One-electron calculation

It is apparent that the molecular orbital theory is a very useful method of classifying the ground and excited states of small molecules. The transition metal complexes occupy a special place here, and the last chapter is devoted entirely to this subject. We believe that modem inorganic chemists should be acquainted with the methods of the theory, and that they will find approximate one-electron calculations as helpful as the organic chemists have found simple Hiickel calculations. For this reason, we have included a calculation of the permanganate ion in Chapter 8. On the other hand, we have not considered conjugated pi systems because they are excellently discussed in a number of books. 

The right to exist of the H2 molecule should be apparent from a glance at the MO energy diagram the molecule should be more stable than the separated atoms (disregarding interelectron repulsion, which, of course, is not allowed for in one-electron calculations). We now address the question of the nonexistence and instability of the He2 molecule. 

In the relativistic DVME method, eqs. (27) and (28) are calculated numerically in a similar way to the one-electron calculation such as... 

On returning to the I l-J one-electron calculation, the assumed linear vibrating structure, shown in Figure 2.18, has to be abandoned in favour of a quantized nuclear framework, and the calculated cylindrically symmetrical structure as suggested by contour maps of electron density should be rotated about all Eulerian angles to reveal the full spherical symmetry of the... 

This section will contain less analytical and more straightforward presentation of current experimental and theoretical results and ideas. The field is developing extremely rapidly and fairly little has been done so far in terms of quantitative calculations on real systems. Most of the theoretical investigations have been in the form of one-electron calculations for molecules and clusters, or many-body calculations on model systems. We shall discuss a number of cases where hole spectra in extended systems are strongly influenced or, in certain regions, even dominated by atomic-like, local excitations, with the particular aim of presenting an overview of unifying aspects. 

In our previous report, however, the calculated multiplet energies tend to be overestimated especially for the doublets. This is due to the underestimation of the effect of electron correlations. Recently, we have developed a simple method to take into account the remaining effect of electron correlations. In this method, the electron-electron repulsion integrals are multiplied by a certain reduction factor (correlation correction factor), c, and the value of c is determined by the consistency between the spin-unrestricted one-electron calculations and the multiplet calculations. The details of this method will be described in another paper (5). In the present paper, the effect of electron correlations on the multiplet structure of ruby is investigated by the comparison between the results with and without the correlation corrections. 

One-electron calculations were carried out self-consistently based on the local density functional approach using the Slater s Xa potential (7). In the present calculation, a was fixed at 0.7, which was found to be the most appropriate value in many cases (8). The molecular orbitals were constructed as Hnear combination of the atomic orbitals (LCAO). The most remarkable feature of our program is that the atomic orbitals are created numerically in each iteration and flexible to the chemical environment. The details of this program have been described by Adachi et al. (9). 

In the DV-ME method, only the electrons occupying the impurity states are considered directly. After the one-electron calculation based on the DV-Xa cluster calculations, the Slater determinants for all the possible choice of the impurity-state orbitals axe constructed and are used as basis functions for diagonalization of the effective many-electron Hamiltonian. For the calculation of the matrix elements, we adopt the approach proposed by Fazzio et al. (10). First we classify all the Slater determinants into several electronic configurations. Then the diagonal matrix element of the i-th Slater determinant belonging to the n-th electronic configuration is expressed as,... 

As we know, a few first-principles calculations for multiplet structure have been tried by several researchers. Ohnishi and Sugano calculated the energy positions of the (R line) and Ti (U band) states in ruby, under one-electron approximation (12). Xia et al. carried out similar calculations using more realistic model cluster (13). They could, however, only consider the energies of lower-lying two states in multiplet structure. Watanabe and Kamimura combined one-electron calculations with ligand field theory, and carried out first-principles calculation for the "full" multiplet structure of several transition metal impurities... 

After the one-electron calculation, the Slater determinants corresponding to all the possible selections of impurity-state orbitals are constructed as,... 

Besides these one-electron calculations we have performed all-electron calculations for the noble gas atoms up to element 118 (Eka-Rn). All calculations were done with the same basis set of the form 32s29p20dl5/. The results are presented in Table 3. The importance of a relativistic treatment in order to get exact total energies is obvious for all systems under consideration. As for the one-electron systems, DKH3 yields always the lowest total energy value of all DKH calculations. 

The extension of the one-electron calculations to the case of the simplest molecule dihydrogen with two electrons is beyond our numerical methodology on a spreadsheet... 

Thus, the one-electron calculation part of subroutine genint may be modified to use symmetry in an almost identical way to the earlier two-electron code ... 

The specific feature for L-shell orbitals is the nondiagonal matrix element (18b). It accounts for the Stark interactions of the 2s and 2p orbitals located at one center when a charged particle approaches. It is noted that no attempt was made to simplify expression (18b). Nevertheless, the adjustable parameter d was introduced to achieve some flexibility in its application to multielectron systems. (The one-electron calculation yields d = 2). It is noted that the terms with the parameter d vanish at large intemuclear distances. There, asymptotically, expression (18b) approaches a value that is governed by the 2s-2p dipole matrix element obtained within the hydrogenic approximation. ... 

The transition probability amplitude, t Vp), in the ID case, has been calculated using non-Hermitian scattering theory [20]. It shows a series of resonance peaks (see Fig. 6. The phase of t Vp) changes by n in resonances and accumulates between resonances (see Fig. 7. The sharp phase drop is not observed. Our calculations show that this phase drop can not be explained by ID one-electron calculations, even when interference between different paths (different y cuts) is taken into consideration. Therefore, it is clear that two dimensions are needed to obtain the sharp phase drop phenomenon when using an effective one-electron model. 

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