Wave Function Theories and Electronic Structure Methods: Quantum Chemistry, from Atoms to Molecules

Abstract

The chapter continues the discussion of the atomic structure, at the level of clarifying concreteness and details, presenting also the basic methods of electronic structure theories, constituting the keystone of the developments debated in the next chapters. The Slater determinants are introduced as primitives for the construction of many-electrons wave functions, using the one-electron orbital functions as basic ingredients. The orbitals, known from the introductory part of atom theory, are generalized at the molecular level. The so-called Slater rules for handling the Hamiltonian matrix elements of the Slater determinants constructed from orthogonal orbitals are presented as the basic algorithm for the practical approach of quantum chemistry. Besides, the rules are generalizable to other operators working with one- and two-electron terms. The Slater rules are generalized, at the end of the chapter, for the case of non-orthogonal orbitals, implying non-orthogonal Slater determinants bases. With the help of Slater rules and symmetry dichotomy of the two-electron integrals in the atomic shells (the Slater–Condon parameters), several poly-electronic atoms are analyzed in quite advanced detail, writing down analytical formulas for the energies of their states and relating to the experimental data on spectral terms. This exercise is important, beyond the domain of free atoms, since in large classes of compounds and materials with technical applicability one deals with embedded ions, whose properties are approachable by an atom-alike phenomenology. An exemplification on lanthanide emission spectra, used in domestic lighting devices, occasions a quick excursus in the open challenges of current materials sciences and the rational design of properties. Going from atoms to molecules, the use of atomic basis sets as the background of quantum chemistry is debated in detail, with hands-on illustration of the various options: Slater-type orbitals, Gaussian-type bases, plane waves, and numerical bases. A critical eye is turned upon Gaussian-type orbitals, signaling certain significant failures of bases rated as rich and accurate. The underperformance is determined by the lack of appropriate polynomial factors in the definition of atomic orbitals assignable to high quantum numbers, an intrinsic design deficiency in the customary implementation and use of Gaussians. The common belief is that the limitation of Gaussians consists in the assumed use of \( \text{exp}(-\alpha r^2)\) –type functions instead of more physical \( \text{exp}(-\alpha r)\) ones, while the lack of proper polynomial factors can cause more severe drawbacks. In turn, numeric basis sets are observed as surprisingly good performers and possible alternative technical options. The final part of the chapter presents the fundamental electronic structure methods based on wave functions and first principles operators: the Hartree–Fock technique, introducing self-consistency, brought to a higher level by the multi-configurational approach, and the Valence Bond theory. The so-called Complete Active Space Self-Consistent Field methods are near the top of powerfulness among actually available methods, which, with broad conceptual scope and flexible technical leverages, allow the approach to a large number of problems, with the right picture of their mechanisms and manifestations. Certain other methodological varieties, such as second-order perturbation corrections to self-consistent Hartree–Fock or multi-configurational techniques, or the Coupled Cluster expansion are discarded from the actual synopsis. From our perspective, such procedures can bring only incremental changes to the physical picture, sometimes not in a well-tempered manner, while their non-variational nature is a hidden drawback, at least in conceptual respects, and a heavy burden to the computation routines. Somewhat greater attention is paid to the Valence Bond frame, acknowledging its merit as a foundational model of the chemical bond and also its potential in a modern methodological reshaping.