Coulomb integral, reduced

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

With very, very large systems, fast-multipole methods analogous to those described in Section 2.4.2 can be used to reduce the scaling of Coulomb integral evaluation to linear... 

Due to the problems discussed at the end of section 7.2.2., the integration in the two center case has been done using the methodology outlined by O-Ohata and Ruedenberg [26b]. The integral to be computed now is a reduced Coulomb integral written as ... 

The same problems encovmtered when discussing Coulomb integrals may be fotmd in this case, even worsened by the presence of higher negative powers of the distance. 0-ohata and Ruedenberg transformation [26b] can be also used here, and thus the problem reduces to a first inteipration over an overlap-like expression, followed by a spherical integration of a potential-like function. 

A is the Coulomb integral and j8 is the resonance integral . Equation 13 derives from equation 10a in the same way that equation 11 derives from 10 and it shows that one effect of conjugation is to reduce the ionization energy of the uppermost occupied orbital, e.g. of X, by the amount A/(X). 

It is clear that the least-squares equations for the model density require only the Coulomb integrals (pv I fj) and (fi I fj), which are 0(N3) and 0(N2) in number, respectively, and therefore the integral evaluation problem is formally reduced by one order to 0(N3). 

LCAO, and the MO energies. To get numbers for H the SHM reduces all the Fock matrix elements to a (the coulomb integral, for AOs on the same atom) and (the bond integral or resonance integral, for AOs not on the same atom for nonadjacent atoms P is set = 0). To get actual numbers for the Fock elements, a and jS are defined as energies relative to a, in units of P this makes the Fock matrix consist of just Os and - Is, where the Os represent same-atom interactions and nonadjacent-atom interactions, and the -Is represent adjacent-atom interactions. The use of just two Fock elements is a big approximation. The SHM Fock matrix is easily written down just by looking at the way the atoms in the molecule are connected. Applications of the SHM include predicting ... 

As in atomic spectra, the form of Eq. (128) shows why the semi-empirical parameters of w-electron theory tend to include correlation. Zero differential overlap allows the Jif integrals in the H.F. part of Eq. (128) to be broken down into one- and two-center Coulomb integrals. The large difference between the non-and semi-empirical values of these is accounted for by introduc-ingi >i correlation into (2 J . This is justified because the "zero differential overlap can equally well be made in the e fs, reducing them to one- and two-center Coulomb correlation energies. 

The first step in reducing the MBPT expressions into a form suitable for numerical evaluation is a decomposition of the Coulomb integrals Uy w into sums of products of angular momentum coupling coefficients and radial integrals. To accomplish this decomposition, we first expand the kernel of the Coulomb integrals l/ri2 as... 

The sum over K is strongly reduced (often to just one term, the HOMO-LUMO interaction), with K corresponding to the replacement of the occupied MO ([)° of A with the virtual MO ([)[ of B (A is the donor, B the acceptor). The expression is then simplified the numerator is reduced to a combination of two-electron Coulomb integrals multiplied by the opportune overlap. The CT contribution rapidly decays with increasing R. 

As shown in [3], this cannot explain the large rates because the decreased interchromophoric distance is counteracted by a reduced orientation factor due to the deviation from the optimal collinear arrangement. As a further point, the vahdity of the dipole approximation, Eq. (7), was investigated [3], To this end, the electronic coupling matrix element in Eq. (6) was calculated from the Coulomb integral over the one-particle transition densities of the donor yofr.r ) and the acceptor yA(<">> ) ... 

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