6.9.1 Introduction

A useful $𝒪(N^4)$ 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. ⁷⁴⁰ Lee J., Lin L., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 243. Link RI-dRPA has been applied to the thermochemistry ³²⁴ Dohm S. et al.
J. Chem. Theory Comput.
(2018), 14, pp. 2596. Link and non-covalent interaction problems ⁹²⁶ Nguyen B. D. et al.
J. Chem. Theory Comput.
(2020), 16, pp. 2258. Link 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. ¹¹⁴² Scuseria G. E., Henderson T. M., Sorensen D. C.
J. Chem. Phys.
(2008), 129, pp. 231101. Link In a nutshell, we define the dRPA energy as

where using the plasmon formula we compute ³⁴⁹ Eshuis H., Yarkony J., Furche F.
J. Chem. Phys.
(2010), 132, pp. 234114. Link

where

with

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.