Ground State Electronic Structure Theory

We shall in this section give a historic overview of how the electronic structure theory for transition metal complexes in their ground state has evolved from the 1950s to the present time. The account will include a discussion of wave function methods based on Hartree Fock and post-Hartree Fock approaches as well as Kohn-Sham density functional theory (KS-DFT). 

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X-Ray absorption data in combination with atomic theory and solid-state band-structure theory can yield detailed information about the ground-state electronic structure of solids on an energy scale on the order of meV. This holds particularly true for correlated narrow-band systems, such as the rare-earth and transition-metal compounds. In broad-band materials, such as the... 

Fig. 18. Ground state electronic structures for cresyl and o-(methylthio)cresyl phenoxyl radicals. Isosurface representations of molecular orbitals solved by ab initio density functional theory methods for cresyl (ere) and o-(methylthio)cresyl (mtc) phenoxyl radicals. Eigenvalues are listed and for each the SOMO is identihed with an asterisk ( ).

We determine the ground-state electronic structure of solids within Density Functional Theory (DFT) and the usual KS variational procedure, all implemented in the computational package gtoff [14]. The results of the all-electron, full-potential calculations are Bloch eigenfunctions p,k(r), expressed as linear combinations of Gaussian Type Orbitals (GTOs), and KS eigenvalues p k-... 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

The ab initio methods used by most investigators include Hartree-Fock (FFF) and Density Functional Theory (DFT) [6, 7]. An ab initio method typically uses one of many basis sets for the solution of a particular problem. These basis sets are discussed in considerable detail in references [1] and [8]. DFT is based on the proof that the ground state electronic energy is determined completely by the electron density [9]. Thus, there is a direct relationship between electron density and the energy of a system. DFT calculations are extremely popular, as they provide reliable molecular structures and are considerably faster than FFF methods where correlation corrections (MP2) are included. Although intermolecular interactions in ion-pairs are dominated by dispersion interactions, DFT (B3LYP) theory lacks this term [10-14]. FFowever, DFT theory is quite successful in representing molecular structure, which is usually a primary concern. 

Figure 14. The absolute value of the average disrotatory angle as a function of time in femtoseconds. (The disrotatory angle is defined in the upper left inset.) Lower inset A onedimensional cut of the excited-state potential energy surface along the disrotatory and conrotatory coordinates. All other coordinates are kept at their ground-state equilibrium value, and the full and dashed lines correspond to two levels of electronic structure theory (see text for details). (Figure adapted from Ref. 216.)...

The development of the method started in the mid 1920 s with the work of Thomas and Fermi [8, 9]. The aim was to formulate an electronic structure theory for the solid state, based on the properties of a homogeneous electron gas, to which we introduce a set of external potentials (i.e. the atomic nuclei). The original formulation, with later additions by Dirac [10] and Slater [11], was, however, inadequate for accurate description of atomic and molecular properties, and it was not until the ground-breaking work of Kohn and coworkers in the mid 1960 s that the theory was put in a form more suited to computational chemistry [12,... 

The electronic chemical potential is constant for a system in its electronic ground state, which led Parr et al. to associate the chemical potential with minus one times the electronegativity, since the electronegativity is also equalized in the ground state [7]. This equalization of the chemical potential also suggests that electronic structure theory can be expressed in a way that resembles classical thermodynamics. Ergo, Parr et al. wrote the total differential of the energy as... 

The central problem in electronic structure theory is to determine the ground state of a system of electrons, which is typically done variationally by minimizing the energy. The lower bound method can be invoked to achieve a feth-order approximation by replacing the variation minpgq5 (p,/ )g by the semidefinite program... 

Density functional theory purists are apt to argue that the Hohenberg-Kohn theorem [1] ensures that the ground-state electron density p(r) determines all the properties of the ground state. In particular, the electron momenmm density n( ) is determined by the electron density. Although this is true in principle, there is no known direct route from p to IT. Thus, in practice, the electron density and momentum density offer complementary approaches to a qualitative understanding of electronic structure. 

We shall now examine recent theoretical results concerning those ground state properties of metals and compounds which in the previous section, we have called bond and valence indicators. In this section, those properties are not taken as starting points for correlations aiming at the composition of the bond. They are instead the final point of electronic structure theories in which the different contributions to cohesion are analysed. 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

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