Nuclear attraction functional

The P matrix involves the HF-LCAO coefficients and the hi matrix has elements that consist of the one-electron integrals (kinetic energy and nuclear attraction) over the basis functions Xi - Xn - " h matrix contains two-electron integrals and elements of the P matrix. If we differentiate with respect to parameter a which could be a nuclear coordinate or a component of an applied electric field, then we have to evaluate terms such as... 

Fh p) = Ec p) + Fxc p) + FM + FEAPhFT p) (3.15) (with subscripts C, XC, eN, Ext, and T denoting Coulomb, exchange-correlation, electron-nuclear attraction, external, and kinetic energies respectively). It is CTucial to remark that (3,15) is not the Kohn-Sham decomposition familiar in conventional presentations of DFT. There is no reference, model, nor auxiliary system involved in (3.15). From the construction presented above it is clear that in order to maintain consistency and to define functional derivatives properly all... 

As for atoms, it is assumed that if the wave function for a molecule is a single product of orbitals, then the energy is the sum of the one-electron energies (kinetic energy and electron-nuclear attractions) and Coulomb interactions... 

The method ofmany-electron Sturmian basis functions is applied to molecules. The basis potential is chosen to be the attractive Coulomb potential of the nuclei in the molecule. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nuclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

In agreement with theoretical prediction, the experimental analysis shows the more positive atoms to be contracted. This is explained by the decrease in electron-electron repulsions, or, in a somewhat different language, the decreased screening of the nuclear attraction forces by a smaller number of electrons. This contraction is incorporated in Slater s rules for approximate, single exponential (and therefore nodeless), hydrogen-like orbital functions (Slater 1932). For a 2px orbital of a second-row atom, for example, the orbital function is given by... 

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... 

In Eq. (2.2), Z0 is called a shielding constant. It measures the effect of the other electrons in reducing the nuclear attraction for the electron in question. It is a function of n and Z, increasing from practically zero for the lowest n values to a value only slightly less than Z for the outermost orbits within the atom. For levels outside the atom, on the other hand, the energy is given approximately by... 

The analytic form of the first two terms in the Kohn-Sham effective potential (Vrff [p](r)) is known. They represent the external potential (vext which is the nuclear attraction potential in most cases) and Coulomb repulsion between electrons. The second term is an explicit functional of electron density. The last term, however, represents the quantum many-body effects and has a traditional name of exchange-correlation potential. vxc is the functional derivative of the component of the total energy functional called conventionally exchange-correlation energy (Exc[p]) ... 

Because of the many center nature of the fourth integral case, a detailed analysis of three center nuclear attraction integral problem is given. Using the ideas developed in Sections 4 and 5, it is described how the three center integrals become expressible in terms of one and tv/o center ones. An example involving s-type WO-CETO functions is presented as a test of the developed theory of the preceding chapters. 

In this sense, electronic spin-spin contact integrals can be computed as simple bilinear functions of overlap integrals, and electron repulsion integrals are expressible as bilinear fimctions of some sort of two electron nuclear attraction integrals. 

Table 7.1 shows how repulsion integrals over CETO functions can be constructed with reliable accuracy, as three center nuclear attraction integrals were computed. 

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