Low-scaling GW: The space-time formalism: Difference between revisions

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Available as of VASP.6 are low-scaling algorithms for ACFDT/RPA.<ref name="kaltak2"/> This page describes the formalism of the corresponding low-scaling GW approach.<ref name="liu"/>  
Available as of VASP.6 are low-scaling algorithms for [[Groundstate in the Random Phase Approximation|ACFDT/RPA]].<ref name="kaltak2"/> This page describes the formalism of the corresponding low-scaling GW approach.<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]].  
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]].  


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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"/>
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  
Following the [[Groundstate in the Random Phase Approximation#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>  
<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>  

Revision as of 15:43, 25 July 2019

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \chi and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Sigma in reciprocal space and results in a relatively high computational cost that scales with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): N^{4} (number of electrons).

The scaling with system size can, however, be reduced to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): N^{3} by performing a so-called Wick-rotation to imaginary time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Theta is the step function and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f_{{n{{\bf {k}}}}} the occupation number of the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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 algorithm is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Sigma (z) from the calculated set of NOMEGA points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Sigma (i\omega _{n}) and solves the non-linear eigenvalue problem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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

on the real frequency axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): z=\omega .

Because, preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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.

References

  1. ↑ M. Kaltak, J. Klimeš, and G. Kresse, Phys. Rev. B 90, 054115 (2014)
  2. ↑ a b c P. Liu, M. Kaltak, J. Klimes and G. Kresse, Phys. Rev. B 94, 165109 (2016).
  3. ↑ M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).
  4. ↑ M. Shishkin and G. Kresse, Phys. Rev. B 75, 235102 (2007).
  5. ↑ H. N. Rojas, R. W. Godby and R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995)
  6. ↑ a b M. Kaltak, J. Klimeš, and G. Kresse, Journal of Chemical Theory and Computation 10, 2498-2507 (2014).
  7. ↑ M. Grumet, P. Liu, M. Kaltak, J. Klimeš and Georg Kresse, Phys. Ref. B 98, 155143 (2018).
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