Approximation of the electron density of Aluminium clusters in tensor-product format

The tensor-structured methods developed recently for the accurate calculation of the Hartree and the non-local exchange operators have been applied successfully to the ab initio numerical solution of the Hartree–Fock equation for some molecules. In the present work, we show that the rank-structured representation can be gainfully applied to the accurate approximation of the electron density of large Aluminium clusters. We consider the Tucker-type decomposition of the electron density of certain Aluminium clusters originating from finite element calculations in the framework of the orbital-free density functional theory. Numerical investigations of the Tucker approximation of the corresponding electron density reveal the exponential decay of the approximation error with respect to the Tucker rank. The resulting low-rank tensor representation reduces dramatically the storage needs and the computational complexity of the consequent tensor operations on the electron density. As main result, the rank of the Tucker approximation for the accurate representation of the electron density is small and only weakly dependent on the system size for the systems studied here. This shows good promise for resolving the electronic structure of materials using tensor-structured techniques.