The Coulomb Integral

To a zeroth-order approximation is the Coulomb energy of an electron, with the wave function in the field of atom i, and might be regarded as but little affected by any other nuclei farther away. This approximation, of course, will be most valid where the surrounding atoms have no net electrical charges. We shall take 

At one time q was rather widely used as a symbol for the Coulomb integral, but this usage has generally been abandoned because qj is now used to denote the charge on the ith atom (see later). 

---

In fact, the Coulomb integrals discussed in Section IV.C are available in contemporary quantum chemistry packages. We do not really need to develop our own method to calculate them. However, it is necessary to master the algebra so that we can calculate the matrix elements of the derivatives of the Coulomb potential. In the following, we shall demonstrate the evaluation of these matrix elements. 

ZlNDO/S is differen t from ZINDO/1 because th ey use differen t algorithms in computing the Coulomb integrals. Hence the two et uation s used in th e rn ixed m odel in ZINDO/1 are also employed... 

One convention (Dickson. 1968) for oxygen heterocycles sets the coulomb integral at z 2f) and the resonance integral at Eor the oxirane moiety,... 

The problem of electrophilic substitution into the anilinium ion has been examined by the methods of m.o. theory. Attempts to simulate the --inductive effect in Hiickel M.o. theory by varying the Coulomb integral of C(j) (the carbon atom to which the NH3+ group is attached) remove 7r-electrons from the o- and -positions and add them to the... 

Hi2 is the resonance integral, usually symbolized by p. In a homonuclear diatomic molecule Hi I = H22 = a, which is known as the Coulomb integral, and the secular determinant becomes... 

When m = n the Coulomb integral is assumed to be the same for each atom and is given the symbol a ... 

This term describes a shift in energy by Acim rn, for an orbital with quantum numbers I — 2, mi and that is proportional to the average orbital angular momentum (/z) for the TOj-spin subsystem and the so-called Racah parameters Bm, that in turn can be represented by the Coulomb integrals and The operator that corresponds to this energy shift is given by... 

In the original, elementary treatment governed by Eq. 4 above, one might initially expect contributions to the barrier from several sources. There is first the Coulomb integral Q, which will contain angle dependent terms from the electrostatic interaction of the electrons and protons ar the two ends of the molecule. In this treatment the only orbitals used are Is on each H atom and tetra-... 

To find the coefficient of the Coulomb integral for two structures, superimpose their vector-bond patterns to form the superposition pattern (Fig. 1). The Coulomb coefficient is 2 " times the sum (— 1)E for the different patterns in which each orbit serves either as the head or as the tail... 

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]. 

The exchange integrals, / are evaluated by representing them as functions of the Coulomb integrals, Hii and the overlap integrals. One such approximation is known as the Wolfsberg-Helmholtz approximation, which is written as... 

As we saw earlier, the first two terms on the right-hand side of this equation represent the electron binding energies in atoms 1 and 2, respectively, which are Hu and H22, the Coulomb integrals. The last two terms represent the exchange integrals, H12 and H21. In this case, Hu = H22 and Hn = H21 because the nuclei are identical. Therefore, the energy of the orbital is... 

---