Density functional approximations for the exchange‐correlation energy EDFAxc of an electronic system are often improved by admixing some exact exchange Ex: Exc≊EDFAxc+(1/n)(Ex−EDFAx). This procedure is justified when the error in EDFAxc arises from the λ=0 or exchange end of the coupling‐constant integral ∫10 dλ EDFAxc,λ. We argue that the optimum integer n is approximately the lowest order of Gorling–Levy perturbation theory which provides a realistic description of the coupling‐constant dependence Exc,λ in the range 0≤λ≤1, whence n≊4 for atomization energies of typical molecules. We also propose a continuous generalization of n as an index of correlation strength, and a possible mixing of second‐order perturbation theory with the generalized gradient approximation.

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Rationale for mixing exact exchange with density functional approximations

Publisher Summary This chapter discusses statistical exchange-correlation in the self-consistent field. There are two sides to a self-consistent field calculation: the determination of the potential and the solution of Schrodinger's equation for the one-electron problem. The solution of Schrodinger's equation has fortunately advanced far enough through the application of the electronic digital computer so that it can be regarded for most purposes as being a standardized technique. The main point of the method of Gaspar, Kohn, and Sham is that they derive the approximate exchange correction from an approximate Hamiltonian for the system by varying the spin-orbitals to minimize the average value of this Hamiltonian for the ground state. The approximate Hamiltonian has many important and valuable features that are described in the chapter.

Statistical Exchange-Correlation in the Self-Consistent Field

A detailed discussion is given of the methods of calculating atomic polarizabilities and shielding factors and the relationships between them are demonstrated. The formulation of the uncoupled Hartree-Fock approximation is presented and it is shown that the methods are all approximate versions of it. A more accurate procedure, the coupled Hartree-Fock approximation, is described. Comprehensive tables of dipole and quadrupole polarizabilities and of quadrupole shielding factors are included together with an assessment of the probable accuracy.

Atomic polarizabilities and shielding factors

Publisher Summary The purpose of this chapter is to provide information on the recent developments in perturbation theory. In recent years, there is a great increase of interest in the application of perturbation theory to the fundamental problems of quantum chemistry. Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects of the perturbations are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known. The new applications have been mainly to atoms where the reciprocal of the atomic number, l/Z, provides a natural perturbation parameter. These may be divided into two groups. The first consists of calculations of energy levels, and is a natural outgrowth of Hylleraas's classic work on the 1/Z expansion for two-electron atoms. The applications in the second group are to the calculation of expectation values and other properties of atoms and molecules, and are of much more recent origin. There are two principal reasons for the success of these new applications: (1) sufficient accuracy is frequently obtained from knowledge of a first-order perturbed wave function, and (2) a great advantage of perturbation theory is that the functional form of the perturbed wave function is shaped by the perturbation itself.

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Recent Developments in Perturbation Theory

Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations Trygve Helgaker,* Sonia Coriani, Poul Jørgensen, Kasper Kristensen, Jeppe Olsen, and Kenneth Ruud Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway Dipartimento di Scienze Chimiche e Farmaceutiche, Universit a degli Studi di Trieste, Via Giorgieri 1, I-34127 Trieste, Italy Lundbeck Center for Theoretical Chemistry, Department of Chemistry, Aarhus University, 8000 Aarhus C, Denmark Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø, N-9037 Tromsø, Norway

Recent advances in wave function-based methods of molecular-property calculations.

Propagators in Quantum Chemistry: Linderberg/Quantum Chemistry

Electronic correlation energy in 3- and 4-electron atoms

Approximate natural orbitals for four‐electron wavefunctions for Be as calculated by Boys, Matsen, Watson, and Weiss and LiH as calculated by Ebbing are obtained by diagonalization of the appropriate first‐order reduced density matrices. The formalism for carrying out the process is extended to the case of a nonorthogonal basis. Occupation numbers are presented and compared for the functions reported. In addition, truncated natural expansions reproducing the original functions are tabulated. Natural‐orbital functions are tabulated. Natural‐orbital occupation numbers, the orbitals themselves, and the natural expansions are shown to have considerable value in comparing these functions. The nature of the approximations used in the individual functions is clarified by the analysis, and considerable insight is afforded to the nature of the physical system.

Jan Linderberg

Papers

Propagators in Quantum Chemistry: Linderberg/Quantum Chemistry

Electronic correlation energy in 3- and 4-electron atoms

Approximate Natural Orbitals for Four‐Electron Systems

Doubly excited states and natural orbitals in the study of atoms by perturbation theory

Penetration Effects in the 2 F-Series of CsI†