Coulomb repulsion integral

The Coulomb repulsion integrals are evaluated using the Mataga-Nishimoto formula The resonance integral is assumed to be of exponential form p=Be , the value of exponent a being taken as 1.7 A... 

Figure 15, Plot of the logarithms of disproportionation constants against the coulomb repulsion integrals for a series of 1,3,3-trimethylindolenin violenes (174).

The first simplification in the TDAN model is to consider only a few electronic orbitals on the scattered atom. For many applications, it is sufficient to consider one only, that from which, or into which, an electron is transferred. Let the ket 10 > denote the spatial part of the orbital. When far from the surface, suppose its energy is So> let Uq be the Coulomb repulsion integral associated with the energy change when it is occupied by two electrons of opposite spin. In terms of creation and annihilation operators and Co for 0>, with ff( = aorfi)a spin index, that part ofJt which refers to the free atom is... 

To complete the specification of Jf, it is necessary to introduce time-dependent terms. These are of two types. The first allows for many-body effects which, to a first approximation, correspond to an interaction of the atom with its image in the solid. As a result, the ionization level and the Coulomb repulsion integral for 0> become functions, o(z) and U(z), of the perpendicular distance z of the atom from the surface. The classical electrostatic forms for these functions are... 

In this section, we write down the explicit forms for the one-electron TDAN equations of motion for the situation where the Coulomb repulsion integral is set equal to zero. Also, we consider only the case where the band energy levels contain 2n electrons and the valence orbital of the scattered atom is initially empty. It will be obvious how to modify the equations to deal with other possible cases mostly this will involve just a change in the initial conditions. 

Table 8. Potentials Ej, E2 or E ,(V) and KgEM of vinylogous systems 22-32 in acetonitrile and Coulomb repulsion integral f...

Quantitatively the Coulomb repulsion integral of SCF-calculations for SEM correlates linearly with Ig Ksem Equation (3)... 

Fig. 6. Correlation between Ig Ksem of different vinylogous redox systems and the Coulomb repulsion integral...

Assume that n electrons on the same site (= atom) repel each other with energy a (the one-site Coulomb repulsion integral, usually positive). 

Also, Coulomb repulsion integrals will be simplified to a combination of two electron integrals like ... 

Cr(C6Hii)4 is tetrahedral and a triplet however, square-planar discriminating factor. Another factor at work in these molecules is the spin pairing energies. As indicated previously, 3d valence orbitals are considerably more contracted that 4d and 5d thus the Coulomb repulsion integrals are much larger for first-row transition metal complexes and high-spin states are more favored than is the case for second and third row complexes. 

Here P denotes the density matrix, h contains kinetic energy and electron-nuclear attraction operators [pqllr] and [rllt] are Coulomb repulsion integrals with 3 and 2 indices, respectively, [pqs] denote one-electron 3-index integrals and is the nuclear-nuclear repulsion term. The form of Equation 4 ensures (11b) that the Coulomb energies are accurate up to second order in the difference between the fitted density and the "exact" density obtained directly from the wavefunctioa... 

One-center coulomb repulsion integral at bridgehead nitrogen. s Variable electronegativity SCF method (57TFS397). 

Theoretical studies of the redox system formed by TTF and its radical-cation, using repulsion integrals calculated by two different methods, showed the possibilities and limitations of the Coulomb repulsion integrals to estimate formation constants of radicals. 

MOyi) Ij. and Zg are the ionization potential of the rth AO in the valence state and the effective core charge of the sth AO,respectively. One center Coulomb repulsion integrals (rrlrr)=Ij.-Ej,(Ei. is the electron affinity of the rth AO in the valence state), the values of which adopted in this paper are collected in Table 1 with the values Ij.. Two center Coulomb repulsion integral are evaluated by the for-... 

The authors revised the PPP method in a few points, especially in the formulation for the two center Coulomb repulsion integrals, in order to calculate the correct values of the triplet state energies (5) The application of the revised method to the calculation of the electronic states of clnnamoyloxy group was found to reproduce well the experimental values of its electronic transitions. The results are also shown in Fig.3. 

In the PPP case we have to deal with a three-dimensional parameter space, namely the resonance integral B0, the electron-phonon coupling constant a, and the on-site Coulomb repulsion integral y0. Thus we varied the values of B and a and calculated the energy of butadiene in the model at u,/u0 = 0 (p=l), 0.5 (p=2), 1.0 (p=3) and 1.5 (p=4). Note, that p is not a site index here. The values obtained (Eppp(up)) are compared with the corresponding Ecc Uj) energies. The energies are computed relative to Up=0. Then we calculated... 

Integration over all space (dv = dv1dv2 = d x 1dy1dz1dx2dy2dz2) yields the Coulomb repulsion integral Jy for two electrons in MOs and ij/j (Equation 4.28). 

The integral J can be simplified to the double sum of two-centre Coulombic repulsion integrals -yM (Equation 4.29) by adopting the zero differential overlap (ZDO) approximation, that is, we assume that atomic orbitals located on different atoms do not overlap, 0 for pffv. 

B2u, B1u, and. Elu) were separated by electron repulsion both from one another and from the corresponding triplets, and that these energy differences could be interpreted in terms of reasonable values for the Coulomb repulsion integrals between atomic orbitals. They did not find it possible to evaluate all the many-center integrals required, and errors crept into their numerical calculations, but subsequent work11 12-58 68 has left little doubt that the 1800 A band of benzene has an Elu upper state and that the upper states of the 2600 A and 2100 A bands are Biu and Bltt, respectively. There was, therefore, even at that time clear evidence that electron repulsion must be included in any final theory, though aromatic molecules with less symmetry than benzene clearly presented a much more difficult problem. 

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