Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference...

A hybrid stochastic-deterministic approach for computing the second-order perturbative contribution E⁽²⁾ within multireference perturbation theory (MRPT) is presented. The idea at the heart of our hybrid scheme-based on a reformulation of E⁽²⁾ as a sum of elementary contributions associated with each determinant of the MR wave function-is to split E⁽²⁾ into a stochastic and a deterministic part. During the simulation, the stochastic part is gradually reduced by dynamically increasing the deterministic part until one reaches the desired accuracy. In sharp contrast with a purely stochastic Monte Carlo scheme where the error decreases indefinitely as t^-1/2 (where t is the computational time), the statistical error in our hybrid algorithm displays a polynomial decay ∼t^-n with n = 3-4 in the examples considered here. If desired, the calculation can be carried on until the stochastic part entirely vanishes. In that case, the exact result is obtained with no error bar and no noticeable computational overhead compared to the fully deterministic calculation. The method is illustrated on the F₂ and Cr₂ molecules. Even for the largest case corresponding to the Cr₂ molecule treated with the cc-pVQZ basis set, very accurate results are obtained for E⁽²⁾ for an active space of (28e, 176o) and a MR wave function including up to 2×10⁷ determinants.