Slater determinants, open-shell molecules

In FIF calculations on open-shell molecules, a complication arises because in addition to doubly occupied and virtual MOs, there are now also singly occupied MOs (one in the case of radicals and two in the case of triplet diradicals). In a first attempt to construct a ground state wavefunction Pq for such a molecule, a Slater determinant may be set up that contains the appropriate number of singly occupied MOs and leaves all the others occupied by two electrons of opposite spin. This type of wavefunction is called restricted open-shell Hartree-Fock (ROHF) restricted because the electrons of opposite spin in the closed shells are constrained to occupy MOs / that are spatially identical for electrons of a ( up ) and p ( down ) spin. A procedure to variationally optimize such a wavefunction for open-shell systems has been devised. ... 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

As we have seen previously (Chapter 5), the eigenfunction for a polyelec-tronic atom is antisymmetric with respect to the exchange of the coordinates of any two electrons, and can be expressed as a Slater determinant whose elements are the various occupied spin-orbitals (or a linear combination of Slater determinants, in the case of open-shell atoms). The same appfies to polyelectronic molecules, the atomic orbitals being replaced by the various occupied molecular orbitals associated with the a and /3 spin-functions spin molecular orbitals. Thus, for the molecules H2O, NH3 or CH4 having five doubly occupied m.o.s (one core s orbital and four valence m.o.s), we have... 

The HF method works best with normal closed-shell molecules, because it uses a single Slater determinant, but ozone has open-shell diradical character it is, or at least resembles, a species with two half-filled orbitals, one with a single a electron and the other with a single P electron there are various ways of handling this molecule [91]. The cause of the problems with FOOF are harder to explain, but fluorine is known to be a somewhat troublesome element [92], although not all fluorine-containing species require very large basis sets [93]. 

Most polyatomic molecules have a closed-shell ground state here, the MO wave function is a single Slater determinant. For states arising from open-shell configurations, the MO wave function may require a linear combination of a few Slater determinants. 

The concept of localized MOs is not as widely applicable as that of delocalized canonical MOs. Delocalized MOs are valid for any molecule. However (as noted in Section 11.5), the Hartree-Fock wave functions of nonclosed-shell electronic states are, in most cases, linear combinations of several Slater determinants [for example, see (10.44) and (10.45)], and the above localization procedure does not apply to the open-shell orbitals in these wave functions. Thus, in a molecule in an excited electronic state with an open-shell configuration, the electrons in the incompletely filled MOs are delocalized over much of the molecule. 

Abstract In this chapter we examine some basic concepts of quantum chemistry to give a solid foundation for the other chapters. We do not pretend to review all the basics of quantum mechanics but rather focus on some specific topics that are central in the theoretical description of magnetic phenomena in molecules and extended systems. First, we will shortly review the Slater-Condon rules for the matrix elements between Slater determinants, then we will extensively discuss the generation of spin functions. Perturbation theory and effective Hamiltonians are fundamental tools for understanding and to capture the complex physics of open shell systems in simpler concepts. Therefore, the last three sections of this introductory chapter are dedicated to standard Rayleigh-Schrddinger perturbation theory, quasi-degenerate perturbation theory and the construction of effective Hamiltonians. 

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