Algebraic solution of the Hartree equation based on a tensor eigenvalue approach.

Hartree's self-consistent determination of the ground state of the Helium atom marks the beginning of computational many-electron theory. The nonlinear nature of the problem seems to suggest that an iterative self-consistent field (SCF) type of approach is the only numerical approach to solve this problem. We will show that such an assumption is wrong. To this end we will re-interpret the Hartree equation as a tensor eigenvalue problem, and then we systematically guide the reader through an algebraic solution of the Hartree equation for the trivial example of a Helium atom with a single Gaussian as a basis set. In a second step we will study the Helium atom based on a basis set with two Gaussians, where we will compare numerical results obtained from an SCF calculation with results obtained from an algebraic solution of the corresponding Hartree equation. We will see that both solutions turn out to be identical. Our results suggest that the use of algebraic rather than iterative numerical techniques could serve as an alternative numerical approach to study Hartree-type systems, as well as a whole variety of similar problems throughout the physical and chemical literature. [ABSTRACT FROM AUTHOR]