Direct Coulomb interaction

Asm is an antisymmetrizer operator between electrons from these two groups s and m which is usually expressed as a sum of the identity operator (1) and normalized permuting operator Pms Asm =l+pms. The total Hamiltonian is symmetric to any electron permutation. The interaction energy Vsm can be cast in terms of a direct Coulomb interaction and an exchange Coulomb interaction ... 

On Fig. 3 we plot the probabilities of different configurations versus the direct Coulomb interaction U. It can be seen that the system undergoes a metal-insulator transition for a sufficiently high value of U, close to 9. It is easy to perform the same kind of calculation in the case of triply degenerate... 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

Another point of considerable importance is the nonlocal character of the exchange. The latter always favours non-uniform distributions of electrons (or holes) among the different d orbitals, e.g. eg and t2g orbitals, when the system is cubic. Direct Coulomb interactions as well as correlations favour uniform occupations of the different atomic orbitals and therefore counteract the effect of nonlocality of the exchange. Despite this, the anisotropies caused by exchange are important, in particular for bulk Ni (10) as well as for its surface (15). 

To[ (r)] defines the kinetic energy of a non-interaoting electron gas which requires the density />(r). (r) is the classical (direct) Coulomb interaction which is equivalent to the Hartree potential. Using the constraints that the total number of electrons are conserved, Kohn and Sham [20] reduced the many electron Schrodinger equations to a set of one-electron equations known as the Kohn-Sham equations... 

On the other hand, fluorescence yields for the XI lines in the case of the ion impacts were assumed to be Yxi only for k L K L through the shake process. Here Yxi was determined from the photon induced X-ray spectra. However, for the K L state produced through the direct Coulomb interaction, Y=0.013 was adopted because the direct Coulomb ionization is not accompanied with the orbital reairangeinent. These considerations give the total X-ray production cross sections for the XO and XI lines, ctxo and Oxi,... 

Fig. 2. Schematic K X-ray spectra induced by 1.4 MeV/amu H and He impact. Here white lines indicate observed spectra [9] and black lines spectra calculated in the direct Coulomb interaction frame.

Fig. 11. Calculated F Ka spectra (solid lines) excited by 5.5 MeV He impact, where the direct Coulomb interaction and the shake Prcoesses are taken into account.

Later, a more concrete case of calculation of perturbing molecule 2 with a multipole moment Qtl on molecule 1 with the multipole moment is considered. Substituting in Eq. (226) an appropriate operator of intermolecular Coulomb interaction (Gray, 1968 Armstrong et al., 1968) and averaging over the orientations of the molecules, we find the following expression of the contribution in (t2) due to the direct Coulomb interaction ... 

For the model of rigid solvent molecules and ions, Ucn is linear in the ionic charges (excluding the direct Coulomb interaction). If the work of charging the ions is expanded in powers of the ionic charges, the first semi-invariant is of the form... 

This shift in redox potential could arise from (i) a direct coulombic interaction between B and the polar component of the bound herbicide and/or (ii) inhibition of the relaxation of the B-binding protein during the B B transformation, and/or (iii) inhibition of proton uptake by the protein on the reducing side of PS II during the B B transformation. 

Dipoles and Polarization Phenomena. Many molecules do not carry formal electrical charges, so that their mutual interactions do not involve the direct coulombic interactions discussed above. However, if one examines the structures of many useful chemical species, including polymers, proteins, and drugs, it is apparent that they often include bonds that can impart an overall polar nature to the molecule as permanent dipoles, or they can be polarized by the effect of neighboring electric fields producing induced dipoles. The presence of permanent or induced dipoles means that the molecules can become involved in specific interactions with charged species, other dipoles, or nonpolar molecules, and those interactions can significantly affect the physical characteristics of the system. 

We may interpret the terms in eqn (2.19) as follows. The first term on the right-hand side represents the transfer of an electron from the spin-orbital Xj(r, ct) to the spin-orbital Xi(r, vice versa), with an energy scale The terms i = j in the sum represent the single-particle on-site energy, while the other terms represent the hybridization of the electrons between different orbitals. The second term on the right-hand side represents electron-electron interactions, the most important being the direct Coulomb interaction when i = j and k = I, as we discuss in Section 2.6. For readers not famihar with the second quantization approach, Appendix A describes a first quantization representation of the first term on the right-hand side of eqn (2.19). 

Equation (9.7) shows why the direct Coulomb interaction only mediates singlet exciton transfer. Since the ground state is a singlet and since the operator Ni — 1) preserves total spin, the excited state connected to the ground state in each of the square brackets must necessarily be a singlet. 

It can be seen that correction terms to the unperturbed Fock operator involve generalized one- and two-electron integrals with the effective interaction kernel, G(r, r ) in place of the usual direct Coulomb interaction r —... 

The first term is the contribution from HS repulsion, and again it can be addressed by MFMT. represents the direct Coulomb interaction contribution... 

The comparison ofEq. (51)fromDFT andEq. (53) from PB clearly indicates the physical essence of the crude approximation involved in PB theory, i.e., both the ion size effect and correlation contribution are completely ignored, and therefore PB theory is applicable only for dilute electrolyte systems in which the direct Coulombic interaction dominates and the size effect is neghgible. 

Other theoretical work has emphasized the role of the direct Coulomb interaction between the 4f hole and the valence electrons (rather than hybridization) in providing the double-peaked structure in the photoemission spectra of Ce. Liu and Ho (1982) calculated 4f photoemission spectra, omitting the hybridization term in the Hamiltonian and treating the Coulomb interaction between the 4f hole and the conduction electrons as the crucial term. Later, they investigated the effect of the hybridization term (Liu and Ho 1983). Kotani and Toyozawa (1974, 1979) studied the problem of the creation of a core hole in the presence of a partly-filled band. 

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