Wave function analysis localized molecular orbitals

The two fundamental building blocks of Hartree-Fock theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital carries two electrons, with opposite spin. The occupied orbitals themselves are only defined as an occupied one-electron subspace of the full space spanned by the eigenfunctions of the Fock operator. Transformations between them leave the total HF wave function invariant. Normally the orbitals are obtained in a delocalized form as the solutions to the HF equations. This formulation is the most relevant one in studies of spectroscopic properties of the molecule, that is, excitation and ionization. The invariance property, however, makes a transformation to locahzed orbitals possible. Such localized orbitals can be valuable for an analysis of the chemical bonds in the system. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

If a vector space representation of electronic states is chosen, that is, a basis-set expansion, two types of basis sets are needed. One for the many-electron states and one for the one-particle states. For the latter, two choices became popular, the molecular orbital (MO) [9] and valence bond (VB) [10] expansions. Both influenced the understanding and interpretation of the chemical bond. A bonding analysis can then be performed in terms of their basic quantities. Although both representations of the wave function can be transformed (at least partially) into each other [11,12], most commonly an MO analysis is employed in electronic structure calculations for practical reasons. Besides, a VB description is often limited to small atomic basis sets as (semi-)localized orbitals are required to generate the VB structures [13]. If, however, diffuse functions with large angular momenta are included in the atomic orbital basis, a VB analysis suffers from their delocalization tails. As a consequence, the application of VB methods can often be limited to organic molecules. 

While seminal works intended to reveal SOC effects on the bonding schemes were discussed in term of spinors [9-13], canonical molecular spinors are not suited for the bonding analysis in complex systems, as opposed to small and/or symmetric model systems. Some have promoted the use of localized spinors [14], and in order to recover some chemical significance in terms of bonding, lone pairs and core orbitals, natural spinors similar to natural orbitals in the non-relativistic firameworks have been derived and implemented [15, 16]. It is worth noting that the concept of bond order in the context of multiconfigurational wave functions have been extended recently to two-step spin-orbit coupling approaches [17]. 

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