Atomic-molecular system, electronic energy

The energy of the electrons may be broken into component parts. Those which raise the total energy of the atomic molecular system, and hence raise the energy content of one gram atom or one mol of the substance, will be termed positive, whereas those which lower the energy... 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

Although equation (2) appears concise in the form above, it cannot be solved exactly for a molecular system. Electron-electron repulsions present a practical problem in the same way that they do in many-electron atoms the repulsive interaction between two electrons is a function of the position coordinates of each electron and cannot be separated into two functions dependent on each set of coordinates individually. However, even if a comphcated wavefunction that was expressed in the coordinates of aU the electrons in a molecule could be developed, it would have to be reported as an extensive grid of points in 3-D space, each associated with a different potential energy with unique solutions for... 

HgH.—Das and Wahl have carried out a calculation on the HgH molecule which has many points of interest for the practical implementation of pseudopotentials on heavy-atomic molecular systems. As the nuclear charge increases so does the importance of the relativistic terms in the hamiltonian, and their influence is not only confined to the core orbitals (e.g. the Hg Is) where the kinetic energy of the electron is comparable with its rest mass, but even afiects the valence (Hg 6s) orbitals (Grant °) and the binding energy of Hgj (Grant and Pyper ),... 

Fig. 2.33 Total energy E of an atomic/molecular system ((a) atomic P (b) atomic Pb (c) arbitrary system X) as a function of the electron number N.

Unlike in molecular systems, semiconductor energy levels are so dense that they form, instead of discrete molecular orbital energy levels, broad energy bands. Consider a solid composed of N atoms. Its frontier band will have IN energy eigenstates, each with an occupancy of two electrons of paired spin. Thus, a solid having atoms with odd number of valence... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

With a = 2/3 this is identical to the Dirac expression. The original method used a = 1, but a value of 3/4 has been shown to give better agreement for atomic and molecular systems. The name Slater is often used as a synonym for the L(S)DA exchange energy involving die electron density raised to the 4/3 power (1/3 power for the energy density). 

This description results from the fact that the optimum orbitals are essentially determined in the region surrounding each atom by the compensation between the kinetic energy T of the electron and the Coulomb attraction of the electron by the nucleus of that atom. This compensation implies that the orbital is very weakly dependent of the environment of the atom in the molecular system so that it is essentially determined by atomic conditions (Valley theorem). 

In the presence of a static, spatially uniform electric field Ea, the electronic cloud of atomic and molecular systems gets polarized. The energy, W, can be written as a Taylor series [1-3]... 

Let us recall that the Hohenberg-Kohn theorems allow us to construct a rigorous many-body theory using the electron density as the fundamental quantity. We showed in the previous chapter that in this framework the ground state energy of an atomic or molecular system can be written as... 

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