By performing a stochastic dynamic in a space of Slater determinants, theFull Configuration Interaction Quantum Monte Carlo (FCIQMC) method has beenable to obtain energies which are essentially free from systematic error to thebasis set correlation energy, within small and systematically improvableerrorbars. However, the weakly exponential scaling with basis size makesconverging the energy with respect to basis set costly and in larger systems,impossible. To ameliorate these basis set issues, here we use perturbationtheory to couple the FCIQMC wave function to an explicitly correlated stronglyorthogonal basis of geminals, following the [2]_{\textrm{R12}} approach ofValeev {\em et al.}. The required one- and two-particle density matrices arecomputed on-the-fly during the FCIQMC dynamic, using a sampling procedure whichincurs relatively little additional computation expense. The F12 energycorrections are shown to converge rapidly as a function of sampling, both inimaginary time, and number of walkers. Our pilot calculations on the bindingcurve for carbon dimer, which exhibits strong correlation effects as well assubstantial basis set dependence, demonstrate that the accuracy of theFCIQMC-F12 method surpasses that of all previous FCIQMC calculations, and thatthe F12 correction improves accuracy equivalent to increasing the quality ofthe one-electron basis by two cardinal numbers.
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