Integrals Coulomb energy

The density fitting functions may or may not be the same as those used in expanding the orbitals. The fitting constants a are chosen so that the Coulomb energy arising from the difference between the exact and fitted densities is minimized, subject to the constraint of charge conservation. The J integrals then become... 

Here q represents the coulomb energy of an electron occupying a definite p]h orbital in unsubstituted benzene its value has been estimated to be about —2.7 v. e. = —60 kcal./mole.5 /3 is a resonance integral between adjacent orbitals its value has been estimated to be about —0.85 v. e. = — 20 kcal./mole.6 Sk is a constant, the purpose of which is to allow for the different electron affinities of the different atoms. For Sk > 0, the... 

The GEM force field follows exactly the SIBFA energy scheme. However, once computed, the auxiliary coefficients can be directly used to compute integrals. That way, the evaluation of the electrostatic interaction can virtually be exact for an perfect fit of the density as the three terms of the coulomb energy, namely the nucleus-nucleus repulsion, electron-nucleus attraction and electron-electron repulsion, through the use of p [2, 14-16, 58],... 

The expression for quantity g(Rs) in (19.52) follows directly from (5.40). Thus, for p- and d-shells we have simple algebraic formulas for the coefficients of radial integrals (actually, for matrix elements) of all relativistic corrections of the order a2 to the Coulomb energy. For the /-shell such a formula exists only in the case of the orbit-orbit interaction. For the almost filled shell we find... 

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... 

Figure 31 A schematic representation of the HF result for (TMTCF)2X. Transfer integrals along the TMTCF chains, t and t2, and the on-site and intersite Coulomb energies, U and V, are indicated, while 8 denotes the CO and the length of the arrows schematically represent the amount of magnetic moment on the corresponding site.

In particular, for negative energies the action integral Ic = jp dr of unperturbed Coulombic oscillations corresponds to the principal quantum number o> which is related to the Coulomb energy via Eq= — 1/(2mo)- We may identify the... 

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