GW approximation of Hedin's equations: Difference between revisions

Revision as of 17:54, 23 July 2019

Green's functions

The GW method can be understood in terms of the following eigenvalue equation^[1]

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+V_{{ext}}+V_{h})\phi _{{n{{\bf {k}}}}}({{\bf {r}}})+\int d{{\bf {r}}}\Sigma ({{\bf {r}}},{{\bf {r}}}',\omega =E_{{n{{\bf {k}}}}})\phi _{{n{{\bf {k}}}}}({{\bf {r}}}')=E_{{n{{\bf {k}}}}}\phi _{{n{{\bf {k}}}}}({{\bf {r}}})

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/":): T is the kinetic 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/":): V_{{ext}} the external potential of the nuclei, 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/":): V_{h} the Hartree potential 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/":): E_{{n{{\bf {k}}}}} the quasiparticle energies with orbitals 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}}}}} . In contrast to DFT, the exchange-correlation potential is replaced by the many-body 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 and should be obtained together with 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 , the irreducible 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/":): \chi , the screened Coulomb interaction 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 and the irreducible vertex 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/":): \Gamma in a self-consistent procedure. For completeness, these equations are^[2]

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(1,2)=G_{0}(1,2)+\int d(3,4)G_{0}(1,3)\Sigma (3,4)G(4,2)

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 (1,2)=\int d(3,4)G(1,3)G(4,1)\Gamma (3,4;2)

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(1,2)=V(1,2)+\int d(3,4)V(1,3)\chi (3,4)W(4,2)

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 (1,2)=\int d(3,4)G(1,3)\Gamma (3,2;4)W(4,1)

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/":): \Gamma (1,2;3)=\delta (1,2)\delta (1,3)+\int d(4,5,6,7){\frac {\delta \Sigma (1,2)}{\delta G(4,5)}}G(4,6)G(7,5)\Gamma (6,7;3)

Here the common notation 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/":): 1=({{\bf {r}}}_{1},t_{1}) was adopted 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/":): V denotes the bare Coulomb interaction. Note, that these equations are exact and provide an alternative to the Schrödinger equation for the many-body problem. Nevertheless, approximations are necessary for realistic systems. The most popular one is the GW approximation and is obtained by neglecting the equation for the vertex function and using the bare vertex 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/":): \Gamma (1,2;3)=\delta (1,2)\delta (1,3) instead.

This means that the equations for the polarizability and self-energy reduce 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/":): \chi (1,2)=G(1,2)G(2,1)

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 (1,2)=G(1,2)W(2,1)

while the equations for the Green's function and the screened potential remain the same. However, in practice, these equations are usually solved in reciprocal space in the frequency domain

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

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 {G}}}{{\bf {G}}}'}}({{\bf {q}}},\omega )=\left[\delta _{{{{\bf {G}}}{{\bf {G}}}'}}-\Sigma _{{{{\bf {G}}}{{\bf {G}}}'}}({{\bf {q}}},\omega )G_{{{{\bf {G}}}{{\bf {G}}}'}}^{{(0)}}({{\bf {q}}})\right]^{{-1}}G_{{{{\bf {G}}}{{\bf {G}}}'}}^{{(0)}}({{\bf {q}}})

In principle Hedin's equations have to be solved self-consistently, where in the first iteration 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^{{(0)}} is the non-interacting 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^{{(0)}}({{\bf {r}}},{{\bf {r}}}',\omega )=\sum _{{n{{\bf {k}}}}}{\frac {\phi _{{n{{\bf {k}}}}}^{{*(0)}}({{\bf {r}}})\phi _{{n{{\bf {k}}}}}^{{(0)}}({{\bf {r}}}')}{\omega -E_{{n{{\bf {k}}}}}^{{(0)}}}}

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/":): \phi _{{n{{\bf {k}}}}}^{{(0)}} being a set of one-electron orbitals 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/":): E_{{n{{\bf {k}}}}}^{{(0)}} the corresponding energies. Afterwards 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/":): \chi ^{{(0)}} is determined, followed by 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^{{(0)}} and 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 ^{{(0)}} . This means that GW calculations require a first guess for the one-electron eigensystem, which is usually taken from a preceding DFT step.

In principle, one has to repeat all steps by the updating the Green's function with the Dyson equation given above in each iteration cycle until self-consistency is reached. In practice, this is hardly ever done due to computational complexity on the one hand (in fact fully self-consistent GW calculations are available as of VASP 6 only).

On the other hand, one observes that by keeping 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 in the first iteration to the DFT level one benefits from error cancelling,^[3] which is the reason why often the screening is kept on the DFT level and one aims at self-consistency in Green's function only.

Following possible approaches are applied in practice.

Single Shot: G₀W₀

Partially self-consistent: GW₀ or EVGW₀

Self-consistent Quasi-particle approximation: QPGW₀

Low-scaling GW: The Space-time Formalism

The scaling with system size 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 (number of electrons) of GW calculations can be reduced^[4] 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 .

References

[HybertsenLouie-1] M. S. Hybertsen, S. G. Louie Phys. Ref. B 34, 5390 (1986)

[Hedin-2] L. Hedin, Phys. Rev. 139, A796 (1965)

[shishkin:prl:07-3] M. Shishkin, M. Marsman, and G. Kresse, Phys. Rev. Lett. 99, 246403 (2007).

[rojas-4] H. N. Rojas, R. W. Godby and R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995)

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@@ Line 98: / Line 98: @@
 == Low-scaling GW: The Space-time Formalism ==
-The scaling with system size <math>N</math> (number of electrons) of GW calculations can be reduced by performing a so-called Wick-rotation to imaginary time <math>t\to -i\tau</math>.
+The scaling with system size <math>N</math> (number of electrons) of GW calculations can be reduced<ref name="rojas"/> by performing a so-called Wick-rotation to imaginary time <math>t\to -i\tau</math>.
 == References ==