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6.8 Direct Random Phase Approximation Methods

6.8.1 Introduction

(December 20, 2021)

A useful 𝒪(N4) 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. 651 Lee J., Lin L., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 243.
Link
RI-dRPA has been applied to the thermochemistry 285 Dohm S. et al.
J. Chem. Theory Comput.
(2018), 14, pp. 2596.
Link
and non-covalent interaction problems 817 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. 1012 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

E=EHF+EcdRPA (6.27)

where using the plasmon formula we compute 310 Eshuis H., Yarkony J., Furche F.
J. Chem. Phys.
(2010), 132, pp. 234114.
Link

EcdRPA=-dω4πtr[ln(𝐈+𝐐(ω))-𝐐(ω)] (6.28)

where

𝐐(ω)=2𝐁T𝐃(𝐃2+ω2𝐈)-1𝐁 (6.29)

with

Bia,P =Q(ia|Q)(Q|P)-1/2 (6.30)
Dia,jb =δijδab(ϵa-ϵi) (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.