One-electron approaches

This set of first-order differential equations can be solved, approximately or numerically, for a specific system. The theory has been applied to Li scattered from Cs/W, and gives more satisfactory agreement with experiment than does the one-electron approach. 

Nevertheless, the one-electron approach does have its deHciencies, and we believe that a major theoretical effort must now be devoted to improving on it. This is not only in order to obtain better quantitative results but, perhaps more importantly, to develop a framework which can encompass all types of charge-transfer processes, including Auger and quasi-resonant ones. To do so is likely to require the use of many-electron multi-configurational wavefunctions. There have been some attempts along these lines and we have indicated, in detail, how such a theory might be developed. The few many-electron calculations which have been made do differ qualitatively from the one-electron results for the same systems and, clearly, further calculations on other systems are required. 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

The information system in the (condensed) orbital resolution involves the AO events / in its input a = xi] and output b = / . It represents the effective promotion of these basis functions in the molecule via the probability/information scattering described by the conditional probabilities of AO outputs given AO inputs, with the input (row) and output (column) indices, respectively. In the one-electron approach [46-48], these AO-communication connections P(XjlXi) = P(j i) result from the appropriately generalized superposition principle of quantum mechanics [51],... 

Recent advances in the techniques of photoelectron spectroscopy (7) are making it possible to observe ionization from incompletely filled shells of valence elctrons, such as the 3d shell in compounds of first-transition-series elements (2—4) and the 4/ shell in lanthanides (5, 6). It is certain that the study of such ionisations will give much information of interest to chemists. Unfortunately, however, the interpretation of spectra from open-shell molecules is more difficult than for closed-shell species, since, even in the simple one-electron approach to photoelectron spectra, each orbital shell may give rise to several states on ionisation (7). This phenomenon has been particularly studied in the ionisation of core electrons, where for example a molecule (or complex ion in the solid state) with initial spin Si can generate two distinct states, with spin S2=Si — or Si + on ionisation from a non-degenerate core level (8). The analogous effect in valence-shell ionisation was seen by Wertheim et al. in the 4/ band of lanthanide tri-fluorides, LnF3 (9). More recent spectra of lanthanide elements and compounds (6, 9), show a partial resolution of different orbital states, in addition to spin-multiplicity effects. Different orbital states have also been resolved in gas-phase photoelectron spectra of transition-metal sandwich compounds, such as bis-(rr-cyclo-pentadienyl) complexes (3, 4). 

One basic assumption almost universally made in interpreting XANES is the validity of the single-particle picture, which assumes that a single-particle density of states can explain the various spectral features. This assumption has been strengthened by recent calculations (83, 126, 155, 205, 206). The appealing result of 15 years of both theoretical and experimental work is that a one-electron approach is usually really relevant. However, in some instances this simple single-particle picture for XANES should be used with caution (13, 273). 

A first approximation to the total quasi-particle spectrum, is given by the DOS. As was the case for 7, A and G, a large part of the LDOS modifications on under-coordinated atoms is well accounted for by an effective one electron approach, because the shape of the LDOS reveals the pecu-... 

Although this theory explains theoretically the experimental observations in the case of ReOj, TiO, and VO, it fails to verify the conductivity characteristics of transition metal oxides such as TiO, VO, MnO, and NiO. Band theory explains the metallic characteristics but fails to account for the electrical properties of insulators or semiconductors and metal-nonmetal transitions because of neglect of electronic correlation inherent in the one-electron approach to the problem. Although there is no universal model for description of the conductivity, magnetic and optical properties of a wide range of materials (e.g., simple and complex oxides, sulfides, phosphides), several models have been proposed (for details, see Refs. 447-453). Of these, a generally accepted one is that described by Goodenough (451). 

The diagram teUs us that no traces of LVI in a particular experiment do not evidence against the fundamental one-electron approach. A lower threshold of thermodynamic stability exists where LVI cannot be registered experimentally. The magnitude of this detection threshold depends on the method applied. 

DFT is an intrinsically one-electron approach and, as such, orbitals are a natural feature of the method. In contrast to HF theory, all the orbitals, both occupied and virtual, are defined with respect to the full molecular potential which confers greater physical meaning to both their energies and compositions. However, DFT orbitals and the tZ-orbitals of a LFT calculation are not the same and cannot be directly compared although they may share qualitatively similar features and, with care, common descriptions of the bonding in metal complexes can be developed. 

The geminal ansatz still requires more effort than the standard one-electron approach of the independent particle model. It is therefore usually restricted to small molecules for feasibility reasons. As an example how the nonlinear optimization problem can be handled we refer to the stochastic variational approach [340]. However, the geminal ansatz as presented above has the useful feature that all elementary particles can be treated on the same footing. This means that we can actually use such an ansatz for total wave functions without employing the Born-Oppenheimer approximation, which exploits the fact that nuclei are much heavier than electrons. Hence, electrons and nuclei can be treated on the same footing [340-342] and even mixed approaches are possible, where protons and electrons are treated in the external field of heavier nuclei [343-346]. The integrals required for the matrix elements are hardly more complicated than those over one-electron Gaussians [338,339,347]. 

It will be noted that the approach we are using is a one-electron approach in that w e are focusing attention on the contribution to optical activity associated with the promotion of a single electron from a lower to a higher orbital. Hence, a slight digression to the one-electron theory of rotatory power of Condon, Altar, and E5ning is in order here. 

Before we return to the quantitative calculation of rotational strengths in saturated ketones, one further point is worth mentioning here. So far we have emphasized the utility of the one-electron approach for symmetric chromophores. However, it should be kept in mind by the reader that from a broader point of view, a one-electron approach to optical activity is always appropriate for the calculation of rotational strengths of single electronic transitions to the same extent that the orbital approach is applicable for the calculation of frequencies and intensities of such transitions. We shall elaborate on this last statement in Section V-D. 

When two atoms with appropriate orbitals-i.e. orbitals of not very different energy and of the same symmetry, and each with one electron-approach one another closely, electron pairing will be produced. In this model electron pair sharing is mathematically represented as a product of the atom wave functions. 

The most popular approximation, which allows to approach the solution of the problem, is the one-electron approach where electrons, according to the Pauli principle, are distributed over the system of levels in some effective potential of nuclei and electrons (in the self-consistent field, SCF). This distribution is named the electronic configuration. There are different variants of the one-electron approximation, and the range of their applicability depends on intemuclear distances. The method of molecular orbitals (MO) is the most popular and highly developed method to present time. In this method, molecular orbitals are arranged as a linear combination (LC) of atomic orbitals (AO) fix>m which the method is named MO LCAO. 

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