Min–max and max–min approaches to the solution of wave equation in relativistic quantum chemistry

Abstract

The variation problem associated with the solution of Dirac’s relativistic electron equation is reviewed here. Derivations of the min–max and max–min theorems are discussed: A new observation is that the spurious roots of negative energy satisfy a max–min theorem. Min–max principle (MMP) for solution of Dirac equation, extendable to the Dirac–Fock case, is concisely inspected and illustrated. MMP for two-electron Dirac–Coulomb equation is also considered. The min–max relation is physically interpreted for both Dirac and Dirac–Coulomb problems. Applications to chemical systems are investigated. Mathematical advances are reviewed. Limitations of MMP are spelt out, recent mathematical developments are collated, and an outline is given of the associated theoretical and computational developments. For a complementary purpose, the case of spin-zero is briefly examined. The variation principle for Dirac fermions in nanoscience is discussed. A brief outline of future prospects is given.