# Random-Phase Approximation and Laplace-Transformed Scaled-Opposite-Spin-MP2

The direct RI-RPA method computes

\[\begin{split} E^{RI-dRPA} &= - \frac{1}{4\pi}\int_{-\infty}^{\infty}d\omega\text{Tr}\left(\text{log}(1+Q(\omega))-Q(\omega)\right)\\ Q_{RS}(\omega) &= 2\sum_{ia}B_{iaR}\frac{\epsilon_a-\epsilon_i}{\left(\epsilon_a-\epsilon_i\right)^2+\omega^2}B_{iaS} \end{split}\]

whereas the Laplace-transformed Scaled-Opposite-Spin-MP2 (LT-RI-SOS-MP2) method computes

\[\begin{split} E^{LT-RI-SOS-MP2} &= - \int_{0}^{\infty}d\tau\text{Tr}\left(\overline{Q}(\tau)^2\right)\\ \overline{Q}_{RS}(\tau) &= \sum_{ia}B_{iaR}e^{-\left(\epsilon_a-\epsilon_i\right)\tau}B_{iaS} \end{split}\]

as implemented in DelBen2013. The integration is performed numerically. Double-hybrid functionals are available in CP2K.

CP2K provides two different implementations: a quartically-scaling and a low-scaling cubically-scaling implementation. Different functionals require rescaling which is achieved using the keywords SCALE_RPA in case of RPA and SCALE_S in case of LT-RI-SOS-MP2.

Here, we will focus on the quartically-scaling implementation only.

## RI-dRPA

CP2K implements two quadrature schemes: Clenshaw-Curtis and Minimax. The first requires 30-40 quadrature points, the latter 6-8 quadrature points. Minimax quadrature rules have to be preoptimized such that not all possible numbers of quadrature points are available.

RPA correlation energies are usually combined with exact exchange energies. In CP2K, this is available by activating the HF section which is setup like an ordinary HF section.

Several Beyond-RPA schemes are available: The Renormalized Screened Exchange (RSE) correction, the Approximate Exchange Kernel (AXK) and the Second-Order Screened Exchange corrections (SOSEX). These are enabled with the RSE keyword and the EXCHANGE_CORRECTION section. This results into the following possible RPA section

      &RI_RPA
        # Choose it as large as necessary and as small as possible
        QUADRATURE_POINTS 6
        # Choose it as large as possible (must be a divisor of the number of quadrature points and the number of processes)
        # -1 is default and let CP2K decide on that value
        # Larger values increase the memory demands but reduce communication
        NUM_INTEG_GROUPS -1
        # The RSE correction is only relevant for a non-HF reference
        RSE .TRUE.
        # Exchange corrections may be quite costly
        &EXCHANGE_CORRECTION [NONE|AXK|SOSEX]
          # The Hartree-Fock-based implementation scales better for larger systems but introduces more noise
          USE_HFX_IMPLEMENTATION F
          # This parameter is ignored if USE_HFX_IMPLEMENTATION is set to T
          # Larger values improve performance but increase the memory demands
          BLOCK_SIZE 16
        &END
      &END

## RI-dRPA Gradient Calculations

Analytical gradients are only available for RPA calculations using a minimax grid, but not for the beyond-RPA methods (RSE, AXK, SOSEX).Stein2024 The general setup is similar to RI-MP2 gradients. In addition, there are two more relevant keywords: DOT_PRODUCT_BLKSIZE and MAX_PARALLEL_COMM. The first splits the contraction along the auxiliary index to improve numerical stability. By default, this feature is turned off. The second keyword determines the number of parallel communication channels for non-blocking communication. Larger numbers allow more overlap but increase the memory requirements. Larger values than 3 are commonly not necessary.

## LT-RI-SOS-MP2

CP2K implements two quadrature schemes only the Minimax scheme requiring usually 6-8 quadrature points.DelBen2013 Minimax quadrature rules have to be preoptimized such that not all possible numbers of quadrature points are available. Scaling of the energy contribution is possible with the SCALE_S keyword which is also used by MP2 calculations. The input is simpler than in the RPA-case

      &RI_SOS_MP2
        # Works similar than in RPA
        NUM_INTEG_GROUPS -1
        # Larger values improve the accuracy but increase the costs
        QUADRATURE_POINTS 6
      &END

## LT-RI-SOS-MP2 Gradient Calculations

Analytical gradients are available and work similarly to RI-MP2 calculations and RI-RPA calculations. As in case of RI-MP2, it makes use of the EPS_CANONICAL keyword. Larger values improve the numerical accuracy but increase the computational costs significantly because CP2K assumes that the number of relevant pairs is negligible.

## Performance Considerations

Without employing the low-scaling implementation, RI-RPA and RI-SOS-MP2 calculations scale quartically with respect to the number of atoms. The implementation relies on parallel matrix-matrix multiplications. Their costs can be reduced with the COSMA library and accelerated on GPUs if COSMA was configured accordingly.

HF calculations may be accelerated using ADMM which is recommended if very diffuse basis functions are employed. The RSE correction employs the same HF section as for the calculation of the HF exchange energy. The costs of AXK and SOSEX depend on the implementation in use. The HF-based implementation is cheaper for larger systems. Its performance depends on the available memory per process group (see GROUP_SIZE and MAX_MEMORY). The costs becomes neglegible for large systems. Without the HF implementation, CP2K relies on SpLA-accelerated local matrix-matrix multiplications. These become the bottleneck for larger systems but introduce less numerical noise than the HF-based implementation.