Ground-state energy of an electron

The sum of terms (a), (c), and (d) is just the ground-state energy of an electron gas in a positive continuum terms (b) and (e) arise from the lattice. 

We now calculate the ground-state energy of an electron gas interacting with the lattice charges through the modified Coulomb potential (131). The calculations follow the same steps as for the case of point charges. We shall therefore merely state the points at which differences must be introduced. This, of course, only happens in the contributions where the lattice plays some role, i.e., the Madelung term and the polarization term. 

This section will be devoted to the calculation of the ground-state energy of an electron gas in a lattice formed by two types of point charges, and Vs, so arranged as to form a superlattice. The various sites of the A lattice will be denoted by a,. . ., while the various sites of the B lattice will be denoted by i,j,. The interaction between an electron at r and the whole lattice is given by the sum ... 

The starting point of the local electron density functional theory is the fundamental theorem stating that the total ground-state energy of an electron many-body system is a functional of the electronic charge density. 

Electronegativity has recently been redefined [36] as the quantum potential of the atomic valence state, calculated from the ground-state energy of an electron, confined to the ionization sphere of radius ro ... 

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... 

According to the HK theorem, the ground state energy of an interacting inhomogeneus electron gas in a static external potential u(r) is given by [26] ... 

In DFT, the ground-state energy of an atom or a molecule is written in terms of the electronic density p(r), and the external potential v(r), in the form [1,6]... 

The ground state energy for an electron that is confined to a potential well with a width of 0.4 nm ... 

Usually one assumes that the electrons stay in their ground state. The ground-state energy of the electronic Hamiltonian as a function of the nuclear coordinates then represents the potential energy surface describing the interaction of a molecule with a substrate. Of course all electronic transitions are neglected in this ansatz. Often this is a reasonable assumption although its validity is hard to prove. Still, there is an important class of dynamical processes on surfaces which involve electronic transitions. I will also briefly discuss such processes later in this review. 

The density functional method is based on a theorem of Hohenberg and Kohn, which states that the ground-state energy of an atom or molecule is a functional of the electron probability density. Schemes exist to obtain a usable approximation to the desired functional and the necessary electron probability density. 

Equation (5.21) is based on the electrochemical way of counting the energy difference between zero (defined throughout this book as the potential energy of an electron at its ground state at "infinite" distance from the metal) and the Fermi level Ep (Eq. 5.15). The latter quantity must not be confused with the Fermi energy go which is the energy difference between... 

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