# 6.8 Direct Random Phase Approximation Methods

(June 30, 2021)

A useful $\mathcal{O}(N^{4})$ approach called the direct random phase approximation (dRPA) based on the RI approximation is available. This particular implementation was added by Joonho Lee working with Martin Head-Gordon. RI-dRPA has been applied to the thermochemistry and non-covalent interaction problems and often demonstrated superior performance over RI-MP2. In terms of the computational cost, RI-dRPA should be compared to the scaled-opposite-spin MP2 while theoretically it involves diagrams far beyond second-order and includes infinite-order diagrams similarly to coupled-cluster theory. In fact, one can view dRPA as a reduced coupled-cluster with doubles approach. In a nutshell, we define the dRPA energy as

 $E=E_{\text{HF}}+E_{\text{c}}^{\text{dRPA}}$ (6.27)

where using the plasmon formula we compute

 $E_{\text{c}}^{\text{dRPA}}=\int_{-\infty}^{\infty}\frac{\mathrm{d}\omega}{4\pi% }\mathrm{tr}\left[\ln\left(\mathbf{I}+\mathbf{Q}(\omega)\right)-\mathbf{Q}(% \omega)\right]$ (6.28)

where

 $\mathbf{Q}(\omega)=2\mathbf{B}^{T}\mathbf{D}(\mathbf{D}^{2}+\omega^{2}\mathbf{% I})^{-1}\mathbf{B}$ (6.29)

with

 $\displaystyle B_{ia,P}$ $\displaystyle=\sum_{Q}(ia|Q)(Q|P)^{-1/2}$ (6.30) $\displaystyle D_{ia,jb}$ $\displaystyle=\delta_{ij}\delta_{ab}(\epsilon_{a}-\epsilon_{i})$ (6.31)

In this form, the cost of computing the dRPA correlation is quartic-scaling which is comparable to SOS-MP2. To use this method, one must set METHOD = RIDRPA along with AUXBASIS.