Low-scaling GW: The space-time formalism

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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 algorithms 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[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 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.

Finite temperature formalism

The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.[8] This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature 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/":): {\displaystyle T>0} , which may be understood by an analytical continuation of the real-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 the imaginary time 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/":): {\displaystyle -i\tau } . Matsubara has shown that this Wick rotation in 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/":): {\displaystyle t\to -i\tau } reveals an intriguing connection to the inverse temperature 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/":): {\displaystyle \beta =1/T} of the system.[9] More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability 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/":): {\displaystyle \chi (-i\tau )} ) over the fundamental interval 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/":): {\displaystyle -\beta \leq \tau \leq \beta } .

As a consequence, one decomposes imaginary time quantities into a Fourier series with period 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/":): \beta that determines the spacing of the Fourier modes. For instance the imaginary polarizability can be written as

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/":): {\displaystyle \chi (-i\tau )={\frac {1}{\beta }}\sum _{m=-\infty }^{\infty }{\tilde {\chi }}(i\nu _{m})e^{-i\nu _{m}\tau },\quad \nu _{m}={\frac {2m}{\beta }}\pi }

and the corresponding random-phase approximation of the correlation energy at finite temperature becomes a series over (in this case, bosonic) Matsubara frequencies

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/":): {\displaystyle \Omega _{c}^{\rm {RPA}}={\frac {1}{2}}{\frac {1}{\beta }}\sum _{m=-\infty }^{\infty }{\rm {Tr}}\left\lbrace \ln \left[1-{\tilde {\chi }}(i\nu _{m})V\right]-{\tilde {\chi }}(i\nu _{m})V\right\rbrace ,\quad \nu _{m}={\frac {2m}{\beta }}\pi }

The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical 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/":): {\displaystyle \epsilon _{n{\bf {k}}}\approx \mu } , such that Matsubara series also converge for metallic systems.

Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method.[10] This approach converges exponentially with the number of considered frequency points.

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).
  8. ↑ W. Kohn and J. M. Luttinger, Phys. Rev. 118, 41 (1960).
  9. ↑ T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).
  10. ↑ M. Kaltak and G. Kresse, Phys. Rev. B. 101, 205145 (2020).
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