General Many-Electron Systems

The structure of the general many-electron system with i electrons is obviously much more difficult to calculate. As a starting point the following approximate Hamiltonian is used 

Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmtum Erbium Thulium Ytterbium 

In Table 2.2 the ground configurations of the atoms are listed. Obviously, the field in atoms is only approximately central. Besides a central part, the electrostatic repulsion between electrons causes a non-central contribution, which can be treated as a perturbation. The spin-orbit interaction must also be taken into account. If the non-central electrostatic part strongly dominates over the spin-orbit interaction, the latter is neglected as a first approximation. A coupling between the individual angular momenta is then obtained 

Depending on which values are possible for the corresponding quantum numbers L and S for a certain configuration, a number of electrostatically split terms are obtained. Such terms are designated 

The quantity 2S+1 is called the multiplicity. In analogy with the one electron case we have 

Here Vij is the distance between the electrons i and j. As an approximation to (2.17) we assume that every electron moves independently of the other electrons in an average field, generated by the nucleus and the other electrons (the independent particle model). The field is assumed to be central (dependent only on r). This is the centralr-field approximation. The assumption of a central field combined with the Pauli exclusion principle results in a shell structure for the electrons and successively heavier elements can be constructed using the building-up principle (the total energy is minimized). The atom can be characterized by its electron configuration, e.g. for the lowest state of sodium we have 

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Fast computers led the interest of many researchers to general many-electron systems like Cl expansions based on an orbital description and Slater determinants. The main advantage of these methods is the reduction of n-electron Hamiltonian matrix elements to one- and two-electron integrals, as stated in the Slater-Condon rules, but also showing a slow convergence. There are two sources of the slow convergence of the Cl expansion. (1) The combinatorial problem . For an n-electron system and a basis of m spin-free one-electron functions the number of... 

In principle, the coupled-cluster ansatz for the wave function is exact if the excitation operator in Eq. (8.234) is not truncated. But this defines an FQ approach, which is unfeasible in actual calculations on general many-electron systems. A truncation of the CC expansion at a predefined order in the excitation operator T is necessary from the point of view of computational practice. Truncation after the single and double excitations, for instance, defines the CCSD scheme. However, in contrast with the linear Cl ansatz, a truncated CC wave function is still size consistent, because all disconnected cluster amplitudes which can be constructed from a truncated set of connected ones are kept [407]. The maximum excitation in T determines the maximum connected... 

For two-electron systems (He, H2) the method with different orbitals for different electrons was thoroughly discussed at the Shelter Island Conference in 1951 (Kotani 1951, Taylor and Parr 1952, Mulliken 1952). A generalization of this method to many-electron systems has now been given (Lowdin 1954, 1955, Itoh and Yoshizumi 1955) and is called the method with different orbitals for different spins. 

The purpose of this bibliography is to give a brief survey of the development of the methods for treating the correlation effects in many-electron systems by listing the most important papers in this field year by year. In accordance with the general outline used in Part I, Section III.C, the following methods will be included ... 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

Once more, free-electron models correctly predict many qualitative trends and demonstrate the appropriateness of the general concept of electron delocalization in molecules. Free electron models are strictly one-electron simulations. The energy levels that are used to predict the distribution of several delocalized electrons are likewise one-electron levels. Interelectronic effects are therefore completely ignored and modelling the behaviour of many-electron systems in the same crude potential field is ndt feasible. Whatever level of sophistication may be aimed for when performing more realistic calculations, the basic fact of delocalized electronic waves in molecular systems remains of central importance... 

In what follows, we present in this short review, the basic formalism of TDDFT of many-electron systems (1) for periodic TD scalar potentials, and also (2) for arbitrary TD electric and magnetic fields in a generalized manner. Practical schemes within the framework of quantum hydrodynamical approach as well as the orbital-based TD single-particle Schrodinger-like equations are presented. Also discussed is the linear response formalism within the framework of TDDFT along with a few miscellaneous aspects. 

The similarity-transformed Hamiltonian method has so far been applied only to two-electron systems. Using closure (i.e., RI) approximations, this technique will be generalized to many-electron systems (IS). 

In previous chapters we considered the wave functions and matrix elements of some operators without specifying their explicit expressions. Now it is time to discuss this question in more detail. Having in mind that our goal is to consider as generally as possible the methods of theoretical studies of many-electron systems, covering, at least in principle, any atom or ion of the Periodical Table, we have to be able to describe the main features of the structure of electronic shells of atoms. In this chapter we restrict ourselves to a shell of equivalent electrons in non-relativistic and relativistic cases. 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

As we have seen in Chapter 4, relativistic operators of the Ek-transitions in the general case have several forms and are dependent on the gauge condition of the electromagnetic field potential. These forms are equivalent and do not depend on gauge for exact wave functions. Unfortunately, we are always dealing with the more or less approximate wave functions of many-electron systems, therefore we need general expressions for the appropriate matrix elements. 

Earlier calculation on many electron atomic systems under plasma was performed by Stewart and Pyatt [58], who estimated the variation of IP of several atoms using a finite temperature TF model. Applications of the density functional theory on these systems were reviewed by Gupta and Rajagopal [57], The calculations on many electron systems are mostly concerned with the hot and dense plasmas with the application of the IS model, or from general solutions of the Poisson equation for the potential function. The discussions using the average atom model in Section 3.3, Inferno model of Liberman in 3.4, STA model in 3.5, hydrodynamic model in... 

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