The historical development of the electron correlation problem

Per-Olov Löwdin

 

 

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Abstract

A brief review is given of the historical development of the treatment of the correlation problem in solving the Schrödinger equation in modern quantum theory from the early 1930s up to now. The correlation energy for a specific state of a system is defined as the difference E_corr = E − E_HF between the exact eigenvalue E and the Hartree–Fock energy E_HF of the same Hamiltonian for the state under consideration. From the concepts of the “Different Orbitals for Different Spins” (DODS) and the “Alternant Molecular Orbital” (AMO) methods introduced in the 1950s, the study goes to the use of transition formulas—instead of expectation values—for the Hamiltonian and the possibility to express the correlation energy exactly in terms of “double excitations” or pair functions. The main emphasis is then put on the concept of wave and reaction operators and the formulation of the results of infinite-order perturbation theory in terms of such operators. The partitioning technique offers a simple way to derive these operators and to explore the resolvent or propagator methods in greater detail, and it replaces the original Schrödinger equation with its degeneracies and multiple roots by a reduced characteristic equation having only single roots, which is often a great simplification. Special attention is given the treatment of Schrödinger's perturbation theory in view of the importance of the linked-cluster theorem in the applications. In the study of the splitting of degenerate levels due to a perturbation, the use of the multidimensional partitioning technique utilizing the concept of an energy-independent wave operator is also discussed. Some developments in the coupled-cluster methods are further briefly reviewed. Since all quantum mechanical calculations going beyond the Hartree-Fock method imply a certain treatment of the correlation problem, a brief survey is also given of the progress in computational quantum theory, particularly in current quantum chemistry.

 https://sci-hub.se/https://doi.org/10.1002/qua.560550203