Two-electron repulsion term

Hf , is the matrix element over the one-electron components in the Hamiltonian and the remaining integrals depend on the two-electron repulsion term, e.g. 

Figure 5.6b The worksheet for the calculation of the two-electron repulsion term, equation 5.52. The values, calculated in column R, follow from the intrinsic Error function defined in the Engineering function library in EXCEL.

Semiempirical MO theories fall into two categories those using a Hamiltonian that is the sum of one-electron terms, and those using a Hamiltonian that includes two-electron repulsion terms, as well as one-electron terms. The Hiickel method is a one-electron theory, whereas the Pariser-Parr-Pople method is a two-electron theory. 

We see now that in the two-electron spin space, the zero-fieldsplitting two-electron terms dictate the principal axes x,y,z in the molecular framework. By the same token, in the two-electron geminal space, the two-electron repulsion terms dictate the principal axes in the space of orbital transformations and thus the "principal" orbital choices. The three directions correspond to the most localized orbital choice A, B, the most delocalized "complex choice A, B, and to the most delocalized "real" choice A defined in Table 1. Our insistence on defining the orbitals A, B as one of the principal choices from the start (the choice of the most localized as opposed to the most delocalized "real" ones was arbitrary), well before electron-repulsion terms were considered, is thus understandable in retrospect. 

The term ( iv X.o) in Equation 32 signifies the two-electron repulsion integrals. Under the Hartree-Fock treatment, each electron sees all of the other electrons as an average distribution there is no instantaneous electron-electron interaction included. Higher level methods attempt to remedy this neglect of electron correlation in various ways, as we shall see. 

In this latter formula, the two electron repulsion integral is written following Mulliken convention and the one electron integrals are grouped in the matrix e. In this way, the one-electron terms of the Hamiltonian are grouped together with the two electron ones into a two electron matrix. Here, the matrix is used only in order to render a more compact formalism. 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

Equation 11.9 certainly implies that the local hardness should be used. It is known that this depends only on the functional dependence of the kinetic energy and electron repulsion terms upon the value of p [3]. However, it is difficult to calculate local values. In spite of this uncertainty, Equation 11.9 or its equivalent has often been used to calculate the interaction between two chemical systems [4]. 

In the previous section we examined the variational result of the two-term wave function consisting of the covalent and ionic functions. This produces a 2 x 2 Hamiltonian, which may be decomposed into kinetic energy, nuclear attraction, and electron repulsion terms. Each of these operators produces a 2 x 2 matrix. Along with the overlap matrix these are... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

Why do these calculations yield results so far from the ub initio curve There are two reasons. First, atomic orbitals are used that are appropriate for isolated atoms, but are hardly expected to be the best orbitals for the electrons when two or more atoms are in close proximity. It is convenient to use atomic orbitals in simple calculations because they are mathematically simple, but more complicated orbitals are known to give better results. Second, neither treatment properly takes into account electron-electron repulsions. For two electrons, a term of the form e2lr2n (in which e is the electronic charge and r12 is the distance between the electrons) is required to describe the repulsion between electrons. The exact calculations avoid both difficulties but are so complex mathematically as to be devoid of any capability for providing qualitative understanding. 

In crystal field theory calculations the direction of the axial distortion is along the z-axis. Therefore, the dz2 orbitals in iron atoms in Fig. 15 are along the line adjoining the two iron atoms. Remembering that the dz2 orbital lies lowest in this symmetry, the effect of reducing the complex is to add electron density to the dz% orbitals of the iron atoms. Since the dz2 iron-orbitals in Fig. 15 overlap, this structure results in an electron repulsion term between the iron atoms which increases as the iron atoms in these proteins are reduced. Thus, the negative reduction potentials (Table 1) of the plant-type ferredoxins can be accounted for by this model. 

There are two types of electron correlation static and dynamic. The static correlation is related to the behavior of HF method at the dissociation limit of the molecule and deals with the long range behavior of this approach. On the other hand dynamic electron correlation is related to the electron repulsion term and is the reciprocal function of a distance between two electrons and thus represents short range phenomena. However, it should be noted that the electron correlation in the HF method is included in the indirect manner by the consideration of an electronic motion in an effective potential field due to the nuclei and the rest of the electrons and due to the inclusion of electron spin. Therefore, despite the known shortcomings, HF method has been extensively used in chemical calculations and has been quite successful for systems which are not extensive for electron correlation. 

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