Slater’s transition state

Koopmans s theorem is not valid for Xa calculations, but Slater s transition state concept applies. This approximation allows the interpretation of electronic transitions in terms of 1-electron orbitals and yet includes electronic relaxation effects (208). The virtual (unoccupied) orbitals of Xa theory have a physical significance and can be used to discuss excitations of the electronic system, because the same potential due to the iV - 1 other electrons affects both occupied and unoccupied... 

In the Xa method, the dipole matrix element is often calculated using the Slater s transition state (TS) concept [38]. The TS for x-ray emission corresponds to the state where the electron concerned with the transition stays half in the initial state and half in the final state. The electron configurations for the ground state, the initial state, the transition state, and the final state are shown in Fig. 1. 

The 3p DOS in Fig. 14A is the local DOS of the sulfur atom which has the Is hole. Comparing Fig. 13c and 14A, the bonding electrons are attracted by the core hole, and the low-energy structure (2461-2466 eV) becomes stronger in the core hole state. If the core hole is 0.5 (Slater s transition state), interpolation of Figs. 

Figure 2 Schematic illustration of ground state (GS), Slater s transition state (TS) and final state (FS) electronic configurations.

Figure 3 The sum of Mg-3s and 3d unoccupied partial density of state (PDOS) calculated for MgO using the (MgiaOn) " cluster at (a) ground state (GS), (b) Slater s transition state (TS) and (c) final state (FS). (d) Experimental ELNES and (e) experimental XANES reported in Refs. [25] and [11].

Table 1 The list of theoretical and experimental values of the transition energy at peak A for MgO, a-Al203 and Si02 (a-quartz). AE and are the values calculated by the Slater s transition state with and without spin polarization, respectively. AE and AE are the veilues estimated from the XANES and the ELNES, respectively.

The absolute transition energy obtained by the present transition state calculation overestimates the experimental XANES by 0.9 eV at peak A. If the spin polarization is taken into account, however, the discrepancy decreases. In this case, the electronic configuration in the Slater s transition state is expressed as (2p ) (72at)° (2pJ.) (72aJ.)° and calculated transition energy at peak A becomes 79.0 eV, which is in excellent agreement with experimental values estimated from the XANES spectrum (78.5 eV for 2pi/2 component and 78.9 eV for 2pa/2 component) as listed in Table 1. 

The electronic state calculation by discrete variational (DV) Xa molecular orbital method is introduced to demonstrate the usefulness for theoretical analysis of electron and x-ray spectroscopies, as well as electron energy loss spectroscopy. For the evaluation of peak energy. Slater s transition state calculation is very efficient to include the orbital relaxation effect. The effects of spin polarization and of relativity are argued and are shown to be important in some cases. For the estimation of peak intensity, the first-principles calculation of dipole transition probability can easily be performed by the use of DV numerical integration scheme, to provide very good correspondence with experiment. The total density of states (DOS) or partial DOS is also useful for a rough estimation of the peak intensity. In addition, it is necessary lo use the realistic model cluster for the quantitative analysis. The... 

Fig.2 Concept of Slater s transition state for ionization and electronic transitions...

Fig.9. In the figure, two types of calculated values for the ground state(GS) and Slater s transition state(TS) are indicated and compared with the experimental values. These values are referred to that of H2S. As previously mentioned, the absolute energy of S 2s level calculated systematically deviates from the measurement. However, the chemical shifts, namely the relative changes of the level energy both for GS and TS show good agreement with the experiment.

DV-Xa molecular orbital calculation is demonstrated to be very efficient for theoretical analysis of the photoelectron and x-ray spectroscopies. For photoelectron spectroscopy, Slater s transition state calculation is very effective to give an accurate peak energy, taking account of the orbital relaxation effect. The more careful analysis including the spin-polarized and the relativistic effects substantially improves the theoretical results for the core level spectrum. By consideration of the photoionization cross section, better theoretical spectrum can be obtained for the valence band structure than the ordinary DOS spectrum. The realistic model cluster reproduce very well the valence state spectrum in details. 

We used the C24 model cluster, which consists of seven carbon hexagons, to represent the graphite plane (Fig. 1 (a)). To perform the ground state calculations, we classified the carbon atoms into three types, and performed the calculations under symmetry. To avoid model cluster termination effects, we treated the electronic structure of the inner carbon atoms as that of graphite. When we performed Slater s transition state calculations, however, we took four types of carbon atom into account. This was because one of the inner carbon atoms, in which the Is electron transits to an unoccupied level, should be distinguished from the other inner carbon atoms. Further we didn t use any symmetry orbitals for the transition state calculations. 

The unoccupied K p-state DOS, shown in Fig. 2 (b), which we calculated for KC48 with Slater s transition state, is in good agreement with the observed K A-edge XANES spectra. [9] The calculation reproduced the edge energy to within 1.5 eV. 

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