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# Low-scaling GW: The space-time formalism

## Contents

Available as of VASP.6 are low-scaling algorithms for ACFDT/RPA.[1] This page describes the formalism of the corresponding low-scaling GW approach.[2] 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.[3][4] avoid storage of the Green's function ${\displaystyle G}$ 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 ${\displaystyle \chi }$ and ${\displaystyle \Sigma }$ in reciprocal space and results in a relatively high computational cost that scales with ${\displaystyle N^{4}}$ (number of electrons).

The scaling with system size can, however, be reduced to ${\displaystyle N^{3}}$ by performing a so-called Wick-rotation to imaginary time ${\displaystyle t\to i\tau }$.[5]

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

${\displaystyle G({{\bf {r}}},{{\bf {r}}}',i\tau )=-\sum _{{n{{\bf {k}}}}}\phi _{{n{{\bf {k}}}}}^{{*(0)}}({{\bf {r}}})\phi _{{n{{\bf {k}}}}}^{{(0)}}({{\bf {r}}}')e^{{-(\epsilon _{{n{{\bf {k}}}}}-\mu )\tau }}\left[\Theta (\tau )(1-f_{{n{{\bf {k}}}}})-\Theta (-\tau )f_{{n{{\bf {k}}}}}\right]}$

Here ${\displaystyle \Theta }$ is the step function and ${\displaystyle f_{{n{{\bf {k}}}}}}$ the occupation number of the state ${\displaystyle \phi _{{n{{\bf {k}}}}}^{{(0)}}}$. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid ${\displaystyle \tau _{{m}}}$, 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.[6]

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

${\displaystyle \chi ({{\bf {r}}},{{\bf {r}}}',i\tau _{m})=-G({{\bf {r}}},{{\bf {r}}}',i\tau _{m})G({{\bf {r}}}',{{\bf {r}}},-i\tau _{m})}$

Afterwards, the same compressed Fourier transformation as for the low scaling ACFDT/RPA algorithms is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis ${\displaystyle \chi ({{\bf {r}}},{{\bf {r}}}',i\tau _{m})\to \chi _{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}},i\omega _{n})}$.[6][2]

The next step is the computation of the screened potential

${\displaystyle W_{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}},i\omega _{m})=\left[\delta _{{{{\bf {G}}}{{\bf {G}}}'}}-\chi _{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}},i\omega _{m})V_{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}})\right]^{{-1}}V_{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}})}$

followed by the inverse Fourier transform ${\displaystyle W_{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}},i\omega _{n})\to \chi ({{\bf {r}}},{{\bf {r}}}',i\tau _{m})}$ and the calculation of the self-energy

${\displaystyle \Sigma ({{\bf {r}}},{{\bf {r}}}',i\tau _{m})=-G({{\bf {r}}},{{\bf {r}}}',i\tau _{m})W({{\bf {r}}}',{{\bf {r}}},i\tau _{m})}$

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 ${\displaystyle \Sigma (z)}$ from the calculated set of NOMEGA points ${\displaystyle \Sigma (i\omega _{n})}$ and solves the non-linear eigenvalue problem

${\displaystyle \left[T+V_{{ext}}+V_{h}+\Sigma (z)\right]\left|\phi _{{n{\bf {k}}}}\right\rangle =z\left|\phi _{{n{\bf {k}}}}\right\rangle }$

on the real frequency axis ${\displaystyle z=\omega }$.

Because, preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation ${\displaystyle \Sigma (z)}$ 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.[2]

In addition, the space-time formulation allows to solve the full Dyson equation for ${\displaystyle G({{\bf {r,r'}}},i\tau )}$ with decent computational cost.[7] This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.