Table of Contents

1. Brief description

This algorithm calculates the electronic transition structure factor and performs a tricubic interpolaton to estimate a finite size correlation energy correction.

2. Algorithm call

A typical input file snippet to call the FiniteSizeCorrection algorithm is given below.

- name: FiniteSizeCorrection
    slicedCoulombVertex: CoulombVertex
    amplitudes: Amplitudes
    coulombVertexSingularVectors: CoulombVertexSingularVectors
    coulombPotential: CoulombPotential
    gridVectors: GridVectors
    transitionStructureFactor: SF

3. Algorithm input

Table 1: Input keywords for CoupledCluster
Keyword Value
slicedCoulombVertex Sliced Coulomb vertex
amplitudes Singles and doubles amplitudes
coulombVertexSingularVectors coulombVertexSingularVectors
coulombPotential Coulomb potential
gridVectors Grid Vectors
interpolationGridSize Interpolation grid size

3.1. interpolationGridSize

The interpolationGridSize keyword can be used to control the density of the mesh used to interpolate the electronic transition structure factor in each spatial direction. A value of \(N\) for the interpolationGridSize means that \(2 N\) sampling points are used to sample the interpolated structure factor in each direction between neighbouring Grid Vectors.

4. Algorithm output

Table 2: Output keywords for CoupledCluster
Keyword Value
transitionStructureFactor Transition Structure factor

The output of the FiniteSizeCorrection algorithm includes transition structure factor \(S(G)\) and an estimate of the finitie size error correction to the correlation energy. The finite size error is estimated using a tricubic interpolation algorithm of the electronic transition structure factor.

4.1. Sample stdout

Below an example standard output stream is shown for a successful FiniteSizeCorrection algorithm run.

step: 15, FiniteSizeCorrection
Finite-size energy correction:    -1.1152868081
realtime 0.373321931 s

5. Sample yaml output

Below an example yaml output stream is shown for a successful FiniteSizeCorrection algorithm run.

floatingPointOperations: 49641291048
flops: 38600052331.414169
  amplitudes: 0x24cd038
  coulombPotential: 0x2487fa8
  coulombVertexSingularVectors: 0x248b168
  gridVectors: 0x2474ce8
  interpolationGridSize: 20
  slicedCoulombVertex: 0x24aee28
name: FiniteSizeCorrection
    corrected: -26.560663044130632
    correction: -1.1152868080903175
    uncorrected: -25.445376236040314
    unit: 0.03674932217686841
  transitionStructureFactor: 0x2bb8ff8
realtime: 0.373321931

6. Computational complexity

No considerable memory footprints or computational cost bottle necks are expected for this algorithm compared to preceding Coupled Cluster theory calculations.

7. Theory

The methods employed in this algorithm are discussed in Refs. (Liao and Grüneis 2016) and (Gruber et al. 2018) . We employ a tricubic interpolation technique to reduce quadrature errors in the numerical expression of the electronic correlation energy: \(\sum_{ G}S({ G}){\tilde{v}}({ G})\). In particular, the quadrature errors around \({G}=0\) are large and result in significant finite-size errors for small simulation cells. In order to obtain a more accurate estimate of the correlation energy in the thermodynamic limit, we proceed as follows. We introduce additional sampling points at an arbitrarily dense grid, which we choose to be \(2N\times 2N \times 2N\) times denser than the original one. The interpolated transition structure factor and the analytic expression of the Coulomb kernel can be used to calculate the correlation energy with the usual expression given above. The difference between the correlation energies computed with the interpolated and non-interpolated structure factor yields an estimate of the finite correction to the electronic correlation energy that is returned by this algorithm. We find that \(N=20\) yields well converged correlation energies for sufficiently large unit cells. However, we recommend to check convergence with respect to this parameter for each system separetely.

8. Literature

Gruber, Thomas, Ke Liao, Theodoros Tsatsoulis, Felix Hummel, and Andreas Grüneis. 2018. “Applying the Coupled-Cluster Ansatz to Solids and Surfaces in the Thermodynamic Limit.” Physical Review X 8 (2): 021043.
Liao, Ke, and Andreas Grüneis. 2016. “Communication: Finite Size Correction in Periodic Coupled Cluster Theory Calculations of Solids.” The Journal of Chemical Physics 145 (14): 141102.

Created: 2022-09-19 Mon 15:00