Reduced Density Matrices in Full Configuration Interaction Quantum Monte Carlo

Reduced density matrices are a powerful construct in quantum chemistry, providing a compact representation of highly multi-determinantal wavefunctions, from which the expectation values of important physical properties can be extracted, including multipole moments, polarizabilities and nuclear forces$^{1,2}$. Full configuration interaction quantum Monte Carlo (FCIQMC)$^3$ and its initiator extension (\emph{i}-FCIQMC)$^4$ perform a stochastic propagation of signed walkers within a space of Slater determinants to achieve FCI-quality energies \emph{without} the need to store the complete wavefunction. We present here a method for a stochastic calculation of the 1- and 2-body reduced density matrices within the framework of (\emph{i})-FCIQMC, and apply this formulation to a range of archetypal molecular systems. Consideration is also given to the source and nature of systematic and stochastic error, and regimes to effectively alleviate these errors are discussed$^5$. $^1$ P.-O. L\"{o}wdin, Phys. Rev. 97, 1474 (1955). $^2$ C. A. Coulson, Rev. Mod. Phys. 32, 170 (1960). $^3$ G. H. Booth, A. Thom, and A. Alavi, J. Chem. Phys. 131, 054106 (2009). $^4$ D. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 132, 041103 (2010). $^5$ D. Cleland, PhD thesis, University of Cambridge, 2012.