Poly , electronic

To that end, we will start with the same equation as the one used above in the case of poly electronic atoms, v/z. the eq.(lO), and we will try to use the equivalent of the compensation between the kinetic energy and the nuclear attraction (T and -Z/r) found in the atomic case. 

The presence or nodes and small subnodal maxima" docs have a profound effect on the energy of electrons in different orbitals. An electron in an orbital with these subnodal maxima (particularly s orbitals with higher values of m are said to be penetrating, that is. they have considerable electron density in the region cl the nucleus. This is the fundamental reason for the ordering of the energy levels in poly-electronic atoms lj. 2s, 2p. 3. ip, etc. (see pages 20-22). 

The basic idea at the foundations of the method of molecular orbitals consists, as is well known, of constructing the wave function of a poly-electronic system as a suitable combination of individual one-electron wave functions. 

In obtaining P(b, v), a one-electron treatment is used even when A is a poly electronic system, and is treated as a point centre of positive... 

The Fock operator is an effective one-electron Hamiltonian for the electron in the poly-electronic system However, written in this form of Equation (2.130), the Hartree-Fock... 

Table VIII summarizes the idealized electron accounting associated with electrochemical reduction of the divalent metal ion complexes of TAAB (Figure 11). For each metal ion, a distinct set of one-electron reduction waves occurs (31). In methanol solution, these multiple one-electron processes are followed by a very complex poly-electron wave...

Although the rigorous description of poly electronic atoms is quite complicated, our simple qualitative ideas about electrons in independent orbitals are often very useful when we try to understand why atoms behave the way they do. We will consider some specific atomic properties in the next section. 

However, even though it is formulated rather easily, this problem cannot be solved exactly. The difficulty is the same as that encountered in dealing with poly electronic atoms— the electron correlation problem. Since we cannot account for the details of the electron movements, we cannot deal with the electron-electron interactions in a specific way. We need to make approximations that allow the solution of the problem but that do not destroy the model s physical integrity. The success of these approximations can be measured only by comparing predictions from the theory with experimental observations. In this case we will see that the simplified model works well. 

The correlation contribution was developed using a somewhat different algorithm, namely, employing its definition as the difference between the exact and Hartree-Fock (HE) total energy of a poly-electronic system [182]. Without reproducing the results (more details are provided in the dedicated review of [92]), we mention only the Lee-Yang-Parr (LYP) correlation functional [183-185] altmg the Vosko-WDk-Nusair (VWN) local correlation density functional [186]. 

Evaluation of the energy in the time-independent Schrodinger equation requires the solution of an eigenvalue-eigenvector problem [22]. For an electronic wave function satisfying Eq. (2.9), an eigenvector— the total electronic energy— wiU be found. A possible poly electronic wave function for n electrons could have the form of a Hartree product ... 

The first and immediate definition of energy correlation may be given by the difference between the exact and Hartree-Fock (HF) total energy of a poly-electronic system (Senatore March, 1994) ... 

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