where H ij = D i Ĥ D j , and S has replaced E to take the role of a population control parameter. Coefficients, C i , are represented by populations of 'walkers' on Slater determinants, and the goal of population control is to make sure that the number of walkers is controlled to be below a certain value. FCIQMC then uses the corresponding spawning and death steps to perform this propagation stochastically, resulting in the FCI solution. Several important modifications exist to improve the performance of FCIQMC, including the initiator method (i-FCIQMC), introduced in 2010 by Cleland et al. made it possible to converge the calculation with a lower number of walkers, and a further modification known as the adaptive shift method is able to reduce initiator error and allows one to obtain near-FCI quality results for systems that have a Hilbert space size up to 10 35 , which is 10 25 times larger than the largest FCI calculation reported to date. Additionally, semi-stochastic adaptations of FCIQMC make it possible to deterministically propagate a number of determinants (usually the most populated ones), which dramatically reduces the stochastic noise. Bootstrap Embedding (BE), attempts to break down the calculation of a whole molecule into overlapping fragments. For a chemical system described by a second-quantized Hamiltonian: