Wave Function Methods

Abstract

A hierarchy of the Self-Consistent Field (SCF) theories of the molecular electronic structure is surveyed. First, the rudiments of the Hartree approach using the trial wave function in the form of the product of the occupied Molecular Orbitals (MO) describing independent one-electron states and providing the reference in defining the electron exchange-correlation effects are given. The Hartree–Fock (HF) method adopting the Slater determinant (antisymmetrized product) as the variational wave function, which constitutes a natural reference for determining the electron Coulomb correlation effects, and its analytical implementation in the finite basis set of AO, called SCF LCAO MO theory, are summarized. The Koopmans theorem is discussed and the concepts of Slater’s transition state (TS) in electronic excitations and of the local pseudopotential of Phillips and Kleinman (PK) are introduced. Typical errors in SCF calculations are identified and the electron correlation problem is formulated in terms of the conditional two-electron densities and the associated correlation holes, the sum-rules of which are examined. Alternative Configuration Interaction (CI) strategies for determining the static and/or dynamic Coulomb correlation effects, formally based upon the MO expansion theorem for molecular electronic states, are reviewed. Both the Single-Reference (SR) SCF (HF) and Multireference (MR) SCF (MR SCF) or Multiconfigurational (MC) SCF (MC SCF) and the Complete-Active-Space (CAS) SCF (CAS SCF) wave functions can be used to generate the excited configurations to be included in the subsequent CI expansion, giving rise to SR CI and MR CI approaches, respectively. Several single- and multi reference CI methods are identified, including the alternative variants using either the full CI (FCI) or a limited expansion in terms of the single (S), double (D), triple (T), quadruple (Q), and in general n-tuple electron excitations from the HF/SCF or MR SCF wave functions, e.g., the variational SR techniques: CID, CISD, CISTQ, etc. The size-consistency and size-extensivity requirements of such approximate variational treatments are commented upon and the problem of choosing an effective orbital set for subsequent CI calculations is addressed. The reduced density matrices are introduced and the associated concepts of the correlated one- and two-electron functions, called the Natural Orbitals (NO) and Natural Geminals (NG), respectively, are defined together with their pseudoapproximations in the limited CI approaches.