Frozen orbital state

When many-body interactions are weak, l f,o(N — l,j) Fi(N — l,j) and the states i f s (N - l,j) have essentially no spectral weight for s > 0, that is, the (N — 1) electron state is close to the frozen orbital state. The many-body matrix element then reduces to the one-electron matrix element mj [ < f(cf,lr) A(t) Pj < j(cj)> with the vector potential A(f) = Aoexp(—2 rivt) from the harmonic long wavelength radiation field (dipole approximation). The time-dependent factors in the wave functions and vector potential produce, after integration, the factor 5(cf — ej — hv), which is the one-electron approximation to the 5-function already anticipated in Eq. (3.2.2.4) and which reflects the conservation of energy. We have then for the matrix element (p(( (,k) Ao Pj j(cj)) where all the time-dependent factors have been removed. Owing to commutation relations, the operator Ao pj can be replaced by the operator A0 rj [19]. The practical form ofthe dipole matrix element for the emission from localized core levels is then... 

Xlie correction due to electron correlation would be expected to be greater for the unionised state than for the ionised state, as the former has more electrons. Fortunately, therefore, the t-tfect of electron correlation often opposes the effect of the frozen orbitals, resulting in many cases in good agreement between experimentally determined ionisation potentials and caU Lila ted values. 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

The X-ray photoelectron spectrum of the core ionization of an atom in a molecule consists of peaks and bands corresponding to transitions to various excited states. None of these transitions corresponds to the formation of the Koopmans theorem frozen-orbital ionic state, which is a completely hypothetical state. However, the center of gravity of the various peaks and bands lies at the energy corresponding... 

In the Hartree-Fock, frozen-orbital case, Pp acquires its maximum value, unity. Final states with large correlation effects are characterized by low pole strengths. Transition intensities, such as those in Eq. (2.7), are proportional to Pp. 

The most accurate theoretical results for positronium formation in positron-helium collisions in the energy range 20-150 eV are probably those of Campbell et al. (1998a), who used the coupled-state method with the lowest three positronium states and 24 helium states, each of which was represented by an uncorrelated frozen orbital wave function... 

Figure 2. Relaxation of t states. Left the frozen orbitals of the 4Agg ground state. Right full-scale SCF calculations. The energies are not drawn to scale.

In the actual calculation94, 95) no attempt was made to include the detailed level structure. The 5 pfod ionic excitations were approximated by 5 pav (5 pmd P RPAE) (cf. Sect. 6.1) and the potential was based on the frozen ground-state orbitals. The resulting spectral strength distribution is given in Fig. 32 a. The shift of the 5 s.i/2 core level of 2.5 eV is of the right order but somewhat too small. Inclusion of the dipole polarizability of the 4d-shell through... 

Photoelectron spectral measurements have prompted high-accuracy near-Hartree-Fock calculations on the Is hole states of 02. 261 Calculations were reported at Re for molecular O2. The frozen-orbital approximation evaluated the energy of Oj from the RHF calculations of Schaefer250 reported above. Then the IP are the difference between the O2 ground-state energy and the Ot energy. The IP obtained was 563.5 eV. Direct hole-state calculations for the relevant states of OJ, with the MO constrained to be of g or a symmetry, were also carried out. For the orbital occupancy (16), the computed IP was 554.4 eV. Finally, the restriction to g and u... 

The ECP s are constructed based on the frozen orbital ECP technique (12). In this technique some of the core orbitals are expanded in the valence basis set and frozen in atomic shapes. This reduces the demand on the accuracy of the ECP potentials and the projection operators. One-electron ECP s constructed by this technique for nickel and copper have been shown to give results of quantitative accuracy for surface problems, particularly for hydrogen chemisorption which is treated here (13,14). In the previous studies the one-electron ECP s included a frozen 3s orbital. In the present case, states with a large occupancy of 4p appeared for the s type configurations in a cluster surrounding. 

The A -shell x-ray emission rates of molecules have been calculated with the DV-Xa method. The x-ray transition probabilites are evaluated in the dipole approximation by the DV-integration method using molecular wave functions. The validity of the DV-integration method is tested. The calculated values in the relaxed-orbital approximation are compared with those of the frozen-orbital approximation and the transition-state method. The contributions from the interatomic transitions are estimated. The chemical effect on the KP/Ka ratios for 3d elements is calculated and compared with the experimental data. The excitation mode dependence on the Kp/Ka ratios for 3d elements is discussed. 

Most theoretical calculations of x-ray emission rates in atoms and molecules have been performed in the frozen-orbital (FR) approximation, where the same atomic or molecular potential is used before and after the transition. It is usual to use the ground state configuration for this purpose. This approximation is convenient because we need only one atomic or molecular calculations and the wave functions for the initial and final states are orthogonal. However, the presence of vacancy is not taken into consideration. On the other hand, we have shown that the TS method is useful to predict x-ray transition energies, but is not so good approximation to the absolute x-ray transition probabilities [39]. 

While the multiconfiguration methods lead to large and accurate descriptions of atomic states, formal insight that can lead to a productive understanding of structure-related reaction problems can be obtained from first-order perturbation theory. We consider the atomic states as perturbed frozen-orbital Hartree—Fock states. It is shown in chapter 11 on electron momentum spectroscopy that the perturbation is quite small, so it is sensible to consider the first order. Here the term Hartree—Fock is used to describe the procedure for obtaining the unperturbed determi-nantal configurations pk). The orbitals may be those obtained from a Hartree—Fock calculation of the ground state. A refinement would be to use natural orbitals. 

The measured E value is directly proportional to the difference Eb(IE) = Ef — E,. The final state in PES consists of an ion and the outgoing photoelectron. The electronic structure of material is often described by approximate, one-electron wavefunctions (MO theory). MO approximation neglects electron correlation in both the initial and final states, but fortunately this often leads to a cancellation of errors when Ef, is calculated. A related problem arises when one tries to use the same wavefunctions to describe 4q and I f. This frozen orbital approximation is embedded in the Koopmans approximation (or the Koopmans theorem as it is most inappropriately called), equation 4,... 

Far from the ionisation threshold, where the escaping photoelectron has a large kinetic energy, the normal method of calculation for the continuum states is to compute the orbitals in a field determined by using the frozen orbitals of the neutral atom with one electron removed. However, if one is calculating a resonance which lies close to the threshold, this approach may fail. This happens because the escaping photoelectron moves slowly, so that the residual ion has time to relax as it escapes. It is then better to compute the orbitals in the relaxed field of the ion. This approximation is called the GRPAE or the RPAER, and is referred to as the RPAE with relaxation. 

Excitation energies calculated with the RPA and TDA approaches for N2 with a moderately large basis set are listed in Table 23. Both the RPA and TDA excitation energies are significantly lower than those obtained with the simplest frozen orbital approximation. All these approaches differ only in their treatment of the final state, and the pattern of predicted excitation energies shows this in a rather dramatic way. Both the RPA and TDA allow for a limited amount of relaxation and provide much improved predictions. Inclusion of the... 

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