Low-scaling GW: The space-time formalism

Available as of VASP.6 are low scaling algorithms for ACFDT/RPA and GW calculations. A theoretical description of the ACFDT/RPA total energies is found here. A brief summary regarding GW theory is given below, while a practical guide can be found here.

Theory

The GW implementations in VASP described in the papers of Shishkin et al. avoid storage of the Green's function $G$ as well as Fourier transformations between time and frequency domain entirely. That is, all calculations are performed solely on the real frequency axis using Kramers-Kronig transformations for convolutions in the equation of $\chi$ and $\mathrm{\Sigma }$ in reciprocal space and results in a relatively high computational cost that scales with $N^4$ (number of electrons).

The scaling with system size can, however, be reduced to $N^3$ by performing a so-called Wick-rotation to imaginary time $t\to i\tau$.

Following the low scaling ACFDT/RPA algorithms the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space

$G(\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i\tau )=-\sum _{n\mathbf{𝐤}}{\varphi }_{n\mathbf{𝐤}}^{*(0)}(\mathbf{𝐫}){\varphi }_{n\mathbf{𝐤}}^{(0)}({\mathbf{𝐫}}^{\prime })e^{-({ϵ}_{n\mathbf{𝐤}}-\mu )\tau }[\mathrm{\Theta }(\tau )(1-f_{n\mathbf{𝐤}})-\mathrm{\Theta }(-\tau )f_{n\mathbf{𝐤}}]$

Here $\mathrm{\Theta }$ is the step function and $f_{n\mathbf{𝐤}}$ the occupation number of the state ${\varphi }_{n\mathbf{𝐤}}^{(0)}$. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid ${\tau }_m$, where the number of time points can be selected in VASP via the NOMEGA tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.

Subsequently, the irreducible polarizability is calculated from a contraction of two imaginary time Green's functions

$\chi (\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i{\tau }_m)=-G(\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i{\tau }_m)G({\mathbf{𝐫}}^{\prime },\mathbf{𝐫},-i{\tau }_m)$

Afterwards, the same compressed Fourier transformation as for the low scaling ACFDT/RPA algorithm is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis $\chi (\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i{\tau }_m)\to {\chi }_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},i{\omega }_n)$.

The next step is the computation of the screened potential

$W_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},i{\omega }_m)={[{\delta }_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}-{\chi }_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},i{\omega }_m)V_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪})]}^{-1}{V}_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪})$

followed by the inverse Fourier transform $W_{\mathbf{𝐆}{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},i{\omega }_n)\to \chi (\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i{\tau }_m)$ and the calculation of the self-energy

$\mathrm{\Sigma }(\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i{\tau }_m)=-G(\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i{\tau }_m)W({\mathbf{𝐫}}^{\prime },\mathbf{𝐫},i{\tau }_m)$

From here, several routes are possible including all approximations mentioned above, that is the single-shot, EVG₀ and QPEVG₀ approximation. All approximations have one point in common.

In contrast to the real-frequency implementation, the low-scaling GW algorithms require an analytical continuation of the self-energy from the imaginary frequency axis to the real axis. In general, this is an ill-defined problem and usually prone to errors, since the self-energy is known on a finite set of points. VASP determines internally a Padé approximation of the self-energy $\mathrm{\Sigma }(z)$ from the calculated set of NOMEGA points $\mathrm{\Sigma }(i{\omega }_n)$ and solves the non-linear eigenvalue problem

$⟨{\varphi }_{n\mathbf{𝐤}}|T+V_{ext}+V_h+\mathrm{\Sigma }(z)|{\varphi }_{n\mathbf{𝐤}}⟩=z|{\varphi }_{n\mathbf{𝐤}}⟩$

on the real frequency axis $z=\omega$.

Because, preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation $\mathrm{\Sigma }(z)$ of the self-energy can be determined with high accuracy. The analytical continuation typically yields energies that differ less than 20 meV from quasi-particle energies obtained from the real-frequency calculation.

In addition, the space-time formulation allows to solve the full Dyson equation for $G(\mathbf{𝐫},{\mathbf{𝐫}}^{\prime },i\tau )$ with decent computational cost. This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.

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