View source for Low-scaling GW: The space-time formalism

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Available as of VASP.6 are low scaling algorithms for ACFDT/RPA<ref name="kaltak2"/> and GW calculations.<ref name="liu"/> 
A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW|here]]. 

== Theory == 
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The GW implementations in VASP described in the papers of Shishkin ''et al.''<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/> avoid storage of the Green's function <math>G</math> 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 <math>\chi</math> and <math>\Sigma</math> in reciprocal space and results in a relatively high computational cost that scales with <math>N^4</math> (number of electrons).

The scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.<ref name="rojas"/>

Following the [[ACFDT/RPA calculations#ACFDTR/RPAR| 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 

<math>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]</math> 

Here <math>\Theta</math> is the step function and <math>f_{n{\bf k}}</math> the occupation number of the state <math>\phi_{n{\bf k}}^{(0)}</math>. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid <math>\tau_{m}</math>, where the number of time points can be selected in VASP via the {{TAG|NOMEGA}} tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.<ref name="kaltak"/>

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

<math>
 \chi({\bf r},{\bf r}',i\tau_m) = -G({\bf r},{\bf r}',i\tau_m)G({\bf r}',{\bf r},-i\tau_m)
</math>

Afterwards, the same compressed Fourier transformation as for the [[ACFDT/RPA calculations#ACFDTR/RPAR| low scaling ACFDT/RPA algorithm]] is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis <math>\chi({\bf r},{\bf r}',i\tau_m) \to \chi_{{\bf G}{\bf G}'}({\bf q},i \omega_n) </math>.<ref name="kaltak"/><ref name="liu"/>

The next step is the computation of the screened potential

<span id="W2_reci">
<math>
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})
</math>


followed by the inverse Fourier transform <math>W_{{\bf G}{\bf G}'}({\bf q},i \omega_n) \to \chi({\bf r},{\bf r}',i\tau_m)</math> and the calculation of the self-energy 

<math>
 \Sigma({\bf r},{\bf r}',i\tau_m) = -G({\bf r},{\bf r}',i\tau_m)W({\bf r}',{\bf r},i\tau_m)
</math>

From here, several routes are possible including all approximations mentioned above, that is the single-shot, EVG<sub>0</sub> and QPEVG<sub>0</sub> 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 <math>\Sigma(z)</math> from the calculated set of {{TAG|NOMEGA}} points <math>\Sigma(i\omega_n)</math> and solves the non-linear eigenvalue problem 

<math>
\left\langle \phi_{n\bf k} \right|  T+V_{ext}+V_h+\Sigma(z)   \left|\phi_{n\bf k}\right\rangle  = z\left| \phi_{n\bf k}\right\rangle
</math>

on the real frequency axis <math>z=\omega</math>. 

Because, preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation <math>\Sigma(z)</math> 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.<ref name="liu"/>

In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.<noinclude>

== References ==
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<ref name="kaltak2">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.054115 M. Kaltak, J. Klimeš, and G. Kresse, Phys. Rev. B 90, 054115 (2014)]</ref>
<ref name="liu">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.165109 P. Liu, M. Kaltak, J. Klimes and G. Kresse, Phys. Rev. B 94, 165109 (2016).]</ref>
<ref name="rojas">[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.1827 H. N. Rojas, R. W. Godby and R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995)]</ref>
<ref name="shishkin-PRB74">[http://link.aps.org/doi/10.1103/PhysRevB.74.035101 M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).]</ref>
<ref name="shishkin-PRB75">[http://link.aps.org/doi/10.1103/PhysRevB.75.235102 M. Shishkin and G. Kresse, Phys. Rev. B 75, 235102 (2007).]</ref>
<ref name="liu">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.165109 P. Liu, M. Kaltak, J. Klimes and G. Kresse, Phys. Rev. B 94, 165109 (2016).]</ref>
<ref name="kaltak">[http://pubs.acs.org/doi/abs/10.1021/ct5001268 M. Kaltak, J. Klimeš, and G. Kresse, Journal of Chemical Theory and Computation 10, 2498-2507 (2014). ]</ref>
<ref name="grumet">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.155143 M. Grumet, P. Liu, M. Kaltak, J. Klimeš and Georg Kresse, Phys. Ref. B 98, 155143 (2018). ]</ref>
</references>
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[[Category:VASP]][[Category:Many-Body Perturbation Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]]
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