6 Wave Function-Based Correlation Methods

6.8 Direct Random Phase Approximation Methods

(June 30, 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. 633 Lee J., Lin L., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 243.
RI-dRPA has been applied to the thermochemistry 280 Dohm Sebastian et al.
J. Chem. Theory Comput.
(2018), 14, pp. 2596.
and non-covalent interaction problems 794 Nguyen Brian D. et al.
J. Chem. Theory Comput.
(2020), 16, pp. 2258.
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. 980 Scuseria Gustavo E., Henderson Thomas M., Sorensen Danny C.
J. Chem. Phys.
(2008), 129, pp. 231101.
In a nutshell, we define the dRPA energy as

E=EHF+EcdRPA (6.27)

where using the plasmon formula we compute

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


𝐐(ω)=2𝐁T𝐃(𝐃2+ω2𝐈)-1𝐁 (6.29)


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.