Quantum Monte Carlo calculation of reduced density matrices

Catherine Overy, Deidre M. Cleland, George H. Booth +2 more

Norman Henry March

Nick S. Blunt, George H. Booth, Ali Alavi

We present developments in the calculation of reduced density matrices (RDMs) in the full configuration interaction quantum Monte Carlo (FCIQMC) method. An efficient scheme is described to allow storage of RDMs across distributed memory, thereby allowing their calculation and storage in large basis sets. We demonstrate the calculation of RDMs for general states by using the recently introduced excited-state FCIQMC approach [N. S. Blunt et al., J. Chem. Phys. 143, 134117 (2015)] and further introduce calculation of transition density matrices in the method. These approaches are combined to calculate excited-state dipole and transition dipole moments for heteronuclear diatomic molecules, including LiH, BH, and MgO, and initiator error is investigated in these quantities.

Catherine Overy, George H. Booth, Nick S. Blunt +3 more

Properties that are necessarily formulated within pure (symmetric) expectation values are difficult to calculate for projector quantum Monte Carlo approaches, but are critical in order to compute many of the important observable properties of electronic systems. Here, we investigate an approach for the sampling of unbiased reduced density matrices within the full configuration interaction quantum Monte Carlo dynamic, which requires only small computational overheads. This is achieved via an independent replica population of walkers in the dynamic, sampled alongside the original population. The resulting reduced density matrices are free from systematic error (beyond those present via constraints on the dynamic itself) and can be used to compute a variety of expectation values and properties, with rapid convergence to an exact limit. A quasi-variational energy estimate derived from these density matrices is proposed as an accurate alternative to the projected estimator for multiconfigurational wavefunctions, while its variational property could potentially lend itself to accurate extrapolation approaches in larger systems.

Matthias Troyer, Philipp Werner, Adolfo Avella +1 more

Norm M. Tubman, David ChangMo Yang

Several algorithms have been proposed to calculate the spatial entanglement spectrum from high order R\'enyi entropies and maximum entropy techniques. We present an alternative approach for computing the entanglement spectrum with quantum Monte Carlo for both continuum and lattice Hamiltonians. This method provides direct access to the matrix elements of the spatially reduced density matrix and we determine an estimator that can be used in variational Monte Carlo as well as other Monte Carlo methods. The algorithm is based on using a generalization of the swap operator, which can be extended to calculate a general class of density matrices that can include combinations of spin, space, particle, and even momentum degrees of freedom. We demonstrate the method by applying it to the ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$ molecules and describe how the spatial entanglement spectrum encodes a covalent bond that includes all the many body correlations.