Time-frequency component of the GreenX library: minimax grids for efficient RPA and GW calculations

random-phase

With larger pre-factor, low-scaling algorithms are typically more expensive for smaller systems and only become more cost-effective than canonical implementations for larger systems thanks to their reduced scaling (Wilhelm et al., 2018).Furthermore, the numerical precision of low-scaling GW algorithms is strongly coupled to the time-frequency treatment (Wilhelm et al., 2021).Early low-scaling GW algorithms did not reach the same precision as canonical implementations (Förster & Visscher, 2020;Vlcek et al., 2017;Wilhelm et al., 2018).Although appropriate Fourier transforms and corresponding time-frequency grids have been implemented (Duchemin & Blase, 2021;Förster & Visscher, 2021a;Liu et al., 2016;Wilhelm et al., 2021), these implementations and grids are tied to particular codes and are often buried deeply inside the code.Furthermore, reuse of such implementations elsewhere is often restricted by license requirements or dependencies on definitions made in the host code.
In this work, we present the GX-TimeFrequency component of the GreenX library, an opensource package distributed under the Apache license (Version 2.0).GX-TimeFrequency provides time and frequency grids and corresponding integration weights to compute correlation energies for Green's function implementations.It also provides Fourier weights to convert between imaginary time and imaginary frequency.The library can be used for low-scaling RPA and GW implementations, or BSE codes, which use (low-scaling) GW as input.The minimax grids are also suitable for RPA implementations with conventional scaling (Del Ben et al., 2015): they are more compact than, e.g., Gauss-Legendre grids, resulting in a reduction of the computational prefactor, while yielding same accuracy (Del Ben et al., 2015).However, minimax grids are not recommended for conventional imaginary-frequency-only GW implementations (Ren, Rinke, Blum, et al., 2012) since they have not been optimized for the frequency integral of the self-energy.
While not being the main target of the library, the minimax time grids can also be utilized to calculate the LT-dMP2 correlation energy (Almlöf, 1991;Glasbrenner et al., 2020;Jung et al., 2004;Kaltak et al., 2014b;Takatsuka et al., 2008).The dMP2 term is one of two terms of the MP2 correlation energy and, in a diagrammatic representation, corresponds to the lowest order of the RPA correlation energy (Ren, Rinke, Joas, et al., 2012).The dMP2 correlation energy can be reformulated using the Laplace transform to obtain the LT-dMP2 expression which scales cubically in contrast to the ( 5 ) scaling of standard MP2.

Mathematical framework
The single-particle Green's function  and the non-interacting susceptibility  0 are starting points for several many-body perturbation theory methods.In canonical implementations,  0 (r, r ′ , ) is often expressed in the Adler-Wiser form (Adler, 1962;Wiser, 1963), where the sums over occupied (index ) and unoccupied (index ) single-particle states  are coupled via their corresponding energies .The Adler-Wiser expression of  0 () can be transformed into the imaginary time domain, χ 0 (r, r ′ ,  ) = −(r, r ′ ,  )(r ′ , r, − ), yielding the equation in the yellow box in Fig. 1, where the two sums are separated, leading to a favorable ( 3 ) scaling.The polarizability χ 0 ( ) is the starting point for LT-dMP2 and low-scaling RPA and GW.The low-scaling GW procedure shown in Fig. 1 is known as the space-time method and given here in its original formulation for planewave codes (Rojas et al., 1995).
The time-frequency integrals in Fig. 1 are performed numerically.All three methods in Fig. 1 require a discrete time grid {  }  =1 , where  is the number of grid points.RPA and GW additionally need the discrete frequency grid {  }  =1 .Since χ 0 (r, r ′ ,  ) is sharply peaked around the origin and then decays slowly, homogeneous time and frequency grids are inefficient.For this reason, non-uniform grids like Gauss-Legendre (Rieger et al., 1999), modified Gauss-Legendre (Ren, Rinke, Blum, et al., 2012) and the here presented minimax (Kaltak et al., 2014b) grids are used.The minimax grids include also integration weights for the computation of the correlation energies.For the calculation of the LT-dMP2 correlation energy  dMP2  (Kaltak et al., 2014b;Takatsuka et al., 2008), a time quadrature is performed, for which our library provides the integration weights {  }  =1 .Similarly, the RPA correlation energy  RPA  (Del Ben et al., 2015;Kaltak et al., 2014b) is computed from frequency quadrature using integration weights {  }  =1 .Low-scaling RPA and GW algorithms include the Fourier transform of χ 0 ( ) to  0 () (blue dashed box in Fig. 1).The GW space-time method performs two additional Fourier transforms: The screened Coulomb interaction  () is transformed to imaginary time (red dashed box), and the self-energy Σ( ) is Fourier transformed back to imaginary frequency (green dashed box).

Required input and output
GX-TimeFrequency requires as input the grid size , the minimal eigenvalue difference Δ min , and the maximal eigenvalue difference Δ max .For the output parameters, see with  being the identity matrix.Inputs and outputs are in atomic units.

Figure 1 :
Figure 1: Sketch of the methods supported by GX-TimeFrequency which start from χ 0 ( ).In addition to the discrete time and frequency grids {  } and {  }, the library provides the corresponding weights {  } and {  } for the integration of the correlation energy   as well as the Fourier weights   ,   and   defined in Equations 2-4.The bare and screened Coulomb interactions are indicated by (r, r ′ ) = 1/|r − r| ′ and  (), respectively.() is the dynamical dielectric function, Σ the GW self-energy, and AC stands for analytic continuation.

Table 1 :
Output returned by the GX-TimeFrequency component of GreenX.We abbreviate low-scaling as ls, and least-squares optimization as L2 opt.