Systems”

  5. Golze, D., Dvorak, M. & Rinke, P. (2019). Front. Chem. 7, 377. “The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy”