Effective potential theory basic formulations

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

The basic concepts of the one-electron Kohn-Sham theory have been presented and the structure, properties and approximations of the Kohn-Sham exchange-correlation potential have been overviewed. The discussion has been focused on the most recent developments in the theory, such as the construction of from the correlated densities, the methods to obtain total energy and energy differences from the potential, and the orbital dependent approximations to v. The recent achievements in analysis of the atomic shell and molecular bond midpoint structure of have been summarized. The consistent formulation of the discontinuous dependence of on the particle number and its effect on the spatial form of and charge transfer within the system have been presented. The recently developed direct approximations of the long- and short-range components of have been overviewed. 

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