Integral nucleus-electron attraction

The nucleus-electron attraction integrals simplify as well ... 

We shall call /3 the exchange integral in this text. In other sources, however, you may find 8 referred to as a resonance or covalent integral. We have seen that an electron in the molecular orbital spends most of its time in the overlap region common to both nuclei. Thus the electron is stabilized in this favorable position for nucleus < -electron-nucleus b attractions. The exchange integral jS simply represents this added covalent-bonding stability. 

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 Fock operator determines three sets of information for each electron i (1) the kinetic energy term of the electron (—1/2V ), (2) an attraction term with each nucleus, A, (—EZA/r,A), and (3) the interaction of the electron with all the other electrons in the molecule. This average force is treated by the (IJjj — Kjj) term and can be described as the potential felt by a single electron in the field of the other i — 1 electrons in the molecule. A few words about the components of this last term in the Fock operator are in order. J is called the coulomb operator and is identified as the classical repulsion between electrons. The exchange integral K is due to the quantum mechanical effect of spin correlation, an intrinsic property of the electron that keeps apart electrons of the same spin. This operator has a stabilizing effect on the energy of the system. 

The application of a modified electron-nucleus potential together with analytical basis functions requires the evaluation of appropriate matrix elements (nuclear attraction integrals) ... 

Obviously, if i = j, = 1 and the approximate expression for the electron-nucleus attraction integral become the simple point-charge formula 1 /R. 

The central problem in both the STO nuclear-attraction integrals and the STO electron-repulsion integrals is the fact that the expression for the inverse-distance operator is intractable. If fa and n, are two position vectors (either for one electron and a nucleus or for two electrons) then the inter-particle distance may be expressed ... 

The one-electron Hamiltonian only involves differences of geometrical parameters the nuclear attraction terms involve the electron-nucleus distance not the absolute position of either particle. Likewise the molecular integrals dependence on molecular geometry is only via inter-centre distance the fact that the basis functions are atom-centred does not induce any dependence of the integrals on absolute position of the integrals. 

In order to interpret this integral (in atomic units) as the Coulombic attraction of the electronic charge Xp (I)/ (1) by the nucleus (of charge Z. located at we have to multiply the integral by —Z. 

The first and the second integrals in equation (2.16) are equal to the sum of the kinetic energy and the electron-nucleus attraction energy for electrons 1 and 2 respectively. Each of the two integrals is equal to (Z — IZZ ). The third integral in (2.16) represents the electron-electron repulsion energy and is equal to 1.25 Z R. 

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