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D. Krisiloff, J. M.! Dieterich, F. Libisch, E Carter:
"Numerical Challenges in a Cholesky-decomposed Local Correlation Quantum Chemistry Framework";
in: "Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts", R. Melnik (Hrg.); Wiley-VCH, 2015, ISBN: 978-1-118-85398-6, S. 59 - 91.

For more than 60 years, chemists have used variants of molecular orbital (MO) theory as a framework to model and predict chemical behavior. Variational MO theory, otherwise known as Hartree-Fock (HF) theory, solves Schrödinger´s equation using a wavefunction where each electron´s spatial distribution is described by a single, one-electron wavefunction called an MO. The electron is said to occupy the MO. In HF theory, the motion of the electrons is uncorrelated-the spatial probability distribution of each electron is independent of all other electrons. This lack of correlation is unphysical since electrons interact via a distance-dependent Coulomb potential: each electron´s probability distribution should be a function of the coordinates of all the other electrons. So-called correlated wavefunction methods improve upon HF theory by including electron correlation. They typically express electron correlation in the many-electron wavefunction via consideration of electronic excitations from occupied to unoccupied MOs. The resulting electron configurations are allowed to mix into the ground state configuration to obtain a better description of the many-body wavefunction. It can be shown that such configuration interaction reduces electron-electron repulsion, thereby lowering the energy toward the exact solution of the Schrödinger equation. Correlated wavefunction methods deliver highly accurate molecular properties at a high computational cost. For instance, the"gold standard" of quantum chemistry for molecules at their equilibrium geometries is Coupled-Cluster theory [1] at the single and double excitation levels with perturbative triple excitations, CCSD(T). However, CCSD(T) scales painfully as O(N 7) with the size of the basis set. In addition to the scaling of the floating point operations, correlated wavefunctions typically also require large intermediary data structures stored in memory or on disk, greatly adding to their computational cost. Hence, recent research efforts have focused on approximations to correlated wave-function approaches that reduce their computational cost. These techniques hope to exploit sophisticated strategies to reduce the computational demands of solving Schrödinger´s equation, allowing the treatment of larger chemical systems than currently feasible.

Cholesky decomposition, Quantum Chemistry